0. Then, (ρ c , z c ) is found to satisfy\n\nθ(ρ c , z c ) = -ϕ/2, (ρ c -ρ 0 ) 2 + z 2 c = r 0 , ( 33\n\n)\n\ni.e., the trap center lies on the surface of the \"resonance toroid\" at ρ = ρ 0 with a radius r 0 as shown in Fig. 1 . The relative angle of the trap center with respect to the center of the toroid cross-section is given by -ϕ/2. On this \"resonance toroid,\" the rf-field is resonant with the static field, i.e., B eff z vanishes. As a result, the \"effective B-field\" lies again in the x-y plane on the \"resonance toroid,\" which gives cos β eff (ρ c , z c ) = 0 and leads to the result A (φ) n (ρ c , z c ) = γ n = 0 as shown in the trap center for the storage ring considered before in Ref. [18] .\n\nFrom the expression (29) of the static field and the definition of the angle θ(ρ, z), we find a simple relationship β s (ρ, z) = π/2 + θ(ρ, z), with which the gauge potential A (φ) n (ρ, φ) in (28) can be further simplified as\n\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z) sin θ(ρ, z) 7 ≈ n ρ cos β eff (ρ, z) sin θ(ρ c , z c ), ( 34\n\n)\n\nnear the trap center. Thus, the spatial fluctuation of the gauge potential A (φ) n (ρ, z) in the region around the trap center is closely related to the angle θ(ρ c , z c ) of the trap center, or the parameter ϕ of the oscillating field B o . When ϕ = 0, the atom is trapped in the region with θ ≈ 0 or π, where the fluctuation of A (φ) n (ρ, z) is suppressed significantly due to the small value of sin θ. On the other hand, if the angle ϕ is set to π with the trap center located in the region with θ ≈ ±π/2, the fluctuation of the gauge potential becomes amplified.\n\nIn Fig. 2 , we illustrate numerical results for the distribution of the geometric phase γ 1 (ρ, z) = 2πρA φ 1 (ρ, z) in the region near the trap center at ϕ = 0, π/2, π. We see clearly decreased fluctuations of γ 1 when the absolute value of sin θ(ρ c , z c ) = -sin(ϕ/2) is decreased.\n\nNext we turn to the storage ring of Ref. [19] constructed from a quadrupole static B-field B s ( r) = B ′ (x, y, -2z) and an r-independent rf field B o = B rf cos(ωt)ê z along the z direction. The resulting ARFP provides a 2D ring shaped trap in the x-y plane. In addition, a 1D optical potential along the z direction is employed to confine atoms in the transverse plane at z = 0 [19]. The \"effective B-field\" takes the form\n\nB eff ( r) = B ′ (ρ -ρ 0 )ê z - 1 2 B rf êρ , ( 35\n\n)\n\nin the plane at z = 0, with\n\nρ 0 = ω/|µ B g F B ′ |.\n\nBecause the strength of B eff is near minimum at the ring ρ = ρ 0 , the trap center for this storage ring is located at ρ c = ρ 0 and z c = 0. At the trap center, the \"effective Bfield\" is along the direction of êρ . Thus, according to Eq. ( 28 ), the geometric phase γ (c) n at the trap center again vanishes.\n\nIn Fig. 3 , we show the distribution of the geometric phase γ 1 in the region near the trap center for B rf = 0.05|B ′ |ρ 0 and B rf = 0.15|B ′ |ρ 0 . We see that the fluctuation is relatively small when the strength of the rf-field is large. This can be explained by Eq. (28) , which shows that A (φ) n is proportional to cos β eff and can be approximated as 2B eff z /B rf near the trap center. When B rf is large, the gauge potential becomes a relatively slow varying function of ρ and z. In this case, the presence of a 1D optical potential allows for the possibility of tuning the trap center position to a nonzero value of z, with the storage ring remaining in the x-y plane. Then cos β s is assumed to a nonzero value, leading to increased fluctuations for the geometric phase.\n\nFinally, we discuss the geometric phase in the \"time averaged\" ARFP storage ring proposed in Ref. [20] . Unlike previously considered ARFP based storage rings, the time dependence now exists in both the \"static B-field\" and the frequency of the rf field given by Color online) The distribution of the geometric phase γ1 near the trap center (ρc, zc) of the storage ring proposed in Ref. [18] at (a) ϕ = 0, (b) ϕ = π/2, and (c) ϕ = π, clearly displaying the sin(ϕ/2) dependence.\n\nB s ( r, t) = B ′ ρê ρ -2B ′ zê z + B m sin(ω m t)ê z , -0.1 0 0.1 -0.1 0 0.1 -1 0 1 (z-z c )/ρ 0 γ 1 (a) (ρ-ρ c )/ρ 0 -0.1 0 0.1 -0.1 0 0.1 -5 0 5 (z-z c )/ρ 0 (ρ-ρ c )/ρ 0 γ 1 (b) -0.1 0 0.1 -0.1 0 0.1 -10 0 10 (z-z c )/ρ 0 γ 1 (c) (ρ-ρ c )/ρ 0 FIG. 2: (\n\nB o (t) = B rf sin[ω(t)t]ê z , ω(t) = ω 0 1 + (B m /B ′ ρ 0 ) 2 sin 2 (ω m t) . (36)\n\nThe frequency ω m is assumed to be much smaller than ω 0 but much larger than the trap frequency. The radius ρ 0 is now defined as\n\nρ 0 = ω 0 /|µ B g F B ′ |,\n\nand the \"effective 8 -0.05 0 0.05 -0.05 0 0.05 -0.2 0 0.2 (ρ-ρ 0 )/ρ 0 z/ρ 0 γ 1 (a) -0.05 0 0.05 -0.05 0 0.05 -0.5 0 0.5 (b) z/ρ 0 (ρ-ρ 0 )/ρ 0 γ 1 FIG. 3: (Color online) The geometric phase γ1 for the storage ring of Ref. [19] with (a) B rf = 0.15B ′ ρ0 and (b) B rf = 0.05B ′ ρ0.\n\nB-field\" takes the form\n\nB eff ( r, t) = ∆( r, t)ê z - B ′ ρ |2 B s ( r, t)| B rf êφ . ( 37\n\n)\n\nThe operating principle for the time averaged storage ring of Ref. [20] is similar to the well-known TOP [2] and TORT traps [3, 4] . The effective trap potential experienced by the atom is proportional to the time averaged value of the \"effective B-field\"\n\n2π/ωm 0 | B eff ( r, t)|dt.\n\nWhen B rf and B m are much smaller than B ′ ρ 0 , the center of the storage ring is located approximately at ρ c = ρ 0 , z c = 0. Using the earlier result [33] , we find that in the time averaged storage ring, the effective gauge potential Ã(φ) n (ρ, z) is reduced simply to the time averaged instantaneous gauge potential 28) . The geometric phase then is given approximately by γ n (ρ, z) = 2π Ã(φ) n (ρ, z). In this case, we find that the geometric phase always vanishes at -0.05 0 0.05 -0.05 0 0.05 -0.2 0 0.2 z/ρ 0 (ρ-ρ 0 )/ρ 0 (a) γ 1 -0.05 0 0.05 -0.05 0 0.05 -0.5 0 0.5 Color online) The geometric phase γ1 for the storage ring of Ref. [20] at (a) B rf = 0.3B ′ ρ0 and (b) B rf = 0.1B ′ ρ0. Bm = 0.05B ′ ρ0.\n\nÃ(φ) n (ρ, z) = ω m 2π 2π/ωm 0 A (φ) n (ρ, z, t)dt, ( 38\n\n) with A (φ) n (ρ, z, t) given in (\n\n(b) (ρ-ρ 0 )/ρ 0 z/ρ 0 γ 1 FIG. 4: (\n\nthe trap center (ρ c , z c ). Figure 4 illustrates the distribution of the geometric phase in the region near the trap center for two different values of the rf-field amplitude B rf . Similar to the storage ring of Ref. [19] , the fluctuation of the geometric phase is suppressed in this case for large B rf .\n\nB. The storage ring proposals of Refs. [21, 22] Next we consider the ARFP based storage ring proposed in Refs. [21, 22] . In this case, the static B-field is that of a Ioffe-Pritchard trap on an atom chip. In the Cartesian coordinate (x, y, z), it takes the form\n\nB s = B ′ xê x -B ′ yê y + B ′ Lê z , ( 39\n\n)\n\nwhere B ′ is the B-field gradient and the bias field along the z-direction is denoted as B ′ L. The amplitudes B\n\n(a) rf and B (b) rf (z) of the rf field are B\n\n(a) rf = [B rf (z)/ √ 2]ê x and B (b) rf = [B rf (z)/ √ 2]ê y with B rf (z) = B ( 0\n\n) rf + B ′′ z 2 . ( 40\n\n)\n\n9 In the schemes of Ref. [21, 22] considered earlier, the phase η of the rf field is assumed to be κπ/2. The xand y-components of the \"effective B-field\" B eff ( r) then become\n\nB eff x ( r) = B rf (z) 2 √ 2 (1 + cos β s (ρ, z)), B eff y ( r) = 0, ( 41\n\n)\n\naccording to Eq. (10). Then the strength of the \"effective B-field\" B eff has its minimum along a circle with a nonzero radius ρ c , provided a positive detuning ∆ exists at the origin (0, 0, 0) [21, 22] . The \"effective B-field\" B eff is easily shown to lie in the x-z plane along the trap bottom mapped out by the atomic center of mass motion. This gives rise to a vanishing γ (I) F . With a proper choice for the local phase of |n( r) eff , the gauge potential A (φ) n takes the form\n\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z) (1 -cos β s (ρ, z)) . ( 42\n\n)\n\nFigure 5 displays the geometric phase along a closed path for a spin-1 atom as a function of ρ c for the ARFP storage ring proposed in Refs. [21, 22] . The parameter λ is defined as\n\nλ = √ 2 ∆[ r = 0] |g F µ B B (0) rf | . ( 43\n\n)\n\nTo assure the validity of the rotating wave approximation, we find that the maximal values of ∆[ r = 0]/(|g F |µ B ) and B (0) rf / √ 2 must be restricted to the region of λ ∈ [0, 0.15].\n\nAs shown in Fig. 1 , the geometric phase remains much smaller than 2π in this situation. This fact can be appreciated easily if we look at the distribution of the \"effective B-field\" B eff . According to Eq. (41), the component B eff x has a nonzero minimal value B rf /(2 √ 2), while |B eff z | can become arbitrarily small, although not necessarily zero in general. Therefore, at the trap center where | B eff | is a minimum, the value of cos β eff = B eff z /| B eff | can become very small, leading to small geometric phases. Yet, despite the relatively small geometric phase found here, our result remains important because it could represent a systematic error if not properly included in a Sagnac interference experiment.\n\nIn Fig. 6 , we show the spatial distribution of the geometric phase γ 1 around the trap center with λ = 1/3 and λ = 3. The fluctuation is found to be relatively small when λ is small or when the rf-field amplitude B rf is large.\n\nAlthough not discussed in Refs. [21, 22] , a ring shaped trap also can be realized if we take η = -κπ/2. The \"effective B-field\" B eff still lies in the x-y plane\n\nB eff x ( r) = - B rf (z) 2 √ 2 cos(2φ)(1 -cos β s (ρ, z)), B eff y ( r) = B rf (z) 2 √ 2 sin(2φ)(1 -cos β s (ρ, z)), ( 44\n\n) 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 -0.08 -0.06 -0.04 -0.02 0 λ=1/3 λ=3 λ=1 ρ c /L γ (c) 1\n\nFIG. 5: (Color online) The geometric phase γ1 is plotted against the radius ρc for the ARFP storage ring of Ref. [21, 22] with η = κπ/2 at λ = 3, λ = 1, and λ = 1/3. To assure the validity of the rotating wave approximation, in the solid lines, the maximal value of ∆[ r = 0]/|gF µBB ′ ρ0| or B rf /( √ 2B ′ ρ0) are restricted to be smaller than 0.15. The extending dashed line is beyond the rotating wave approximation for λ = 1/3 and B rf /( √ 2B ′ ρ0) ∈ [0.15, 0.3].\n\nclearly giving rise to a non-zero solid angle with respect to a closed path along the storage ring. Therefore, the term γ (eff) F is non-zero in this case. We choose the eigenstates |n( r\n\n) s and |n( r) eff as |n( r) s = exp[-i F • ns ( r)β s (ρ, z)]|n z , |n( r) eff = exp[-i F • n eff ⊥ ( r)β eff (ρ, z)]|n z . ( 45\n\n)\n\nwith the unit vector ns ⊥ ( r) in the x-y plane orthogonal to B s ( r). In this case, the gauge potential A (φ) n can be expressed as\n\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z)[1 + cos β s (ρ, z)]. ( 46\n\n)\n\nIn Figure 7 , we show the fluctuation of the geometric phase γ 1 for a closed path with a new parameter\n\nλ ′ = 6 √ 2 ∆[ r = 0] |g F µ B B (0) rf | , ( 47\n\n)\n\nequal to 3 and 1/3. The fluctuation for γ 1 is found to be much larger than the case of η = κπ/2, which can be explained by the transverse components B eff x,y of the \"effective B-field.\" Because cos β s is always close to unity.\n\nIn the case of η = -κπ/2, B eff x,y can take only small positive values. Therefore, at the minimum of the ARFP\n\ntrap ρ = ρ 0 of | B eff |, both B eff z\n\nand B eff x,y have to be close to zero. In this case the value for cos β eff becomes a rapidly changing function of ρ in the region near ρ c .\n\nOur above calculations have obtained analytical expressions of the geometric phases in an ARFP based storage ring for η = ±κπ/2. We have further investigated the 10 -0.05 0 0.05 -0.05 0 0.05 -0.06 -0.02 0.02 (z-z c )/L (a) (ρ-ρ c )/L γ 1 -0.05 0 0.05 -0.05 0 0.05 -0.6 0 0.6 (z-z c )/L (b) (ρ-ρ c )/L γ 1 FIG. 6: (Color online) The spatial distribution of the geometric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ = 1/3 and (b) λ = 3. η = κπ/2. B (0) rf = 0.08B ′ L and B ′′ = 10 -12 B ′ /L are assumed.\n\nfluctuations of the geometric phase for the two cases of η = ±κπ/2. It seems one benefits from implementing a Sagnac interferometer in the discussed ARFP storage ring with η = κπ/2 and operating at a relatively large λ.\n\nBefore proceeding onto the concluding section, we will discuss the geometric phase in an ARFP based beam splitter created via a double potential [14, 21] . In such an implementation, the static field B s is created from a Ioffe-Pritchard trap, while the oscillating rf field components are B (a) rf = B rf [z]ê x and B (b) rf = 0. By spatially tuning the amplitude of B rf from zero to a significant value, in the x-y plane, an ARFP can be tuned from a single well centered near the origin to a double well with two minimal points at the point with nonzero radius ρ 0 and φ = 0, π. Therefore, a Y-shaped atom beam splitter can be accomplished when the B rf [z] initially is increased along the z-axis to a large value, and then decreased to zero. In such an arrangement, the atom beam moving along the z direction can be separated into two beams that move along the z-axis at φ = 0, π for a while, and then can be recombined again into a single beam.\n\n-0.05 0 0.05 -0.05 0 0.05 -10 0 10 (z-z c )/L (a) (ρ-ρ c )/L γ 1 -0.05 0 0.05 -0.05 0 0.05 -15 0 15 (z-z c )/L (b) (ρ-ρ c )/L γ 1 FIG. 7: (Color online) The spatial distribution of the geometric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ ′ = 3 and (b)λ ′ = 1/3. η = -κπ/2. B (0) rf = 0.08B ′ L and B ′′ = 10 -12 B ′ /L are assumed.\n\nIn the atom interferometer considered above, both the static field B s and the \"effective B-field\" B eff are limited to the x-z plane. Therefore, for motion along the closed path of the trap bottom, the solid angle enclosed by the trajectory of B eff is zero. Thus, the geometric phase in (16) can be expressed as\n\nγ n (t) = -i t 0 l | eff n( r)|l z | 2 s l( r)|∇|l( r) s • vdt ′ . ( 48\n\n)\n\nWe can show that the product r) s is a function of ρ c and is independent of z. Thus, the geometric phase can be expressed as an integral of this function with respect to ρ c , from zero to a large value and then back to zero. Therefore, the value of the geometric phase would be zero in the end.\n\neff n( r)|l z | 2 s l( r)|∇|l(\n\n11 V. CONCLUSION\n\nIn this study, we develop theoretical formalisms for the calculation of the atomic geometric phase inside an ARFP. We show that, due to the complexity of the ARFP, the geometric phase depends on the spatial variation of both the static field and an \"effective B-field\" B eff . We provide general expressions for the geometric phase and the corresponding adiabatic gauge potential in Eqs. (16) and (23), respectively.\n\nTo shed light on actual applications of the atomic geometric phase, we investigate the distribution of atomic geometric phases for several proposed or ongoing experiments with ARFP based storage rings and atom beam splitters. We prove rigorously that the geometric phase in the center of the storage rings proposed in Refs. [18, 19] is always zero. In addition, we find that in the storage ring of Ref. [18] , the spatial fluctuation of the geometric phase sensitively depends on the position of the trap center on the \"resonance toroid.\" In the proposals of Refs. [19, 20, 21, 22] , the fluctuation for the geometric phase becomes significantly suppressed when the amplitude B rf of the rf-field is large. In the proposals of [21, 22] , the fluctuations of the geometric phase also is suppressed if the angle η is set to be κ2π. In the beam splitter realized with the double well potential ARFP [14, 21], the geometric phase is shown to be zero.\n\nOur work helps to clarify the working principle of trapping neutral atoms in an ARFP and the validity conditions for the various approximations involved. We hope our results will shine new light on the proposed inertial sensing experiments based on trapped atoms in ARFP."},{"section_type":"OTHER","section_title":"Acknowledgments","text":"We thank Dr. T. Uzer and Dr. B. Sun for helpful discussions. This work is supported by NASA, NSF, CNSF, and the 863 and 973 programs of the MOST of China.\n\n[1] D. E. Pritchard, Phys. Rev. Lett. 51, 1336 (1983). [2] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A.\n\nCornell, Phys. Rev. Lett. 74, 3352 (1995). [3] A.S. Arnold and E. Riis, J. Mod. Opt. 49, 959 (2002); C. S. Garvie, E. Riis, and A. S. Arnold, Laser Spectroscopy XVI, edited by P. Hannaford et al. 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For instance, the widely used quadrupole trap and the Ioffe-Pritchard trap [1] are usually created with static B-fields, while the time averaged orbiting potential (TOP) [2] and time orbiting ring trap (TORT) [3, 4] are created using oscillating B-fields with frequencies larger than the effective trap frequencies. Atom chips [5] have brought further developments to magnetic trap technology, as they can provide larger B-fields and gradients at reduced power-consumptions or electric currents using micro-fabricated coils. Today, magnetic trapping is a versatile tool used in many laboratories around the world for controlling atomic spatial motion in regions of different scales and geometric shapes, e.g., 3D or 2D traps, double well traps, and storage ring traps [1, 2, 3, 4, 5, 6, 7] .\\n\\nRecently, magnetic traps based on adiabatic microwave [8, 9] and adiabatic radio frequency (rf) potentials (ARFP) [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] have attracted considerable attention. An ARFP is typically created with the combination of a static B-field and an rf field. The idea for an ARFP has been around for some time [10, 11, 12] , and experimental demonstrations recently have been carried out for confining both thermal [13] and Bose condensed atoms [14, 15, 16, 17] . Further development of improved ARFP with atom chip technology likely will assist in practical applications of atom interferometry. For instance, a double well potential was constructed recently using low order multi-poles capable of atomic beam splitting while maintaining tight spatial confinement [21] . Several interesting recent proposals outline the construction of small storage rings with radii of the order 1µm [18, 19, 20, 21, 22] , which could become useful if implemented for atom Sagnac interferometry [27] setups on atom chips.\\n\\nWhen a neutral atom is confined in a magnetic potential, its hyperfine spin is assumed to follow adiabatically the spatial variation of the B-field direction during its spatial translational motion. As a result of this adiabatic approximation, the center of mass motion for the atom experiences an induced gauge field [28] , giving rise to a geometric phase (or Berry's phase) to the atomic internal spin state [29, 30] . The effect of this geometric phase is widely known, and is first addressed carefully in a meaningful way for atomic quantum gases by an explicit calculation of the resulting geometric phase in a static or a time averaged magnetic trap in Ref. [31] . Several important consequences are predicted to occur for a magnetically trapped atomic condensate in a quadrupole trap, a Ioffe-Pritchard trap [31, 32] , or a TORT based storage ring [33] . To our knowledge, this geometric phase effect has not been investigated in any detail for an atom inside an ARFP.\\n\\nIn a recent paper, we show that this geometric phase causes an effective Aharonov-Bohm-type [34] phase shift in a magnetic storage ring based atom interferometer [33] . In addition, our studies imply that the spatial fluctuation of the geometric phase can lead to a reduction of the visibility of the interference contrast. In view of this, we decided to carry out this study as reported here for the atomic geometric phase in an ARFP in order to shed light on the proposed high precision atom Sagnac interference experiment [27] . Analytical derivations for this study at some places become rather tedious and complicated. We therefore first will summarize our major results here for readers who may not be interested in the intricate details. We find that the geometric phase in an ARFP generally takes a more complicated form in comparison to the case of a static trap or a time averaged trap. In an ARFP, this phase factor is found to be determined by the trajectory of the time independent component of the trap field as well as an \\\"effective B-field\\\" that depends on the total B-field. In contrast to the earlier result found for a static trap or a time averaged trap [31, 33] , the final result turns out to be not expressible as a functional of the trajectory 2 for the direction of the total B-field in the parameter space. This paper is organized as follows. In sec. II, we generalize the semi-classical approach as outlined in Ref. [21] for the operating principle of an ARFP to a form more convenient for discussing the geometric phase. Section III parallels that of sec. II by reformulating a full quantum theory for discussing the geometric phase inside an ARFP [22] . The explicit expression for the geometric phase inside an ARFP is given in a readily adaptable form for specifical calculations. In sec. IV, we discuss the effect of the geometric phase in several types of ARFP recently proposed for atomic splitters and storage rings [18, 19, 20, 21, 22] and beam splitters [14, 21] . Finally, concluding remarks are given in sec. V.\"\n },\n {\n \"section_type\": \"METHOD\",\n \"section_title\": \"II. A SEMI-CLASSICAL APPROACH\",\n \"text\": \"In this section, we provide a semi-classical formulation for calculating the atomic geometric phase inside an ARFP. The semi-classical working principle for an ARFP is described in Ref. [21] , although only for the special case when the atomic center of mass is assumed at a fixed location. In order to calculate the geometric phase, our formulation allows for the explicit consideration of atomic center of mass motion classically. In our approach, the geometric phase is obtained naturally, and the validity conditions for both the adiabatic and the rotating wave approximations are clearly shown for an ARFP.\\n\\nInside an ARFP [21] , the total B-field B( r, t) is the sum of a static field component B s ( r) and an oscillatory rf field B o ( r, t), which conveniently is expressed as B o ( r, t) = B (a) rf ( r, t) cos(ωt) + B (b) rf ( r, t) cos(ωt + η). (1)\\n\\nwhere r is the spatial position vector of the atom, ω is frequency of the rf field, and η is a relative phase factor.\\n\\nIn this section, we will assume that the atomic spatial motion is pre-determined, i.e., r(t) is given (as a slowly varying function of time t). For weak B-fields, the system Hamiltonian is simply the linear Zemman interaction\\n\\nH(t) = g F µ B F • B[ r(t), t], ( 2\\n\\n)\\n\\nwhere g F is the corresponding Lande g-factor and µ B denotes the Bohr magneton. = 1 is assumed. For a static or a time averaged magnetic trap, the Hamiltonian (2) varies slowly over time scales of the Larmor precession of the atomic spin in the total B-field.\\n\\nDuring the effectively slow trapped motion, the atomic hyperfine spin is assumed to be fixed at the instantaneous eigenstate of the Hamiltonian (2). The geometric phase then can be calculated straightforwardly from the variation of the B-field direction in the parameter space [31, 33] .\\n\\nIn an ARFP, the situation is more complicated. Although the variation of B s [ r(t)] remains much slower than the Larmor precession, the rf frequency ω usually is assumed to be nearly resonant with the precession frequency. Thus, the Hamiltonian (2) contains both fast and slow time varying components, making the direct calculation of the geometric phase a more involved task.\\n\\nIn the following, we will proceed step by step, clarifying the various approximations adopted along the way.\\n\\nTo understand the working principle for an ARFP, we first decompose the Hamiltonian H(t) (2) into the following form\\n\\nH(t) = H s [ r(t)] + H + [ r(t)]e -iωt + H -[ r(t)]e iωt , ( 3\\n\\n)\\n\\nwhere H s and H ± are all slow varying functions of time and are given by\\n\\nH s [ r(t)] = g F µ B F • B s [ r(t)], H + [ r(t)] = 1 2 g F µ B F • B (a) rf [ r(t)] + e -iη B (b) rf [ r(t)] , H -[ r(t)] = H † + [ r(t)]. ( 4\\n\\n)\\n\\nH s is diagonal in the spin angular momentum basis defined along the local direction of the static Bfield B s [ r(t)]. The eigenstate takes the familiar form |m F [ r(t)] s , quantized along the direction of B s [ r(t)], with the eigenvalue\\n\\nm F |B s [ r(t)]| for B s [ r(t)] • F and m F ∈ [-F, F ], in analogy with the usual case of the z- quantized representation result of F z |m F z = m F |m F z .\\n\\nNext we introduce a unitary transformation\\n\\nU (t) = F mF =-F |m F z s m F [ r(t)]|e imF κωt , ( 5\\n\\n)\\n\\nwith κ = sign(g F ) for the rotating wave approximation. The quantum state in the interaction picture |Ψ(t) I = U (t)|Ψ(t) defined by U (t) is governed by the Schroedinger equation\\n\\ni∂ t |Ψ(t) I = H I (t)|Ψ(t) I , with\\n\\nthe Hamiltonian in the interaction picture given by\\n\\nH I (t) = U HU † + i(∂ t U )U † = F m=-F mκ∆[ r(t)]|m z z m| -i F m,n=-F |m z s m[ r(t)]| d dt |n[ r(t)] s z n|e i(m-n)κωt\\n\\n3 + F m=-F +1 h (+) m [ r(t)]|m z z m -1| + h (-) m [ r(t)]|m z z m -1|e 2iκωt + h.c.\\n\\n+ F m=-F h m [ r(t)]|m z z m|e iκωt + h.c. , ( 6\\n\\n)\\n\\nwhere the time dependent parameters are defined as\\n\\n∆[ r(t)] = µ B |g F B[ r(t)]| -ω, h (±) m [ r(t)] = s m[ r(t)]|H ± [ r(t)]|(m -1)[ r(t)] s , and (7) h m [ r(t)] = s m[ r(t)]|H ± [ r(t)]|m[ r(t)] s .\\n\\nThe above result is obtained easily if we note that the matrix element s m[ r(t)|H ± (t)|m ′ [ r(t) s is non-zero only when m -m ′ = 0, ±1. So far, we have always assumed that |m[ r(t)] s is a single valued function of the atomic position r. A careful examination shows that the eigenstate |m[ r(t)] s cannot be determined uniquely because of the presence of the U (1) gauge freedom for selecting a local phase factor exp{iφ[ r(t)]}, which consequently affects the resulting expressions for h\\n\\n(±) m (t) and s m[ r(t)]|d/dt|m ′ [ r(t)] s .\\n\\nThe rotating wave approximation neglects of the oscillating terms proportional to e imωt (m = 0) in the Hamiltonian H I (6) . The error for this approximation is estimated easily from a time dependent perturbation calculation. The sufficient condition for its validity requires that all factors such as\\n\\nt 0 dt ′ h m (t ′ ) exp[iκωt ′ ], t 0 dt ′ h (-) m (t ′ )ξ m,m-1 (t ′ ) exp[iκ(2ω + ∆)t ′ ], and t 0 dt ′ m[ r(t ′ )]|d/dt ′ |n[ r(t ′ )] ξ mn (t ′ ) exp[i(m -n)κ(ω + ∆)t ′ ] are negligible, where ξ mn (t) = exp t 0 dt ′ s m[ r(t ′ )]| d dt ′ |m[ r(t ′ )] s × exp - t 0 dt ′ s n[ r(t ′ )]| d dt ′ |n[ r(t ′ )] s . ( 8\\n\\n)\\n\\nThus, the gauge independent factors h (-)\\n\\nm ξ m,m-1 , m s |d/dt|n s ξ mn , h m ,\\n\\nand ∆ should all vary slowly with time and with the modulus of their amplitudes much less than ω.\\n\\nThe effective Hamiltonian in the interaction picture under the rotating wave approximation then becomes\\n\\nH (I) eff (t) = µ B g F F • B eff [ r(t)] -i F m=-F |m z s m[ r(t)]| d dt |m[ r(t)] s z m|, ( 9\\n\\n)\\n\\nwhere the first term resembles a coupling between the atomic spin and an \\\"effective B-field\\\" B eff ( r), whose components in real space are given by\\n\\nB eff x ( r) = Re 2 s m( r)|H ± ( r)|(m -1)( r) s µ B g F (F + m)(F -m -1) , B eff y ( r) = -Im 2 s m( r)|H ± ( r)|(m -1)( r) s µ B g F (F + m)(F -m -1) , and\\n\\nB eff z ( r) = B s ( r) - ω µ B |g F | . ( 10\\n\\n)\\n\\nClearly, the x-and y-components of the effective field B eff ( r) depend on the explicit form of the eigenstate |m( r) s . In fact, it easily can be seen that different choices of the local phase factor for the |m( r) s actually lead to different values of B eff ( r) related to each other through r-dependent rotations in the x-y plane. In practice, the eigenstate |n( r) s and the effective field B eff can sometimes be constructed more simply, as in Ref. [21] . For any spatial position r, we first choose a rotation R[ m( r), χ( r)] along the axis m( r) with an angle χ( r\\n\\n) that satisfies R[ n( r), χ( r)] B s ( r) = | B s ( r)|ê z . It is then easy to show that the eigenstate |n[ r(t)] s can be chosen as |n( r) s = exp i F • m( r)χ( r) |n z . ( 11\\n\\n)\\n\\nUnfortunately, the choice for R is not unique in a given static field B s ( r), an analogous result to the U (1) gauge freedom for the the egienstate |n( r) s . Corresponding to the choice (11) given above for |n( r) s , the unitary transformation U defined in (5) would become\\n\\nU (t) = exp(-iF z ωt) • exp -i F • m( r)χ( r) , ( 12\\n\\n)\\n\\nand the transverse components of the \\\"effective B-field\\\" given by\\n\\nB eff x,y ( r) = B x,y ( r)/2 [21] with B( r) = R[ m( r), χ( r)] B (a) rf ( r) +R[ê z , -κη]R[ m( r), χ( r)] B (b) rf ( r). (13)\\n\\nIn earlier discussions of an ARFP [21, 22] , the atomic internal state is assumed uniformly to remain adiabatically in a certain eigenstate of the first term of H (I) eff (t). To fully appreciate this adiabatic approximation and to calculate the geometric phase, we expand |Ψ(t) I into the instantaneous eigenstate basis |n[ r(t)] eff quantized along the direction of the effective B-field\\n\\nB eff accord- ing to |Ψ(t) I = n C n (t)|n[ r(t)] eff .\\n\\nThe first term of H (I) eff (t) is simply the effective Zemman interaction between the atomic hyperfine spin and the effective B-field. The corresponding Schroedinger equation for the Hamiltonian H (I) eff (t) of (9) then becomes\\n\\ni d dt C n (t) = [ǫ (n) I (t) + ν nn (t)]C n (t) + m =n ν nm (t)C m (t), 4 with ǫ (n) I (t) = nµ B g F | B eff (t)|, ν pq (t) = -i l eff p[ r(t)]|l z s l[ r(t)]| d dt |l[ r(t)] s z l|q[ r(t)] eff -i eff p[ r(t)]| d dt |q[ r(t)] eff . ( 14\\n\\n)\\n\\nUnder the adiabatic approximation, the atomic internal state remains in a given eigenstate |n[ r(t)] eff with transitions to states |m[ r(t)] eff (m = n) being negligibly small. Thus, the transition probability, as estimated from the first order perturbation theory,\\n\\nt 0 dt ′ ν nm (t ′ )e i R t ′ 0 dt ′′ [ǫ (n) I (t ′′ )+νnn(t ′′ )-ǫ m I (t ′′ )-νmm(t ′′ )]\\n\\nshould be much less than one. As before, we find the sufficient condition for the validity of the adiabatic approximation is given by\\n\\n|ν mn (t)| |ǫ m I (t ′ ) -ǫ (n) I (t ′ )| ≪ 1, ( 15\\n\\n) provided that ν mn (t) exp[i t ′ 0 [ν nn (t ′′ ) -ν mm (t ′′\\n\\n)]dt ′ ], which is independent of the local phase factor for |n[ r(t)] eff and |n[ r(t)] s , remains a slowly varying function of time.\\n\\nA straight forward calculation from the effective Hamiltonian (9) then gives the general expression for the geometric phase in an ARFP\\n\\nγ n (t) = t 0 ν nn (t ′ )dt ′ = -i t 0 l | eff n[ r(t ′ )|l z | 2 × s l[ r(t ′ )]| d dt ′ |l[ r(t ′ )] s dt ′ +γ (I) n (t), ( 16\\n\\n) and γ (I) n (t) = -i t 0 dt ′ eff n[ r(t ′ )|d/dt ′ |n[ r(t ′ ) eff .\\n\\nDuring the adiabatic motion in a given internal state, the time evolution of the coefficient C n (t) takes the form (16) is the central result of this work. The geometric phase of an atom inside an ARFP is shown to contain two parts. The second part, γ 9) , with its value determined by the trajectory of the direction for the \\\"effective B-field\\\" B eff . The first part arises from the second term of H (I) eff (9). It is determined by the trajectories of both the static field B s and the effective B-field B eff . The expression for γ n in an ARFP is complicated because the internal quantum state in an ARFP is assumed to be adiabatically kept in an eigenstate of µ B g F F • B eff , rather than an eigenstate of the total interaction Hamiltonian H (I) eff (t). In section IV, we will perform explicit calculations for several examples of ARFP proposed for various applications: e.g., as atomic storage rings or atomic beam splitters. Most often we find that only the first part of Eq. (16) contributes a non-zero value to the geometric phase.\\n\\nC n (t) = C n (0)e -i R t 0 ǫ (n) I (t ′ )dt ′ e -iγn(t) . ( 17\\n\\n) Equation\\n\\n(I) n (t), is clearly due to the interaction term µ B g F F • B eff in H (I) eff (\\n\\nBefore proceeding to the next section for a quantal treatment of the geometric phase, we find the time evolution of the atomic spin state in the Schroedinger picture\\n\\n|Ψ(t) = ml C l (0) z m|l[ r(t)] eff × e -i R t 0 ǫ l I (t ′ )dt ′ e -iγ l (t) e -imωt |m[ r(t)] s , ( 18\\n\\n) obtained directly from |Ψ(t) = U † (t)|Ψ(t) I after the\\n\\napplications of the rotating wave and adiabatic approximations. When the atom is prepared initially in a specific adiabatic state |n[ r(t)] eff of the interaction picture, we arrive at the simple case of C l (0) = δ ln .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. A QUANTUM MECHANICAL TREATMENT\",\n \"text\": \"In the previous section, we provided the result for the geometric phase γ n (t) in an ARFP based on a semiclassical approach, where the atomic center of mass motion is described classically. A clear physical picture exists in this case for the appearance of the geometric phase in a certain parameter space. The validity conditions for the rotating wave and the adiabatic approximations as obtained above are all formulated in terms of gauge independent forms. However, if the influence of the geometric phase on the atomic spatial motion is to be included, e.g., as in the Aharonov-Bohm-type, phase shift, interference arrangement in an atomic Sagnac interferometer discussed earlier [33] , we would need an improved description where both the atomic spin and its center of mass motion are treated quantum mechanically.\\n\\nIn a full quantum treatment of the atomic motion, the quantum state of an atom can be expressed as |Φ(t) = φ l ( r, t)|l z , where φ l ( r, t) is the atomic spatial wave function for the internal state |l z of F z . The state |Φ(t) then satisfies the Schroedinger equation governed by the Hamiltonian\\n\\nH = P 2 2M + g F µ B F • B( r, t), ( 19\\n\\n)\\n\\nwith P being the kinetic momentum and M the atomic mass. The rotating wave and adiabatic approximations can be introduced now by defining the interaction picture 5 with the unitary transformation\\n\\nU(t) = F m=-F |m z eff m( r)| × F n=-F |n z s n( r)|e inκωt . ( 20\\n\\n)\\n\\nThe state in the interaction picture |Φ(t) I = U(t)|Φ(t) now is governed by the Schroedinger equation with the Hamiltonian H eff = UHU † . Under the rotating wave and adiabatic approximations, we neglect transitions between states |m z and |n z (m = n) as well as the rapidly oscillating terms. We then obtain\\n\\nH eff ≈ n |n z z n|H eff |n z z n| ≈ n H (n) ad |n z z n|, ( 21\\n\\n)\\n\\nwhere the adiabatic Hamiltonian H (n) ad for the n-th adiabatic branch is defined as\\n\\nH (n) ad = P -A n 2 2M + ǫ (n) I ( r), ( 22\\n\\n)\\n\\nwith the effective gauge potential\\n\\nA n ( r) = -i l | eff n( r)|l z | 2 s l( r)|∇|l( r) s -i eff n( r)|∇|n( r) eff . ( 23\\n\\n)\\n\\nIn this form, it is well known that the geometric phase γ n can be expressed as the integral of the gauge potential A n along the spatial trajectory for the atomic center of mass in an ARFP, i.e., one would expect generally that γ n = A n • d r. Similar to the result of the semiclassical approach, the gauge potential A n ( r) can be expressed as the sum of two parts. The first part in Eq. (23) is the weighted sum of the atomic gauge potential -i s l( r)|∇|l( r) s from the static field B s , while the second term is the atomic gauge potential from the \\\"effective B-field\\\" B eff .\\n\\nA full quantum treatment for atomic motion in an ARFP has been attempted earlier [22] . In fact, many of our formulations are identical to the results of Ref. [22] . For instance, it is easy to show that the unitary transformations U S , U R , and U F in [22]\\n\\nare related directly to ours as U † F U † R U † S = U.\\n\\nThe only difference concerns the gauge potential A n that was neglected in Ref. [22] . Thus, they did not give the expression for the gauge potential, and the result for the geometric phase was not obtained either [22] . Our study shows that the neglect of the adiabatic gauge potential potentially can give rise to a final result, dependent on the choice of the local phase factors for the internal eigenstate.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. GEOMETRIC PHASES IN ARFP BASED APPLICATIONS\",\n \"text\": \"In the above two sections, we obtain the expression for the atomic geometric phase in an ARFP. This section is devoted to the calculations of the geometric phases for several proposed applications of ARFP, such as storage rings or beam splitters for neutral atoms [18, 19, 20, 21, 22] .\\n\\nBefore presenting our results for the more specific cases, we provide some general discussions of the geometric phases in several ARFP based storage rings. As was pointed out earlier, the geometric phase γ n is given by the line integral of the gauge potential A n along the trajectory for the atomic center of mass motion. For a closed path in the storage ring at a fixed ρ = ρ c and z = z c , this can be further reduced to\\n\\nγ n = q 2π 0 A (φ) n (ρ, φ, z)ρdφ, ( 24\\n\\n)\\n\\nwhere the integer q is the winding number of the path and A (φ) n is the component of A n along the azimuthal direction êφ of the familiar cylindrical coordinate system (ρ, φ, z). Without loss of generality, we take q = 1 in this paper. For the storage rings proposed in Refs. [18, 19, 20, 21, 22] , the gauge potentials A (φ) n (ρ, φ, z) are actually independent of the angle φ. Therefore, the geometric phase is simply given by\\n\\nγ (c) n = 2πρ c A (φ) n (ρ c , z c ), ( 25\\n\\n)\\n\\ngiven out in explicit forms for different storage ring schemes [18, 19, 20, 21, 22] .\\n\\nIn reality, because of thermal motion or when the atomic transverse motional state is considered, the center of mass for an atom can deviate from (ρ c , z c ) even for a closed trajectory. This uncertainty in the exact shape of the closed trajectory gives rise to a fluctuating geometric phase and is usually difficult to study. Assuming a simple closed path at fixed ρ and z, we have found previously that the subsequently fluctuations could decrease the visibility of the interference pattern [33] . Quantum mechanically, such destructive interference can be explained as resulting from entanglement between the freedoms for φ and (ρ, z) because of the dependence of the gauge potential A φ n on ρ and z. Therefore, it is important to investigate this dependence near the trap center.\\n\\nFor simplicity, our discussions below will focus on the closed loops where ρ and z are φ-independent constants. In this case, the geometric phase can be expressed as ρ, z) . We will show numerically the distributions for γ n (ρ, z) obtained this way near the central region of (ρ c , z c ). If needed, a more rigorous approach can be developed to investigate the fluctuations of the resulting geometric phase from the gauge potential A (φ) n (ρ, z).\\n\\nγ n (ρ, z) = 2πρA (φ) n (\\n\\nA. The storage ring proposals of Refs. [18, 19, 20] This subsection is devoted to a detailed calculation of the geometric phases for the ARFP storage ring proposals of Refs. [18, 19, 20] . We will derive the analytical expressions for the azimuthal component A (φ) n of the gauge potential that arises in both cases from cylindrically symmetric static B-field and rf fields. Because of the cylindrical symmetry, the angle β s (ρ, z) between the local static B-field and the z-axis is required to be analytical in the region near the storage ring. Therefore, the eigenstate |n( r) s of F • B s can be chosen as\\n\\n|n( r) s = exp{-i[ F • êφ β s (ρ, z) + nφ]}|n z . ( 26\\n\\n)\\n\\nConsequently, B eff ( r) is also cylindrically symmetric, which leads to the eigenstate |n( r\\n\\n) eff of F • B eff as |n( r) eff = exp{-i[ F • n eff ⊥ ( r)β eff (ρ, z) + nφ]}|n z , ( 27\\n\\n)\\n\\nwith the unit vector n eff ⊥ ( r) in the x-y plane orthogonal to B eff ( r) and β eff (ρ, z) denoting the angle between B eff ( r) and the z-axis. We note that the unit vector field n eff ⊥ ( r) also possesses cylindrical symmetry, i.e., remains invariant under rotation around the z-axis. The expressions of (26) and (27) allow us to obtain the simple expression of the gauge potential\\n\\nA (φ) n (ρ, z) = - n ρ cos β eff (ρ, z) cos β s (ρ, z), ( 28\\n\\n)\\n\\nafter straightforward calculations. In the scheme of Ref. [18] , the static B-field is a \\\"ringshaped quadrupole field\\\" that vanishes along a circle of a radius ρ 0 in the x-y plane. Near ρ = ρ 0 , the B-field is given approximately by\\n\\nB s ( r) = B ′ (ρ -ρ 0 )ê ρ -B ′ zê z , ( 29\\n\\n) like a quadrupole field, while the rf-field takes a complicated form\\n\\nB o ( r, t) = a √ 2 cos(ωt) + b √ 2 cos(ωt + ϕ) êρ + - a √ 2 sin(ωt) + b √ 2 sin(ωt + ϕ) êz , ( 30\\n\\n)\\n\\nwith constants a and b independent of r. From the expression of (26) for the eigenstate |n( r) s , the \\\"effective B-field\\\" B eff becomes\\n\\nB eff ( r) = B ′ [ (ρ -ρ 0 ) 2 + z 2 -r 0 ]ê z - b √ 2 cos(θ + ϕ) + a √ 2 cos θ êρ + b √ 2 sin(θ + ϕ) - a √ 2 sin θ êφ , ( 31\\n\\n)\\n\\nFIG. 1: (Color online) A cross-sectional view for the storage ring of Ref. [18] . The static field is zero in the ring at the fixed radius ρ0. The addition of rf-fields creates an ARFP centered at a ring through (ρc, zc). The distance from the trap center to the ring with radius ρ0 in the plane z = 0 is r0.\\n\\nwhere r 0 and θ are given by\\n\\nr 0 = ω |µ B g F B ′ | , cos θ(ρ, z) = ρ -ρ 0 (ρ -ρ 0 ) 2 + z 2 , sin θ(ρ, z) = z (ρ -ρ 0 ) 2 + z 2 . ( 32\\n\\n)\\n\\nIn an ARFP, as discussed here, the trap center at (ρ c , z c ) is determined by minimizing both the zcomponent and the transverse component of B eff . Without loss of generality, we will assume a, b > 0. Then, (ρ c , z c ) is found to satisfy\\n\\nθ(ρ c , z c ) = -ϕ/2, (ρ c -ρ 0 ) 2 + z 2 c = r 0 , ( 33\\n\\n)\\n\\ni.e., the trap center lies on the surface of the \\\"resonance toroid\\\" at ρ = ρ 0 with a radius r 0 as shown in Fig. 1 . The relative angle of the trap center with respect to the center of the toroid cross-section is given by -ϕ/2. On this \\\"resonance toroid,\\\" the rf-field is resonant with the static field, i.e., B eff z vanishes. As a result, the \\\"effective B-field\\\" lies again in the x-y plane on the \\\"resonance toroid,\\\" which gives cos β eff (ρ c , z c ) = 0 and leads to the result A (φ) n (ρ c , z c ) = γ n = 0 as shown in the trap center for the storage ring considered before in Ref. [18] .\\n\\nFrom the expression (29) of the static field and the definition of the angle θ(ρ, z), we find a simple relationship β s (ρ, z) = π/2 + θ(ρ, z), with which the gauge potential A (φ) n (ρ, φ) in (28) can be further simplified as\\n\\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z) sin θ(ρ, z) 7 ≈ n ρ cos β eff (ρ, z) sin θ(ρ c , z c ), ( 34\\n\\n)\\n\\nnear the trap center. Thus, the spatial fluctuation of the gauge potential A (φ) n (ρ, z) in the region around the trap center is closely related to the angle θ(ρ c , z c ) of the trap center, or the parameter ϕ of the oscillating field B o . When ϕ = 0, the atom is trapped in the region with θ ≈ 0 or π, where the fluctuation of A (φ) n (ρ, z) is suppressed significantly due to the small value of sin θ. On the other hand, if the angle ϕ is set to π with the trap center located in the region with θ ≈ ±π/2, the fluctuation of the gauge potential becomes amplified.\\n\\nIn Fig. 2 , we illustrate numerical results for the distribution of the geometric phase γ 1 (ρ, z) = 2πρA φ 1 (ρ, z) in the region near the trap center at ϕ = 0, π/2, π. We see clearly decreased fluctuations of γ 1 when the absolute value of sin θ(ρ c , z c ) = -sin(ϕ/2) is decreased.\\n\\nNext we turn to the storage ring of Ref. [19] constructed from a quadrupole static B-field B s ( r) = B ′ (x, y, -2z) and an r-independent rf field B o = B rf cos(ωt)ê z along the z direction. The resulting ARFP provides a 2D ring shaped trap in the x-y plane. In addition, a 1D optical potential along the z direction is employed to confine atoms in the transverse plane at z = 0 [19]. The \\\"effective B-field\\\" takes the form\\n\\nB eff ( r) = B ′ (ρ -ρ 0 )ê z - 1 2 B rf êρ , ( 35\\n\\n)\\n\\nin the plane at z = 0, with\\n\\nρ 0 = ω/|µ B g F B ′ |.\\n\\nBecause the strength of B eff is near minimum at the ring ρ = ρ 0 , the trap center for this storage ring is located at ρ c = ρ 0 and z c = 0. At the trap center, the \\\"effective Bfield\\\" is along the direction of êρ . Thus, according to Eq. ( 28 ), the geometric phase γ (c) n at the trap center again vanishes.\\n\\nIn Fig. 3 , we show the distribution of the geometric phase γ 1 in the region near the trap center for B rf = 0.05|B ′ |ρ 0 and B rf = 0.15|B ′ |ρ 0 . We see that the fluctuation is relatively small when the strength of the rf-field is large. This can be explained by Eq. (28) , which shows that A (φ) n is proportional to cos β eff and can be approximated as 2B eff z /B rf near the trap center. When B rf is large, the gauge potential becomes a relatively slow varying function of ρ and z. In this case, the presence of a 1D optical potential allows for the possibility of tuning the trap center position to a nonzero value of z, with the storage ring remaining in the x-y plane. Then cos β s is assumed to a nonzero value, leading to increased fluctuations for the geometric phase.\\n\\nFinally, we discuss the geometric phase in the \\\"time averaged\\\" ARFP storage ring proposed in Ref. [20] . Unlike previously considered ARFP based storage rings, the time dependence now exists in both the \\\"static B-field\\\" and the frequency of the rf field given by Color online) The distribution of the geometric phase γ1 near the trap center (ρc, zc) of the storage ring proposed in Ref. [18] at (a) ϕ = 0, (b) ϕ = π/2, and (c) ϕ = π, clearly displaying the sin(ϕ/2) dependence.\\n\\nB s ( r, t) = B ′ ρê ρ -2B ′ zê z + B m sin(ω m t)ê z , -0.1 0 0.1 -0.1 0 0.1 -1 0 1 (z-z c )/ρ 0 γ 1 (a) (ρ-ρ c )/ρ 0 -0.1 0 0.1 -0.1 0 0.1 -5 0 5 (z-z c )/ρ 0 (ρ-ρ c )/ρ 0 γ 1 (b) -0.1 0 0.1 -0.1 0 0.1 -10 0 10 (z-z c )/ρ 0 γ 1 (c) (ρ-ρ c )/ρ 0 FIG. 2: (\\n\\nB o (t) = B rf sin[ω(t)t]ê z , ω(t) = ω 0 1 + (B m /B ′ ρ 0 ) 2 sin 2 (ω m t) . (36)\\n\\nThe frequency ω m is assumed to be much smaller than ω 0 but much larger than the trap frequency. The radius ρ 0 is now defined as\\n\\nρ 0 = ω 0 /|µ B g F B ′ |,\\n\\nand the \\\"effective 8 -0.05 0 0.05 -0.05 0 0.05 -0.2 0 0.2 (ρ-ρ 0 )/ρ 0 z/ρ 0 γ 1 (a) -0.05 0 0.05 -0.05 0 0.05 -0.5 0 0.5 (b) z/ρ 0 (ρ-ρ 0 )/ρ 0 γ 1 FIG. 3: (Color online) The geometric phase γ1 for the storage ring of Ref. [19] with (a) B rf = 0.15B ′ ρ0 and (b) B rf = 0.05B ′ ρ0.\\n\\nB-field\\\" takes the form\\n\\nB eff ( r, t) = ∆( r, t)ê z - B ′ ρ |2 B s ( r, t)| B rf êφ . ( 37\\n\\n)\\n\\nThe operating principle for the time averaged storage ring of Ref. [20] is similar to the well-known TOP [2] and TORT traps [3, 4] . The effective trap potential experienced by the atom is proportional to the time averaged value of the \\\"effective B-field\\\"\\n\\n2π/ωm 0 | B eff ( r, t)|dt.\\n\\nWhen B rf and B m are much smaller than B ′ ρ 0 , the center of the storage ring is located approximately at ρ c = ρ 0 , z c = 0. Using the earlier result [33] , we find that in the time averaged storage ring, the effective gauge potential Ã(φ) n (ρ, z) is reduced simply to the time averaged instantaneous gauge potential 28) . The geometric phase then is given approximately by γ n (ρ, z) = 2π Ã(φ) n (ρ, z). In this case, we find that the geometric phase always vanishes at -0.05 0 0.05 -0.05 0 0.05 -0.2 0 0.2 z/ρ 0 (ρ-ρ 0 )/ρ 0 (a) γ 1 -0.05 0 0.05 -0.05 0 0.05 -0.5 0 0.5 Color online) The geometric phase γ1 for the storage ring of Ref. [20] at (a) B rf = 0.3B ′ ρ0 and (b) B rf = 0.1B ′ ρ0. Bm = 0.05B ′ ρ0.\\n\\nÃ(φ) n (ρ, z) = ω m 2π 2π/ωm 0 A (φ) n (ρ, z, t)dt, ( 38\\n\\n) with A (φ) n (ρ, z, t) given in (\\n\\n(b) (ρ-ρ 0 )/ρ 0 z/ρ 0 γ 1 FIG. 4: (\\n\\nthe trap center (ρ c , z c ). Figure 4 illustrates the distribution of the geometric phase in the region near the trap center for two different values of the rf-field amplitude B rf . Similar to the storage ring of Ref. [19] , the fluctuation of the geometric phase is suppressed in this case for large B rf .\\n\\nB. The storage ring proposals of Refs. [21, 22] Next we consider the ARFP based storage ring proposed in Refs. [21, 22] . In this case, the static B-field is that of a Ioffe-Pritchard trap on an atom chip. In the Cartesian coordinate (x, y, z), it takes the form\\n\\nB s = B ′ xê x -B ′ yê y + B ′ Lê z , ( 39\\n\\n)\\n\\nwhere B ′ is the B-field gradient and the bias field along the z-direction is denoted as B ′ L. The amplitudes B\\n\\n(a) rf and B (b) rf (z) of the rf field are B\\n\\n(a) rf = [B rf (z)/ √ 2]ê x and B (b) rf = [B rf (z)/ √ 2]ê y with B rf (z) = B ( 0\\n\\n) rf + B ′′ z 2 . ( 40\\n\\n)\\n\\n9 In the schemes of Ref. [21, 22] considered earlier, the phase η of the rf field is assumed to be κπ/2. The xand y-components of the \\\"effective B-field\\\" B eff ( r) then become\\n\\nB eff x ( r) = B rf (z) 2 √ 2 (1 + cos β s (ρ, z)), B eff y ( r) = 0, ( 41\\n\\n)\\n\\naccording to Eq. (10). Then the strength of the \\\"effective B-field\\\" B eff has its minimum along a circle with a nonzero radius ρ c , provided a positive detuning ∆ exists at the origin (0, 0, 0) [21, 22] . The \\\"effective B-field\\\" B eff is easily shown to lie in the x-z plane along the trap bottom mapped out by the atomic center of mass motion. This gives rise to a vanishing γ (I) F . With a proper choice for the local phase of |n( r) eff , the gauge potential A (φ) n takes the form\\n\\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z) (1 -cos β s (ρ, z)) . ( 42\\n\\n)\\n\\nFigure 5 displays the geometric phase along a closed path for a spin-1 atom as a function of ρ c for the ARFP storage ring proposed in Refs. [21, 22] . The parameter λ is defined as\\n\\nλ = √ 2 ∆[ r = 0] |g F µ B B (0) rf | . ( 43\\n\\n)\\n\\nTo assure the validity of the rotating wave approximation, we find that the maximal values of ∆[ r = 0]/(|g F |µ B ) and B (0) rf / √ 2 must be restricted to the region of λ ∈ [0, 0.15].\\n\\nAs shown in Fig. 1 , the geometric phase remains much smaller than 2π in this situation. This fact can be appreciated easily if we look at the distribution of the \\\"effective B-field\\\" B eff . According to Eq. (41), the component B eff x has a nonzero minimal value B rf /(2 √ 2), while |B eff z | can become arbitrarily small, although not necessarily zero in general. Therefore, at the trap center where | B eff | is a minimum, the value of cos β eff = B eff z /| B eff | can become very small, leading to small geometric phases. Yet, despite the relatively small geometric phase found here, our result remains important because it could represent a systematic error if not properly included in a Sagnac interference experiment.\\n\\nIn Fig. 6 , we show the spatial distribution of the geometric phase γ 1 around the trap center with λ = 1/3 and λ = 3. The fluctuation is found to be relatively small when λ is small or when the rf-field amplitude B rf is large.\\n\\nAlthough not discussed in Refs. [21, 22] , a ring shaped trap also can be realized if we take η = -κπ/2. The \\\"effective B-field\\\" B eff still lies in the x-y plane\\n\\nB eff x ( r) = - B rf (z) 2 √ 2 cos(2φ)(1 -cos β s (ρ, z)), B eff y ( r) = B rf (z) 2 √ 2 sin(2φ)(1 -cos β s (ρ, z)), ( 44\\n\\n) 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 -0.08 -0.06 -0.04 -0.02 0 λ=1/3 λ=3 λ=1 ρ c /L γ (c) 1\\n\\nFIG. 5: (Color online) The geometric phase γ1 is plotted against the radius ρc for the ARFP storage ring of Ref. [21, 22] with η = κπ/2 at λ = 3, λ = 1, and λ = 1/3. To assure the validity of the rotating wave approximation, in the solid lines, the maximal value of ∆[ r = 0]/|gF µBB ′ ρ0| or B rf /( √ 2B ′ ρ0) are restricted to be smaller than 0.15. The extending dashed line is beyond the rotating wave approximation for λ = 1/3 and B rf /( √ 2B ′ ρ0) ∈ [0.15, 0.3].\\n\\nclearly giving rise to a non-zero solid angle with respect to a closed path along the storage ring. Therefore, the term γ (eff) F is non-zero in this case. We choose the eigenstates |n( r\\n\\n) s and |n( r) eff as |n( r) s = exp[-i F • ns ( r)β s (ρ, z)]|n z , |n( r) eff = exp[-i F • n eff ⊥ ( r)β eff (ρ, z)]|n z . ( 45\\n\\n)\\n\\nwith the unit vector ns ⊥ ( r) in the x-y plane orthogonal to B s ( r). In this case, the gauge potential A (φ) n can be expressed as\\n\\nA (φ) n (ρ, z) = n ρ cos β eff (ρ, z)[1 + cos β s (ρ, z)]. ( 46\\n\\n)\\n\\nIn Figure 7 , we show the fluctuation of the geometric phase γ 1 for a closed path with a new parameter\\n\\nλ ′ = 6 √ 2 ∆[ r = 0] |g F µ B B (0) rf | , ( 47\\n\\n)\\n\\nequal to 3 and 1/3. The fluctuation for γ 1 is found to be much larger than the case of η = κπ/2, which can be explained by the transverse components B eff x,y of the \\\"effective B-field.\\\" Because cos β s is always close to unity.\\n\\nIn the case of η = -κπ/2, B eff x,y can take only small positive values. Therefore, at the minimum of the ARFP\\n\\ntrap ρ = ρ 0 of | B eff |, both B eff z\\n\\nand B eff x,y have to be close to zero. In this case the value for cos β eff becomes a rapidly changing function of ρ in the region near ρ c .\\n\\nOur above calculations have obtained analytical expressions of the geometric phases in an ARFP based storage ring for η = ±κπ/2. We have further investigated the 10 -0.05 0 0.05 -0.05 0 0.05 -0.06 -0.02 0.02 (z-z c )/L (a) (ρ-ρ c )/L γ 1 -0.05 0 0.05 -0.05 0 0.05 -0.6 0 0.6 (z-z c )/L (b) (ρ-ρ c )/L γ 1 FIG. 6: (Color online) The spatial distribution of the geometric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ = 1/3 and (b) λ = 3. η = κπ/2. B (0) rf = 0.08B ′ L and B ′′ = 10 -12 B ′ /L are assumed.\\n\\nfluctuations of the geometric phase for the two cases of η = ±κπ/2. It seems one benefits from implementing a Sagnac interferometer in the discussed ARFP storage ring with η = κπ/2 and operating at a relatively large λ.\\n\\nBefore proceeding onto the concluding section, we will discuss the geometric phase in an ARFP based beam splitter created via a double potential [14, 21] . In such an implementation, the static field B s is created from a Ioffe-Pritchard trap, while the oscillating rf field components are B (a) rf = B rf [z]ê x and B (b) rf = 0. By spatially tuning the amplitude of B rf from zero to a significant value, in the x-y plane, an ARFP can be tuned from a single well centered near the origin to a double well with two minimal points at the point with nonzero radius ρ 0 and φ = 0, π. Therefore, a Y-shaped atom beam splitter can be accomplished when the B rf [z] initially is increased along the z-axis to a large value, and then decreased to zero. In such an arrangement, the atom beam moving along the z direction can be separated into two beams that move along the z-axis at φ = 0, π for a while, and then can be recombined again into a single beam.\\n\\n-0.05 0 0.05 -0.05 0 0.05 -10 0 10 (z-z c )/L (a) (ρ-ρ c )/L γ 1 -0.05 0 0.05 -0.05 0 0.05 -15 0 15 (z-z c )/L (b) (ρ-ρ c )/L γ 1 FIG. 7: (Color online) The spatial distribution of the geometric phase γ1 for the storage ring of Refs. [21, 22] at (a) λ ′ = 3 and (b)λ ′ = 1/3. η = -κπ/2. B (0) rf = 0.08B ′ L and B ′′ = 10 -12 B ′ /L are assumed.\\n\\nIn the atom interferometer considered above, both the static field B s and the \\\"effective B-field\\\" B eff are limited to the x-z plane. Therefore, for motion along the closed path of the trap bottom, the solid angle enclosed by the trajectory of B eff is zero. Thus, the geometric phase in (16) can be expressed as\\n\\nγ n (t) = -i t 0 l | eff n( r)|l z | 2 s l( r)|∇|l( r) s • vdt ′ . ( 48\\n\\n)\\n\\nWe can show that the product r) s is a function of ρ c and is independent of z. Thus, the geometric phase can be expressed as an integral of this function with respect to ρ c , from zero to a large value and then back to zero. Therefore, the value of the geometric phase would be zero in the end.\\n\\neff n( r)|l z | 2 s l( r)|∇|l(\\n\\n11 V. CONCLUSION\\n\\nIn this study, we develop theoretical formalisms for the calculation of the atomic geometric phase inside an ARFP. We show that, due to the complexity of the ARFP, the geometric phase depends on the spatial variation of both the static field and an \\\"effective B-field\\\" B eff . We provide general expressions for the geometric phase and the corresponding adiabatic gauge potential in Eqs. (16) and (23), respectively.\\n\\nTo shed light on actual applications of the atomic geometric phase, we investigate the distribution of atomic geometric phases for several proposed or ongoing experiments with ARFP based storage rings and atom beam splitters. We prove rigorously that the geometric phase in the center of the storage rings proposed in Refs. [18, 19] is always zero. In addition, we find that in the storage ring of Ref. [18] , the spatial fluctuation of the geometric phase sensitively depends on the position of the trap center on the \\\"resonance toroid.\\\" In the proposals of Refs. [19, 20, 21, 22] , the fluctuation for the geometric phase becomes significantly suppressed when the amplitude B rf of the rf-field is large. In the proposals of [21, 22] , the fluctuations of the geometric phase also is suppressed if the angle η is set to be κ2π. In the beam splitter realized with the double well potential ARFP [14, 21], the geometric phase is shown to be zero.\\n\\nOur work helps to clarify the working principle of trapping neutral atoms in an ARFP and the validity conditions for the various approximations involved. We hope our results will shine new light on the proposed inertial sensing experiments based on trapped atoms in ARFP.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgments\",\n \"text\": \"We thank Dr. T. Uzer and Dr. B. Sun for helpful discussions. This work is supported by NASA, NSF, CNSF, and the 863 and 973 programs of the MOST of China.\\n\\n[1] D. E. Pritchard, Phys. Rev. Lett. 51, 1336 (1983). [2] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A.\\n\\nCornell, Phys. Rev. Lett. 74, 3352 (1995). [3] A.S. Arnold and E. Riis, J. Mod. Opt. 49, 959 (2002); C. S. Garvie, E. Riis, and A. S. Arnold, Laser Spectroscopy XVI, edited by P. Hannaford et al. (World Scientific, Singapore, 2004), p. 178, see also www.photonics.phys.strath.ac.uk . [4] S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, and D. M. Stamper-Kurn, Phys. Rev. Lett. 95, 143201 (2005); K. W. Murch, K. L. Moore, S. Gupta, and D.M. Stamper-Kurn, Phys. Rev. Lett. 96, 013202 (2005). [5] R. Folman, P. Krüger, J. Schmiedmayer, J. Denschlag, and C. Henkel, Advances in Atomic, Molecular, and Optical Physics, vol. 48, 263 (2002). [6] J. A. Sauer, M. D. Barrett, and M. S. Chapman, Phys.\\n\\nRev. Lett. 87, 270401 (2001). [7] A.S. Arnold, C.S. Garvie, and E. Riis, Phys. Rev. A 73, 041606(R) (2006). [8] C. C. Agosta, I. F. Silvera, H. T. C. Stoof, and B. J. Verhaar, Phys. Rev. Lett. 62, 2361 (1989). [9] Z. Zhao, I. F. Silvera, and M. Reynolds, Jour. Low. Temp. Phys. 89, 703 (1992). [10] A. J. Moerdijk, B. J. Verhaar, and T. M. Nagtegaal, Phys. Rev. A 53, 4343 (1996). [11] H. Zhang, P. Zhang, X. Xu, J. Han, and Y. Wang, Chin. Phys. Lett. 22, 83 (2001). [12] O. Zobay and B. M. 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Rev. 115, 485 (1959).\"\n }\n]","isConversational":false}}},{"rowIdx":85,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0482"},"arxiv_id":{"kind":"string","value":"0704.0482"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevA.76.062311"},"content_hash":{"kind":"string","value":"4caec5185f281f351422ba4711e16fc0b298b4789732d3ff316689474088ac07"},"title":{"kind":"string","value":"Implementation of holonomic quantum computation through engineering and\n manipulating environment"},"abstract":{"kind":"string","value":"We consider an atom-field coupled system, in which two pairs of four-level\natoms are respectively driven by laser fields and trapped in two distant\ncavities that are connected by an optical fiber. First, we show that an\neffective squeezing reservoir can be engineered under appropriate conditions.\nThen, we show that a two-qubit geometric CPHASE gate between the atoms in the\ntwo cavities can be implemented through adiabatically manipulating the\nengineered reservoir along a closed loop. This scheme that combines engineering\nenvironment with decoherence-free space and geometric phase quantum computation\ntogether has the remarkable feature: a CPHASE gate with arbitrary phase shift\nis implemented by simply changing the strength and relative phase of the\ndriving fields."},"authors":{"kind":"list like","value":["Zhang-qi Yin","Fu-li Li","Peng Peng"],"string":"[\n \"Zhang-qi Yin\",\n \"Fu-li Li\",\n \"Peng Peng\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["quant-ph"],"string":"[\n \"quant-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"Quantum computation, attracting much current interest since Shor's algorithm [1] was proposed, depends on two key factors: quantum entanglement and precision control of quantum systems. Unfortunately, quantum systems are inevitably coupled to their environment so that entanglement is too fragile to be retained. This makes the realization of quantum computation extremely difficult in the real world. In order to overcome this difficulty, one proposed the decoherence-free space concept [2, 3] . It is found that when qubits involved in quantum computation collectively interact with a same environment there exists a \"protected\" subspace in the entire Hilbert space, in which the qubits are immune to the decoherence effects induced by the environment. This subspace is called decoherence-free space (DFS). To perform quantum computation in a DFS, one has to design the specific Hamiltonian containing controlling parameters, which eigenspace is spanned by DFS states and the state-unitary manipulation related to quantum computation goal is implemented by changing the controlling parameters [4] .\n\nAs well known, instantaneous eigenstates of a quantum system with the time-dependent Hamiltonian may acquire a geometric phase when the time-dependent parameters adiabatically undergo a closed loop in the parameter space [5] . The phase depends only on the swept solid angle by the parameter vector in the parameter space. This feature can be utilized to implement geometric quantum computation (GQC) which is resilient to stochastic control errors [6, 7, 8] . On combining the DFS approach with the GQC scheme, one may build quantum gates which may be immune to both the environment-induced decoherence effects and the control-led errors [9] . In the scheme, quantum logical bits are represented by degenerate eigenstates of the parameterized Hamiltonian. These states have the features: they belong to DFS, and unitarily evolve in time and acquire a geometric phase when the controlling parameters adiabatically vary and undergo a closed loop.\n\nIn the recent paper [10], Carollo and coworkers showed that a cascade three-level atom interacting with a broadband squeezed vacuum bosonic bath can be prepared in a state which is decoupled to the environment. This state depends on the reservoir parameters such as squeezing degree and phase angle. As the squeezing parameters smoothly vary, the atomic state can unitarily evolve in time and always be in the manifold of the DFS. Moreover, after a cyclic evolution of the squeezing parameters, the state acquires a geometric phase. This investigation has been generalized to cases where both quantum systems and manipulated reservoir under consideration are not restricted to cascade three-level atoms and squeezed vacuum [11] . These results strongly inspire us that instead of engineering Hamiltonian one may implement the decoherence-free GQC by engineering and manipulating reservoir.\n\nIn this paper, we propose a scheme in which the quantum-reservoir engineering [12, 13, 14] is combined with DFS and Berry phase together to realize a twoqubit CPHASE gate [15] . We show that atomic states can unitarily evolve in time in a DFS if the change rate of reservoir parameters is much smaller than the characteristic relaxation time of an atom-reservoir coupled system. Moreover, we find that as the reservoir parameters adiabatically change in time along an appropriate closed loop, the atomic state in the DFS acquires a Berry phase and a CPHASE gate with arbitrary phase shift can be realized. To our knowledge, it is the first proposal for the realization of quantum gates by engineering and steering the environment.\n\nThis paper is organized as follows. In Sec. II, we introduce a cavity-atom coupling model in which two pairs of four-level atoms are respectively trapped in two distant cavities that are connected by an optical fiber. In the model, each of pairs of the atoms are simultaneously driven by laser fields and coupled to the local cavity 2 modes through the double Raman transition configuration. Under large detuning and bad cavity limits, we investigate to engineer an effective broadband squeezing reservoir for the atoms. In Sec. III, we analyze how to realize controlling gates between the atoms trapped in the two cavities by steering the squeezing reservoir. Section IV contains conclusions of our investigations.\n\nFIG. 1. Atom-field coupling scheme.\n\n|g jn |r jn |s jn ∆ r jn g s jn ∆ s jn δ jn |e jn g r jn Ω r jn Ω s jn\n\nFIG. 2. Atomic level configuration for atom j in cavity n.\n\nOur scheme is shown in Fig. 1 . A pair of four-level atoms are trapped in each of two distant cavities, respectively, which are connected through an optical fiber. In the short fiber limit [16, 17, 18] , only one fiber mode b is excited and coupled to cavity modes a 1 and a 2 with strength ν [19] . We assume that the cavity modes and the fiber mode have the same frequency ω. The level scheme of atoms is shown in Fig. 2 . Atom j in cavity n is labeled by the index jn with j, n = 1, 2. The distance between the atoms in the same cavity is assumed to be large enough that there is no direct interaction between the atoms. The levels |g jn and |e jn of atom j in cavity n, with j, n = 1, 2 are stable with a long life time. The energy of the level |g jn is taken to be zero as the energy reference point. The lower lying level |e jn , and upper levels |r jn and |s jn have the energy δ jn , and ω r jn and ω s jn , respectively, in the unit with = 1. Transitions |g jn ↔ |s jn and |e jn ↔ |r jn are driven by laser fields of frequencies ω Ls jn and ω Lr jn with Rabi frequencies Ω s jn and Ω r jn and relative phase ϕ, respectively. Transitions |g jn ↔ |r jn and |e jn ↔ |s jn are coupled to the cavity mode a n with the strengths g r jn and g s jn , respectively. Here, we set ∆ r jn = ω r jn -ω = ω r jn -ω Lr jn -δ jn , and ∆ s jn = ω s jn -ω -δ jn = ω s jn -ω Ls jn . Under the Markovian approximation, the master equation of the density matrix for the whole system under consideration can be written as [14]\n\nρT = -i[H, ρ T ] + L cav1 ρ T + L cav2 ρ T + L f iber ρ T , (1) where H = H 0 + H d + H ac + H cf with H 0 = 2 j,n=1 ω r jn |r jn r jn | + ω s jn |s jn s jn | + δ jn |e jn e jn |) + ω( 2 n=1 a † n a n + b † b), H d = 2 j,n=1 Ω s jn 2 e -iω Ls jn t |s jn g jn | + Ω r jn 2 e -i(ω Lr jn t+ϕ) |r jn e jn | + H.c. , H ac = 2 j,n=1 (g r jn |r jn g jn |a n + g s jn |s jn e jn |a n + H.c.), H cf =ν b(a † 1 + a † 2 ) + H.c. . (2)\n\nHere, H 0 is the free energy of atoms and cavity fields, H d is the interaction energy between the atoms and laser fields, H ac is the interaction energy between the atoms and the cavity fields, and H cf describes the interaction between the cavity modes and the fiber mode. The last three terms in (1) describe the relaxation processes of the cavity and fibre modes in the usual vacuum reservoir, taking the forms\n\nL cavn ρ T =κ n (2a n ρ T a † n -a † n a n ρ T -ρ T a † n a n ), L f iber ρ T =κ f (2bρ T b † -b † bρ T -ρ T b † b), ( 3\n\n)\n\nwhere κ n is the leakage rate of photons from cavity n, and κ f is the decay rate of the fiber mode.\n\nLet's introduce collective basis:\n\n|a n = (|g 1n |e 2n - |e 1n |g 2n )/ √ 2, | -1 n = |g 1n |g 2n , |0 n = (|g 1n |e 2n + |e 1n |g 2n )/ √ 2, |1 n = |e 1n |e 2n .\n\nThe states |a n and | -1 n are taken as a qubit n for quantum computation. In the large detuning limit, adiabatically eliminating the excited states and setting 2 ), we obtain the effective interaction Hamiltonian\n\nΩ 1n g r 1n 2∆ r 1n = Ω r 2n g r 2n 2∆ r 2n = β r n and Ω s 1n g s 1n 2∆ s 1n = Ω s 2n g s 2n 2∆ s 2n = β s n , from (\n\nH ef f = n √ 2 a n (β r n e iϕ S + n + β s n S n ) + H.c. + H cf , ( 4\n\n) where S + n = |0 nn -1|+|1 nn 0|.\n\nIn the derivation of (4), we have assumed the resonant condition\n\ng s jn 2 ∆ s jn a † n a n + 3 Ω r jn 2 4∆ r jn = Ω s jn 2 4∆ s jn + g r jn 2 ∆ r jn a † n a n + δ ′ jn .\n\nIn order to satisfy the condition with the flexible choice of Ω r jn , Ω s jn , ∆ r jn and ∆ s jn , we have introduced additional ac-Stark shifts δ ′ jn to states |g jn , which can be generated by using a laser field to couple the level |g jn to an ancillary level.\n\nWe now introduce three normal modes c and c ± with frequencies ω and ω ± √ 2ν by use of the unitary transfor-\n\nmation a 1 = 1 2 (c + + c -+ √ 2c), a 2 = 1 2 (c + + c -- √ 2c), b = 1 √ 2 (c + -c -) [17, 18]. In the limit ν ≫ |β r j |, |β s j |,\n\nneglecting the far off-resonant modes c ± and setting β p 1 = -β p 2 = β p with p = r, s, we can approximately write the effective Hamiltonian (4) as\n\nH ef f = (β r e iϕ S + + β s S)c + H.c., ( 5\n\n)\n\nwhere\n\nS + = S + 1 + S + 2 .\n\nSince the modes c ± are nearly not excited and decoupled with the resonant mode c, the fiber mode b is mostly in the vacuum state, therefore, L f iber ρ T can be neglected, and L cav1 ρ T + L cav2 ρ T can be approximated as\n\nL cav ρ T = κ(2cρ T c † -c † cρ T -ρ T c † c), ( 6\n\n)\n\nwhere κ = (κ 1 + κ 2 )/2. In the bad cavity limit, κ ≫ β, adiabatically eliminating the mode c [12, 14] , from Eq. ( 1 ) with the replacement of the Hamiltonian (2) and the relaxation terms (3) by the effective Hamiltonian (5) and the relaxation term (6), respectively, we can obtain the master equation for the density matrix of the atoms\n\nρ = - Γ 2 (R + Rρ + ρR + R -2RρR + ), ( 7\n\n)\n\nwhere ρ = Tr f (ρ T ), R = S cosh r + e iϕ S † sinh r, r = cosh -1 (β r / β r 2 -β s 2\n\n) and Γ = 2(β r 2 -β s2 )/κ. Eq. ( 7 ) describes the collective interaction of two cascade threelevel atoms with the effective squeezed vacuum reservoir [10] . The parameters β r , β s and ϕ are easily changed and controlled at will by varying the strength and phase of the driving lasers [8] . We will show that a geometric phase gate can be realized through changing these parameters.\n\nThe DFS of the atomic system is spanned by the states which satisfy the equation R(r, ϕ\n\n)|ψ DF (r, ϕ) = 0 [10]. In terms of basis states |e 1 = |a 1 |a 2 , |e 2 = |a 1 | -1 2 , |e 3 = | -1 1 |a 2 , |e 4 = |a 1 |0 2 , |e 5 = |0 1 |a 2 , |e 6 = |a 1 |1 2 , |e 7 = |1 1 |a 2 , |e 8 = |1 1 |1 2 , |e 9 = 1 √ 2 (|1 1 |0 2 + |0 1 |1 2 ), |e 10 = | - 1 1 | -1 2 , |e 11 = 1 √ 2 (|0 1 | -1 2 + | -1 1 |0 2 ), |e 12 = 1 √ 6 (|1 1 | -1 2 + | -1 1 |1 2 ) + 2 √ 6 |0 1 |0 2 )\n\n, the DFS states can be written as\n\n|ψ DF (r, ϕ) 1 =|e 1 , |ψ DF (r, ϕ) j = cosh r √ cosh 2r |e j -e iϕ sinh r √ cosh 2r |e j+4 , j = 2, 3, |ψ DF (r, ϕ) 4 = e 2iϕ (tanh r) 2 |e 8 -2 3 e iϕ tanh r|e 12 + |e 10 (tanh r) 4 + 2 3 (tanh r) 2 + 1 . ( 8\n\n)\n\nLet's introduce a unitary transformation O(r, ϕ) 1, 2, 3, 4. For the transformed density matrix ρ = O † ρO, we have\n\nby |φ i = 12 i=1 O ij (r, ϕ)|e j , where |φ i = |ψ DF i for i =\n\ndρ dt = i[G, ρ] + O † dρ dt O, ( 9\n\n)\n\nwhere G(r, ϕ) = iO † dO dt = iO † [ ṙ dO dr + φ dO dϕ ]. To solve Eq. (9) in the DFS, let's define the time-independent projector Π(0 9) , we obtain the equation of motion for ρDF = Π(0)ρΠ(0)\n\n) = O † Π(r, ϕ)O = 4 i=1 O † |φ i φ i |O = 3 j=1 |e j e j | + |e 10 e 10 | onto the DFS. From (\n\ndρ DF dt =i[G DF , ρDF ] + iΠ(0)GΠ ⊥ (0)ρΠ( 0\n\n)\n\n-iΠ(0)ρΠ ⊥ (0)GΠ(0) + Π(0)O † dρ dt OΠ(0), (10) where Π ⊥ (0) = 1 -Π(0) and G DF = Π(0)GΠ(0). In the limit of ṙ, φ ≪ Γ, the last three terms in Eq. (10) can be neglected [11] . In this way, Eq. (10) is reduced to\n\ndρ DF dt = i[G DF , ρDF ]. ( 11\n\n)\n\nTherefore, in the frame dragged adiabatically by the reservoir, the state of the atoms in the DFS unitarily evolves in time.\n\nIn this section, we investigate how to realize a CPHASE gate through manipulating the engineered reservoir. Suppose that at the initial time the laser field driving the transition |g ↔ |s is switched off but the laser field driving the transition |r ↔ |e is switched on and the atoms are in the DFS state |Ψ(0\n\n) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ) = 4 j=1 |ψ DF (0, 0) j /2.\n\nTo generate a geometric phase for the atomic state, we smoothly change the parameters of the engineered reservoir along a closed loop, which is divided into the following three steps: (1) From time 0 to T 1 , hold on ϕ = 0, and adiabatically increase the parameter r from 0 to r 0 ; (2) From time T 1 to T 2 , hold on r = r 0 , and adiabatically change the phase ϕ from 0 to ϕ 0 ; (3) From time T 2 to T 3 , hold on ϕ = ϕ 0 , and adiabatically decrease r from r 0 to 0. When the cyclic evolution ends, the atomic state becomes\n\n|Ψ(T 3 ) a = 1 2 (|e 1 + e iχ1 |e 2 + e iχ1 |e 3 + e iχ12 |e 10 ), (12) where geometric phases χ 1 = -ν 1 ϕ 0 , χ 12 = -ν 12 ϕ 0 with ν 1 = sinh 2 r0 sinh 2 r0+cosh 2 r0 , ν 12 = 2 tanh 4 r0+ 2 3 tanh 2 r0 tanh 4 r0+ 2 3 tanh 2 r0+1 . By per- forming local transformations U 1 = e -iχ1 |-1 11 -1| and 4 U 2 = e -iχ1 | -1 22 -1|,\n\nthe state (12) can be written as\n\n|Ψ ′ (T 3 ) a = U 1 U 2 |Ψ(T 3 ) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + e i∆ | -1 1 | -1 2 ), where ∆ = χ 12 -2χ 1 = (2ν 1 -ν 12 )ϕ 0 .\n\nThus, the CPHASE gate with the phase shift ∆ is realized. If both the atoms in cavity 1 and the atoms in cavity 2 \"see\" different environments, |χ 12 | must be equal to |2χ 1 | and ∆ = 0. Therefore, the phase shift ∆ results from the collective coupling of the atoms in both cavities with the same engineered environment.\n\nIf r 0 = atanh( 4/3 -1) ≃ 0.4157, |ν 12 | = |ν 1 |. Under this condition with ϕ 0 = π/ν 1 ,\n\nthe state of the atoms at the time T\n\n3 is |Ψ ′′ (T 3 ) a = -1 2 (-|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ).\n\nIn this case, the Controlled-Z gate between the two qubits is realized without local transformations. 0 0.2 0.4 0.6 0.8 0.9965 0.997 0.9975 0.998 0.9985 0.999 0.9995 1 r 0 F r T=100/Γ T=200/Γ T=400/Γ (a)\n\nFIG. 3. Fidelity F r of the atomic state. 0 0.2 0.4 0.6 0.8 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 r 0 F p T = 200/Γ T = 400/Γ T = 1000/Γ (b) FIG. 4. Fidelity F p of the atomic state.\n\nThe above results depend on the adiabatical approximation. To check the adiabatical condition, we numerically simulate the following two examples. In the first example, we suppose that at the initial time the atoms are in the state |Ψ 1 a = (|a 1 |a 2 + |ψ DF (0, 0) 2 )/ √ 2 and the laser field driving the transition |e ↔ |r are turned on. Then, by slowly switching the laser field driving the transition |g ↔ |s , we increase the parameter r from 0 to r 0 according to the linear function r(t) = r 0 t/T . In the adiabatical limit (T ≫ Γ -1 ), the atomic state becomes\n\n|Ψ ′ 1 a = (|a 1 |a 2 + |ψ DF (r 0 , 0) 2 )/ √ 2\n\nat the time T . On the other hand, in the Hilbert space spanned by the basis states {|e i } for i = 1, 2, • • • , 12 , we can numerically solve Eq. ( 7 ) and obtain the density matrix ρ 1 (T ) of the atoms. Let's define\n\nF r = a Ψ ′ 1 |ρ 1 (T )|Ψ ′ 1 a as\n\nthe fidelity for this process. As shown in Fig. 3 , if T > 100/Γ, F r is always bigger than 0.997 if r ∈ (0, 0.8), corresponding to the almost perfect evolution.\n\nIn the second example, we suppose that the atoms are initially in the state\n\n|Ψ 2 a = (|a 1 |a 2 + |ψ DF (r, 0) 4 )/\n\n√ 2 and all the driving fields are turned on to hold the parameters r = r 0 and ϕ = 0. By adiabatically changing the phase ϕ from 0 to 2π at the rate φ = 2π/T , the atomic state at the time\n\nT becomes |Ψ ′ 2 a = (|a 1 |a 2 + e iχ12 |ψ DF (r, 2π) 4 )/ √ 2.\n\nLet's define the fidelity for this example as\n\nF p = a Ψ ′ 2 |ρ(T )|Ψ ′ a ,\n\nwhere ρ(T ) is the numerical solution of Eq. ( 7 ). As shown in Fig. 4 , F p increases as T increases but decreases as the parameter r 0 increases. If T > 1000/Γ, F p is larger than 0.992 for 0 < r 0 < 0.8. From these two examples, we find that to fulfill the adiabatical condition the time used in the step 2 should be much longer than in the steps 1 and 3.\n\nA controlled-Z gate has been numerically simulated by directly solving Eq. ( 7 ) with r 0 = 0.5, and ϕ\n\n0 = π/|2ν 1 -ν 12 |.\n\nIn the simulation, we set ṙ = r 0 /T 1 in the steps 1 and 3, and φ = ϕ 0 /(T 2 -T 1 ) in the step 2 with T 1 = 0.05T 3 and T 2 -T 1 = 0.90T 3 . If T 3 > 1100/Γ, we find that the fidelity F = a Ψ(T 3 )|ρ(T 3 )|Ψ(T 3 ) a is larger than 0.95. For an almost perfect controlled-Z gate with F > 0.99, we find that T 3 must be longer than 6000/Γ. Now let's briefly discuss the effects of the atomic spontaneous emission, the fiber mode decay and cavity photon leakage. For simplicity but without the loss of generality, we suppose that atomic spontaneous emission rates of the excited levels are equal to γ. In the large detunig limit, the characteristic spontaneous emission rate of the atoms is γ ef f = γ(Ω 2 /2∆ 2 ) [14, 16] and the effective decay rate of the fiber mode is\n\nκ ef f = κ f Ω 2 g 2 /(4∆ 2 ν 2 ). If κ f ≤ γ and g 2 ≪ ν 2 , κ\n\nef f be much smaller than γ ef f . Under this condition, the present scheme is feasible if Γ ≫ γ ef f . In the current cavity quantum dynamic (CQED) experiment, the parameters (g, κ, γ) = (2000, 10, 10) MHz could be available [20] . If setting Ω/(2∆) = 1 √ 2 × 10 -3 , we have Γ ≃ 4 × 10 4 γ ef f . The condition is held. In the present scheme, the large cavity decay rate is required to ensure that the cavity modes are in a broadband squeezed vacuum reservoir and then the atoms always \"see\" the broadband squeezed vacuum reservoir during the dynamic evolution. For an arbitrary small but nonzero value of the squeezing degree of the reservoir, a CPHASE gate with arbitrary high fidelity can always be realized in the represent scheme. The cavity decay does 5 not directly affect the fidelity of the realized CPHASE gates. However, the larger the decay rate is, the longer the operation time of the CPHASE gates is. Thus, we have the condition κ >> β, γ ef f for realizing the reliable CPHASE gates. Based on the parameters quoted above, this condition can be well satisfied. With the parameters of the current CQED experiment, we find that the operation time of the controlled-Z gate, with fidelity larger than 0.95, is about 2.8 ms. It is much shorter than both 1/γ ef f and the single-atom trapping time in cavity [21] . On the other hand, the present scheme needs a strong coupling between the cavity and the fiber. This could be realized at the current experiment [22] . Therefore, the requirement for the realization of the present scheme can be satisfied with the current technology.\n\nWe propose a cavity-atom coupled scheme for the realization of quantum controlling gates, in which each of two pairs of four-level atoms in two distant cavities connected by a short optical fibre are simultaneously driven by laser fields and coupled to the local cavity modes through the double Raman transition configuration. We show that an effective squeezing reservoir coupled to the multilevel atoms can be engineered under appropriate driving condition and bad cavity limit. We find that in the scheme a CPHASE gate with arbitrary phase shift can be implemented through adiabatically changing the strength and phase of driving fields along a closed loop. It is also noticed that the larger the effective coupling strength between the environment and the atoms is, the more reliable the realized CPHASE gate is.\n\nWe thank Yun-feng Xiao and Wen-ping He for valuable discussions and suggestions. This work was supported by the Natural Science Foundation of China (Grant Nos. 10674106, 60778021 and 05-06-01).\n\n[1] P. W. Shor, in Proceeings, 35 th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Press, Los Almitos, CA, 1994), p. 124. [2] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953 (1997). [3] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997). [4] D. A. Lidar and et al., Phys. Rev. Lett. 81, 2594 (1998). [5] M. V. Berry, Proc. R. Soc. A 392, 45 (1984). [6] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999). [7] S.-L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301 (2005). [8] L.-M. Duan and et al., Science 292, 1695 (2001). [9] L.-A. Wu and et al., Phys. Rev. Lett. 95, 130501 (2005). [10] Angelo Carollo et al., Phys. Rev. Lett. 96, 150403 (2006). [11] Angelo Carollo et al., Phys. Rev. Lett. 96, 020403 (2006). [12] J. I. Cirac, Phys. Rev. A 46, 4354 (1992). [13] C. J. Myatt and et al., Nature (London) 403, 269 (2000). [14] S. G. Clark and A. S. Parkins, Phys. Rev. Lett. 90, 047905 (2003). [15] S. Lloyd, Phys. Rev. Lett. 75, 346 (1995). [16] T. Pellizzari, Phys. Rev. Lett. 79, 5242 (1997). [17] A. Serafini and et al., Phys. Rev. Lett. 96, 010503 (2006) . [18] Zhang-qi Yin and Fu-li Li, Phys. Rev. A 75, 012324 (2007). [19] S. J. van Enk and et al., Phys. Rev. A 59, 2659 (1999). [20] S. M. Spillane et al., Phys. Rev. A 71, 013817 (2005). [21] Stefan Nuβmann and et al., Nature Phys. 1, 122 (2005). [22] S. M. Spillane et al., Phys. Rev. Lett. 91, 043902 (2003)."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"Quantum computation, attracting much current interest since Shor's algorithm [1] was proposed, depends on two key factors: quantum entanglement and precision control of quantum systems. Unfortunately, quantum systems are inevitably coupled to their environment so that entanglement is too fragile to be retained. This makes the realization of quantum computation extremely difficult in the real world. In order to overcome this difficulty, one proposed the decoherence-free space concept [2, 3] . It is found that when qubits involved in quantum computation collectively interact with a same environment there exists a \"protected\" subspace in the entire Hilbert space, in which the qubits are immune to the decoherence effects induced by the environment. This subspace is called decoherence-free space (DFS). To perform quantum computation in a DFS, one has to design the specific Hamiltonian containing controlling parameters, which eigenspace is spanned by DFS states and the state-unitary manipulation related to quantum computation goal is implemented by changing the controlling parameters [4] .\n\nAs well known, instantaneous eigenstates of a quantum system with the time-dependent Hamiltonian may acquire a geometric phase when the time-dependent parameters adiabatically undergo a closed loop in the parameter space [5] . The phase depends only on the swept solid angle by the parameter vector in the parameter space. This feature can be utilized to implement geometric quantum computation (GQC) which is resilient to stochastic control errors [6, 7, 8] . On combining the DFS approach with the GQC scheme, one may build quantum gates which may be immune to both the environment-induced decoherence effects and the control-led errors [9] . In the scheme, quantum logical bits are represented by degenerate eigenstates of the parameterized Hamiltonian. These states have the features: they belong to DFS, and unitarily evolve in time and acquire a geometric phase when the controlling parameters adiabatically vary and undergo a closed loop.\n\nIn the recent paper [10], Carollo and coworkers showed that a cascade three-level atom interacting with a broadband squeezed vacuum bosonic bath can be prepared in a state which is decoupled to the environment. This state depends on the reservoir parameters such as squeezing degree and phase angle. As the squeezing parameters smoothly vary, the atomic state can unitarily evolve in time and always be in the manifold of the DFS. Moreover, after a cyclic evolution of the squeezing parameters, the state acquires a geometric phase. This investigation has been generalized to cases where both quantum systems and manipulated reservoir under consideration are not restricted to cascade three-level atoms and squeezed vacuum [11] . These results strongly inspire us that instead of engineering Hamiltonian one may implement the decoherence-free GQC by engineering and manipulating reservoir.\n\nIn this paper, we propose a scheme in which the quantum-reservoir engineering [12, 13, 14] is combined with DFS and Berry phase together to realize a twoqubit CPHASE gate [15] . We show that atomic states can unitarily evolve in time in a DFS if the change rate of reservoir parameters is much smaller than the characteristic relaxation time of an atom-reservoir coupled system. Moreover, we find that as the reservoir parameters adiabatically change in time along an appropriate closed loop, the atomic state in the DFS acquires a Berry phase and a CPHASE gate with arbitrary phase shift can be realized. To our knowledge, it is the first proposal for the realization of quantum gates by engineering and steering the environment.\n\nThis paper is organized as follows. In Sec. II, we introduce a cavity-atom coupling model in which two pairs of four-level atoms are respectively trapped in two distant cavities that are connected by an optical fiber. In the model, each of pairs of the atoms are simultaneously driven by laser fields and coupled to the local cavity 2 modes through the double Raman transition configuration. Under large detuning and bad cavity limits, we investigate to engineer an effective broadband squeezing reservoir for the atoms. In Sec. III, we analyze how to realize controlling gates between the atoms trapped in the two cavities by steering the squeezing reservoir. Section IV contains conclusions of our investigations."},{"section_type":"OTHER","section_title":"laser laser fibre","text":"FIG. 1. Atom-field coupling scheme.\n\n|g jn |r jn |s jn ∆ r jn g s jn ∆ s jn δ jn |e jn g r jn Ω r jn Ω s jn\n\nFIG. 2. Atomic level configuration for atom j in cavity n."},{"section_type":"OTHER","section_title":"II. ENGINEERING A SQUEEZING ENVIRONMENT AND GENERATING A DECOHERENCE-FREE SUBSPACE","text":"Our scheme is shown in Fig. 1 . A pair of four-level atoms are trapped in each of two distant cavities, respectively, which are connected through an optical fiber. In the short fiber limit [16, 17, 18] , only one fiber mode b is excited and coupled to cavity modes a 1 and a 2 with strength ν [19] . We assume that the cavity modes and the fiber mode have the same frequency ω. The level scheme of atoms is shown in Fig. 2 . Atom j in cavity n is labeled by the index jn with j, n = 1, 2. The distance between the atoms in the same cavity is assumed to be large enough that there is no direct interaction between the atoms. The levels |g jn and |e jn of atom j in cavity n, with j, n = 1, 2 are stable with a long life time. The energy of the level |g jn is taken to be zero as the energy reference point. The lower lying level |e jn , and upper levels |r jn and |s jn have the energy δ jn , and ω r jn and ω s jn , respectively, in the unit with = 1. Transitions |g jn ↔ |s jn and |e jn ↔ |r jn are driven by laser fields of frequencies ω Ls jn and ω Lr jn with Rabi frequencies Ω s jn and Ω r jn and relative phase ϕ, respectively. Transitions |g jn ↔ |r jn and |e jn ↔ |s jn are coupled to the cavity mode a n with the strengths g r jn and g s jn , respectively. Here, we set ∆ r jn = ω r jn -ω = ω r jn -ω Lr jn -δ jn , and ∆ s jn = ω s jn -ω -δ jn = ω s jn -ω Ls jn . Under the Markovian approximation, the master equation of the density matrix for the whole system under consideration can be written as [14]\n\nρT = -i[H, ρ T ] + L cav1 ρ T + L cav2 ρ T + L f iber ρ T , (1) where H = H 0 + H d + H ac + H cf with H 0 = 2 j,n=1 ω r jn |r jn r jn | + ω s jn |s jn s jn | + δ jn |e jn e jn |) + ω( 2 n=1 a † n a n + b † b), H d = 2 j,n=1 Ω s jn 2 e -iω Ls jn t |s jn g jn | + Ω r jn 2 e -i(ω Lr jn t+ϕ) |r jn e jn | + H.c. , H ac = 2 j,n=1 (g r jn |r jn g jn |a n + g s jn |s jn e jn |a n + H.c.), H cf =ν b(a † 1 + a † 2 ) + H.c. . (2)\n\nHere, H 0 is the free energy of atoms and cavity fields, H d is the interaction energy between the atoms and laser fields, H ac is the interaction energy between the atoms and the cavity fields, and H cf describes the interaction between the cavity modes and the fiber mode. The last three terms in (1) describe the relaxation processes of the cavity and fibre modes in the usual vacuum reservoir, taking the forms\n\nL cavn ρ T =κ n (2a n ρ T a † n -a † n a n ρ T -ρ T a † n a n ), L f iber ρ T =κ f (2bρ T b † -b † bρ T -ρ T b † b), ( 3\n\n)\n\nwhere κ n is the leakage rate of photons from cavity n, and κ f is the decay rate of the fiber mode.\n\nLet's introduce collective basis:\n\n|a n = (|g 1n |e 2n - |e 1n |g 2n )/ √ 2, | -1 n = |g 1n |g 2n , |0 n = (|g 1n |e 2n + |e 1n |g 2n )/ √ 2, |1 n = |e 1n |e 2n .\n\nThe states |a n and | -1 n are taken as a qubit n for quantum computation. In the large detuning limit, adiabatically eliminating the excited states and setting 2 ), we obtain the effective interaction Hamiltonian\n\nΩ 1n g r 1n 2∆ r 1n = Ω r 2n g r 2n 2∆ r 2n = β r n and Ω s 1n g s 1n 2∆ s 1n = Ω s 2n g s 2n 2∆ s 2n = β s n , from (\n\nH ef f = n √ 2 a n (β r n e iϕ S + n + β s n S n ) + H.c. + H cf , ( 4\n\n) where S + n = |0 nn -1|+|1 nn 0|.\n\nIn the derivation of (4), we have assumed the resonant condition\n\ng s jn 2 ∆ s jn a † n a n + 3 Ω r jn 2 4∆ r jn = Ω s jn 2 4∆ s jn + g r jn 2 ∆ r jn a † n a n + δ ′ jn .\n\nIn order to satisfy the condition with the flexible choice of Ω r jn , Ω s jn , ∆ r jn and ∆ s jn , we have introduced additional ac-Stark shifts δ ′ jn to states |g jn , which can be generated by using a laser field to couple the level |g jn to an ancillary level.\n\nWe now introduce three normal modes c and c ± with frequencies ω and ω ± √ 2ν by use of the unitary transfor-\n\nmation a 1 = 1 2 (c + + c -+ √ 2c), a 2 = 1 2 (c + + c -- √ 2c), b = 1 √ 2 (c + -c -) [17, 18]. In the limit ν ≫ |β r j |, |β s j |,\n\nneglecting the far off-resonant modes c ± and setting β p 1 = -β p 2 = β p with p = r, s, we can approximately write the effective Hamiltonian (4) as\n\nH ef f = (β r e iϕ S + + β s S)c + H.c., ( 5\n\n)\n\nwhere\n\nS + = S + 1 + S + 2 .\n\nSince the modes c ± are nearly not excited and decoupled with the resonant mode c, the fiber mode b is mostly in the vacuum state, therefore, L f iber ρ T can be neglected, and L cav1 ρ T + L cav2 ρ T can be approximated as\n\nL cav ρ T = κ(2cρ T c † -c † cρ T -ρ T c † c), ( 6\n\n)\n\nwhere κ = (κ 1 + κ 2 )/2. In the bad cavity limit, κ ≫ β, adiabatically eliminating the mode c [12, 14] , from Eq. ( 1 ) with the replacement of the Hamiltonian (2) and the relaxation terms (3) by the effective Hamiltonian (5) and the relaxation term (6), respectively, we can obtain the master equation for the density matrix of the atoms\n\nρ = - Γ 2 (R + Rρ + ρR + R -2RρR + ), ( 7\n\n)\n\nwhere ρ = Tr f (ρ T ), R = S cosh r + e iϕ S † sinh r, r = cosh -1 (β r / β r 2 -β s 2\n\n) and Γ = 2(β r 2 -β s2 )/κ. Eq. ( 7 ) describes the collective interaction of two cascade threelevel atoms with the effective squeezed vacuum reservoir [10] . The parameters β r , β s and ϕ are easily changed and controlled at will by varying the strength and phase of the driving lasers [8] . We will show that a geometric phase gate can be realized through changing these parameters.\n\nThe DFS of the atomic system is spanned by the states which satisfy the equation R(r, ϕ\n\n)|ψ DF (r, ϕ) = 0 [10]. In terms of basis states |e 1 = |a 1 |a 2 , |e 2 = |a 1 | -1 2 , |e 3 = | -1 1 |a 2 , |e 4 = |a 1 |0 2 , |e 5 = |0 1 |a 2 , |e 6 = |a 1 |1 2 , |e 7 = |1 1 |a 2 , |e 8 = |1 1 |1 2 , |e 9 = 1 √ 2 (|1 1 |0 2 + |0 1 |1 2 ), |e 10 = | - 1 1 | -1 2 , |e 11 = 1 √ 2 (|0 1 | -1 2 + | -1 1 |0 2 ), |e 12 = 1 √ 6 (|1 1 | -1 2 + | -1 1 |1 2 ) + 2 √ 6 |0 1 |0 2 )\n\n, the DFS states can be written as\n\n|ψ DF (r, ϕ) 1 =|e 1 , |ψ DF (r, ϕ) j = cosh r √ cosh 2r |e j -e iϕ sinh r √ cosh 2r |e j+4 , j = 2, 3, |ψ DF (r, ϕ) 4 = e 2iϕ (tanh r) 2 |e 8 -2 3 e iϕ tanh r|e 12 + |e 10 (tanh r) 4 + 2 3 (tanh r) 2 + 1 . ( 8\n\n)\n\nLet's introduce a unitary transformation O(r, ϕ) 1, 2, 3, 4. For the transformed density matrix ρ = O † ρO, we have\n\nby |φ i = 12 i=1 O ij (r, ϕ)|e j , where |φ i = |ψ DF i for i =\n\ndρ dt = i[G, ρ] + O † dρ dt O, ( 9\n\n)\n\nwhere G(r, ϕ) = iO † dO dt = iO † [ ṙ dO dr + φ dO dϕ ]. To solve Eq. (9) in the DFS, let's define the time-independent projector Π(0 9) , we obtain the equation of motion for ρDF = Π(0)ρΠ(0)\n\n) = O † Π(r, ϕ)O = 4 i=1 O † |φ i φ i |O = 3 j=1 |e j e j | + |e 10 e 10 | onto the DFS. From (\n\ndρ DF dt =i[G DF , ρDF ] + iΠ(0)GΠ ⊥ (0)ρΠ( 0\n\n)\n\n-iΠ(0)ρΠ ⊥ (0)GΠ(0) + Π(0)O † dρ dt OΠ(0), (10) where Π ⊥ (0) = 1 -Π(0) and G DF = Π(0)GΠ(0). In the limit of ṙ, φ ≪ Γ, the last three terms in Eq. (10) can be neglected [11] . In this way, Eq. (10) is reduced to\n\ndρ DF dt = i[G DF , ρDF ]. ( 11\n\n)\n\nTherefore, in the frame dragged adiabatically by the reservoir, the state of the atoms in the DFS unitarily evolves in time."},{"section_type":"OTHER","section_title":"III. REALIZING CONTROLLING PHASE GATES THROUGH MANIPULATING THE SQUEEZING ENVIRONMENT","text":"In this section, we investigate how to realize a CPHASE gate through manipulating the engineered reservoir. Suppose that at the initial time the laser field driving the transition |g ↔ |s is switched off but the laser field driving the transition |r ↔ |e is switched on and the atoms are in the DFS state |Ψ(0\n\n) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ) = 4 j=1 |ψ DF (0, 0) j /2.\n\nTo generate a geometric phase for the atomic state, we smoothly change the parameters of the engineered reservoir along a closed loop, which is divided into the following three steps: (1) From time 0 to T 1 , hold on ϕ = 0, and adiabatically increase the parameter r from 0 to r 0 ; (2) From time T 1 to T 2 , hold on r = r 0 , and adiabatically change the phase ϕ from 0 to ϕ 0 ; (3) From time T 2 to T 3 , hold on ϕ = ϕ 0 , and adiabatically decrease r from r 0 to 0. When the cyclic evolution ends, the atomic state becomes\n\n|Ψ(T 3 ) a = 1 2 (|e 1 + e iχ1 |e 2 + e iχ1 |e 3 + e iχ12 |e 10 ), (12) where geometric phases χ 1 = -ν 1 ϕ 0 , χ 12 = -ν 12 ϕ 0 with ν 1 = sinh 2 r0 sinh 2 r0+cosh 2 r0 , ν 12 = 2 tanh 4 r0+ 2 3 tanh 2 r0 tanh 4 r0+ 2 3 tanh 2 r0+1 . By per- forming local transformations U 1 = e -iχ1 |-1 11 -1| and 4 U 2 = e -iχ1 | -1 22 -1|,\n\nthe state (12) can be written as\n\n|Ψ ′ (T 3 ) a = U 1 U 2 |Ψ(T 3 ) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + e i∆ | -1 1 | -1 2 ), where ∆ = χ 12 -2χ 1 = (2ν 1 -ν 12 )ϕ 0 .\n\nThus, the CPHASE gate with the phase shift ∆ is realized. If both the atoms in cavity 1 and the atoms in cavity 2 \"see\" different environments, |χ 12 | must be equal to |2χ 1 | and ∆ = 0. Therefore, the phase shift ∆ results from the collective coupling of the atoms in both cavities with the same engineered environment.\n\nIf r 0 = atanh( 4/3 -1) ≃ 0.4157, |ν 12 | = |ν 1 |. Under this condition with ϕ 0 = π/ν 1 ,\n\nthe state of the atoms at the time T\n\n3 is |Ψ ′′ (T 3 ) a = -1 2 (-|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ).\n\nIn this case, the Controlled-Z gate between the two qubits is realized without local transformations. 0 0.2 0.4 0.6 0.8 0.9965 0.997 0.9975 0.998 0.9985 0.999 0.9995 1 r 0 F r T=100/Γ T=200/Γ T=400/Γ (a)\n\nFIG. 3. Fidelity F r of the atomic state. 0 0.2 0.4 0.6 0.8 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 r 0 F p T = 200/Γ T = 400/Γ T = 1000/Γ (b) FIG. 4. Fidelity F p of the atomic state.\n\nThe above results depend on the adiabatical approximation. To check the adiabatical condition, we numerically simulate the following two examples. In the first example, we suppose that at the initial time the atoms are in the state |Ψ 1 a = (|a 1 |a 2 + |ψ DF (0, 0) 2 )/ √ 2 and the laser field driving the transition |e ↔ |r are turned on. Then, by slowly switching the laser field driving the transition |g ↔ |s , we increase the parameter r from 0 to r 0 according to the linear function r(t) = r 0 t/T . In the adiabatical limit (T ≫ Γ -1 ), the atomic state becomes\n\n|Ψ ′ 1 a = (|a 1 |a 2 + |ψ DF (r 0 , 0) 2 )/ √ 2\n\nat the time T . On the other hand, in the Hilbert space spanned by the basis states {|e i } for i = 1, 2, • • • , 12 , we can numerically solve Eq. ( 7 ) and obtain the density matrix ρ 1 (T ) of the atoms. Let's define\n\nF r = a Ψ ′ 1 |ρ 1 (T )|Ψ ′ 1 a as\n\nthe fidelity for this process. As shown in Fig. 3 , if T > 100/Γ, F r is always bigger than 0.997 if r ∈ (0, 0.8), corresponding to the almost perfect evolution.\n\nIn the second example, we suppose that the atoms are initially in the state\n\n|Ψ 2 a = (|a 1 |a 2 + |ψ DF (r, 0) 4 )/\n\n√ 2 and all the driving fields are turned on to hold the parameters r = r 0 and ϕ = 0. By adiabatically changing the phase ϕ from 0 to 2π at the rate φ = 2π/T , the atomic state at the time\n\nT becomes |Ψ ′ 2 a = (|a 1 |a 2 + e iχ12 |ψ DF (r, 2π) 4 )/ √ 2.\n\nLet's define the fidelity for this example as\n\nF p = a Ψ ′ 2 |ρ(T )|Ψ ′ a ,\n\nwhere ρ(T ) is the numerical solution of Eq. ( 7 ). As shown in Fig. 4 , F p increases as T increases but decreases as the parameter r 0 increases. If T > 1000/Γ, F p is larger than 0.992 for 0 < r 0 < 0.8. From these two examples, we find that to fulfill the adiabatical condition the time used in the step 2 should be much longer than in the steps 1 and 3.\n\nA controlled-Z gate has been numerically simulated by directly solving Eq. ( 7 ) with r 0 = 0.5, and ϕ\n\n0 = π/|2ν 1 -ν 12 |.\n\nIn the simulation, we set ṙ = r 0 /T 1 in the steps 1 and 3, and φ = ϕ 0 /(T 2 -T 1 ) in the step 2 with T 1 = 0.05T 3 and T 2 -T 1 = 0.90T 3 . If T 3 > 1100/Γ, we find that the fidelity F = a Ψ(T 3 )|ρ(T 3 )|Ψ(T 3 ) a is larger than 0.95. For an almost perfect controlled-Z gate with F > 0.99, we find that T 3 must be longer than 6000/Γ. Now let's briefly discuss the effects of the atomic spontaneous emission, the fiber mode decay and cavity photon leakage. For simplicity but without the loss of generality, we suppose that atomic spontaneous emission rates of the excited levels are equal to γ. In the large detunig limit, the characteristic spontaneous emission rate of the atoms is γ ef f = γ(Ω 2 /2∆ 2 ) [14, 16] and the effective decay rate of the fiber mode is\n\nκ ef f = κ f Ω 2 g 2 /(4∆ 2 ν 2 ). If κ f ≤ γ and g 2 ≪ ν 2 , κ\n\nef f be much smaller than γ ef f . Under this condition, the present scheme is feasible if Γ ≫ γ ef f . In the current cavity quantum dynamic (CQED) experiment, the parameters (g, κ, γ) = (2000, 10, 10) MHz could be available [20] . If setting Ω/(2∆) = 1 √ 2 × 10 -3 , we have Γ ≃ 4 × 10 4 γ ef f . The condition is held. In the present scheme, the large cavity decay rate is required to ensure that the cavity modes are in a broadband squeezed vacuum reservoir and then the atoms always \"see\" the broadband squeezed vacuum reservoir during the dynamic evolution. For an arbitrary small but nonzero value of the squeezing degree of the reservoir, a CPHASE gate with arbitrary high fidelity can always be realized in the represent scheme. The cavity decay does 5 not directly affect the fidelity of the realized CPHASE gates. However, the larger the decay rate is, the longer the operation time of the CPHASE gates is. Thus, we have the condition κ >> β, γ ef f for realizing the reliable CPHASE gates. Based on the parameters quoted above, this condition can be well satisfied. With the parameters of the current CQED experiment, we find that the operation time of the controlled-Z gate, with fidelity larger than 0.95, is about 2.8 ms. It is much shorter than both 1/γ ef f and the single-atom trapping time in cavity [21] . On the other hand, the present scheme needs a strong coupling between the cavity and the fiber. This could be realized at the current experiment [22] . Therefore, the requirement for the realization of the present scheme can be satisfied with the current technology."},{"section_type":"CONCLUSION","section_title":"IV. CONCLUSIONS","text":"We propose a cavity-atom coupled scheme for the realization of quantum controlling gates, in which each of two pairs of four-level atoms in two distant cavities connected by a short optical fibre are simultaneously driven by laser fields and coupled to the local cavity modes through the double Raman transition configuration. We show that an effective squeezing reservoir coupled to the multilevel atoms can be engineered under appropriate driving condition and bad cavity limit. We find that in the scheme a CPHASE gate with arbitrary phase shift can be implemented through adiabatically changing the strength and phase of driving fields along a closed loop. It is also noticed that the larger the effective coupling strength between the environment and the atoms is, the more reliable the realized CPHASE gate is."},{"section_type":"OTHER","section_title":"Acknowledgments","text":"We thank Yun-feng Xiao and Wen-ping He for valuable discussions and suggestions. This work was supported by the Natural Science Foundation of China (Grant Nos. 10674106, 60778021 and 05-06-01).\n\n[1] P. W. Shor, in Proceeings, 35 th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Press, Los Almitos, CA, 1994), p. 124. [2] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953 (1997). [3] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997). [4] D. A. Lidar and et al., Phys. Rev. Lett. 81, 2594 (1998). [5] M. V. Berry, Proc. R. Soc. A 392, 45 (1984). [6] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999). [7] S.-L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301 (2005). [8] L.-M. Duan and et al., Science 292, 1695 (2001). [9] L.-A. Wu and et al., Phys. Rev. Lett. 95, 130501 (2005). [10] Angelo Carollo et al., Phys. Rev. Lett. 96, 150403 (2006). [11] Angelo Carollo et al., Phys. Rev. Lett. 96, 020403 (2006). [12] J. I. Cirac, Phys. Rev. A 46, 4354 (1992). [13] C. J. Myatt and et al., Nature (London) 403, 269 (2000). [14] S. G. Clark and A. S. Parkins, Phys. Rev. Lett. 90, 047905 (2003). [15] S. Lloyd, Phys. Rev. Lett. 75, 346 (1995). [16] T. Pellizzari, Phys. Rev. Lett. 79, 5242 (1997). [17] A. Serafini and et al., Phys. Rev. Lett. 96, 010503 (2006) . [18] Zhang-qi Yin and Fu-li Li, Phys. Rev. A 75, 012324 (2007). [19] S. J. van Enk and et al., Phys. Rev. A 59, 2659 (1999). [20] S. M. Spillane et al., Phys. Rev. A 71, 013817 (2005). [21] Stefan Nuβmann and et al., Nature Phys. 1, 122 (2005). [22] S. M. Spillane et al., Phys. Rev. Lett. 91, 043902 (2003)."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"Quantum computation, attracting much current interest since Shor's algorithm [1] was proposed, depends on two key factors: quantum entanglement and precision control of quantum systems. Unfortunately, quantum systems are inevitably coupled to their environment so that entanglement is too fragile to be retained. This makes the realization of quantum computation extremely difficult in the real world. In order to overcome this difficulty, one proposed the decoherence-free space concept [2, 3] . It is found that when qubits involved in quantum computation collectively interact with a same environment there exists a \\\"protected\\\" subspace in the entire Hilbert space, in which the qubits are immune to the decoherence effects induced by the environment. This subspace is called decoherence-free space (DFS). To perform quantum computation in a DFS, one has to design the specific Hamiltonian containing controlling parameters, which eigenspace is spanned by DFS states and the state-unitary manipulation related to quantum computation goal is implemented by changing the controlling parameters [4] .\\n\\nAs well known, instantaneous eigenstates of a quantum system with the time-dependent Hamiltonian may acquire a geometric phase when the time-dependent parameters adiabatically undergo a closed loop in the parameter space [5] . The phase depends only on the swept solid angle by the parameter vector in the parameter space. This feature can be utilized to implement geometric quantum computation (GQC) which is resilient to stochastic control errors [6, 7, 8] . On combining the DFS approach with the GQC scheme, one may build quantum gates which may be immune to both the environment-induced decoherence effects and the control-led errors [9] . In the scheme, quantum logical bits are represented by degenerate eigenstates of the parameterized Hamiltonian. These states have the features: they belong to DFS, and unitarily evolve in time and acquire a geometric phase when the controlling parameters adiabatically vary and undergo a closed loop.\\n\\nIn the recent paper [10], Carollo and coworkers showed that a cascade three-level atom interacting with a broadband squeezed vacuum bosonic bath can be prepared in a state which is decoupled to the environment. This state depends on the reservoir parameters such as squeezing degree and phase angle. As the squeezing parameters smoothly vary, the atomic state can unitarily evolve in time and always be in the manifold of the DFS. Moreover, after a cyclic evolution of the squeezing parameters, the state acquires a geometric phase. This investigation has been generalized to cases where both quantum systems and manipulated reservoir under consideration are not restricted to cascade three-level atoms and squeezed vacuum [11] . These results strongly inspire us that instead of engineering Hamiltonian one may implement the decoherence-free GQC by engineering and manipulating reservoir.\\n\\nIn this paper, we propose a scheme in which the quantum-reservoir engineering [12, 13, 14] is combined with DFS and Berry phase together to realize a twoqubit CPHASE gate [15] . We show that atomic states can unitarily evolve in time in a DFS if the change rate of reservoir parameters is much smaller than the characteristic relaxation time of an atom-reservoir coupled system. Moreover, we find that as the reservoir parameters adiabatically change in time along an appropriate closed loop, the atomic state in the DFS acquires a Berry phase and a CPHASE gate with arbitrary phase shift can be realized. To our knowledge, it is the first proposal for the realization of quantum gates by engineering and steering the environment.\\n\\nThis paper is organized as follows. In Sec. II, we introduce a cavity-atom coupling model in which two pairs of four-level atoms are respectively trapped in two distant cavities that are connected by an optical fiber. In the model, each of pairs of the atoms are simultaneously driven by laser fields and coupled to the local cavity 2 modes through the double Raman transition configuration. Under large detuning and bad cavity limits, we investigate to engineer an effective broadband squeezing reservoir for the atoms. In Sec. III, we analyze how to realize controlling gates between the atoms trapped in the two cavities by steering the squeezing reservoir. Section IV contains conclusions of our investigations.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"laser laser fibre\",\n \"text\": \"FIG. 1. Atom-field coupling scheme.\\n\\n|g jn |r jn |s jn ∆ r jn g s jn ∆ s jn δ jn |e jn g r jn Ω r jn Ω s jn\\n\\nFIG. 2. Atomic level configuration for atom j in cavity n.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. ENGINEERING A SQUEEZING ENVIRONMENT AND GENERATING A DECOHERENCE-FREE SUBSPACE\",\n \"text\": \"Our scheme is shown in Fig. 1 . A pair of four-level atoms are trapped in each of two distant cavities, respectively, which are connected through an optical fiber. In the short fiber limit [16, 17, 18] , only one fiber mode b is excited and coupled to cavity modes a 1 and a 2 with strength ν [19] . We assume that the cavity modes and the fiber mode have the same frequency ω. The level scheme of atoms is shown in Fig. 2 . Atom j in cavity n is labeled by the index jn with j, n = 1, 2. The distance between the atoms in the same cavity is assumed to be large enough that there is no direct interaction between the atoms. The levels |g jn and |e jn of atom j in cavity n, with j, n = 1, 2 are stable with a long life time. The energy of the level |g jn is taken to be zero as the energy reference point. The lower lying level |e jn , and upper levels |r jn and |s jn have the energy δ jn , and ω r jn and ω s jn , respectively, in the unit with = 1. Transitions |g jn ↔ |s jn and |e jn ↔ |r jn are driven by laser fields of frequencies ω Ls jn and ω Lr jn with Rabi frequencies Ω s jn and Ω r jn and relative phase ϕ, respectively. Transitions |g jn ↔ |r jn and |e jn ↔ |s jn are coupled to the cavity mode a n with the strengths g r jn and g s jn , respectively. Here, we set ∆ r jn = ω r jn -ω = ω r jn -ω Lr jn -δ jn , and ∆ s jn = ω s jn -ω -δ jn = ω s jn -ω Ls jn . Under the Markovian approximation, the master equation of the density matrix for the whole system under consideration can be written as [14]\\n\\nρT = -i[H, ρ T ] + L cav1 ρ T + L cav2 ρ T + L f iber ρ T , (1) where H = H 0 + H d + H ac + H cf with H 0 = 2 j,n=1 ω r jn |r jn r jn | + ω s jn |s jn s jn | + δ jn |e jn e jn |) + ω( 2 n=1 a † n a n + b † b), H d = 2 j,n=1 Ω s jn 2 e -iω Ls jn t |s jn g jn | + Ω r jn 2 e -i(ω Lr jn t+ϕ) |r jn e jn | + H.c. , H ac = 2 j,n=1 (g r jn |r jn g jn |a n + g s jn |s jn e jn |a n + H.c.), H cf =ν b(a † 1 + a † 2 ) + H.c. . (2)\\n\\nHere, H 0 is the free energy of atoms and cavity fields, H d is the interaction energy between the atoms and laser fields, H ac is the interaction energy between the atoms and the cavity fields, and H cf describes the interaction between the cavity modes and the fiber mode. The last three terms in (1) describe the relaxation processes of the cavity and fibre modes in the usual vacuum reservoir, taking the forms\\n\\nL cavn ρ T =κ n (2a n ρ T a † n -a † n a n ρ T -ρ T a † n a n ), L f iber ρ T =κ f (2bρ T b † -b † bρ T -ρ T b † b), ( 3\\n\\n)\\n\\nwhere κ n is the leakage rate of photons from cavity n, and κ f is the decay rate of the fiber mode.\\n\\nLet's introduce collective basis:\\n\\n|a n = (|g 1n |e 2n - |e 1n |g 2n )/ √ 2, | -1 n = |g 1n |g 2n , |0 n = (|g 1n |e 2n + |e 1n |g 2n )/ √ 2, |1 n = |e 1n |e 2n .\\n\\nThe states |a n and | -1 n are taken as a qubit n for quantum computation. In the large detuning limit, adiabatically eliminating the excited states and setting 2 ), we obtain the effective interaction Hamiltonian\\n\\nΩ 1n g r 1n 2∆ r 1n = Ω r 2n g r 2n 2∆ r 2n = β r n and Ω s 1n g s 1n 2∆ s 1n = Ω s 2n g s 2n 2∆ s 2n = β s n , from (\\n\\nH ef f = n √ 2 a n (β r n e iϕ S + n + β s n S n ) + H.c. + H cf , ( 4\\n\\n) where S + n = |0 nn -1|+|1 nn 0|.\\n\\nIn the derivation of (4), we have assumed the resonant condition\\n\\ng s jn 2 ∆ s jn a † n a n + 3 Ω r jn 2 4∆ r jn = Ω s jn 2 4∆ s jn + g r jn 2 ∆ r jn a † n a n + δ ′ jn .\\n\\nIn order to satisfy the condition with the flexible choice of Ω r jn , Ω s jn , ∆ r jn and ∆ s jn , we have introduced additional ac-Stark shifts δ ′ jn to states |g jn , which can be generated by using a laser field to couple the level |g jn to an ancillary level.\\n\\nWe now introduce three normal modes c and c ± with frequencies ω and ω ± √ 2ν by use of the unitary transfor-\\n\\nmation a 1 = 1 2 (c + + c -+ √ 2c), a 2 = 1 2 (c + + c -- √ 2c), b = 1 √ 2 (c + -c -) [17, 18]. In the limit ν ≫ |β r j |, |β s j |,\\n\\nneglecting the far off-resonant modes c ± and setting β p 1 = -β p 2 = β p with p = r, s, we can approximately write the effective Hamiltonian (4) as\\n\\nH ef f = (β r e iϕ S + + β s S)c + H.c., ( 5\\n\\n)\\n\\nwhere\\n\\nS + = S + 1 + S + 2 .\\n\\nSince the modes c ± are nearly not excited and decoupled with the resonant mode c, the fiber mode b is mostly in the vacuum state, therefore, L f iber ρ T can be neglected, and L cav1 ρ T + L cav2 ρ T can be approximated as\\n\\nL cav ρ T = κ(2cρ T c † -c † cρ T -ρ T c † c), ( 6\\n\\n)\\n\\nwhere κ = (κ 1 + κ 2 )/2. In the bad cavity limit, κ ≫ β, adiabatically eliminating the mode c [12, 14] , from Eq. ( 1 ) with the replacement of the Hamiltonian (2) and the relaxation terms (3) by the effective Hamiltonian (5) and the relaxation term (6), respectively, we can obtain the master equation for the density matrix of the atoms\\n\\nρ = - Γ 2 (R + Rρ + ρR + R -2RρR + ), ( 7\\n\\n)\\n\\nwhere ρ = Tr f (ρ T ), R = S cosh r + e iϕ S † sinh r, r = cosh -1 (β r / β r 2 -β s 2\\n\\n) and Γ = 2(β r 2 -β s2 )/κ. Eq. ( 7 ) describes the collective interaction of two cascade threelevel atoms with the effective squeezed vacuum reservoir [10] . The parameters β r , β s and ϕ are easily changed and controlled at will by varying the strength and phase of the driving lasers [8] . We will show that a geometric phase gate can be realized through changing these parameters.\\n\\nThe DFS of the atomic system is spanned by the states which satisfy the equation R(r, ϕ\\n\\n)|ψ DF (r, ϕ) = 0 [10]. In terms of basis states |e 1 = |a 1 |a 2 , |e 2 = |a 1 | -1 2 , |e 3 = | -1 1 |a 2 , |e 4 = |a 1 |0 2 , |e 5 = |0 1 |a 2 , |e 6 = |a 1 |1 2 , |e 7 = |1 1 |a 2 , |e 8 = |1 1 |1 2 , |e 9 = 1 √ 2 (|1 1 |0 2 + |0 1 |1 2 ), |e 10 = | - 1 1 | -1 2 , |e 11 = 1 √ 2 (|0 1 | -1 2 + | -1 1 |0 2 ), |e 12 = 1 √ 6 (|1 1 | -1 2 + | -1 1 |1 2 ) + 2 √ 6 |0 1 |0 2 )\\n\\n, the DFS states can be written as\\n\\n|ψ DF (r, ϕ) 1 =|e 1 , |ψ DF (r, ϕ) j = cosh r √ cosh 2r |e j -e iϕ sinh r √ cosh 2r |e j+4 , j = 2, 3, |ψ DF (r, ϕ) 4 = e 2iϕ (tanh r) 2 |e 8 -2 3 e iϕ tanh r|e 12 + |e 10 (tanh r) 4 + 2 3 (tanh r) 2 + 1 . ( 8\\n\\n)\\n\\nLet's introduce a unitary transformation O(r, ϕ) 1, 2, 3, 4. For the transformed density matrix ρ = O † ρO, we have\\n\\nby |φ i = 12 i=1 O ij (r, ϕ)|e j , where |φ i = |ψ DF i for i =\\n\\ndρ dt = i[G, ρ] + O † dρ dt O, ( 9\\n\\n)\\n\\nwhere G(r, ϕ) = iO † dO dt = iO † [ ṙ dO dr + φ dO dϕ ]. To solve Eq. (9) in the DFS, let's define the time-independent projector Π(0 9) , we obtain the equation of motion for ρDF = Π(0)ρΠ(0)\\n\\n) = O † Π(r, ϕ)O = 4 i=1 O † |φ i φ i |O = 3 j=1 |e j e j | + |e 10 e 10 | onto the DFS. From (\\n\\ndρ DF dt =i[G DF , ρDF ] + iΠ(0)GΠ ⊥ (0)ρΠ( 0\\n\\n)\\n\\n-iΠ(0)ρΠ ⊥ (0)GΠ(0) + Π(0)O † dρ dt OΠ(0), (10) where Π ⊥ (0) = 1 -Π(0) and G DF = Π(0)GΠ(0). In the limit of ṙ, φ ≪ Γ, the last three terms in Eq. (10) can be neglected [11] . In this way, Eq. (10) is reduced to\\n\\ndρ DF dt = i[G DF , ρDF ]. ( 11\\n\\n)\\n\\nTherefore, in the frame dragged adiabatically by the reservoir, the state of the atoms in the DFS unitarily evolves in time.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. REALIZING CONTROLLING PHASE GATES THROUGH MANIPULATING THE SQUEEZING ENVIRONMENT\",\n \"text\": \"In this section, we investigate how to realize a CPHASE gate through manipulating the engineered reservoir. Suppose that at the initial time the laser field driving the transition |g ↔ |s is switched off but the laser field driving the transition |r ↔ |e is switched on and the atoms are in the DFS state |Ψ(0\\n\\n) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ) = 4 j=1 |ψ DF (0, 0) j /2.\\n\\nTo generate a geometric phase for the atomic state, we smoothly change the parameters of the engineered reservoir along a closed loop, which is divided into the following three steps: (1) From time 0 to T 1 , hold on ϕ = 0, and adiabatically increase the parameter r from 0 to r 0 ; (2) From time T 1 to T 2 , hold on r = r 0 , and adiabatically change the phase ϕ from 0 to ϕ 0 ; (3) From time T 2 to T 3 , hold on ϕ = ϕ 0 , and adiabatically decrease r from r 0 to 0. When the cyclic evolution ends, the atomic state becomes\\n\\n|Ψ(T 3 ) a = 1 2 (|e 1 + e iχ1 |e 2 + e iχ1 |e 3 + e iχ12 |e 10 ), (12) where geometric phases χ 1 = -ν 1 ϕ 0 , χ 12 = -ν 12 ϕ 0 with ν 1 = sinh 2 r0 sinh 2 r0+cosh 2 r0 , ν 12 = 2 tanh 4 r0+ 2 3 tanh 2 r0 tanh 4 r0+ 2 3 tanh 2 r0+1 . By per- forming local transformations U 1 = e -iχ1 |-1 11 -1| and 4 U 2 = e -iχ1 | -1 22 -1|,\\n\\nthe state (12) can be written as\\n\\n|Ψ ′ (T 3 ) a = U 1 U 2 |Ψ(T 3 ) a = 1 2 (|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + e i∆ | -1 1 | -1 2 ), where ∆ = χ 12 -2χ 1 = (2ν 1 -ν 12 )ϕ 0 .\\n\\nThus, the CPHASE gate with the phase shift ∆ is realized. If both the atoms in cavity 1 and the atoms in cavity 2 \\\"see\\\" different environments, |χ 12 | must be equal to |2χ 1 | and ∆ = 0. Therefore, the phase shift ∆ results from the collective coupling of the atoms in both cavities with the same engineered environment.\\n\\nIf r 0 = atanh( 4/3 -1) ≃ 0.4157, |ν 12 | = |ν 1 |. Under this condition with ϕ 0 = π/ν 1 ,\\n\\nthe state of the atoms at the time T\\n\\n3 is |Ψ ′′ (T 3 ) a = -1 2 (-|a 1 |a 2 + |a 1 | -1 2 + | -1 1 |a 2 + | -1 1 | -1 2 ).\\n\\nIn this case, the Controlled-Z gate between the two qubits is realized without local transformations. 0 0.2 0.4 0.6 0.8 0.9965 0.997 0.9975 0.998 0.9985 0.999 0.9995 1 r 0 F r T=100/Γ T=200/Γ T=400/Γ (a)\\n\\nFIG. 3. Fidelity F r of the atomic state. 0 0.2 0.4 0.6 0.8 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 r 0 F p T = 200/Γ T = 400/Γ T = 1000/Γ (b) FIG. 4. Fidelity F p of the atomic state.\\n\\nThe above results depend on the adiabatical approximation. To check the adiabatical condition, we numerically simulate the following two examples. In the first example, we suppose that at the initial time the atoms are in the state |Ψ 1 a = (|a 1 |a 2 + |ψ DF (0, 0) 2 )/ √ 2 and the laser field driving the transition |e ↔ |r are turned on. Then, by slowly switching the laser field driving the transition |g ↔ |s , we increase the parameter r from 0 to r 0 according to the linear function r(t) = r 0 t/T . In the adiabatical limit (T ≫ Γ -1 ), the atomic state becomes\\n\\n|Ψ ′ 1 a = (|a 1 |a 2 + |ψ DF (r 0 , 0) 2 )/ √ 2\\n\\nat the time T . On the other hand, in the Hilbert space spanned by the basis states {|e i } for i = 1, 2, • • • , 12 , we can numerically solve Eq. ( 7 ) and obtain the density matrix ρ 1 (T ) of the atoms. Let's define\\n\\nF r = a Ψ ′ 1 |ρ 1 (T )|Ψ ′ 1 a as\\n\\nthe fidelity for this process. As shown in Fig. 3 , if T > 100/Γ, F r is always bigger than 0.997 if r ∈ (0, 0.8), corresponding to the almost perfect evolution.\\n\\nIn the second example, we suppose that the atoms are initially in the state\\n\\n|Ψ 2 a = (|a 1 |a 2 + |ψ DF (r, 0) 4 )/\\n\\n√ 2 and all the driving fields are turned on to hold the parameters r = r 0 and ϕ = 0. By adiabatically changing the phase ϕ from 0 to 2π at the rate φ = 2π/T , the atomic state at the time\\n\\nT becomes |Ψ ′ 2 a = (|a 1 |a 2 + e iχ12 |ψ DF (r, 2π) 4 )/ √ 2.\\n\\nLet's define the fidelity for this example as\\n\\nF p = a Ψ ′ 2 |ρ(T )|Ψ ′ a ,\\n\\nwhere ρ(T ) is the numerical solution of Eq. ( 7 ). As shown in Fig. 4 , F p increases as T increases but decreases as the parameter r 0 increases. If T > 1000/Γ, F p is larger than 0.992 for 0 < r 0 < 0.8. From these two examples, we find that to fulfill the adiabatical condition the time used in the step 2 should be much longer than in the steps 1 and 3.\\n\\nA controlled-Z gate has been numerically simulated by directly solving Eq. ( 7 ) with r 0 = 0.5, and ϕ\\n\\n0 = π/|2ν 1 -ν 12 |.\\n\\nIn the simulation, we set ṙ = r 0 /T 1 in the steps 1 and 3, and φ = ϕ 0 /(T 2 -T 1 ) in the step 2 with T 1 = 0.05T 3 and T 2 -T 1 = 0.90T 3 . If T 3 > 1100/Γ, we find that the fidelity F = a Ψ(T 3 )|ρ(T 3 )|Ψ(T 3 ) a is larger than 0.95. For an almost perfect controlled-Z gate with F > 0.99, we find that T 3 must be longer than 6000/Γ. Now let's briefly discuss the effects of the atomic spontaneous emission, the fiber mode decay and cavity photon leakage. For simplicity but without the loss of generality, we suppose that atomic spontaneous emission rates of the excited levels are equal to γ. In the large detunig limit, the characteristic spontaneous emission rate of the atoms is γ ef f = γ(Ω 2 /2∆ 2 ) [14, 16] and the effective decay rate of the fiber mode is\\n\\nκ ef f = κ f Ω 2 g 2 /(4∆ 2 ν 2 ). If κ f ≤ γ and g 2 ≪ ν 2 , κ\\n\\nef f be much smaller than γ ef f . Under this condition, the present scheme is feasible if Γ ≫ γ ef f . In the current cavity quantum dynamic (CQED) experiment, the parameters (g, κ, γ) = (2000, 10, 10) MHz could be available [20] . If setting Ω/(2∆) = 1 √ 2 × 10 -3 , we have Γ ≃ 4 × 10 4 γ ef f . The condition is held. In the present scheme, the large cavity decay rate is required to ensure that the cavity modes are in a broadband squeezed vacuum reservoir and then the atoms always \\\"see\\\" the broadband squeezed vacuum reservoir during the dynamic evolution. For an arbitrary small but nonzero value of the squeezing degree of the reservoir, a CPHASE gate with arbitrary high fidelity can always be realized in the represent scheme. The cavity decay does 5 not directly affect the fidelity of the realized CPHASE gates. However, the larger the decay rate is, the longer the operation time of the CPHASE gates is. Thus, we have the condition κ >> β, γ ef f for realizing the reliable CPHASE gates. Based on the parameters quoted above, this condition can be well satisfied. With the parameters of the current CQED experiment, we find that the operation time of the controlled-Z gate, with fidelity larger than 0.95, is about 2.8 ms. It is much shorter than both 1/γ ef f and the single-atom trapping time in cavity [21] . On the other hand, the present scheme needs a strong coupling between the cavity and the fiber. This could be realized at the current experiment [22] . Therefore, the requirement for the realization of the present scheme can be satisfied with the current technology.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"IV. CONCLUSIONS\",\n \"text\": \"We propose a cavity-atom coupled scheme for the realization of quantum controlling gates, in which each of two pairs of four-level atoms in two distant cavities connected by a short optical fibre are simultaneously driven by laser fields and coupled to the local cavity modes through the double Raman transition configuration. We show that an effective squeezing reservoir coupled to the multilevel atoms can be engineered under appropriate driving condition and bad cavity limit. We find that in the scheme a CPHASE gate with arbitrary phase shift can be implemented through adiabatically changing the strength and phase of driving fields along a closed loop. It is also noticed that the larger the effective coupling strength between the environment and the atoms is, the more reliable the realized CPHASE gate is.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgments\",\n \"text\": \"We thank Yun-feng Xiao and Wen-ping He for valuable discussions and suggestions. This work was supported by the Natural Science Foundation of China (Grant Nos. 10674106, 60778021 and 05-06-01).\\n\\n[1] P. W. Shor, in Proceeings, 35 th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Press, Los Almitos, CA, 1994), p. 124. [2] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953 (1997). [3] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997). [4] D. A. Lidar and et al., Phys. Rev. Lett. 81, 2594 (1998). [5] M. V. Berry, Proc. R. Soc. A 392, 45 (1984). [6] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999). [7] S.-L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301 (2005). [8] L.-M. Duan and et al., Science 292, 1695 (2001). [9] L.-A. Wu and et al., Phys. Rev. Lett. 95, 130501 (2005). [10] Angelo Carollo et al., Phys. Rev. Lett. 96, 150403 (2006). [11] Angelo Carollo et al., Phys. Rev. Lett. 96, 020403 (2006). [12] J. I. Cirac, Phys. Rev. A 46, 4354 (1992). [13] C. J. Myatt and et al., Nature (London) 403, 269 (2000). [14] S. G. Clark and A. S. Parkins, Phys. Rev. Lett. 90, 047905 (2003). [15] S. Lloyd, Phys. Rev. Lett. 75, 346 (1995). [16] T. Pellizzari, Phys. Rev. Lett. 79, 5242 (1997). [17] A. Serafini and et al., Phys. Rev. Lett. 96, 010503 (2006) . [18] Zhang-qi Yin and Fu-li Li, Phys. Rev. A 75, 012324 (2007). [19] S. J. van Enk and et al., Phys. Rev. A 59, 2659 (1999). [20] S. M. Spillane et al., Phys. Rev. A 71, 013817 (2005). [21] Stefan Nuβmann and et al., Nature Phys. 1, 122 (2005). [22] S. M. Spillane et al., Phys. Rev. Lett. 91, 043902 (2003).\"\n }\n]","isConversational":false}}},{"rowIdx":86,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0488"},"arxiv_id":{"kind":"string","value":"0704.0488"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1007/s10714-007-0431-5"},"content_hash":{"kind":"string","value":"5164ad34ca4de631f8a4c37f1aba38127c537f07d91eec4dfc4322512c2ad2cd"},"title":{"kind":"string","value":"Teleparallel Version of the Stationary Axisymmetric Solutions and their\n Energy Contents"},"abstract":{"kind":"string","value":"This work contains the teleparallel version of the stationary axisymmetric\nsolutions. We obtain the tetrad and the torsion fields representing these\nsolutions. The tensor, vector and axial-vector parts of the torsion tensor are\nevaluated. It is found that the axial-vector has component only along $\\rho$\nand $z$ directions. The three possibilities of the axial vector depending on\nthe metric function $B$ are discussed. The vector related with spin has also\nbeen evaluated and the corresponding extra Hamiltonian is furnished. Further,\nwe use the teleparallel version of M$\\ddot{o}$ller prescription to find the\nenergy-momentum distribution of the solutions. It is interesting to note that\n(for $\\lambda=1$) energy and momentum densities in teleparallel theory are\nequal to the corresponding quantities in GR plus an additional quantity in\neach, which may become equal under certain conditions. Finally, we discuss the\ntwo special cases of the stationary axisymmetric solutions."},"authors":{"kind":"list like","value":["M. Sharif and M. Jamil Amir"],"string":"[\n \"M. Sharif and M. Jamil Amir\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["gr-qc"],"string":"[\n \"gr-qc\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"gr-qc"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"This work contains the teleparallel version of the stationary axisymmetric solutions. We obtain the tetrad and the torsion fields representing these solutions. The tensor, vector and axial-vector parts of the torsion tensor are evaluated. It is found that the axial-vector has component only along ρ and z directions. The three possibilities of the axial vector depending on the metric function B are discussed. The vector related with spin has also been evaluated and the corresponding extra Hamiltonian is furnished. Further, we use the teleparallel version of Möller prescription to find the energy-momentum distribution of the solutions. It is interesting to note that (for λ = 1) energy and momentum densities in teleparallel theory are equal to the corresponding quantities in GR plus an additional quantity in each, which may become equal under certain conditions. Finally, we discuss the two special cases of the stationary axisymmetric solutions.\n\nThe attempts made by Einstein and his followers to unify gravitation with other interactions led to the investigation of structures of gravitation other than the metric tensor. These structures yield the metric tensor as a by product. Tetrad field is one of these structures which leads to the theory of teleparallel gravity (TPG) [1, 2] . TPG is an alternative theory of gravity which corresponds to a gauge theory of translation group [3, 4] based on Weitzenböck geometry [5] . This theory is characterized by the vanishing of curvature identically while the torsion is taken to be non-zero. In TPG, gravitation is attributed to torsion [4] which plays a role of force [6] . In General Relativity (GR), gravitation geometrizes the underlying spacetime. The translational gauge potentials appear as a non-trivial part of the tetrad field and induce a teleparallel (TP) structure on spacetime which is directly related to the presence of a gravitational field. In some other theories [3] [4] [5] [6] [7] [8] , torsion is only relevant when spins are important [9] . This point of view indicates that torsion might represent additional degrees of freedom as compared to curvature. As a result, some new physics may be associated with it. Teleparallelism is naturally formulated by gauging external (spacetime) translations which are closely related to the group of general coordinate transformations underlying GR. Thus the energy-momentum tensor represents the matter source in the field equations of tetradic theories of gravity like in GR.\n\nThere is a large literature available [10] about the study of TP versions of the exact solutions of GR. Recently, Pereira, et al. [11] obtained the TP versions of the Schwarzschild and the stationary axisymmetric Kerr solutions of GR. They proved that the axial-vector torsion plays the role of the gravitomagnetic component of the gravitational field in the case of slow rotation and weak field approximations. In a previous paper [12] , we have found the TP versions of the Friedmann models and of the Lewis-Papapetrou spacetimes, and also discussed their axial-vectors.\n\nThere has been a longstanding, controversial and still unresolved problem of the localization of energy (i.e., to express it as a unique tensor quantity) in GR [13] . Einstein [14] introduced the energy-momentum pseudo-tensor and then Landau-Lifshitz [15] , Papapetrou [16] , Bergmann [17] , Tolman [18] and Weinberg [19] proposed their own prescriptions to resolve this issue. All these prescriptions can provide meaningful results only in Cartesian coordinates. But Möller [20] introduced a coordinate-independent prescription. The idea of coordinate-independent quasi-local mass was introduced [21] by associ-ating a Hamiltonian term to each gravitational energy-momentum pseudotensor. Later, a Hamiltonian approach in the frame of Schwinger condition [22] was developed, followed by the construction of a Lagrangian density of TP equivalent to GR [4, 6, 23, 24] . Many authors explored several examples in the framework of GR and found that different energy-momentum complexes can give either the same [25] or different [26] results for a given spacetime.\n\nMikhail et al. [27] defined the superpotential in the Moller's tetrad theory which has been used to find the energy in TPG. Vargas [28] defined the TP version of Bergman, Einstein and Landau-Lifshitz prescriptions and found that the total energy of the closed Friedman-Robinson-Walker universe is zero by using the last two prescriptions. This agrees with the results of GR available in literature [29] . Later, many authors [30] used TP version of these prescriptions and showed that energy may be localized in TPG. Keeping these points in mind, this paper is addressed to the following two problems: We obtain TP version of the stationary axisymmetric solutions and then calculate the axial-vector part of the torsion tensor. The energymomentum distribution of the solutions is explored by using the TP version of Möller prescription.\n\nThe scheme adopted in this paper is as follows. In section 2, we shall review the basic concepts of TP theory. Section 3 contains the TP version of the stationary axisymmetric solutions and the tensor, vector and axial-vector parts of the torsion tensor. Section 4 is devoted to evaluate the energymomentum distribution for this family of solutions using the TP version of Möller prescription. In section 5, we present two special solutions for this class of metrics and investigate the corresponding quantities. The last section contains a summary and a discussion of the results obtained.\n\nIn teleparallel theory, the connection is a Weitzenböck connection given as [31] Γ\n\nθ µν = h a θ ∂ ν h a µ , (1)\n\nwhere h a ν is a non-trivial tetrad. Its inverse field is denoted by h a µ and satisfy the relations\n\nh a µ h a ν = δ µ ν ; h a µ h b µ = δ a b . (2)\n\nIn this paper, the Latin alphabet (a, b, c, ... = 0, 1, 2, 3) will be used to denote tangent space indices and the Greek alphabet (µ, ν, ρ, ... = 0, 1, 2, 3) to denote spacetime indices. The Riemannian metric in TP theory arises as a by product [4] of the tetrad field given by\n\ng µν = η ab h a µ h b ν , (3)\n\nwhere η ab is the Minkowski metric η ab = diag(+1, -1, -1, -1). For the Weitzenböck spacetime, the torsion is defined as [2]\n\nT θ µν = Γ θ νµ -Γ θ µν (4)\n\nwhich is antisymmetric w.r.t. its last two indices. Due to the requirement of absolute parallelism, the curvature of the Weitzenböck connection vanishes identically. The Weitzenböck connection also satisfies the relation\n\nΓ 0 θ µν = Γ θ µν -K θ µν , (5)\n\nwhere\n\nK θ µν = 1 2 [T µ θ ν + T ν θ µ -T θ µν ] (6)\n\nis the contortion tensor and Γ 0 θ µν are the Christoffel symbols in GR. The torsion tensor of the Weitzenböck connection can be decomposed into three irreducible parts under the group of global Lorentz transformations [4] : the tensor part\n\nt λµν = 1 2 (T λµν + T µλν ) + 1 6 (g νλ V µ + g νµ V λ ) - 1 3 g λµ V ν , (7)\n\nthe vector part\n\nV µ = T ν νµ ( 8\n\n)\n\nand the axial-vector part\n\nA µ = 1 6 ǫ µνρσ T νρσ . (9)\n\nThe torsion tensor can now be expressed in terms of these irreducible components as follows\n\nT λµν = 1 2 (t λµν -t λνµ ) + 1 3 (g λµ V ν -g λν V µ ) + ǫ λµνρ A ρ , (10)\n\nwhere\n\nǫ λµνρ = 1 √ -g δ λµνρ . (11)\n\nHere δ = {δ λµνρ } and δ * = {δ λµνρ } are completely skew symmetric tensor densities of weight -1 and +1 respectively [4] . TP theory provides an alternate description of the Einstein's field equations which is given by the teleparallel equivalent of GR [24, 31] . Mikhail et al. [27] defined the super-potential (which is antisymmetric in its last two indices) of the Möller tetrad theory as\n\nU µ νβ = √ -g 2κ P τ νβ χρσ [V ρ g σχ g µτ -λg τ µ K χρσ -(1 -2λ)g τ µ K σρχ ], (12)\n\nwhere\n\nP τ νβ χρσ = δ χ τ g νβ ρσ + δ ρ τ g νβ σχ -δ σ τ g νβ χρ ( 13\n\n)\n\nand g νβ ρσ is a tensor quantity defined by\n\ng νβ ρσ = δ ρ ν δ σ β -δ σ ν δ ρ β . ( 14\n\n)\n\nK σρχ is the contortion tensor given by Eq.( 6 ), g is the determinant of the metric tensor g µν , λ is the free dimensionless coupling constant of TPG, κ is the Einstein constant and V µ is the basic vector field given by Eq.( 8 ). The energy-momentum density is defined as\n\nΞ ν µ = U νρ µ , ρ , (15)\n\nwhere comma means ordinary differentiation. The momentum 4-vector of Möller prescription can be expressed as\n\nP µ = Σ Ξ 0 µ dxdydz, (16)\n\nwhere P 0 gives the energy and P 1 , P 2 and P 3 are the momentum components while the integration is taken over the hypersurface element Σ described by x 0 = t = constant. The energy may be given in the form of surface integral [20] as\n\nE = lim r→∞ r=constant U 0 0ρ u ρ dS, (17)\n\nwhere u ρ is the unit three-vector normal to the surface element dS.\n\nTupper [32] found five classes of non-null electromagnetic field plus perfect fluid solutions in which the electromagnetic field does not inherit one of the symmetries of the spacetime. The metric representing the stationary axisymmetric solutions is given by [32 ]\n\nds 2 = dt 2 -e 2K dρ 2 -(F 2 -B 2 )dφ 2 -e 2K dz 2 + 2Bdtdφ, (18)\n\nwhere B = B(ρ, z), K = K(ρ, z) and F = F (ρ) are such functions which satisfy the following relations . The metric given by Eq.( 18 ) represents five classes of non-null electromagnetic field and perfect fluid solutions which possesses a metric symmetry not inherited by the electromagnetic field and admits a homothetic vector field. Two of these classes contain electrovac solutions as special cases, while the other three necessarily contain some fluid. The generalization of this metric is given in [34] .\n\nḂ = F W ′ , B ′ = - 1 4 aF ( Ẇ 2 -W ′2 ), K ′ = - 1 2 aF Ẇ W ′ , Ẅ + Ḟ F -1 Ẇ + W ′′ = 0, (19)\n\nUsing the procedure adopted in the papers [11, 12] , the tetrad components of the above metric can be written as\n\nh a µ = 1 0 B 0 0 e K cos φ -F sin φ 0 0 e K sin φ F cos φ 0 0 0 0 e K (20)\n\nwith its inverse\n\nh a µ = 1 0 0 0 B F sin φ e -K cos φ -1 F sin φ 0 -B F cos φ e -K sin φ 1 F cos φ 0 0 0 0 e -K . ( 21\n\n)\n\nThe non-vanishing components of the torsion tensor are\n\nT 0 12 = Ḃ + B F (e K -Ḟ ), T 0 32 = B ′ , T 1 13 = -K ′ , T 2 12 = 1 F ( Ḟ -e K ), T 3 31 = -K. ( 22\n\n)\n\nUsing these expressions in Eqs.( 7 )-( 9 ), we obtain the following non-zero components of the tensor part\n\nt 001 = 1 3 [ K + 1 F ( Ḟ -e K )], t 003 = 1 3 K ′ , t 010 = 1 6 { 1 F (e K -Ḟ ) -K} = t 100 , t 030 = - 1 6 K ′ = t 300 , t 012 = 1 2 Ḃ + B 6 { 1 F (e K -Ḟ ) -K} = t 102 , t 021 = - 1 2 Ḃ - B 3 { 1 F (e K -Ḟ ) -K} = t 201 , t 023 = - 1 2 B ′ + 1 3 BK ′ = t 203 , t 032 = 1 2 B ′ - 1 6 BK ′ = t 302 , t 122 = 1 2 {F (e K -Ḟ ) + B Ḃ} + 1 6 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K} = t 212 , t 120 = B 6 { 1 F (e K -Ḟ ) -K} = t 210 , t 113 = 2K ′ 3 e 2K , t 131 = - K ′ 3 e 2K = t 311 , t 133 = - e 2K 6 { 1 F (e K -Ḟ ) + 2 K} = t 313 , t 221 = -F (e K -Ḟ ) -B Ḃ - 1 3 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K}, t 223 = -BB ′ + K ′ 3 (B 2 -F 2 ), t 331 = e 2K 3 { 1 F (e K -Ḟ ) + 2 K}, t 322 = 1 2 BB ′ - K ′ 6 (B 2 -F 2 ) = t 232 , t 320 = - 1 6 BK ′ = t 232 , (23)\n\nthe vector part\n\nV 1 = - 1 F ( Ḟ -e K ) -K, (24)\n\nV 3 = -K ′ , (25)\n\nand the axial-vector part\n\nA 1 = B ′ 3F e -2K , (26)\n\nA 3 = Ḃ 3F e -2K , (27)\n\nrespectively. The axial-vector component along the φ-direction vanishes and hence the spacelike axial-vector can be written as\n\nA = √ -g 11 A 1 êρ + √ -g 33 A 3 êz , (28)\n\nwhere êρ and êz are unit vectors along the radial and z-directions respectively. Substituting A 1 , A 3 , g 11 and g 33 in Eq.( 28 ), it follows that\n\nA = e -K 3F (B ′ êρ + Ḃê z ). ( 29\n\n)\n\nThis shows that the axial-vector lies along radial direction if B = B(z), along z-direction if B = B(ρ) and vanishes identically if B is constant. As the axialvector torsion represents the deviation of axial symmetry from cylindrical symmetry, the symmetry of the underlying spacetime will not be affected even for B constant. Also, the torsion plays the role of the gravitational force in TP theory, hence a spinless particle will obey the force equation [11, 24]\n\ndu ρ ds -Γ µρν u µ u ν = T µρν u µ u ν . ( 30\n\n)\n\nThe left hand side of this equation is the Weitzenböck covariant derivative of u ρ along the particle world-line. The appearance of the torsion tensor on its right hand side indicates that the torsion plays the role of an external force in TPG. It has been shown, both in GR and TP theories, by many authors [4, 35] that the spin precession of a Dirac particle in torsion gravity is related to the torsion axial-vector by\n\ndS dt = -b × S, ( 31\n\n)\n\nwhere S is the spin vector of a Dirac particle and b = 3 2 A, with A the spacelike part of the torsion axial-vector. Thus\n\nb = e -K 2F {B ′ êρ + Ḃê z }. (32)\n\nThe corresponding extra Hamiltonian [36] is given by\n\nδH = -b.σ, ( 33\n\n)\n\nwhere σ is the spin of the particle [35] . Using Eq.( 32 ), this takes the form\n\nδH = - e -K 2F (B ′ êρ + Ḃê z ).σ. ( 34\n\n)\n\n4 Teleparallel Energy of the Stationary Axisymmetric Solutions\n\nIn this section we evaluate the component of energy-momentum densities by using the teleparallel version of Möller prescription. Multiplying Eqs.( 24 ) and ( 25 ) by g 11 and g 33 respectively, it follows that\n\nV 1 = Ke -2K + e -2K F ( Ḟ -e K ), (35)\n\nV 3 = K ′ e -2K . (36)\n\nIn view of Eqs.( 6 ) and ( 22 ), the non-vanishing components of the contorsion tensor are\n\nK 100 = -e -2K { B 2 F 3 (e K -Ḟ ) + B Ḃ F 2 } = -K 010 , K 300 = - BB ′ F 2 e -2K = -K 030 , K 122 = - e -2K F 3 (e K -Ḟ ) = -K 212 , K 133 = Ke -4K = -K 313 , K 311 = K ′ e -4K = -K 131 , K 102 = K 120 = e -2K { B F 3 (e K -Ḟ ) + Ḃ 2F 2 } = -K 012 = -K 210 , K 302 = K 320 = K 023 = B ′ 2F 2 e -2K = -K 032 = -K 230 = -K 203 , K 021 = Ḃ 2F 2 e -2K = -K 201 . ( 37\n\n)\n\nIt should be mentioned here that the contorsion tensor is antisymmetric w.r.t. its first two indices. Making use of Eqs.( 35 )- (37) in Eq.( 12 ), we obtain the required independent non-vanishing components of the supperpotential in Möller's tetrad theory as\n\nU 01 0 = 1 κ [e K -Ḟ -F K + 1 2 (1 + λ) B Ḃ F ] = -U 10 0 , U 03 0 = 1 κ [-F K ′ + 1 2 (1 + λ) BB ′ F ] = -U 30 0 , U 21 0 = - 1 2κ (1 + λ) Ḃ F = -U 12 0 , U 23 0 = - 1 2κ (1 + λ) B ′ F = -U 32 0 , U 01 2 = 1 κ [B(e K -Ḟ ) + 1 2 (1 + λ) B 2 Ḃ F + 1 2 (1 -λ) ḂF ] = -U 10 2 , U 03 2 = 1 κ [ 1 2 (1 + λ) B 2 B ′ F + 1 2 (1 -λ)B ′ F ] = -U 30 2 , U 02 1 = 1 2κF (λ -1) Ḃe 2K = -U 20 1 , U 02 3 = 1 2κF (λ -1)B ′ e 2K = -U 30 1 . (38)\n\nIt is worth mentioning here that the supperpotential is skew symmetric w.r.t. its last two indices. When we make use of Eqs.( 15 ), ( 37 ), (38) and take λ = 1, the energy density turns out to be\n\nΞ 0 0 = 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ ) + 1 F 2 {BF ( B + E ′′ ) + ( Ḃ2 + B ′ 2 )F -B Ḃ Ḟ }]. ( 39\n\n)\n\nThis implies that\n\nE d T P T = E d GR + 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ )], (40)\n\nwhere E d stands for energy density. The only non-zero component of momentum density is along φ-direction and (for λ = 1) it takes the form\n\nΞ 0 2 = 1 κF 2 {F 3 ( B + B ′′ ) + B 2 F ( B + B ′′ ) + 2BF ( Ḃ2 + B ′ 2 ) -Ḃ Ḟ (B 2 + F 2 )} + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (41)\n\nthat is,\n\nM d T P T = M d GR + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (42)\n\nwhere M d stands for momentum density.\n\nIn this section, we evaluate the above quantities for some special cases of the non-null Einstein Maxwell solutions.\n\nA special case of the non-null Einstein-Maxwell solutions can be obtained by choosing\n\nB = m n e nρ , F = e nρ , K = 0. ( 43\n\n)\n\nThis is known as electromagnetic generalization of the Gödel solution [32] .\n\nWhen we make use of Eq.( 43 ) in Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42), the corresponding results reduce to\n\nt 001 = 1 3 (n -e -nρ ), t 010 = 1 6 (e -nρ -n) = t 100 , t 012 = m 6n (1 + 2ne nρ ) = t 102 , t 021 = - m 3n (1 + 2ne nρ ) = t 201 , t 122 = e nρ 6n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ } = t 212 , t 120 = m 6n (1 -ne nρ ) = t 210 , t 133 = 1 6 (n -e -nρ ) = t 313 , t 221 = - e nρ 3n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ },\n\nt 331 = - 1 3 (n -e -nρ ), (44)\n\nV 1 = e -nρ -n, V 3 = 0, (45)\n\nA 1 = 0, A 3 = m 3 , (46)\n\nA = m 3 êz , b = m 2 êz , (47)\n\nδH = m 2 êz .σ, (48)\n\nΞ 0 0 = 1 κ (m 2 -n 2 )e nρ , (49)\n\nE d T P T = E d GR - n 2 κ e nρ (50)\n\nΞ 0 2 = 1 κ ( 2m 3 n ) + m κ (1 -2ne nρ )e nρ , (51)\n\nM d T P T = M d GR + m κ (1 -2ne nρ )e nρ . (52)\n\nThe metric (43) reduces to the usual perfect fluid solution when m = √ 2n [32] , i.e., B = √ 2e nρ . The corresponding energy and momentum densities take the form as\n\nE d T P T = E d GR - n 2 κ e nρ (53)\n\nM d T P T = M d GR + √ 2n κ (1 -2ne nρ )e nρ . (54)\n\nWhen we choose B = e aρ , F = e aρ √ 2 and K = 0, the metric given by Eq.( 18 ) reduces to the Gödel metric [32] . The results corresponding to Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42) take the following form\n\nt 001 = 1 3 (a - √ 2e -aρ\n\n), t 010 = -1 6 (a -√ 2e -aρ ) = t 100 , t 012 = 1 6 ( √ 2 + 2ae aρ ) = t 102 , t 021 = -1 6 (2 √ 2 + ae aρ ) = t 102 , t 122 = e aρ 6 (2 √ 2 + ae aρ ) = t 212 , t 120 = 1 6 ( √ 2ae aρ ) = t 210 , t 133 = 1 6 (a -√ 2e -aρ ) = t 313 , t 221 = -e aρ 3 (2 √ 2 + ae aρ ), (55)\n\nt 331 = 1 3 (a - √ 2e -aρ ), ( 56\n\n) V 1 = √ 2e -aρ -a, V 3 = 0, (57)\n\nA 1 = 0, A 3 = √ 2a 3 , (58)\n\nA = √ 2a 3 êz , b = a √ 2 êz , (59)\n\nδH = a √ 2 êz .σ, (60)\n\nΞ 0 0 = √ 2 κ a 2 e aρ - a 2 κ √ 2 e aρ , (61)\n\nE d T P T = E d GR - a 2 κ √ 2 e aρ (62) Ξ 0 2 = a 2 κ √ 2 e 2aρ + a κ (1 - √ 2ae aρ )e aρ , (63)\n\nM d T P T = M d GR + a κ (1 - √ 2ae aρ )e aρ . ( 64\n\n)\n\nThe purpose of this paper is twofold: Firstly, we have found the TP version of the non-null Einstein Maxwell solutions. This provides some interesting features about the axial vector and the corresponding quantities. Secondly, we have used the TP version of Möller prescription to evaluate the energymomentum distribution of the solutions. The axial-vector torsion of these solutions has been evaluated. The only non-vanishing components of the vector part are along the radial and the z-directions due to the cross term dx 0 dx 2 involving in the metric. This corresponds to the case of Kerr metric [11] , which involves the cross term dx 0 dx 3 . We also find the vector b which is related to the spin vector [4] as given by Eq.( 32 ). The axial-vector torsion lies in the ρz-plane, as its component along the φ-direction vanishes everywhere.\n\nThe non-inertial force on the Dirac particle can be represented as a rotation induced torsion of spacetime. There arise three possibilities for the axial-vector, depending upon the metric function B(ρ, z). When B is a function of z only, the axial-vector lies only along the radial direction. When B is a function of ρ only, the axialvector will lie along z-direction. The axial-vector vanishes identically for B to be constant. As the axial-vector represents the deviation from the symmetry of the underlying spacetime corresponding to an inertial field with respect to the Dirac particle, the symmetry of the spacetime will not be affected in the third possibility. Consequently there exists no inertial field with respect to the Dirac particle and the spin vector of the Dirac particle becomes constant. The corresponding extra Hamiltonian is expressed in terms of the vector b which vanishes when the metric function B is constant, i.e., when the axialvector becomes zero.\n\nThe energy-momentum distribution of the non-null Einstein-Maxwell solutions has been explored by using the TP version of Möller prescription. It is found that energy in the TP theory is equal to the energy in GR (as found by Sharif and Fatima [37] ) plus some additional part. If, for a particular case, we have K = 0 and K ′ , Ḟ = constant (or if Ḟ , K = 0 and K\n\n′ = constant), then E d T P G = E d GR . (65)\n\nOn the other hand, the only non-vanishing component of the momentum density lies along φ-direction, similar to the case of Kerr metric [11] , due to the cross term appearing in both the metrics. When we choose λ = 1, it becomes equal to be the momentum in GR [37] plus an additional quantity. If F , Ḃ, B ′′ , K all vanish, then\n\nM d T P G = M d GR . (66)\n\nBy taking particular values of E, F and K, we obtain the electromagnetic generalization of the Gödel solution and the Gödel metric as two special cases. The corresponding results for both the special cases are obtained. It is shown that, for the electromagnetic generalization of the Gödel solution, Eq.(65) does not hold, while Eq.(66) holds when m = 0. However, for the perfect fluid case, i.e., when m = √ 2n, both Eqs.( 65 ) and (66) hold by taking n = 0. In the case of the Gödel metric, these equations hold if we choose the arbitrary constant a = 0. For the special solutions, the vector part lies along the radial direction while the axial-vector part along z-direction.\n\nWe would like to re-iterate here that the tetrad formalism itself has some advantages which comes mainly from its independence from the equivalence principle and consequent suitability to the discussion of quantum issues. In TPG, an energy-momentum gauge current j i µ for the gravitational field can be defined. This is covariant under a spacetime general coordinate transformation and transforms covariantly under a global tangent space Lorentz transformation [38] . It, then, follows that j i µ is a true spacetime tensor but not a tangent space tensor. When we re-write the gauge field equations in a purely spacetime form, they lead to the Einstein field equations and the gauge current j i µ reduces to the canonical energy-momentum pseudo-tensor of the gravitational field. Thus TPG seems to provide a more appropriate environment to deal with the energy problem.\n\nFinally, it is pointed out that we are not claiming that this paper has resolved the problems of GR using the TPG. This is an attempt to touch some issues in TPG with the hope that this alternative may provide more feasible results. Also, it is always an interesting and enriching to look at things from another point of view. This endeavor is in itself commendable."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"This work contains the teleparallel version of the stationary axisymmetric solutions. We obtain the tetrad and the torsion fields representing these solutions. The tensor, vector and axial-vector parts of the torsion tensor are evaluated. It is found that the axial-vector has component only along ρ and z directions. The three possibilities of the axial vector depending on the metric function B are discussed. The vector related with spin has also been evaluated and the corresponding extra Hamiltonian is furnished. Further, we use the teleparallel version of Möller prescription to find the energy-momentum distribution of the solutions. It is interesting to note that (for λ = 1) energy and momentum densities in teleparallel theory are equal to the corresponding quantities in GR plus an additional quantity in each, which may become equal under certain conditions. Finally, we discuss the two special cases of the stationary axisymmetric solutions."},{"section_type":"BACKGROUND","section_title":"Introduction","text":"The attempts made by Einstein and his followers to unify gravitation with other interactions led to the investigation of structures of gravitation other than the metric tensor. These structures yield the metric tensor as a by product. Tetrad field is one of these structures which leads to the theory of teleparallel gravity (TPG) [1, 2] . TPG is an alternative theory of gravity which corresponds to a gauge theory of translation group [3, 4] based on Weitzenböck geometry [5] . This theory is characterized by the vanishing of curvature identically while the torsion is taken to be non-zero. In TPG, gravitation is attributed to torsion [4] which plays a role of force [6] . In General Relativity (GR), gravitation geometrizes the underlying spacetime. The translational gauge potentials appear as a non-trivial part of the tetrad field and induce a teleparallel (TP) structure on spacetime which is directly related to the presence of a gravitational field. In some other theories [3] [4] [5] [6] [7] [8] , torsion is only relevant when spins are important [9] . This point of view indicates that torsion might represent additional degrees of freedom as compared to curvature. As a result, some new physics may be associated with it. Teleparallelism is naturally formulated by gauging external (spacetime) translations which are closely related to the group of general coordinate transformations underlying GR. Thus the energy-momentum tensor represents the matter source in the field equations of tetradic theories of gravity like in GR.\n\nThere is a large literature available [10] about the study of TP versions of the exact solutions of GR. Recently, Pereira, et al. [11] obtained the TP versions of the Schwarzschild and the stationary axisymmetric Kerr solutions of GR. They proved that the axial-vector torsion plays the role of the gravitomagnetic component of the gravitational field in the case of slow rotation and weak field approximations. In a previous paper [12] , we have found the TP versions of the Friedmann models and of the Lewis-Papapetrou spacetimes, and also discussed their axial-vectors.\n\nThere has been a longstanding, controversial and still unresolved problem of the localization of energy (i.e., to express it as a unique tensor quantity) in GR [13] . Einstein [14] introduced the energy-momentum pseudo-tensor and then Landau-Lifshitz [15] , Papapetrou [16] , Bergmann [17] , Tolman [18] and Weinberg [19] proposed their own prescriptions to resolve this issue. All these prescriptions can provide meaningful results only in Cartesian coordinates. But Möller [20] introduced a coordinate-independent prescription. The idea of coordinate-independent quasi-local mass was introduced [21] by associ-ating a Hamiltonian term to each gravitational energy-momentum pseudotensor. Later, a Hamiltonian approach in the frame of Schwinger condition [22] was developed, followed by the construction of a Lagrangian density of TP equivalent to GR [4, 6, 23, 24] . Many authors explored several examples in the framework of GR and found that different energy-momentum complexes can give either the same [25] or different [26] results for a given spacetime.\n\nMikhail et al. [27] defined the superpotential in the Moller's tetrad theory which has been used to find the energy in TPG. Vargas [28] defined the TP version of Bergman, Einstein and Landau-Lifshitz prescriptions and found that the total energy of the closed Friedman-Robinson-Walker universe is zero by using the last two prescriptions. This agrees with the results of GR available in literature [29] . Later, many authors [30] used TP version of these prescriptions and showed that energy may be localized in TPG. Keeping these points in mind, this paper is addressed to the following two problems: We obtain TP version of the stationary axisymmetric solutions and then calculate the axial-vector part of the torsion tensor. The energymomentum distribution of the solutions is explored by using the TP version of Möller prescription.\n\nThe scheme adopted in this paper is as follows. In section 2, we shall review the basic concepts of TP theory. Section 3 contains the TP version of the stationary axisymmetric solutions and the tensor, vector and axial-vector parts of the torsion tensor. Section 4 is devoted to evaluate the energymomentum distribution for this family of solutions using the TP version of Möller prescription. In section 5, we present two special solutions for this class of metrics and investigate the corresponding quantities. The last section contains a summary and a discussion of the results obtained."},{"section_type":"OTHER","section_title":"An Overview of the Teleparallel Theory","text":"In teleparallel theory, the connection is a Weitzenböck connection given as [31] Γ\n\nθ µν = h a θ ∂ ν h a µ , (1)\n\nwhere h a ν is a non-trivial tetrad. Its inverse field is denoted by h a µ and satisfy the relations\n\nh a µ h a ν = δ µ ν ; h a µ h b µ = δ a b . (2)\n\nIn this paper, the Latin alphabet (a, b, c, ... = 0, 1, 2, 3) will be used to denote tangent space indices and the Greek alphabet (µ, ν, ρ, ... = 0, 1, 2, 3) to denote spacetime indices. The Riemannian metric in TP theory arises as a by product [4] of the tetrad field given by\n\ng µν = η ab h a µ h b ν , (3)\n\nwhere η ab is the Minkowski metric η ab = diag(+1, -1, -1, -1). For the Weitzenböck spacetime, the torsion is defined as [2]\n\nT θ µν = Γ θ νµ -Γ θ µν (4)\n\nwhich is antisymmetric w.r.t. its last two indices. Due to the requirement of absolute parallelism, the curvature of the Weitzenböck connection vanishes identically. The Weitzenböck connection also satisfies the relation\n\nΓ 0 θ µν = Γ θ µν -K θ µν , (5)\n\nwhere\n\nK θ µν = 1 2 [T µ θ ν + T ν θ µ -T θ µν ] (6)\n\nis the contortion tensor and Γ 0 θ µν are the Christoffel symbols in GR. The torsion tensor of the Weitzenböck connection can be decomposed into three irreducible parts under the group of global Lorentz transformations [4] : the tensor part\n\nt λµν = 1 2 (T λµν + T µλν ) + 1 6 (g νλ V µ + g νµ V λ ) - 1 3 g λµ V ν , (7)\n\nthe vector part\n\nV µ = T ν νµ ( 8\n\n)\n\nand the axial-vector part\n\nA µ = 1 6 ǫ µνρσ T νρσ . (9)\n\nThe torsion tensor can now be expressed in terms of these irreducible components as follows\n\nT λµν = 1 2 (t λµν -t λνµ ) + 1 3 (g λµ V ν -g λν V µ ) + ǫ λµνρ A ρ , (10)\n\nwhere\n\nǫ λµνρ = 1 √ -g δ λµνρ . (11)\n\nHere δ = {δ λµνρ } and δ * = {δ λµνρ } are completely skew symmetric tensor densities of weight -1 and +1 respectively [4] . TP theory provides an alternate description of the Einstein's field equations which is given by the teleparallel equivalent of GR [24, 31] . Mikhail et al. [27] defined the super-potential (which is antisymmetric in its last two indices) of the Möller tetrad theory as\n\nU µ νβ = √ -g 2κ P τ νβ χρσ [V ρ g σχ g µτ -λg τ µ K χρσ -(1 -2λ)g τ µ K σρχ ], (12)\n\nwhere\n\nP τ νβ χρσ = δ χ τ g νβ ρσ + δ ρ τ g νβ σχ -δ σ τ g νβ χρ ( 13\n\n)\n\nand g νβ ρσ is a tensor quantity defined by\n\ng νβ ρσ = δ ρ ν δ σ β -δ σ ν δ ρ β . ( 14\n\n)\n\nK σρχ is the contortion tensor given by Eq.( 6 ), g is the determinant of the metric tensor g µν , λ is the free dimensionless coupling constant of TPG, κ is the Einstein constant and V µ is the basic vector field given by Eq.( 8 ). The energy-momentum density is defined as\n\nΞ ν µ = U νρ µ , ρ , (15)\n\nwhere comma means ordinary differentiation. The momentum 4-vector of Möller prescription can be expressed as\n\nP µ = Σ Ξ 0 µ dxdydz, (16)\n\nwhere P 0 gives the energy and P 1 , P 2 and P 3 are the momentum components while the integration is taken over the hypersurface element Σ described by x 0 = t = constant. The energy may be given in the form of surface integral [20] as\n\nE = lim r→∞ r=constant U 0 0ρ u ρ dS, (17)\n\nwhere u ρ is the unit three-vector normal to the surface element dS."},{"section_type":"OTHER","section_title":"Teleparallel Version of the Stationary Axisymmetric Solutions","text":"Tupper [32] found five classes of non-null electromagnetic field plus perfect fluid solutions in which the electromagnetic field does not inherit one of the symmetries of the spacetime. The metric representing the stationary axisymmetric solutions is given by [32 ]\n\nds 2 = dt 2 -e 2K dρ 2 -(F 2 -B 2 )dφ 2 -e 2K dz 2 + 2Bdtdφ, (18)\n\nwhere B = B(ρ, z), K = K(ρ, z) and F = F (ρ) are such functions which satisfy the following relations . The metric given by Eq.( 18 ) represents five classes of non-null electromagnetic field and perfect fluid solutions which possesses a metric symmetry not inherited by the electromagnetic field and admits a homothetic vector field. Two of these classes contain electrovac solutions as special cases, while the other three necessarily contain some fluid. The generalization of this metric is given in [34] .\n\nḂ = F W ′ , B ′ = - 1 4 aF ( Ẇ 2 -W ′2 ), K ′ = - 1 2 aF Ẇ W ′ , Ẅ + Ḟ F -1 Ẇ + W ′′ = 0, (19)\n\nUsing the procedure adopted in the papers [11, 12] , the tetrad components of the above metric can be written as\n\nh a µ = 1 0 B 0 0 e K cos φ -F sin φ 0 0 e K sin φ F cos φ 0 0 0 0 e K (20)\n\nwith its inverse\n\nh a µ = 1 0 0 0 B F sin φ e -K cos φ -1 F sin φ 0 -B F cos φ e -K sin φ 1 F cos φ 0 0 0 0 e -K . ( 21\n\n)\n\nThe non-vanishing components of the torsion tensor are\n\nT 0 12 = Ḃ + B F (e K -Ḟ ), T 0 32 = B ′ , T 1 13 = -K ′ , T 2 12 = 1 F ( Ḟ -e K ), T 3 31 = -K. ( 22\n\n)\n\nUsing these expressions in Eqs.( 7 )-( 9 ), we obtain the following non-zero components of the tensor part\n\nt 001 = 1 3 [ K + 1 F ( Ḟ -e K )], t 003 = 1 3 K ′ , t 010 = 1 6 { 1 F (e K -Ḟ ) -K} = t 100 , t 030 = - 1 6 K ′ = t 300 , t 012 = 1 2 Ḃ + B 6 { 1 F (e K -Ḟ ) -K} = t 102 , t 021 = - 1 2 Ḃ - B 3 { 1 F (e K -Ḟ ) -K} = t 201 , t 023 = - 1 2 B ′ + 1 3 BK ′ = t 203 , t 032 = 1 2 B ′ - 1 6 BK ′ = t 302 , t 122 = 1 2 {F (e K -Ḟ ) + B Ḃ} + 1 6 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K} = t 212 , t 120 = B 6 { 1 F (e K -Ḟ ) -K} = t 210 , t 113 = 2K ′ 3 e 2K , t 131 = - K ′ 3 e 2K = t 311 , t 133 = - e 2K 6 { 1 F (e K -Ḟ ) + 2 K} = t 313 , t 221 = -F (e K -Ḟ ) -B Ḃ - 1 3 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K}, t 223 = -BB ′ + K ′ 3 (B 2 -F 2 ), t 331 = e 2K 3 { 1 F (e K -Ḟ ) + 2 K}, t 322 = 1 2 BB ′ - K ′ 6 (B 2 -F 2 ) = t 232 , t 320 = - 1 6 BK ′ = t 232 , (23)\n\nthe vector part\n\nV 1 = - 1 F ( Ḟ -e K ) -K, (24)\n\nV 3 = -K ′ , (25)\n\nand the axial-vector part\n\nA 1 = B ′ 3F e -2K , (26)\n\nA 3 = Ḃ 3F e -2K , (27)\n\nrespectively. The axial-vector component along the φ-direction vanishes and hence the spacelike axial-vector can be written as\n\nA = √ -g 11 A 1 êρ + √ -g 33 A 3 êz , (28)\n\nwhere êρ and êz are unit vectors along the radial and z-directions respectively. Substituting A 1 , A 3 , g 11 and g 33 in Eq.( 28 ), it follows that\n\nA = e -K 3F (B ′ êρ + Ḃê z ). ( 29\n\n)\n\nThis shows that the axial-vector lies along radial direction if B = B(z), along z-direction if B = B(ρ) and vanishes identically if B is constant. As the axialvector torsion represents the deviation of axial symmetry from cylindrical symmetry, the symmetry of the underlying spacetime will not be affected even for B constant. Also, the torsion plays the role of the gravitational force in TP theory, hence a spinless particle will obey the force equation [11, 24]\n\ndu ρ ds -Γ µρν u µ u ν = T µρν u µ u ν . ( 30\n\n)\n\nThe left hand side of this equation is the Weitzenböck covariant derivative of u ρ along the particle world-line. The appearance of the torsion tensor on its right hand side indicates that the torsion plays the role of an external force in TPG. It has been shown, both in GR and TP theories, by many authors [4, 35] that the spin precession of a Dirac particle in torsion gravity is related to the torsion axial-vector by\n\ndS dt = -b × S, ( 31\n\n)\n\nwhere S is the spin vector of a Dirac particle and b = 3 2 A, with A the spacelike part of the torsion axial-vector. Thus\n\nb = e -K 2F {B ′ êρ + Ḃê z }. (32)\n\nThe corresponding extra Hamiltonian [36] is given by\n\nδH = -b.σ, ( 33\n\n)\n\nwhere σ is the spin of the particle [35] . Using Eq.( 32 ), this takes the form\n\nδH = - e -K 2F (B ′ êρ + Ḃê z ).σ. ( 34\n\n)\n\n4 Teleparallel Energy of the Stationary Axisymmetric Solutions\n\nIn this section we evaluate the component of energy-momentum densities by using the teleparallel version of Möller prescription. Multiplying Eqs.( 24 ) and ( 25 ) by g 11 and g 33 respectively, it follows that\n\nV 1 = Ke -2K + e -2K F ( Ḟ -e K ), (35)\n\nV 3 = K ′ e -2K . (36)\n\nIn view of Eqs.( 6 ) and ( 22 ), the non-vanishing components of the contorsion tensor are\n\nK 100 = -e -2K { B 2 F 3 (e K -Ḟ ) + B Ḃ F 2 } = -K 010 , K 300 = - BB ′ F 2 e -2K = -K 030 , K 122 = - e -2K F 3 (e K -Ḟ ) = -K 212 , K 133 = Ke -4K = -K 313 , K 311 = K ′ e -4K = -K 131 , K 102 = K 120 = e -2K { B F 3 (e K -Ḟ ) + Ḃ 2F 2 } = -K 012 = -K 210 , K 302 = K 320 = K 023 = B ′ 2F 2 e -2K = -K 032 = -K 230 = -K 203 , K 021 = Ḃ 2F 2 e -2K = -K 201 . ( 37\n\n)\n\nIt should be mentioned here that the contorsion tensor is antisymmetric w.r.t. its first two indices. Making use of Eqs.( 35 )- (37) in Eq.( 12 ), we obtain the required independent non-vanishing components of the supperpotential in Möller's tetrad theory as\n\nU 01 0 = 1 κ [e K -Ḟ -F K + 1 2 (1 + λ) B Ḃ F ] = -U 10 0 , U 03 0 = 1 κ [-F K ′ + 1 2 (1 + λ) BB ′ F ] = -U 30 0 , U 21 0 = - 1 2κ (1 + λ) Ḃ F = -U 12 0 , U 23 0 = - 1 2κ (1 + λ) B ′ F = -U 32 0 , U 01 2 = 1 κ [B(e K -Ḟ ) + 1 2 (1 + λ) B 2 Ḃ F + 1 2 (1 -λ) ḂF ] = -U 10 2 , U 03 2 = 1 κ [ 1 2 (1 + λ) B 2 B ′ F + 1 2 (1 -λ)B ′ F ] = -U 30 2 , U 02 1 = 1 2κF (λ -1) Ḃe 2K = -U 20 1 , U 02 3 = 1 2κF (λ -1)B ′ e 2K = -U 30 1 . (38)\n\nIt is worth mentioning here that the supperpotential is skew symmetric w.r.t. its last two indices. When we make use of Eqs.( 15 ), ( 37 ), (38) and take λ = 1, the energy density turns out to be\n\nΞ 0 0 = 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ ) + 1 F 2 {BF ( B + E ′′ ) + ( Ḃ2 + B ′ 2 )F -B Ḃ Ḟ }]. ( 39\n\n)\n\nThis implies that\n\nE d T P T = E d GR + 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ )], (40)\n\nwhere E d stands for energy density. The only non-zero component of momentum density is along φ-direction and (for λ = 1) it takes the form\n\nΞ 0 2 = 1 κF 2 {F 3 ( B + B ′′ ) + B 2 F ( B + B ′′ ) + 2BF ( Ḃ2 + B ′ 2 ) -Ḃ Ḟ (B 2 + F 2 )} + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (41)\n\nthat is,\n\nM d T P T = M d GR + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (42)\n\nwhere M d stands for momentum density."},{"section_type":"OTHER","section_title":"Special Solutions of the Non-Null Einstein Maxwell Solutions","text":"In this section, we evaluate the above quantities for some special cases of the non-null Einstein Maxwell solutions."},{"section_type":"OTHER","section_title":"Electromagnetic Generalization of the Gödel Solution","text":"A special case of the non-null Einstein-Maxwell solutions can be obtained by choosing\n\nB = m n e nρ , F = e nρ , K = 0. ( 43\n\n)\n\nThis is known as electromagnetic generalization of the Gödel solution [32] .\n\nWhen we make use of Eq.( 43 ) in Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42), the corresponding results reduce to\n\nt 001 = 1 3 (n -e -nρ ), t 010 = 1 6 (e -nρ -n) = t 100 , t 012 = m 6n (1 + 2ne nρ ) = t 102 , t 021 = - m 3n (1 + 2ne nρ ) = t 201 , t 122 = e nρ 6n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ } = t 212 , t 120 = m 6n (1 -ne nρ ) = t 210 , t 133 = 1 6 (n -e -nρ ) = t 313 , t 221 = - e nρ 3n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ },\n\nt 331 = - 1 3 (n -e -nρ ), (44)\n\nV 1 = e -nρ -n, V 3 = 0, (45)\n\nA 1 = 0, A 3 = m 3 , (46)\n\nA = m 3 êz , b = m 2 êz , (47)\n\nδH = m 2 êz .σ, (48)\n\nΞ 0 0 = 1 κ (m 2 -n 2 )e nρ , (49)\n\nE d T P T = E d GR - n 2 κ e nρ (50)\n\nΞ 0 2 = 1 κ ( 2m 3 n ) + m κ (1 -2ne nρ )e nρ , (51)\n\nM d T P T = M d GR + m κ (1 -2ne nρ )e nρ . (52)\n\nThe metric (43) reduces to the usual perfect fluid solution when m = √ 2n [32] , i.e., B = √ 2e nρ . The corresponding energy and momentum densities take the form as\n\nE d T P T = E d GR - n 2 κ e nρ (53)\n\nM d T P T = M d GR + √ 2n κ (1 -2ne nρ )e nρ . (54)"},{"section_type":"OTHER","section_title":"The Gödel Metric","text":"When we choose B = e aρ , F = e aρ √ 2 and K = 0, the metric given by Eq.( 18 ) reduces to the Gödel metric [32] . The results corresponding to Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42) take the following form\n\nt 001 = 1 3 (a - √ 2e -aρ\n\n), t 010 = -1 6 (a -√ 2e -aρ ) = t 100 , t 012 = 1 6 ( √ 2 + 2ae aρ ) = t 102 , t 021 = -1 6 (2 √ 2 + ae aρ ) = t 102 , t 122 = e aρ 6 (2 √ 2 + ae aρ ) = t 212 , t 120 = 1 6 ( √ 2ae aρ ) = t 210 , t 133 = 1 6 (a -√ 2e -aρ ) = t 313 , t 221 = -e aρ 3 (2 √ 2 + ae aρ ), (55)\n\nt 331 = 1 3 (a - √ 2e -aρ ), ( 56\n\n) V 1 = √ 2e -aρ -a, V 3 = 0, (57)\n\nA 1 = 0, A 3 = √ 2a 3 , (58)\n\nA = √ 2a 3 êz , b = a √ 2 êz , (59)\n\nδH = a √ 2 êz .σ, (60)\n\nΞ 0 0 = √ 2 κ a 2 e aρ - a 2 κ √ 2 e aρ , (61)\n\nE d T P T = E d GR - a 2 κ √ 2 e aρ (62) Ξ 0 2 = a 2 κ √ 2 e 2aρ + a κ (1 - √ 2ae aρ )e aρ , (63)\n\nM d T P T = M d GR + a κ (1 - √ 2ae aρ )e aρ . ( 64\n\n)"},{"section_type":"DISCUSSION","section_title":"Summary and Discussion","text":"The purpose of this paper is twofold: Firstly, we have found the TP version of the non-null Einstein Maxwell solutions. This provides some interesting features about the axial vector and the corresponding quantities. Secondly, we have used the TP version of Möller prescription to evaluate the energymomentum distribution of the solutions. The axial-vector torsion of these solutions has been evaluated. The only non-vanishing components of the vector part are along the radial and the z-directions due to the cross term dx 0 dx 2 involving in the metric. This corresponds to the case of Kerr metric [11] , which involves the cross term dx 0 dx 3 . We also find the vector b which is related to the spin vector [4] as given by Eq.( 32 ). The axial-vector torsion lies in the ρz-plane, as its component along the φ-direction vanishes everywhere.\n\nThe non-inertial force on the Dirac particle can be represented as a rotation induced torsion of spacetime. There arise three possibilities for the axial-vector, depending upon the metric function B(ρ, z). When B is a function of z only, the axial-vector lies only along the radial direction. When B is a function of ρ only, the axialvector will lie along z-direction. The axial-vector vanishes identically for B to be constant. As the axial-vector represents the deviation from the symmetry of the underlying spacetime corresponding to an inertial field with respect to the Dirac particle, the symmetry of the spacetime will not be affected in the third possibility. Consequently there exists no inertial field with respect to the Dirac particle and the spin vector of the Dirac particle becomes constant. The corresponding extra Hamiltonian is expressed in terms of the vector b which vanishes when the metric function B is constant, i.e., when the axialvector becomes zero.\n\nThe energy-momentum distribution of the non-null Einstein-Maxwell solutions has been explored by using the TP version of Möller prescription. It is found that energy in the TP theory is equal to the energy in GR (as found by Sharif and Fatima [37] ) plus some additional part. If, for a particular case, we have K = 0 and K ′ , Ḟ = constant (or if Ḟ , K = 0 and K\n\n′ = constant), then E d T P G = E d GR . (65)\n\nOn the other hand, the only non-vanishing component of the momentum density lies along φ-direction, similar to the case of Kerr metric [11] , due to the cross term appearing in both the metrics. When we choose λ = 1, it becomes equal to be the momentum in GR [37] plus an additional quantity. If F , Ḃ, B ′′ , K all vanish, then\n\nM d T P G = M d GR . (66)\n\nBy taking particular values of E, F and K, we obtain the electromagnetic generalization of the Gödel solution and the Gödel metric as two special cases. The corresponding results for both the special cases are obtained. It is shown that, for the electromagnetic generalization of the Gödel solution, Eq.(65) does not hold, while Eq.(66) holds when m = 0. However, for the perfect fluid case, i.e., when m = √ 2n, both Eqs.( 65 ) and (66) hold by taking n = 0. In the case of the Gödel metric, these equations hold if we choose the arbitrary constant a = 0. For the special solutions, the vector part lies along the radial direction while the axial-vector part along z-direction.\n\nWe would like to re-iterate here that the tetrad formalism itself has some advantages which comes mainly from its independence from the equivalence principle and consequent suitability to the discussion of quantum issues. In TPG, an energy-momentum gauge current j i µ for the gravitational field can be defined. This is covariant under a spacetime general coordinate transformation and transforms covariantly under a global tangent space Lorentz transformation [38] . It, then, follows that j i µ is a true spacetime tensor but not a tangent space tensor. When we re-write the gauge field equations in a purely spacetime form, they lead to the Einstein field equations and the gauge current j i µ reduces to the canonical energy-momentum pseudo-tensor of the gravitational field. Thus TPG seems to provide a more appropriate environment to deal with the energy problem.\n\nFinally, it is pointed out that we are not claiming that this paper has resolved the problems of GR using the TPG. This is an attempt to touch some issues in TPG with the hope that this alternative may provide more feasible results. Also, it is always an interesting and enriching to look at things from another point of view. This endeavor is in itself commendable."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"This work contains the teleparallel version of the stationary axisymmetric solutions. We obtain the tetrad and the torsion fields representing these solutions. The tensor, vector and axial-vector parts of the torsion tensor are evaluated. It is found that the axial-vector has component only along ρ and z directions. The three possibilities of the axial vector depending on the metric function B are discussed. The vector related with spin has also been evaluated and the corresponding extra Hamiltonian is furnished. Further, we use the teleparallel version of Möller prescription to find the energy-momentum distribution of the solutions. It is interesting to note that (for λ = 1) energy and momentum densities in teleparallel theory are equal to the corresponding quantities in GR plus an additional quantity in each, which may become equal under certain conditions. Finally, we discuss the two special cases of the stationary axisymmetric solutions.\"\n },\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction\",\n \"text\": \"The attempts made by Einstein and his followers to unify gravitation with other interactions led to the investigation of structures of gravitation other than the metric tensor. These structures yield the metric tensor as a by product. Tetrad field is one of these structures which leads to the theory of teleparallel gravity (TPG) [1, 2] . TPG is an alternative theory of gravity which corresponds to a gauge theory of translation group [3, 4] based on Weitzenböck geometry [5] . This theory is characterized by the vanishing of curvature identically while the torsion is taken to be non-zero. In TPG, gravitation is attributed to torsion [4] which plays a role of force [6] . In General Relativity (GR), gravitation geometrizes the underlying spacetime. The translational gauge potentials appear as a non-trivial part of the tetrad field and induce a teleparallel (TP) structure on spacetime which is directly related to the presence of a gravitational field. In some other theories [3] [4] [5] [6] [7] [8] , torsion is only relevant when spins are important [9] . This point of view indicates that torsion might represent additional degrees of freedom as compared to curvature. As a result, some new physics may be associated with it. Teleparallelism is naturally formulated by gauging external (spacetime) translations which are closely related to the group of general coordinate transformations underlying GR. Thus the energy-momentum tensor represents the matter source in the field equations of tetradic theories of gravity like in GR.\\n\\nThere is a large literature available [10] about the study of TP versions of the exact solutions of GR. Recently, Pereira, et al. [11] obtained the TP versions of the Schwarzschild and the stationary axisymmetric Kerr solutions of GR. They proved that the axial-vector torsion plays the role of the gravitomagnetic component of the gravitational field in the case of slow rotation and weak field approximations. In a previous paper [12] , we have found the TP versions of the Friedmann models and of the Lewis-Papapetrou spacetimes, and also discussed their axial-vectors.\\n\\nThere has been a longstanding, controversial and still unresolved problem of the localization of energy (i.e., to express it as a unique tensor quantity) in GR [13] . Einstein [14] introduced the energy-momentum pseudo-tensor and then Landau-Lifshitz [15] , Papapetrou [16] , Bergmann [17] , Tolman [18] and Weinberg [19] proposed their own prescriptions to resolve this issue. All these prescriptions can provide meaningful results only in Cartesian coordinates. But Möller [20] introduced a coordinate-independent prescription. The idea of coordinate-independent quasi-local mass was introduced [21] by associ-ating a Hamiltonian term to each gravitational energy-momentum pseudotensor. Later, a Hamiltonian approach in the frame of Schwinger condition [22] was developed, followed by the construction of a Lagrangian density of TP equivalent to GR [4, 6, 23, 24] . Many authors explored several examples in the framework of GR and found that different energy-momentum complexes can give either the same [25] or different [26] results for a given spacetime.\\n\\nMikhail et al. [27] defined the superpotential in the Moller's tetrad theory which has been used to find the energy in TPG. Vargas [28] defined the TP version of Bergman, Einstein and Landau-Lifshitz prescriptions and found that the total energy of the closed Friedman-Robinson-Walker universe is zero by using the last two prescriptions. This agrees with the results of GR available in literature [29] . Later, many authors [30] used TP version of these prescriptions and showed that energy may be localized in TPG. Keeping these points in mind, this paper is addressed to the following two problems: We obtain TP version of the stationary axisymmetric solutions and then calculate the axial-vector part of the torsion tensor. The energymomentum distribution of the solutions is explored by using the TP version of Möller prescription.\\n\\nThe scheme adopted in this paper is as follows. In section 2, we shall review the basic concepts of TP theory. Section 3 contains the TP version of the stationary axisymmetric solutions and the tensor, vector and axial-vector parts of the torsion tensor. Section 4 is devoted to evaluate the energymomentum distribution for this family of solutions using the TP version of Möller prescription. In section 5, we present two special solutions for this class of metrics and investigate the corresponding quantities. The last section contains a summary and a discussion of the results obtained.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"An Overview of the Teleparallel Theory\",\n \"text\": \"In teleparallel theory, the connection is a Weitzenböck connection given as [31] Γ\\n\\nθ µν = h a θ ∂ ν h a µ , (1)\\n\\nwhere h a ν is a non-trivial tetrad. Its inverse field is denoted by h a µ and satisfy the relations\\n\\nh a µ h a ν = δ µ ν ; h a µ h b µ = δ a b . (2)\\n\\nIn this paper, the Latin alphabet (a, b, c, ... = 0, 1, 2, 3) will be used to denote tangent space indices and the Greek alphabet (µ, ν, ρ, ... = 0, 1, 2, 3) to denote spacetime indices. The Riemannian metric in TP theory arises as a by product [4] of the tetrad field given by\\n\\ng µν = η ab h a µ h b ν , (3)\\n\\nwhere η ab is the Minkowski metric η ab = diag(+1, -1, -1, -1). For the Weitzenböck spacetime, the torsion is defined as [2]\\n\\nT θ µν = Γ θ νµ -Γ θ µν (4)\\n\\nwhich is antisymmetric w.r.t. its last two indices. Due to the requirement of absolute parallelism, the curvature of the Weitzenböck connection vanishes identically. The Weitzenböck connection also satisfies the relation\\n\\nΓ 0 θ µν = Γ θ µν -K θ µν , (5)\\n\\nwhere\\n\\nK θ µν = 1 2 [T µ θ ν + T ν θ µ -T θ µν ] (6)\\n\\nis the contortion tensor and Γ 0 θ µν are the Christoffel symbols in GR. The torsion tensor of the Weitzenböck connection can be decomposed into three irreducible parts under the group of global Lorentz transformations [4] : the tensor part\\n\\nt λµν = 1 2 (T λµν + T µλν ) + 1 6 (g νλ V µ + g νµ V λ ) - 1 3 g λµ V ν , (7)\\n\\nthe vector part\\n\\nV µ = T ν νµ ( 8\\n\\n)\\n\\nand the axial-vector part\\n\\nA µ = 1 6 ǫ µνρσ T νρσ . (9)\\n\\nThe torsion tensor can now be expressed in terms of these irreducible components as follows\\n\\nT λµν = 1 2 (t λµν -t λνµ ) + 1 3 (g λµ V ν -g λν V µ ) + ǫ λµνρ A ρ , (10)\\n\\nwhere\\n\\nǫ λµνρ = 1 √ -g δ λµνρ . (11)\\n\\nHere δ = {δ λµνρ } and δ * = {δ λµνρ } are completely skew symmetric tensor densities of weight -1 and +1 respectively [4] . TP theory provides an alternate description of the Einstein's field equations which is given by the teleparallel equivalent of GR [24, 31] . Mikhail et al. [27] defined the super-potential (which is antisymmetric in its last two indices) of the Möller tetrad theory as\\n\\nU µ νβ = √ -g 2κ P τ νβ χρσ [V ρ g σχ g µτ -λg τ µ K χρσ -(1 -2λ)g τ µ K σρχ ], (12)\\n\\nwhere\\n\\nP τ νβ χρσ = δ χ τ g νβ ρσ + δ ρ τ g νβ σχ -δ σ τ g νβ χρ ( 13\\n\\n)\\n\\nand g νβ ρσ is a tensor quantity defined by\\n\\ng νβ ρσ = δ ρ ν δ σ β -δ σ ν δ ρ β . ( 14\\n\\n)\\n\\nK σρχ is the contortion tensor given by Eq.( 6 ), g is the determinant of the metric tensor g µν , λ is the free dimensionless coupling constant of TPG, κ is the Einstein constant and V µ is the basic vector field given by Eq.( 8 ). The energy-momentum density is defined as\\n\\nΞ ν µ = U νρ µ , ρ , (15)\\n\\nwhere comma means ordinary differentiation. The momentum 4-vector of Möller prescription can be expressed as\\n\\nP µ = Σ Ξ 0 µ dxdydz, (16)\\n\\nwhere P 0 gives the energy and P 1 , P 2 and P 3 are the momentum components while the integration is taken over the hypersurface element Σ described by x 0 = t = constant. The energy may be given in the form of surface integral [20] as\\n\\nE = lim r→∞ r=constant U 0 0ρ u ρ dS, (17)\\n\\nwhere u ρ is the unit three-vector normal to the surface element dS.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Teleparallel Version of the Stationary Axisymmetric Solutions\",\n \"text\": \"Tupper [32] found five classes of non-null electromagnetic field plus perfect fluid solutions in which the electromagnetic field does not inherit one of the symmetries of the spacetime. The metric representing the stationary axisymmetric solutions is given by [32 ]\\n\\nds 2 = dt 2 -e 2K dρ 2 -(F 2 -B 2 )dφ 2 -e 2K dz 2 + 2Bdtdφ, (18)\\n\\nwhere B = B(ρ, z), K = K(ρ, z) and F = F (ρ) are such functions which satisfy the following relations . The metric given by Eq.( 18 ) represents five classes of non-null electromagnetic field and perfect fluid solutions which possesses a metric symmetry not inherited by the electromagnetic field and admits a homothetic vector field. Two of these classes contain electrovac solutions as special cases, while the other three necessarily contain some fluid. The generalization of this metric is given in [34] .\\n\\nḂ = F W ′ , B ′ = - 1 4 aF ( Ẇ 2 -W ′2 ), K ′ = - 1 2 aF Ẇ W ′ , Ẅ + Ḟ F -1 Ẇ + W ′′ = 0, (19)\\n\\nUsing the procedure adopted in the papers [11, 12] , the tetrad components of the above metric can be written as\\n\\nh a µ = 1 0 B 0 0 e K cos φ -F sin φ 0 0 e K sin φ F cos φ 0 0 0 0 e K (20)\\n\\nwith its inverse\\n\\nh a µ = 1 0 0 0 B F sin φ e -K cos φ -1 F sin φ 0 -B F cos φ e -K sin φ 1 F cos φ 0 0 0 0 e -K . ( 21\\n\\n)\\n\\nThe non-vanishing components of the torsion tensor are\\n\\nT 0 12 = Ḃ + B F (e K -Ḟ ), T 0 32 = B ′ , T 1 13 = -K ′ , T 2 12 = 1 F ( Ḟ -e K ), T 3 31 = -K. ( 22\\n\\n)\\n\\nUsing these expressions in Eqs.( 7 )-( 9 ), we obtain the following non-zero components of the tensor part\\n\\nt 001 = 1 3 [ K + 1 F ( Ḟ -e K )], t 003 = 1 3 K ′ , t 010 = 1 6 { 1 F (e K -Ḟ ) -K} = t 100 , t 030 = - 1 6 K ′ = t 300 , t 012 = 1 2 Ḃ + B 6 { 1 F (e K -Ḟ ) -K} = t 102 , t 021 = - 1 2 Ḃ - B 3 { 1 F (e K -Ḟ ) -K} = t 201 , t 023 = - 1 2 B ′ + 1 3 BK ′ = t 203 , t 032 = 1 2 B ′ - 1 6 BK ′ = t 302 , t 122 = 1 2 {F (e K -Ḟ ) + B Ḃ} + 1 6 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K} = t 212 , t 120 = B 6 { 1 F (e K -Ḟ ) -K} = t 210 , t 113 = 2K ′ 3 e 2K , t 131 = - K ′ 3 e 2K = t 311 , t 133 = - e 2K 6 { 1 F (e K -Ḟ ) + 2 K} = t 313 , t 221 = -F (e K -Ḟ ) -B Ḃ - 1 3 (B 2 -F 2 ){ 1 F (e K -Ḟ ) -K}, t 223 = -BB ′ + K ′ 3 (B 2 -F 2 ), t 331 = e 2K 3 { 1 F (e K -Ḟ ) + 2 K}, t 322 = 1 2 BB ′ - K ′ 6 (B 2 -F 2 ) = t 232 , t 320 = - 1 6 BK ′ = t 232 , (23)\\n\\nthe vector part\\n\\nV 1 = - 1 F ( Ḟ -e K ) -K, (24)\\n\\nV 3 = -K ′ , (25)\\n\\nand the axial-vector part\\n\\nA 1 = B ′ 3F e -2K , (26)\\n\\nA 3 = Ḃ 3F e -2K , (27)\\n\\nrespectively. The axial-vector component along the φ-direction vanishes and hence the spacelike axial-vector can be written as\\n\\nA = √ -g 11 A 1 êρ + √ -g 33 A 3 êz , (28)\\n\\nwhere êρ and êz are unit vectors along the radial and z-directions respectively. Substituting A 1 , A 3 , g 11 and g 33 in Eq.( 28 ), it follows that\\n\\nA = e -K 3F (B ′ êρ + Ḃê z ). ( 29\\n\\n)\\n\\nThis shows that the axial-vector lies along radial direction if B = B(z), along z-direction if B = B(ρ) and vanishes identically if B is constant. As the axialvector torsion represents the deviation of axial symmetry from cylindrical symmetry, the symmetry of the underlying spacetime will not be affected even for B constant. Also, the torsion plays the role of the gravitational force in TP theory, hence a spinless particle will obey the force equation [11, 24]\\n\\ndu ρ ds -Γ µρν u µ u ν = T µρν u µ u ν . ( 30\\n\\n)\\n\\nThe left hand side of this equation is the Weitzenböck covariant derivative of u ρ along the particle world-line. The appearance of the torsion tensor on its right hand side indicates that the torsion plays the role of an external force in TPG. It has been shown, both in GR and TP theories, by many authors [4, 35] that the spin precession of a Dirac particle in torsion gravity is related to the torsion axial-vector by\\n\\ndS dt = -b × S, ( 31\\n\\n)\\n\\nwhere S is the spin vector of a Dirac particle and b = 3 2 A, with A the spacelike part of the torsion axial-vector. Thus\\n\\nb = e -K 2F {B ′ êρ + Ḃê z }. (32)\\n\\nThe corresponding extra Hamiltonian [36] is given by\\n\\nδH = -b.σ, ( 33\\n\\n)\\n\\nwhere σ is the spin of the particle [35] . Using Eq.( 32 ), this takes the form\\n\\nδH = - e -K 2F (B ′ êρ + Ḃê z ).σ. ( 34\\n\\n)\\n\\n4 Teleparallel Energy of the Stationary Axisymmetric Solutions\\n\\nIn this section we evaluate the component of energy-momentum densities by using the teleparallel version of Möller prescription. Multiplying Eqs.( 24 ) and ( 25 ) by g 11 and g 33 respectively, it follows that\\n\\nV 1 = Ke -2K + e -2K F ( Ḟ -e K ), (35)\\n\\nV 3 = K ′ e -2K . (36)\\n\\nIn view of Eqs.( 6 ) and ( 22 ), the non-vanishing components of the contorsion tensor are\\n\\nK 100 = -e -2K { B 2 F 3 (e K -Ḟ ) + B Ḃ F 2 } = -K 010 , K 300 = - BB ′ F 2 e -2K = -K 030 , K 122 = - e -2K F 3 (e K -Ḟ ) = -K 212 , K 133 = Ke -4K = -K 313 , K 311 = K ′ e -4K = -K 131 , K 102 = K 120 = e -2K { B F 3 (e K -Ḟ ) + Ḃ 2F 2 } = -K 012 = -K 210 , K 302 = K 320 = K 023 = B ′ 2F 2 e -2K = -K 032 = -K 230 = -K 203 , K 021 = Ḃ 2F 2 e -2K = -K 201 . ( 37\\n\\n)\\n\\nIt should be mentioned here that the contorsion tensor is antisymmetric w.r.t. its first two indices. Making use of Eqs.( 35 )- (37) in Eq.( 12 ), we obtain the required independent non-vanishing components of the supperpotential in Möller's tetrad theory as\\n\\nU 01 0 = 1 κ [e K -Ḟ -F K + 1 2 (1 + λ) B Ḃ F ] = -U 10 0 , U 03 0 = 1 κ [-F K ′ + 1 2 (1 + λ) BB ′ F ] = -U 30 0 , U 21 0 = - 1 2κ (1 + λ) Ḃ F = -U 12 0 , U 23 0 = - 1 2κ (1 + λ) B ′ F = -U 32 0 , U 01 2 = 1 κ [B(e K -Ḟ ) + 1 2 (1 + λ) B 2 Ḃ F + 1 2 (1 -λ) ḂF ] = -U 10 2 , U 03 2 = 1 κ [ 1 2 (1 + λ) B 2 B ′ F + 1 2 (1 -λ)B ′ F ] = -U 30 2 , U 02 1 = 1 2κF (λ -1) Ḃe 2K = -U 20 1 , U 02 3 = 1 2κF (λ -1)B ′ e 2K = -U 30 1 . (38)\\n\\nIt is worth mentioning here that the supperpotential is skew symmetric w.r.t. its last two indices. When we make use of Eqs.( 15 ), ( 37 ), (38) and take λ = 1, the energy density turns out to be\\n\\nΞ 0 0 = 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ ) + 1 F 2 {BF ( B + E ′′ ) + ( Ḃ2 + B ′ 2 )F -B Ḃ Ḟ }]. ( 39\\n\\n)\\n\\nThis implies that\\n\\nE d T P T = E d GR + 1 κ [ Ke K -F -Ḟ K -F ( K + K ′′ )], (40)\\n\\nwhere E d stands for energy density. The only non-zero component of momentum density is along φ-direction and (for λ = 1) it takes the form\\n\\nΞ 0 2 = 1 κF 2 {F 3 ( B + B ′′ ) + B 2 F ( B + B ′′ ) + 2BF ( Ḃ2 + B ′ 2 ) -Ḃ Ḟ (B 2 + F 2 )} + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (41)\\n\\nthat is,\\n\\nM d T P T = M d GR + 1 κ { Ḃe K + B( Ke K -F ) -F ( B + B ′′ )}, (42)\\n\\nwhere M d stands for momentum density.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Special Solutions of the Non-Null Einstein Maxwell Solutions\",\n \"text\": \"In this section, we evaluate the above quantities for some special cases of the non-null Einstein Maxwell solutions.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Electromagnetic Generalization of the Gödel Solution\",\n \"text\": \"A special case of the non-null Einstein-Maxwell solutions can be obtained by choosing\\n\\nB = m n e nρ , F = e nρ , K = 0. ( 43\\n\\n)\\n\\nThis is known as electromagnetic generalization of the Gödel solution [32] .\\n\\nWhen we make use of Eq.( 43 ) in Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42), the corresponding results reduce to\\n\\nt 001 = 1 3 (n -e -nρ ), t 010 = 1 6 (e -nρ -n) = t 100 , t 012 = m 6n (1 + 2ne nρ ) = t 102 , t 021 = - m 3n (1 + 2ne nρ ) = t 201 , t 122 = e nρ 6n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ } = t 212 , t 120 = m 6n (1 -ne nρ ) = t 210 , t 133 = 1 6 (n -e -nρ ) = t 313 , t 221 = - e nρ 3n 2 {m 2 + 2n 2 + 2n(m 2 -n 2 )e nρ },\\n\\nt 331 = - 1 3 (n -e -nρ ), (44)\\n\\nV 1 = e -nρ -n, V 3 = 0, (45)\\n\\nA 1 = 0, A 3 = m 3 , (46)\\n\\nA = m 3 êz , b = m 2 êz , (47)\\n\\nδH = m 2 êz .σ, (48)\\n\\nΞ 0 0 = 1 κ (m 2 -n 2 )e nρ , (49)\\n\\nE d T P T = E d GR - n 2 κ e nρ (50)\\n\\nΞ 0 2 = 1 κ ( 2m 3 n ) + m κ (1 -2ne nρ )e nρ , (51)\\n\\nM d T P T = M d GR + m κ (1 -2ne nρ )e nρ . (52)\\n\\nThe metric (43) reduces to the usual perfect fluid solution when m = √ 2n [32] , i.e., B = √ 2e nρ . The corresponding energy and momentum densities take the form as\\n\\nE d T P T = E d GR - n 2 κ e nρ (53)\\n\\nM d T P T = M d GR + √ 2n κ (1 -2ne nρ )e nρ . (54)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The Gödel Metric\",\n \"text\": \"When we choose B = e aρ , F = e aρ √ 2 and K = 0, the metric given by Eq.( 18 ) reduces to the Gödel metric [32] . The results corresponding to Eqs.( 23 )-( 27 ), ( 29 ), ( 32 ), ( 34 ) and ( 39 )-(42) take the following form\\n\\nt 001 = 1 3 (a - √ 2e -aρ\\n\\n), t 010 = -1 6 (a -√ 2e -aρ ) = t 100 , t 012 = 1 6 ( √ 2 + 2ae aρ ) = t 102 , t 021 = -1 6 (2 √ 2 + ae aρ ) = t 102 , t 122 = e aρ 6 (2 √ 2 + ae aρ ) = t 212 , t 120 = 1 6 ( √ 2ae aρ ) = t 210 , t 133 = 1 6 (a -√ 2e -aρ ) = t 313 , t 221 = -e aρ 3 (2 √ 2 + ae aρ ), (55)\\n\\nt 331 = 1 3 (a - √ 2e -aρ ), ( 56\\n\\n) V 1 = √ 2e -aρ -a, V 3 = 0, (57)\\n\\nA 1 = 0, A 3 = √ 2a 3 , (58)\\n\\nA = √ 2a 3 êz , b = a √ 2 êz , (59)\\n\\nδH = a √ 2 êz .σ, (60)\\n\\nΞ 0 0 = √ 2 κ a 2 e aρ - a 2 κ √ 2 e aρ , (61)\\n\\nE d T P T = E d GR - a 2 κ √ 2 e aρ (62) Ξ 0 2 = a 2 κ √ 2 e 2aρ + a κ (1 - √ 2ae aρ )e aρ , (63)\\n\\nM d T P T = M d GR + a κ (1 - √ 2ae aρ )e aρ . ( 64\\n\\n)\"\n },\n {\n \"section_type\": \"DISCUSSION\",\n \"section_title\": \"Summary and Discussion\",\n \"text\": \"The purpose of this paper is twofold: Firstly, we have found the TP version of the non-null Einstein Maxwell solutions. This provides some interesting features about the axial vector and the corresponding quantities. Secondly, we have used the TP version of Möller prescription to evaluate the energymomentum distribution of the solutions. The axial-vector torsion of these solutions has been evaluated. The only non-vanishing components of the vector part are along the radial and the z-directions due to the cross term dx 0 dx 2 involving in the metric. This corresponds to the case of Kerr metric [11] , which involves the cross term dx 0 dx 3 . We also find the vector b which is related to the spin vector [4] as given by Eq.( 32 ). The axial-vector torsion lies in the ρz-plane, as its component along the φ-direction vanishes everywhere.\\n\\nThe non-inertial force on the Dirac particle can be represented as a rotation induced torsion of spacetime. There arise three possibilities for the axial-vector, depending upon the metric function B(ρ, z). When B is a function of z only, the axial-vector lies only along the radial direction. When B is a function of ρ only, the axialvector will lie along z-direction. The axial-vector vanishes identically for B to be constant. As the axial-vector represents the deviation from the symmetry of the underlying spacetime corresponding to an inertial field with respect to the Dirac particle, the symmetry of the spacetime will not be affected in the third possibility. Consequently there exists no inertial field with respect to the Dirac particle and the spin vector of the Dirac particle becomes constant. The corresponding extra Hamiltonian is expressed in terms of the vector b which vanishes when the metric function B is constant, i.e., when the axialvector becomes zero.\\n\\nThe energy-momentum distribution of the non-null Einstein-Maxwell solutions has been explored by using the TP version of Möller prescription. It is found that energy in the TP theory is equal to the energy in GR (as found by Sharif and Fatima [37] ) plus some additional part. If, for a particular case, we have K = 0 and K ′ , Ḟ = constant (or if Ḟ , K = 0 and K\\n\\n′ = constant), then E d T P G = E d GR . (65)\\n\\nOn the other hand, the only non-vanishing component of the momentum density lies along φ-direction, similar to the case of Kerr metric [11] , due to the cross term appearing in both the metrics. When we choose λ = 1, it becomes equal to be the momentum in GR [37] plus an additional quantity. If F , Ḃ, B ′′ , K all vanish, then\\n\\nM d T P G = M d GR . (66)\\n\\nBy taking particular values of E, F and K, we obtain the electromagnetic generalization of the Gödel solution and the Gödel metric as two special cases. The corresponding results for both the special cases are obtained. It is shown that, for the electromagnetic generalization of the Gödel solution, Eq.(65) does not hold, while Eq.(66) holds when m = 0. However, for the perfect fluid case, i.e., when m = √ 2n, both Eqs.( 65 ) and (66) hold by taking n = 0. In the case of the Gödel metric, these equations hold if we choose the arbitrary constant a = 0. For the special solutions, the vector part lies along the radial direction while the axial-vector part along z-direction.\\n\\nWe would like to re-iterate here that the tetrad formalism itself has some advantages which comes mainly from its independence from the equivalence principle and consequent suitability to the discussion of quantum issues. In TPG, an energy-momentum gauge current j i µ for the gravitational field can be defined. This is covariant under a spacetime general coordinate transformation and transforms covariantly under a global tangent space Lorentz transformation [38] . It, then, follows that j i µ is a true spacetime tensor but not a tangent space tensor. When we re-write the gauge field equations in a purely spacetime form, they lead to the Einstein field equations and the gauge current j i µ reduces to the canonical energy-momentum pseudo-tensor of the gravitational field. Thus TPG seems to provide a more appropriate environment to deal with the energy problem.\\n\\nFinally, it is pointed out that we are not claiming that this paper has resolved the problems of GR using the TPG. This is an attempt to touch some issues in TPG with the hope that this alternative may provide more feasible results. Also, it is always an interesting and enriching to look at things from another point of view. This endeavor is in itself commendable.\"\n }\n]","isConversational":false}}},{"rowIdx":87,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0489"},"arxiv_id":{"kind":"string","value":"0704.0489"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.2478/s11534-008-0018-0"},"content_hash":{"kind":"string","value":"05a07c71e5ae114221137d0f5aa9063589e159ec0dabb4c42451345ca0534803"},"title":{"kind":"string","value":"Relativistic treatment in D-dimensions to a spin-zero particle with\n noncentral equal scalar and vector ring-shaped Kratzer potential"},"abstract":{"kind":"string","value":"The Klein-Gordon equation in D-dimensions for a recently proposed Kratzer\npotential plus ring-shaped potential is solved analytically by means of the\nconventional Nikiforov-Uvarov method. The exact energy bound-states and the\ncorresponding wave functions of the Klein-Gordon are obtained in the presence\nof the noncentral equal scalar and vector potentials. The results obtained in\nthis work are more general and can be reduced to the standard forms in\nthree-dimensions given by other works."},"authors":{"kind":"list like","value":["Sameer M. Ikhdair","Ramazan Sever"],"string":"[\n \"Sameer M. Ikhdair\",\n \"Ramazan Sever\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["quant-ph"],"string":"[\n \"quant-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"In various physical applications including those in nuclear physics and high energy physics [1, 2] , one of the interesting problems is to obtain exact solutions of the relativistic equations like Klein-Gordon and Dirac equations for mixed vector and scalar potential. The Klein-Gordon and Dirac wave equations are frequently used to describe the particle dynamics in relativistic quantum mechanics. The Klein-Gordon equation has also been used to understand the motion of a spin-0 particle in large class of potentials. In recent years, much efforts have been paid to solve these relativistic wave equations for various potentials by using different methods. These relativistic equations contain two objects: the four-vector linear momentum operator and the scalar rest mass. They allow us to introduce two types of potential coupling, which are the four-vector potential (V) and the space-time scalar potential (S).\n\nRecently, many authors have worked on solving these equations with physical potentials including Morse potential [3], Hulthen potential [4] , Woods-Saxon potential [5] , Pösch-Teller potential [6] , reflectionless-type potential [7] , pseudoharmonic oscillator [8] , ring-shaped harmonic oscillator [9], V 0 tanh 2 (r/r 0 ) potential [10] , five-parameter exponential potential [11] , Rosen-Morse potential [12] , and generalized symmetrical double-well potential [13] , etc. It is remarkable that in most works in this area, the scalar and vector potentials are almost taken to be equal (i.e., S = V ) [2, 14] . However, in some few other cases, it is considered the case where the scalar potential is greater than the vector potential (in order to guarantee the existence of Klein-Gordon bound states) (i.e., S > V ) [15] [16] [17] [18] [19] . Nonetheless, such physical potentials are very few. The bound-state solutions for the last case is obtained for the exponential potential for the s-wave Klein-Gordon equation when the scalar potential is greater than the vector potential [15] .\n\nThe study of exact solutions of the nonrelativistic equation for a class of non-central potentials with a vector potential and a non-central scalar potential is of considerable interest in quantum chemistry [20] [21] [22] . In recent years, numerous studies [23] have been made in 2 analyzing the bound states of an electron in a Coulomb field with simultaneous presence of Aharanov-Bohm (AB) [24] field, and/or a magnetic Dirac monopole [25] , and Aharanov-Bohm plus oscillator (ABO) systems. In most of these works, the eigenvalues and eigenfunctions are obtained by means of seperation of variables in spherical or other orthogonal curvilinear coordinate systems. The path integral for particles moving in non-central potentials is evaluated to derive the energy spectrum of this system analytically [26] . In addition, the idea of SUSY and shape invariance is also used to obtain exact solutions of such noncentral but seperable potentials [27, 28] . Very recently, the conventional Nikiforov-Uvarov (NU) method [29] has been used to give a clear recipe of how to obtain an explicit exact bound-states solutions for the energy eigenvalues and their corresponding wave functions in terms of orthogonal polynomials for a class of non-central potentials [30] .\n\nAnother type of noncentral potentials is the ring-shaped Kratzer potential, which is a combination of a Coulomb potential plus an inverse square potential plus a noncentral angular part [31, 32] . The Kratzer potential has been used to describe the vibrational-rotational motion of isolated diatomic molecules [33] and has a mixed-energy spectrum containing both bound and scattering states with bound-states have been widely used in molecular spectroscopy [34] . The ring-shaped Kratzer potential consists of radial and angular-dependent potentials and is useful in studying ring-shaped molecules [22] . In taking the relativistic effects into account for spin-0 particle in the presence of a class of noncentral potentials, Yasuk et al [35] applied the NU method to solve the Klein-Gordon equation for the noncentral Coulombic ring-shaped potential [21] for the case V = S. Further, Berkdemir [36] also used the same method to solve the Klein-Gordon equation for the Kratzer-type potential.\n\nRecently, Chen and Dong [37] proposed a new ring-shaped potential and obtained the exact solution of the Schrödinger equation for the Coulomb potential plus this new ringshaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. This type of potential used by Chen and Dong [37] appears to be very similar to the potential used by Yasuk et al [35] . Moreover, Cheng and Dai [38] , proposed a new potential consisting from the modified Kratzer's potential [33] plus the 3 new proposed ring-shaped potential in [37] . They have presented the energy eigenvalues for this proposed exactly-solvable non-central potential in three dimensional (i.e., D = 3)-Schrödinger equation by means of the NU method. The two quantum systems solved by Refs [37, 38] are closely relevant to each other as they deal with a Coulombic field interaction except for a slight change in the angular momentum barrier acts as a repulsive core which is for any arbitrary angular momentum ℓ prevents collapse of the system in any dimensional space due to the slight perturbation to the original angular momentum barrier. Very recently, we have also applied the NU method to solve the Schrödinger equation in any arbitrary Ddimension to this new modified Kratzer-type potential [39, 40] .\n\nThe aim of the present paper is to consider the relativistic effects for the spin-0 particle in our recent works [39, 40] . So we want to present a systematic recipe to solving the D-dimensional Klein-Gordon equation for the Kratzer plus the new ring-shaped potential proposed in [38] using the simple NU method. This method is based on solving the Klein-Gordon equation by reducing it to a generalized hypergeometric equation.\n\nThis work is organized as follows: in section II, we shall present the Klein-Gordon equation in spherical coordinates for spin-0 particle in the presence of equal scalar and vector noncentral Kratzer plus the new ring-shaped potential and we also separate it into radial and angular parts. Section III is devoted to a brief description of the NU method.\n\nIn section IV, we present the exact solutions to the radial and angular parts of the Klein-Gordon equation in D-dimensions. Finally, the relevant conclusions are given in section V.\n\nIn relativistic quantum mechanics, we usually use the Klein-Gordon equation for describing a scalar particle, i.e., the spin-0 particle dynamics. The discussion of the relativistic behavior of spin-zero particles requires understanding the single particle spectrum and the 4 exact solutions to the Klein Gordon equation which are constructed by using the four-vector potential A λ (λ = 0, 1, 2, 3) and the scalar potential (S). In order to simplify the solution of the Klein-Gordon equation, the four-vector potential can be written as A λ = (A 0 , 0, 0, 0).\n\nThe first component of the four-vector potential is represented by a vector potential (V ), i.e., A 0 = V. In this case, the motion of a relativistic spin-0 particle in a potential is described by the Klein-Gordon equation with the potentials V and S [1]. For the case S ≥ V, there exist bound-state (real) solutions for a relativistic spin-zero particle [15] [16] [17] [18] [19] . On the other hand, for S = V, the Klein-Gordon equation reduces to a Schrödinger-like equation and thereby the bound-state solutions are easily obtained by using the well-known methods developed in nonrelativistic quantum mechanics [2] .\n\nThe Klein-Gordon equation describing a scalar particle (spin-0 particle) with scalar S(r, θ, ϕ) and vector V (r, θ, ϕ) potentials is given by [2,14]\n\nP 2 -[E R -V (r, θ, ϕ)/2] 2 + [µ + S(r, θ, ϕ)/2] 2 ψ(r, θ, ϕ) = 0, ( 1\n\n)\n\nwhere E R , P and µ are the relativistic energy, momentum operator and rest mass of the particle, respectively. The potential terms are scaled in (1) by Alhaidari et al [14] so that in the nonrelativistic limit the interaction potential becomes V.\n\nIn this work, we consider the equal scalar and vector potentials case, that is, S(r, θ, ϕ) = V (r, θ, ϕ) with the recently proposed general non-central potential taken in the form of the Kratzer plus ring-shaped potential [38-40]:\n\nV (r, θ, ϕ) = V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ , ( 2\n\n) V 1 (r) = - A r + B r 2 , V 2 (θ) = Cctg 2 θ, V 3 (ϕ) = 0, ( 3\n\n)\n\nwhere A = 2a 0 r 0 , B = a 0 r 2 0 and C is positive real constant with a 0 is the dissociation energy and r 0 is the equilibrium internuclear distance [33] . The potentials in Eq. ( 3 ) introduced by Cheng-Dai [38] reduce to the Kratzer potential in the limiting case of C = 0 [33]. In fact the energy spectrum for this potential can be obtained directly by considering it as special case of the general non-central seperable potentials [30] .\n\nIn the relativistic atomic units (h = c = 1), the D-dimensional Klein-Gordon equation in (1) becomes [41]\n\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 -(E R + µ) V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ + E 2 R -µ 2 ψ(r, θ, ϕ) = 0. ( 4\n\n)\n\nwith ψ(r, θ, ϕ) being the spherical total wave function separated as follows\n\nψ njm (r, θ, ϕ) = R(r)Y m j (θ, ϕ), R(r) = r -(D-1)/2 g(r), Y m j (θ, ϕ) = H(θ)Φ(ϕ). ( 5\n\n)\n\nInserting Eqs (3) and (5) into Eq. ( 4 ) and using the method of separation of variables, the following differential equations are obtained:\n\n1 r D-1 d dr r D-1 dR(r) dr - j(j + D -2) r 2 + α 2 2 α 2 1 - A r + B r 2 R(r) = 0, ( 6\n\n) 1 sin θ d dθ sin θ d dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ + j(j + D -2) H(θ) = 0, ( 7\n\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 8\n\n)\n\nwhere α 2 1 = µ -E R , α 2 2 = µ + E R , m\n\nand j are constants and with m 2 and λ j = j(j + D -2) are the separation constants.\n\nFor a nonrelativistic treatment with the same potential, the Schrödinger equation in spherical coordinates is\n\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 + 2µ E N R -V 1 (r) - V 2 (θ) r 2 - V 3 (ϕ) r 2 sin 2 θ ψ(r, θ, ϕ) = 0. ( 9\n\n)\n\nwhere µ and E N R are the reduced mass and the nonrelativistic energy, respectively. Besides, the spherical total wave function appearing in Eq. (9) has the same representation as in Eq. 6 (5) but with the transformation j → ℓ. Inserting Eq. (5) into Eq. (9) leads to the following differential equations [39,40]:\n\n1 r D-1 d dr r D-1 dR(r) dr - λ D r 2 -2µ E N R + A r - B r 2 R(r) = 0, ( 10\n\n) 1 sin θ d dθ sin θ d dθ - m 2 + 2µC cos 2 θ sin 2 θ + ℓ(ℓ + D -2) H(θ) = 0, ( 11\n\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 12\n\n)\n\nwhere m 2 and λ ℓ = ℓ(ℓ + D -2) are the separation constants. Equations (6)-(8) have the same functional form as Eqs (10)-(12). Therefore, the solution of the Klein-Gordon equation can be reduced to the solution of the Schrödinger equation with the appropriate choice of parameters: j → ℓ, α 2 1 → -E N R and α 2 2 → 2µ.\n\nThe solution of Eq. ( 8 ) is well-known periodic and must satisfy the period boundary condition Φ(ϕ + 2π) = Φ(ϕ) which is the azimuthal angle solution:\n\nΦ m (ϕ) = 1 √ 2π exp(±imϕ), m = 0, 1, 2, ..... ( 13\n\n)\n\nAdditionally, Eqs (6) and (7) are radial and polar angle equations and they will be solved by using the Nikiforov-Uvarov (NU) method [29] which is given briefly in the following section.\n\nThe NU method is based on reducing the second-order differential equation to a generalized equation of hypergeometric type [29] . In this sense, the Schrödinger equation, after employing an appropriate coordinate transformation s = s(r), transforms to the following form:\n\nψ ′′ n (s) + τ (s) σ(s) ψ ′ n (s) + σ(s) σ 2 (s) ψ n (s) = 0, ( 14\n\n) 7\n\nwhere σ(s) and σ(s) are polynomials, at most of second-degree, and τ (s) is a first-degree polynomial. Using a wave function, ψ n (s), of the simple ansatz:\n\nψ n (s) = φ n (s)y n (s), ( 15\n\n)\n\nreduces (14) into an equation of a hypergeometric type\n\nσ(s)y ′′ n (s) + τ (s)y ′ n (s) + λy n (s) = 0, ( 16\n\n)\n\nwhere\n\nσ(s) = π(s) φ(s) φ ′ (s) , ( 17\n\n)\n\nτ (s) = τ (s) + 2π(s), τ ′ (s) < 0, ( 18\n\n)\n\nand λ is a parameter defined as\n\nλ = λ n = -nτ ′ (s) - n (n -1) 2 σ ′′ (s), n = 0, 1, 2, .... ( 19\n\n)\n\nThe polynomial τ (s) with the parameter s and prime factors show the differentials at first degree be negative. It is worthwhile to note that λ or λ n are obtained from a particular solution of the form y(s) = y n (s) which is a polynomial of degree n. Further, the other part y n (s) of the wave function (14) is the hypergeometric-type function whose polynomial solutions are given by Rodrigues relation\n\ny n (s) = B n ρ(s) d n ds n [σ n (s)ρ(s)] , ( 20\n\n)\n\nwhere B n is the normalization constant and the weight function ρ(s) must satisfy the condition [29]\n\nd ds w(s) = τ (s) σ(s) w(s), w(s) = σ(s)ρ(s). ( 21\n\n)\n\nThe function π and the parameter λ are defined as\n\nπ(s) = σ ′ (s) -τ (s) 2 ± σ ′ (s) -τ (s) 2 2 -σ(s) + kσ(s), ( 22\n\n) 8 λ = k + π ′ (s). ( 23\n\n)\n\nIn principle, since π(s) has to be a polynomial of degree at most one, the expression under the square root sign in (22) can be arranged to be the square of a polynomial of first degree [29] . This is possible only if its discriminant is zero. In this case, an equation for k is obtained. After solving this equation, the obtained values of k are substituted in (22) . In addition, by comparing equations (19) and (23), we obtain the energy eigenvalues.\n\nWe seek to solving the radial and angular parts of the Klein-Gordon equation given by Eqs (6) and (7). Equation (6) involving the radial part can be written simply in the following form [39-41]:\n\nd 2 g(r) dr 2 - (M -1)(M -3) 4r 2 -α 2 2 A r - B r 2 + α 2 1 α 2 2 g(r) = 0, ( 24\n\n)\n\nwhere\n\nM = D + 2j. ( 25\n\n)\n\nOn the other hand, Eq. ( 7 ) involving the angular part of Klein-Gordon equation retakes the simple form\n\nd 2 H(θ) dθ 2 + ctg(θ) dH(θ) dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ -j(j + D -2) H(θ) = 0. ( 26\n\n)\n\nThus, Eqs (24) and (26) have to be solved latter through the NU method in the following subsections. 9 B. Eigenvalues and eigenfunctions of the angle-dependent equation\n\nIn order to apply NU method [29, 30, 33, 35, 36, [38] [39] [40] [42] [43] [44] , we use a suitable transformation variable s = cos θ. The polar angle part of the Klein Gordon equation in (26) can be written in the following universal associated-Legendre differential equation form [38-40]\n\nd 2 H(s) ds 2 - 2s 1 -s 2 dH(s) ds + 1 (1 -s 2 ) 2 j(j + D -2)(1 -s 2 ) -m 2 -Cα 2 2 s 2 H(θ) = 0. ( 27\n\n)\n\nEquation (27) has already been solved for the three-dimensional Schrödinger equation through the NU method in [38] . However, the aim in this subsection is to solve with different parameters resulting from the D-space-dimensions of Klein-Gordon equation. Further, Eq.\n\n(27) is compared with (14) and the following identifications are obtained\n\nτ (s) = -2s, σ(s) = 1 -s 2 , σ(s) = -m ′2 + (1 -s 2 )ν ′ , ( 28\n\n)\n\nwhere\n\nν ′ = j ′ (j ′ + D -2) = j(j + D -2) + Cα 2 2 , m ′2 = m 2 + Cα 2 2 . ( 29\n\n)\n\nInserting the above expressions into equation (22), one obtains the following function:\n\nπ(s) = ± (ν ′ -k)s 2 + k -ν ′ + m ′2 , ( 30\n\n)\n\nFollowing the method, the polynomial π(s) is found in the following possible values\n\nπ(s) = m ′ s for k 1 = ν ′ -m ′2 , -m ′ s for k 1 = ν ′ -m ′2 , m ′ for k 2 = ν ′ , -m ′ for k 2 = ν ′ . ( 31\n\n)\n\nImposing the condition τ ′ (s) < 0, for equation (18), one selects\n\nk 1 = ν ′ -m ′2 and π(s) = -m ′ s, ( 32\n\n) which yields 10 τ (s) = -2(1 + m ′ )s. ( 33\n\n)\n\nUsing equations (19) and (23), the following expressions for λ are obtained, respectively,\n\nλ = λ n = 2 n(1 + m ′ ) + n( n -1), ( 34\n\n) λ = ν ′ -m ′ (1 + m ′ ). ( 35\n\n)\n\nWe compare equations (34) and (35), the new angular momentum j values are obtained as\n\nj = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -4Cα 2 2 -1, ( 36\n\n) or j ′ = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -1. ( 37\n\n)\n\nUsing Eqs (15)-(17) and (20)-(21), the polynomial solution of y n is expressed in terms of Jacobi polynomials [39,40] which are one of the orthogonal polynomials:\n\nH n (θ) = N n sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ), ( 38\n\n) where N n = 1 2 m ′ ( n+m ′ )! (2 n+2m ′ +1)( n+2m ′ )! n! 2\n\nis the normalization constant determined by +1 -1 [H n (s)] 2 ds = 1 and using the orthogonality relation of Jacobi polynomials [35, 45, 46] .\n\nBesides n = - (1 + 2m ′ ) 2 + 1 2 (2j + 1) 2 + 4j(D -3) + 4Cα 2 2 , ( 39\n\n)\n\nwhere m ′ is defined by equation (29).\n\nThe solution of the radial part of Klein-Gordon equation, Eq. ( 24 ), for the Kratzer's potential has already been solved by means of NU-method in [39] . Very recently, using the same method, the problem for the non-central potential in (2) has been solved in three 11 dimensions (3D) by Cheng and Dai [36] . However, the aim of this subsection is to solve the problem with a different radial separation function g(r) in any arbitrary dimensions. In what follows, we present the exact bound-states (real) solution of Eq. (24). Letting\n\nε 2 = α 2 1 α 2 2 , 4γ 2 = (M -1)(M -3) + 4Bα 2 2 , β 2 = Aα 2 2 , ( 40\n\n)\n\nand substituting these expressions in equation (24), one gets\n\nd 2 g(r) dr 2 + -ε 2 r 2 + β 2 r -γ 2 r 2 g(r) = 0. ( 41\n\n)\n\nTo apply the conventional NU-method, equation (41) is compared with (14), resulting in the following expressions:\n\nτ (r) = 0, σ(r) = r, σ(r) = -ε 2 r 2 + β 2 r -γ 2 . ( 42\n\n)\n\nSubstituting the above expressions into equation (22) gives\n\nπ(r) = 1 2 ± 1 2 4ε 2 r 2 + 4(k -β 2 )r + 4γ 2 + 1. ( 43\n\n)\n\nTherefore, we can determine the constant k by using the condition that the discriminant of the square root is zero, that is\n\nk = β 2 ± ε 4γ 2 + 1, 4γ 2 + 1 = (D + 2j -2) 2 + 4Bα 2 2 . ( 44\n\n)\n\nIn view of that, we arrive at the following four possible functions of π(r) :\n\nπ(r) = 1 2 + εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 -εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 + εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1, 1 2 -εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1. ( 45\n\n)\n\nThe correct value of π(r) is chosen such that the function τ (r) given by Eq. (18) will have negative derivative [29] . So we can select the physical values to be\n\nk = β 2 -ε 4γ 2 + 1 and π(r) = 1 2 -εr - 1 2 4γ 2 + 1 , ( 46\n\n)\n\n12 which yield τ (r) = -2εr + (1 + 4γ 2 + 1), τ ′ (r) = -2ε < 0. (47) Using Eqs (19) and (23), the following expressions for λ are obtained, respectively,\n\nλ = λ n = 2nε, n = 0, 1, 2, ..., ( 48\n\n) λ = δ 2 -ε(1 + 4γ 2 + 1). ( 49\n\n)\n\nSo we can obtain the Klein Gordon energy eigenvalues from the following relation:\n\n1 + 2n + (D + 2j -2) 2 + 4(µ + E R )B µ -E R = A µ + E R , ( 50\n\n)\n\nand hence for the Kratzer plus the new ring-shaped potential, it becomes\n\n1 + 2n + (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 51\n\n)\n\nwith j defined in (36) . Although Eq. (51) is exactly solvable for E R but it looks to be little complicated. Further, it is interesting to investigate the solution for the Coulomb potential.\n\nTherefore, applying the following transformations: A = Ze 2 , B = 0, and j = ℓ, the central part of the potential in (3) turns into the Coulomb potential with Klein Gordon solution for the energy spectra given by\n\nE R = µ 1 - 2q 2 e 2 q 2 e 2 + (2n + 2ℓ + D -1) 2 , n, ℓ = 0, 1, 2, ..., ( 52\n\n)\n\nwhere q = Ze is the charge of the nucleus. Further, Eq. ( 52 ) can be expanded as a series in the nucleus charge as\n\nE R = µ - 2µq 2 e 2 (2n + 2ℓ + D -1) 2 + 2µq 4 e 4 (2n + 2ℓ + D -1) 4 -O(qe) 6 , ( 53\n\n)\n\nThe physical meaning of each term in the last equation was given in detail by Ref. [36] .\n\nBesides, the difference from the conventional nonrelativistic form is because of the choice of the vector V (r, θ, ϕ) and scalar S(r, θ, ϕ) parts of the potential in Eq. ( 1 ). 13 Overmore, if the value of j obtained by Eq.(36) is inserted into the eigenvalues of the radial part of the Klein Gordon equation with the noncentral potential given by Eq. (51), we finally find the energy eigenvalues for a bound electron in the presence of a noncentral potential by Eq. (2) as\n\n1 + 2n + (2j ′ + D -2) 2 + 4(a 0 r 2 0 -C)(µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 54\n\n)\n\nwhere m ′ = m 2 + C(µ + E R ) and n is given by Eq. (39). On the other hand, the solution of the Schrödinger equation, Eq. ( 9 ), for this potential has already been obtained by using the same method in Ref. [39] and it is in the Coulombic-like form:\n\nE N R = - 8µa 2 0 r 2 0 2n + 1 + (2ℓ ′ + D -2) 2 + 8µ(a 0 r 2 0 -C) 2 , n = 0, 1, 2, ... ( 55\n\n) 2ℓ ′ + D -2 = (D -2) 2 + (2 n + 2m ′ + 1) 2 -1, ( 56\n\n)\n\nwhere m ′ = √ m 2 + 2µC. Also, applying the following appropriate transformation: 54 ) provides exactly the nonrelativistic limit given by Eq. (55).\n\nµ+E R → 2µ, µ -E R → -E N R , j → ℓ to Eq. (\n\nIn what follows, let us now turn attention to find the radial wavefunctions for this potential. Substituting the values of σ(r), π(r) and τ (r) in Eqs (42), (45) and (47) into Eqs.\n\n(17) and (21), we find φ(r) = r (ζ+1)\n\n/2 e -εr , ( 57\n\n) ρ(r) = r ζ e -2εr , ( 58\n\n)\n\nwhere ζ = √ 4γ 2 + 1. Then from equation (20), we obtain\n\ny nj (r) = B nj r -ζ e 2εr d n dr n r n+ζ e -2εr , ( 59\n\n)\n\nand the wave function g(r) can be written in the form of the generalized Laguerre polynomials as 14 g(ρ) = C nj ρ 2ε\n\n(1+ζ)/2\n\ne -ρ/2 L ζ n (ρ), ( 60\n\n)\n\nwhere for Kratzer's potential we have\n\nζ = (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ), ρ = 2εr. ( 61\n\n)\n\nFinally, the radial wave functions of the Klein-Gordon equation are obtained R(ρ) = C nj ρ 2ε\n\n(ζ+2-D)/2\n\ne -ρ/2 L ζ n (ρ), ( 62\n\n)\n\nwhere C nj is the normalization constant to be determined below. Using the normalization condition, ∞ 0 R 2 (r)r D-1 dr = 1, and the orthogonality relation of the generalized Laguerre polynomials, (2n+η+1)(n+η)! n! , we have\n\n∞ 0 z η+1 e -z [L η n (z)] 2 dz =\n\nC nj = 2 µ 2 -E 2 R 1+ ζ 2 n! (2n + ζ + 1) (n + ζ)! . ( 63\n\n)\n\nFinally, we may express the normalized total wave functions as\n\nψ(r, θ, ϕ) = 2 µ 2 -E 2 R 1+ ζ 2 2 m ′ ( n + m ′ )! (2 n + 2m ′ + 1)( n + 2m ′ )! n!n! 2π (2n + ζ + 1) (n + ζ)! ×r (ζ+2-D) 2 exp(-µ 2 -E 2 R r)L ζ n (2 µ 2 -E 2 R r) sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ) exp(±imϕ). ( 64\n\n)\n\nwhere ζ is defined in Eq. ( 61 ) and m ′ is given after the Eq. (54).\n\nThe relativistic spin-0 particle D-dimensional Klein-Gordon equation has been solved easily for its exact bound-states with equal scalar and vector ring-shaped Kratzer potential through the conventional NU method. The analytical expressions for the total energy levels and eigenfunctions of this system can be reduced to their conventional three-dimensional space form upon setting D = 3. Further, the noncentral potentials treated in [30] can be introduced as perturbation to the Kratzer's potential by adjusting the strength of the 15 coupling constant C in terms of a 0 , which is the coupling constant of the Kratzer's potential.\n\nAdditionally, the radial and polar angle wave functions of Klein-Gordon equation are found in terms of Laguerre and Jacobi polynomials, respectively. The method presented in this paper is general and worth extending to the solution of other interaction problems. This method is very simple and useful in solving other complicated systems analytically without given a restiction conditions on the solution of some quantum systems as the case in the other models. We have seen that for the nonrelativistic model, the exact energy spectra can be obtained either by solving the Schrödinger equation in (9) (cf. Ref. [39] or Eq. ( 55 )) or by applying appropriate transformation to the relativistic solution given by Eq. ( 54 ). 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[38] Y.F. Cheng and T.Q. Dai, Phys. Scr. 75 (2007) 274. [39] S.M. Ikhdair and R. Sever, preprint quant-ph/0703008; quant-ph/0703042. [40] S.M. Ikhdair and R. Sever, preprint quant-ph/0703131. [41] S.M. Ikhdair and R. Sever, Z. Phys. C 56 (1992) 155; C 58 (1993) 153; D 28 (1993) 1; Hadronic J. 15 (1992) 389; Int. J. Mod. Phys. A 18 (2003) 4215; A 19 (2004) 1771; A 20 (2005) 4035; A 20 (2005) 6509; A 21 (2006) 2191; A 21 (2006) 3989; A 21 (2006) 6699; Int. J. Mod. Phys. E (in press) (preprint hep-ph/0504176); S. Ikhdair et al, Tr. J. Phys. 16 (1992) 510; 17 (1993) 474. [42] S.M. Ikhdair and R. Sever, Int. J. Theor. Phys. (DOI 10.1007/s10773-006-9317-7; J.\n\nMath. Chem. (DOI 10.1007/s10910-006-9115-8). [43] S.M. Ikhdair and R. Sever, Ann. Phys. (Leipzig) 16 (3) (2007) 218. [44] S.M. Ikhdair and R. Sever, to appear in the Int. J. Mod. Phys. E (preprint quant-19 ph/0611065). [45] G. Sezgo, Orthogonal Polynomials (American Mathematical Society, New York, 1939). [46] N.N. Lebedev, Special Functions and Their Applications (Prentice-Hall, Englewood Cliffs, NJ, 1965). 20"},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"In various physical applications including those in nuclear physics and high energy physics [1, 2] , one of the interesting problems is to obtain exact solutions of the relativistic equations like Klein-Gordon and Dirac equations for mixed vector and scalar potential. The Klein-Gordon and Dirac wave equations are frequently used to describe the particle dynamics in relativistic quantum mechanics. The Klein-Gordon equation has also been used to understand the motion of a spin-0 particle in large class of potentials. In recent years, much efforts have been paid to solve these relativistic wave equations for various potentials by using different methods. These relativistic equations contain two objects: the four-vector linear momentum operator and the scalar rest mass. They allow us to introduce two types of potential coupling, which are the four-vector potential (V) and the space-time scalar potential (S).\n\nRecently, many authors have worked on solving these equations with physical potentials including Morse potential [3], Hulthen potential [4] , Woods-Saxon potential [5] , Pösch-Teller potential [6] , reflectionless-type potential [7] , pseudoharmonic oscillator [8] , ring-shaped harmonic oscillator [9], V 0 tanh 2 (r/r 0 ) potential [10] , five-parameter exponential potential [11] , Rosen-Morse potential [12] , and generalized symmetrical double-well potential [13] , etc. It is remarkable that in most works in this area, the scalar and vector potentials are almost taken to be equal (i.e., S = V ) [2, 14] . However, in some few other cases, it is considered the case where the scalar potential is greater than the vector potential (in order to guarantee the existence of Klein-Gordon bound states) (i.e., S > V ) [15] [16] [17] [18] [19] . Nonetheless, such physical potentials are very few. The bound-state solutions for the last case is obtained for the exponential potential for the s-wave Klein-Gordon equation when the scalar potential is greater than the vector potential [15] .\n\nThe study of exact solutions of the nonrelativistic equation for a class of non-central potentials with a vector potential and a non-central scalar potential is of considerable interest in quantum chemistry [20] [21] [22] . In recent years, numerous studies [23] have been made in 2 analyzing the bound states of an electron in a Coulomb field with simultaneous presence of Aharanov-Bohm (AB) [24] field, and/or a magnetic Dirac monopole [25] , and Aharanov-Bohm plus oscillator (ABO) systems. In most of these works, the eigenvalues and eigenfunctions are obtained by means of seperation of variables in spherical or other orthogonal curvilinear coordinate systems. The path integral for particles moving in non-central potentials is evaluated to derive the energy spectrum of this system analytically [26] . In addition, the idea of SUSY and shape invariance is also used to obtain exact solutions of such noncentral but seperable potentials [27, 28] . Very recently, the conventional Nikiforov-Uvarov (NU) method [29] has been used to give a clear recipe of how to obtain an explicit exact bound-states solutions for the energy eigenvalues and their corresponding wave functions in terms of orthogonal polynomials for a class of non-central potentials [30] .\n\nAnother type of noncentral potentials is the ring-shaped Kratzer potential, which is a combination of a Coulomb potential plus an inverse square potential plus a noncentral angular part [31, 32] . The Kratzer potential has been used to describe the vibrational-rotational motion of isolated diatomic molecules [33] and has a mixed-energy spectrum containing both bound and scattering states with bound-states have been widely used in molecular spectroscopy [34] . The ring-shaped Kratzer potential consists of radial and angular-dependent potentials and is useful in studying ring-shaped molecules [22] . In taking the relativistic effects into account for spin-0 particle in the presence of a class of noncentral potentials, Yasuk et al [35] applied the NU method to solve the Klein-Gordon equation for the noncentral Coulombic ring-shaped potential [21] for the case V = S. Further, Berkdemir [36] also used the same method to solve the Klein-Gordon equation for the Kratzer-type potential.\n\nRecently, Chen and Dong [37] proposed a new ring-shaped potential and obtained the exact solution of the Schrödinger equation for the Coulomb potential plus this new ringshaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. This type of potential used by Chen and Dong [37] appears to be very similar to the potential used by Yasuk et al [35] . Moreover, Cheng and Dai [38] , proposed a new potential consisting from the modified Kratzer's potential [33] plus the 3 new proposed ring-shaped potential in [37] . They have presented the energy eigenvalues for this proposed exactly-solvable non-central potential in three dimensional (i.e., D = 3)-Schrödinger equation by means of the NU method. The two quantum systems solved by Refs [37, 38] are closely relevant to each other as they deal with a Coulombic field interaction except for a slight change in the angular momentum barrier acts as a repulsive core which is for any arbitrary angular momentum ℓ prevents collapse of the system in any dimensional space due to the slight perturbation to the original angular momentum barrier. Very recently, we have also applied the NU method to solve the Schrödinger equation in any arbitrary Ddimension to this new modified Kratzer-type potential [39, 40] .\n\nThe aim of the present paper is to consider the relativistic effects for the spin-0 particle in our recent works [39, 40] . So we want to present a systematic recipe to solving the D-dimensional Klein-Gordon equation for the Kratzer plus the new ring-shaped potential proposed in [38] using the simple NU method. This method is based on solving the Klein-Gordon equation by reducing it to a generalized hypergeometric equation.\n\nThis work is organized as follows: in section II, we shall present the Klein-Gordon equation in spherical coordinates for spin-0 particle in the presence of equal scalar and vector noncentral Kratzer plus the new ring-shaped potential and we also separate it into radial and angular parts. Section III is devoted to a brief description of the NU method.\n\nIn section IV, we present the exact solutions to the radial and angular parts of the Klein-Gordon equation in D-dimensions. Finally, the relevant conclusions are given in section V."},{"section_type":"OTHER","section_title":"II. THE KLEIN-GORDON EQUATION WITH EQUAL SCALAR AND VECTOR POTENTIALS","text":"In relativistic quantum mechanics, we usually use the Klein-Gordon equation for describing a scalar particle, i.e., the spin-0 particle dynamics. The discussion of the relativistic behavior of spin-zero particles requires understanding the single particle spectrum and the 4 exact solutions to the Klein Gordon equation which are constructed by using the four-vector potential A λ (λ = 0, 1, 2, 3) and the scalar potential (S). In order to simplify the solution of the Klein-Gordon equation, the four-vector potential can be written as A λ = (A 0 , 0, 0, 0).\n\nThe first component of the four-vector potential is represented by a vector potential (V ), i.e., A 0 = V. In this case, the motion of a relativistic spin-0 particle in a potential is described by the Klein-Gordon equation with the potentials V and S [1]. For the case S ≥ V, there exist bound-state (real) solutions for a relativistic spin-zero particle [15] [16] [17] [18] [19] . On the other hand, for S = V, the Klein-Gordon equation reduces to a Schrödinger-like equation and thereby the bound-state solutions are easily obtained by using the well-known methods developed in nonrelativistic quantum mechanics [2] .\n\nThe Klein-Gordon equation describing a scalar particle (spin-0 particle) with scalar S(r, θ, ϕ) and vector V (r, θ, ϕ) potentials is given by [2,14]\n\nP 2 -[E R -V (r, θ, ϕ)/2] 2 + [µ + S(r, θ, ϕ)/2] 2 ψ(r, θ, ϕ) = 0, ( 1\n\n)\n\nwhere E R , P and µ are the relativistic energy, momentum operator and rest mass of the particle, respectively. The potential terms are scaled in (1) by Alhaidari et al [14] so that in the nonrelativistic limit the interaction potential becomes V.\n\nIn this work, we consider the equal scalar and vector potentials case, that is, S(r, θ, ϕ) = V (r, θ, ϕ) with the recently proposed general non-central potential taken in the form of the Kratzer plus ring-shaped potential [38-40]:\n\nV (r, θ, ϕ) = V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ , ( 2\n\n) V 1 (r) = - A r + B r 2 , V 2 (θ) = Cctg 2 θ, V 3 (ϕ) = 0, ( 3\n\n)\n\nwhere A = 2a 0 r 0 , B = a 0 r 2 0 and C is positive real constant with a 0 is the dissociation energy and r 0 is the equilibrium internuclear distance [33] . The potentials in Eq. ( 3 ) introduced by Cheng-Dai [38] reduce to the Kratzer potential in the limiting case of C = 0 [33]. In fact the energy spectrum for this potential can be obtained directly by considering it as special case of the general non-central seperable potentials [30] .\n\nIn the relativistic atomic units (h = c = 1), the D-dimensional Klein-Gordon equation in (1) becomes [41]\n\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 -(E R + µ) V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ + E 2 R -µ 2 ψ(r, θ, ϕ) = 0. ( 4\n\n)\n\nwith ψ(r, θ, ϕ) being the spherical total wave function separated as follows\n\nψ njm (r, θ, ϕ) = R(r)Y m j (θ, ϕ), R(r) = r -(D-1)/2 g(r), Y m j (θ, ϕ) = H(θ)Φ(ϕ). ( 5\n\n)\n\nInserting Eqs (3) and (5) into Eq. ( 4 ) and using the method of separation of variables, the following differential equations are obtained:\n\n1 r D-1 d dr r D-1 dR(r) dr - j(j + D -2) r 2 + α 2 2 α 2 1 - A r + B r 2 R(r) = 0, ( 6\n\n) 1 sin θ d dθ sin θ d dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ + j(j + D -2) H(θ) = 0, ( 7\n\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 8\n\n)\n\nwhere α 2 1 = µ -E R , α 2 2 = µ + E R , m\n\nand j are constants and with m 2 and λ j = j(j + D -2) are the separation constants.\n\nFor a nonrelativistic treatment with the same potential, the Schrödinger equation in spherical coordinates is\n\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 + 2µ E N R -V 1 (r) - V 2 (θ) r 2 - V 3 (ϕ) r 2 sin 2 θ ψ(r, θ, ϕ) = 0. ( 9\n\n)\n\nwhere µ and E N R are the reduced mass and the nonrelativistic energy, respectively. Besides, the spherical total wave function appearing in Eq. (9) has the same representation as in Eq. 6 (5) but with the transformation j → ℓ. Inserting Eq. (5) into Eq. (9) leads to the following differential equations [39,40]:\n\n1 r D-1 d dr r D-1 dR(r) dr - λ D r 2 -2µ E N R + A r - B r 2 R(r) = 0, ( 10\n\n) 1 sin θ d dθ sin θ d dθ - m 2 + 2µC cos 2 θ sin 2 θ + ℓ(ℓ + D -2) H(θ) = 0, ( 11\n\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 12\n\n)\n\nwhere m 2 and λ ℓ = ℓ(ℓ + D -2) are the separation constants. Equations (6)-(8) have the same functional form as Eqs (10)-(12). Therefore, the solution of the Klein-Gordon equation can be reduced to the solution of the Schrödinger equation with the appropriate choice of parameters: j → ℓ, α 2 1 → -E N R and α 2 2 → 2µ.\n\nThe solution of Eq. ( 8 ) is well-known periodic and must satisfy the period boundary condition Φ(ϕ + 2π) = Φ(ϕ) which is the azimuthal angle solution:\n\nΦ m (ϕ) = 1 √ 2π exp(±imϕ), m = 0, 1, 2, ..... ( 13\n\n)\n\nAdditionally, Eqs (6) and (7) are radial and polar angle equations and they will be solved by using the Nikiforov-Uvarov (NU) method [29] which is given briefly in the following section."},{"section_type":"METHOD","section_title":"III. NIKIFOROV-UVAROV METHOD","text":"The NU method is based on reducing the second-order differential equation to a generalized equation of hypergeometric type [29] . In this sense, the Schrödinger equation, after employing an appropriate coordinate transformation s = s(r), transforms to the following form:\n\nψ ′′ n (s) + τ (s) σ(s) ψ ′ n (s) + σ(s) σ 2 (s) ψ n (s) = 0, ( 14\n\n) 7\n\nwhere σ(s) and σ(s) are polynomials, at most of second-degree, and τ (s) is a first-degree polynomial. Using a wave function, ψ n (s), of the simple ansatz:\n\nψ n (s) = φ n (s)y n (s), ( 15\n\n)\n\nreduces (14) into an equation of a hypergeometric type\n\nσ(s)y ′′ n (s) + τ (s)y ′ n (s) + λy n (s) = 0, ( 16\n\n)\n\nwhere\n\nσ(s) = π(s) φ(s) φ ′ (s) , ( 17\n\n)\n\nτ (s) = τ (s) + 2π(s), τ ′ (s) < 0, ( 18\n\n)\n\nand λ is a parameter defined as\n\nλ = λ n = -nτ ′ (s) - n (n -1) 2 σ ′′ (s), n = 0, 1, 2, .... ( 19\n\n)\n\nThe polynomial τ (s) with the parameter s and prime factors show the differentials at first degree be negative. It is worthwhile to note that λ or λ n are obtained from a particular solution of the form y(s) = y n (s) which is a polynomial of degree n. Further, the other part y n (s) of the wave function (14) is the hypergeometric-type function whose polynomial solutions are given by Rodrigues relation\n\ny n (s) = B n ρ(s) d n ds n [σ n (s)ρ(s)] , ( 20\n\n)\n\nwhere B n is the normalization constant and the weight function ρ(s) must satisfy the condition [29]\n\nd ds w(s) = τ (s) σ(s) w(s), w(s) = σ(s)ρ(s). ( 21\n\n)\n\nThe function π and the parameter λ are defined as\n\nπ(s) = σ ′ (s) -τ (s) 2 ± σ ′ (s) -τ (s) 2 2 -σ(s) + kσ(s), ( 22\n\n) 8 λ = k + π ′ (s). ( 23\n\n)\n\nIn principle, since π(s) has to be a polynomial of degree at most one, the expression under the square root sign in (22) can be arranged to be the square of a polynomial of first degree [29] . This is possible only if its discriminant is zero. In this case, an equation for k is obtained. After solving this equation, the obtained values of k are substituted in (22) . In addition, by comparing equations (19) and (23), we obtain the energy eigenvalues."},{"section_type":"OTHER","section_title":"IV. EXACT SOLUTIONS OF THE RADIAL AND ANGLE-DEPENDENT EQUATIONS A. Separating variables of the Klein-Gordon equation","text":"We seek to solving the radial and angular parts of the Klein-Gordon equation given by Eqs (6) and (7). Equation (6) involving the radial part can be written simply in the following form [39-41]:\n\nd 2 g(r) dr 2 - (M -1)(M -3) 4r 2 -α 2 2 A r - B r 2 + α 2 1 α 2 2 g(r) = 0, ( 24\n\n)\n\nwhere\n\nM = D + 2j. ( 25\n\n)\n\nOn the other hand, Eq. ( 7 ) involving the angular part of Klein-Gordon equation retakes the simple form\n\nd 2 H(θ) dθ 2 + ctg(θ) dH(θ) dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ -j(j + D -2) H(θ) = 0. ( 26\n\n)\n\nThus, Eqs (24) and (26) have to be solved latter through the NU method in the following subsections. 9 B. Eigenvalues and eigenfunctions of the angle-dependent equation\n\nIn order to apply NU method [29, 30, 33, 35, 36, [38] [39] [40] [42] [43] [44] , we use a suitable transformation variable s = cos θ. The polar angle part of the Klein Gordon equation in (26) can be written in the following universal associated-Legendre differential equation form [38-40]\n\nd 2 H(s) ds 2 - 2s 1 -s 2 dH(s) ds + 1 (1 -s 2 ) 2 j(j + D -2)(1 -s 2 ) -m 2 -Cα 2 2 s 2 H(θ) = 0. ( 27\n\n)\n\nEquation (27) has already been solved for the three-dimensional Schrödinger equation through the NU method in [38] . However, the aim in this subsection is to solve with different parameters resulting from the D-space-dimensions of Klein-Gordon equation. Further, Eq.\n\n(27) is compared with (14) and the following identifications are obtained\n\nτ (s) = -2s, σ(s) = 1 -s 2 , σ(s) = -m ′2 + (1 -s 2 )ν ′ , ( 28\n\n)\n\nwhere\n\nν ′ = j ′ (j ′ + D -2) = j(j + D -2) + Cα 2 2 , m ′2 = m 2 + Cα 2 2 . ( 29\n\n)\n\nInserting the above expressions into equation (22), one obtains the following function:\n\nπ(s) = ± (ν ′ -k)s 2 + k -ν ′ + m ′2 , ( 30\n\n)\n\nFollowing the method, the polynomial π(s) is found in the following possible values\n\nπ(s) = m ′ s for k 1 = ν ′ -m ′2 , -m ′ s for k 1 = ν ′ -m ′2 , m ′ for k 2 = ν ′ , -m ′ for k 2 = ν ′ . ( 31\n\n)\n\nImposing the condition τ ′ (s) < 0, for equation (18), one selects\n\nk 1 = ν ′ -m ′2 and π(s) = -m ′ s, ( 32\n\n) which yields 10 τ (s) = -2(1 + m ′ )s. ( 33\n\n)\n\nUsing equations (19) and (23), the following expressions for λ are obtained, respectively,\n\nλ = λ n = 2 n(1 + m ′ ) + n( n -1), ( 34\n\n) λ = ν ′ -m ′ (1 + m ′ ). ( 35\n\n)\n\nWe compare equations (34) and (35), the new angular momentum j values are obtained as\n\nj = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -4Cα 2 2 -1, ( 36\n\n) or j ′ = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -1. ( 37\n\n)\n\nUsing Eqs (15)-(17) and (20)-(21), the polynomial solution of y n is expressed in terms of Jacobi polynomials [39,40] which are one of the orthogonal polynomials:\n\nH n (θ) = N n sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ), ( 38\n\n) where N n = 1 2 m ′ ( n+m ′ )! (2 n+2m ′ +1)( n+2m ′ )! n! 2\n\nis the normalization constant determined by +1 -1 [H n (s)] 2 ds = 1 and using the orthogonality relation of Jacobi polynomials [35, 45, 46] .\n\nBesides n = - (1 + 2m ′ ) 2 + 1 2 (2j + 1) 2 + 4j(D -3) + 4Cα 2 2 , ( 39\n\n)\n\nwhere m ′ is defined by equation (29)."},{"section_type":"OTHER","section_title":"C. Eigensolutions of the radial equation","text":"The solution of the radial part of Klein-Gordon equation, Eq. ( 24 ), for the Kratzer's potential has already been solved by means of NU-method in [39] . Very recently, using the same method, the problem for the non-central potential in (2) has been solved in three 11 dimensions (3D) by Cheng and Dai [36] . However, the aim of this subsection is to solve the problem with a different radial separation function g(r) in any arbitrary dimensions. In what follows, we present the exact bound-states (real) solution of Eq. (24). Letting\n\nε 2 = α 2 1 α 2 2 , 4γ 2 = (M -1)(M -3) + 4Bα 2 2 , β 2 = Aα 2 2 , ( 40\n\n)\n\nand substituting these expressions in equation (24), one gets\n\nd 2 g(r) dr 2 + -ε 2 r 2 + β 2 r -γ 2 r 2 g(r) = 0. ( 41\n\n)\n\nTo apply the conventional NU-method, equation (41) is compared with (14), resulting in the following expressions:\n\nτ (r) = 0, σ(r) = r, σ(r) = -ε 2 r 2 + β 2 r -γ 2 . ( 42\n\n)\n\nSubstituting the above expressions into equation (22) gives\n\nπ(r) = 1 2 ± 1 2 4ε 2 r 2 + 4(k -β 2 )r + 4γ 2 + 1. ( 43\n\n)\n\nTherefore, we can determine the constant k by using the condition that the discriminant of the square root is zero, that is\n\nk = β 2 ± ε 4γ 2 + 1, 4γ 2 + 1 = (D + 2j -2) 2 + 4Bα 2 2 . ( 44\n\n)\n\nIn view of that, we arrive at the following four possible functions of π(r) :\n\nπ(r) = 1 2 + εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 -εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 + εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1, 1 2 -εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1. ( 45\n\n)\n\nThe correct value of π(r) is chosen such that the function τ (r) given by Eq. (18) will have negative derivative [29] . So we can select the physical values to be\n\nk = β 2 -ε 4γ 2 + 1 and π(r) = 1 2 -εr - 1 2 4γ 2 + 1 , ( 46\n\n)\n\n12 which yield τ (r) = -2εr + (1 + 4γ 2 + 1), τ ′ (r) = -2ε < 0. (47) Using Eqs (19) and (23), the following expressions for λ are obtained, respectively,\n\nλ = λ n = 2nε, n = 0, 1, 2, ..., ( 48\n\n) λ = δ 2 -ε(1 + 4γ 2 + 1). ( 49\n\n)\n\nSo we can obtain the Klein Gordon energy eigenvalues from the following relation:\n\n1 + 2n + (D + 2j -2) 2 + 4(µ + E R )B µ -E R = A µ + E R , ( 50\n\n)\n\nand hence for the Kratzer plus the new ring-shaped potential, it becomes\n\n1 + 2n + (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 51\n\n)\n\nwith j defined in (36) . Although Eq. (51) is exactly solvable for E R but it looks to be little complicated. Further, it is interesting to investigate the solution for the Coulomb potential.\n\nTherefore, applying the following transformations: A = Ze 2 , B = 0, and j = ℓ, the central part of the potential in (3) turns into the Coulomb potential with Klein Gordon solution for the energy spectra given by\n\nE R = µ 1 - 2q 2 e 2 q 2 e 2 + (2n + 2ℓ + D -1) 2 , n, ℓ = 0, 1, 2, ..., ( 52\n\n)\n\nwhere q = Ze is the charge of the nucleus. Further, Eq. ( 52 ) can be expanded as a series in the nucleus charge as\n\nE R = µ - 2µq 2 e 2 (2n + 2ℓ + D -1) 2 + 2µq 4 e 4 (2n + 2ℓ + D -1) 4 -O(qe) 6 , ( 53\n\n)\n\nThe physical meaning of each term in the last equation was given in detail by Ref. [36] .\n\nBesides, the difference from the conventional nonrelativistic form is because of the choice of the vector V (r, θ, ϕ) and scalar S(r, θ, ϕ) parts of the potential in Eq. ( 1 ). 13 Overmore, if the value of j obtained by Eq.(36) is inserted into the eigenvalues of the radial part of the Klein Gordon equation with the noncentral potential given by Eq. (51), we finally find the energy eigenvalues for a bound electron in the presence of a noncentral potential by Eq. (2) as\n\n1 + 2n + (2j ′ + D -2) 2 + 4(a 0 r 2 0 -C)(µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 54\n\n)\n\nwhere m ′ = m 2 + C(µ + E R ) and n is given by Eq. (39). On the other hand, the solution of the Schrödinger equation, Eq. ( 9 ), for this potential has already been obtained by using the same method in Ref. [39] and it is in the Coulombic-like form:\n\nE N R = - 8µa 2 0 r 2 0 2n + 1 + (2ℓ ′ + D -2) 2 + 8µ(a 0 r 2 0 -C) 2 , n = 0, 1, 2, ... ( 55\n\n) 2ℓ ′ + D -2 = (D -2) 2 + (2 n + 2m ′ + 1) 2 -1, ( 56\n\n)\n\nwhere m ′ = √ m 2 + 2µC. Also, applying the following appropriate transformation: 54 ) provides exactly the nonrelativistic limit given by Eq. (55).\n\nµ+E R → 2µ, µ -E R → -E N R , j → ℓ to Eq. (\n\nIn what follows, let us now turn attention to find the radial wavefunctions for this potential. Substituting the values of σ(r), π(r) and τ (r) in Eqs (42), (45) and (47) into Eqs.\n\n(17) and (21), we find φ(r) = r (ζ+1)\n\n/2 e -εr , ( 57\n\n) ρ(r) = r ζ e -2εr , ( 58\n\n)\n\nwhere ζ = √ 4γ 2 + 1. Then from equation (20), we obtain\n\ny nj (r) = B nj r -ζ e 2εr d n dr n r n+ζ e -2εr , ( 59\n\n)\n\nand the wave function g(r) can be written in the form of the generalized Laguerre polynomials as 14 g(ρ) = C nj ρ 2ε\n\n(1+ζ)/2\n\ne -ρ/2 L ζ n (ρ), ( 60\n\n)\n\nwhere for Kratzer's potential we have\n\nζ = (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ), ρ = 2εr. ( 61\n\n)\n\nFinally, the radial wave functions of the Klein-Gordon equation are obtained R(ρ) = C nj ρ 2ε\n\n(ζ+2-D)/2\n\ne -ρ/2 L ζ n (ρ), ( 62\n\n)\n\nwhere C nj is the normalization constant to be determined below. Using the normalization condition, ∞ 0 R 2 (r)r D-1 dr = 1, and the orthogonality relation of the generalized Laguerre polynomials, (2n+η+1)(n+η)! n! , we have\n\n∞ 0 z η+1 e -z [L η n (z)] 2 dz =\n\nC nj = 2 µ 2 -E 2 R 1+ ζ 2 n! (2n + ζ + 1) (n + ζ)! . ( 63\n\n)\n\nFinally, we may express the normalized total wave functions as\n\nψ(r, θ, ϕ) = 2 µ 2 -E 2 R 1+ ζ 2 2 m ′ ( n + m ′ )! (2 n + 2m ′ + 1)( n + 2m ′ )! n!n! 2π (2n + ζ + 1) (n + ζ)! ×r (ζ+2-D) 2 exp(-µ 2 -E 2 R r)L ζ n (2 µ 2 -E 2 R r) sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ) exp(±imϕ). ( 64\n\n)\n\nwhere ζ is defined in Eq. ( 61 ) and m ′ is given after the Eq. (54)."},{"section_type":"CONCLUSION","section_title":"V. CONCLUSIONS","text":"The relativistic spin-0 particle D-dimensional Klein-Gordon equation has been solved easily for its exact bound-states with equal scalar and vector ring-shaped Kratzer potential through the conventional NU method. The analytical expressions for the total energy levels and eigenfunctions of this system can be reduced to their conventional three-dimensional space form upon setting D = 3. Further, the noncentral potentials treated in [30] can be introduced as perturbation to the Kratzer's potential by adjusting the strength of the 15 coupling constant C in terms of a 0 , which is the coupling constant of the Kratzer's potential.\n\nAdditionally, the radial and polar angle wave functions of Klein-Gordon equation are found in terms of Laguerre and Jacobi polynomials, respectively. The method presented in this paper is general and worth extending to the solution of other interaction problems. This method is very simple and useful in solving other complicated systems analytically without given a restiction conditions on the solution of some quantum systems as the case in the other models. We have seen that for the nonrelativistic model, the exact energy spectra can be obtained either by solving the Schrödinger equation in (9) (cf. Ref. [39] or Eq. ( 55 )) or by applying appropriate transformation to the relativistic solution given by Eq. ( 54 ). Finally, we point out that these exact results obtained for this new proposed form of the potential (2) may have some interesting applications in the study of different quantum mechanical systems, atomic and molecular physics."},{"section_type":"OTHER","section_title":"ACKNOWLEDGMENTS","text":"This research was partially supported by the Scientific and Technological Research Council of Turkey. S.M. Ikhdair wishes to dedicate this work to his family for their love and assistance. 16 REFERENCES [1] T.Y. Wu and W.Y. Pauchy Hwang, Relativistic Quantum Mechanics and Quantum Fields (World Scientific, Singapore, 1991). [2] W. Greiner, Relativistic Quantum Mechanics: Wave Equations, 3rd edn (springer, Berlin, 2000). [3] A.D. Alhaidari, Phys. Rev. Lett. 87 (2001) 210405; 88 (2002) 189901.\n\n[4] G. Chen, Mod. Phys. Lett. A 19 (2004) 2009; J.-Y. Guo, J. Meng and F.-X. Xu, Chin.\n\nPhys. Lett. 20 (2003) 602; A.D. Alhaidari, J. Phys. A: Math. Gen. 34 (2001) 9827; 35 (2002) 6207; M. 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J. Phys. 75 (2007) 81. [37] C.Y. Chen and S.H. Dong, Phys. Lett. A 335 (2005) 374. [38] Y.F. Cheng and T.Q. Dai, Phys. Scr. 75 (2007) 274. [39] S.M. Ikhdair and R. Sever, preprint quant-ph/0703008; quant-ph/0703042. [40] S.M. Ikhdair and R. Sever, preprint quant-ph/0703131. [41] S.M. Ikhdair and R. Sever, Z. Phys. C 56 (1992) 155; C 58 (1993) 153; D 28 (1993) 1; Hadronic J. 15 (1992) 389; Int. J. Mod. Phys. A 18 (2003) 4215; A 19 (2004) 1771; A 20 (2005) 4035; A 20 (2005) 6509; A 21 (2006) 2191; A 21 (2006) 3989; A 21 (2006) 6699; Int. J. Mod. Phys. E (in press) (preprint hep-ph/0504176); S. Ikhdair et al, Tr. J. Phys. 16 (1992) 510; 17 (1993) 474. [42] S.M. Ikhdair and R. Sever, Int. J. Theor. Phys. (DOI 10.1007/s10773-006-9317-7; J.\n\nMath. Chem. (DOI 10.1007/s10910-006-9115-8). [43] S.M. Ikhdair and R. Sever, Ann. Phys. (Leipzig) 16 (3) (2007) 218. [44] S.M. Ikhdair and R. Sever, to appear in the Int. J. Mod. Phys. E (preprint quant-19 ph/0611065). [45] G. Sezgo, Orthogonal Polynomials (American Mathematical Society, New York, 1939). [46] N.N. Lebedev, Special Functions and Their Applications (Prentice-Hall, Englewood Cliffs, NJ, 1965). 20"}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"In various physical applications including those in nuclear physics and high energy physics [1, 2] , one of the interesting problems is to obtain exact solutions of the relativistic equations like Klein-Gordon and Dirac equations for mixed vector and scalar potential. The Klein-Gordon and Dirac wave equations are frequently used to describe the particle dynamics in relativistic quantum mechanics. The Klein-Gordon equation has also been used to understand the motion of a spin-0 particle in large class of potentials. In recent years, much efforts have been paid to solve these relativistic wave equations for various potentials by using different methods. These relativistic equations contain two objects: the four-vector linear momentum operator and the scalar rest mass. They allow us to introduce two types of potential coupling, which are the four-vector potential (V) and the space-time scalar potential (S).\\n\\nRecently, many authors have worked on solving these equations with physical potentials including Morse potential [3], Hulthen potential [4] , Woods-Saxon potential [5] , Pösch-Teller potential [6] , reflectionless-type potential [7] , pseudoharmonic oscillator [8] , ring-shaped harmonic oscillator [9], V 0 tanh 2 (r/r 0 ) potential [10] , five-parameter exponential potential [11] , Rosen-Morse potential [12] , and generalized symmetrical double-well potential [13] , etc. It is remarkable that in most works in this area, the scalar and vector potentials are almost taken to be equal (i.e., S = V ) [2, 14] . However, in some few other cases, it is considered the case where the scalar potential is greater than the vector potential (in order to guarantee the existence of Klein-Gordon bound states) (i.e., S > V ) [15] [16] [17] [18] [19] . Nonetheless, such physical potentials are very few. The bound-state solutions for the last case is obtained for the exponential potential for the s-wave Klein-Gordon equation when the scalar potential is greater than the vector potential [15] .\\n\\nThe study of exact solutions of the nonrelativistic equation for a class of non-central potentials with a vector potential and a non-central scalar potential is of considerable interest in quantum chemistry [20] [21] [22] . In recent years, numerous studies [23] have been made in 2 analyzing the bound states of an electron in a Coulomb field with simultaneous presence of Aharanov-Bohm (AB) [24] field, and/or a magnetic Dirac monopole [25] , and Aharanov-Bohm plus oscillator (ABO) systems. In most of these works, the eigenvalues and eigenfunctions are obtained by means of seperation of variables in spherical or other orthogonal curvilinear coordinate systems. The path integral for particles moving in non-central potentials is evaluated to derive the energy spectrum of this system analytically [26] . In addition, the idea of SUSY and shape invariance is also used to obtain exact solutions of such noncentral but seperable potentials [27, 28] . Very recently, the conventional Nikiforov-Uvarov (NU) method [29] has been used to give a clear recipe of how to obtain an explicit exact bound-states solutions for the energy eigenvalues and their corresponding wave functions in terms of orthogonal polynomials for a class of non-central potentials [30] .\\n\\nAnother type of noncentral potentials is the ring-shaped Kratzer potential, which is a combination of a Coulomb potential plus an inverse square potential plus a noncentral angular part [31, 32] . The Kratzer potential has been used to describe the vibrational-rotational motion of isolated diatomic molecules [33] and has a mixed-energy spectrum containing both bound and scattering states with bound-states have been widely used in molecular spectroscopy [34] . The ring-shaped Kratzer potential consists of radial and angular-dependent potentials and is useful in studying ring-shaped molecules [22] . In taking the relativistic effects into account for spin-0 particle in the presence of a class of noncentral potentials, Yasuk et al [35] applied the NU method to solve the Klein-Gordon equation for the noncentral Coulombic ring-shaped potential [21] for the case V = S. Further, Berkdemir [36] also used the same method to solve the Klein-Gordon equation for the Kratzer-type potential.\\n\\nRecently, Chen and Dong [37] proposed a new ring-shaped potential and obtained the exact solution of the Schrödinger equation for the Coulomb potential plus this new ringshaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. This type of potential used by Chen and Dong [37] appears to be very similar to the potential used by Yasuk et al [35] . Moreover, Cheng and Dai [38] , proposed a new potential consisting from the modified Kratzer's potential [33] plus the 3 new proposed ring-shaped potential in [37] . They have presented the energy eigenvalues for this proposed exactly-solvable non-central potential in three dimensional (i.e., D = 3)-Schrödinger equation by means of the NU method. The two quantum systems solved by Refs [37, 38] are closely relevant to each other as they deal with a Coulombic field interaction except for a slight change in the angular momentum barrier acts as a repulsive core which is for any arbitrary angular momentum ℓ prevents collapse of the system in any dimensional space due to the slight perturbation to the original angular momentum barrier. Very recently, we have also applied the NU method to solve the Schrödinger equation in any arbitrary Ddimension to this new modified Kratzer-type potential [39, 40] .\\n\\nThe aim of the present paper is to consider the relativistic effects for the spin-0 particle in our recent works [39, 40] . So we want to present a systematic recipe to solving the D-dimensional Klein-Gordon equation for the Kratzer plus the new ring-shaped potential proposed in [38] using the simple NU method. This method is based on solving the Klein-Gordon equation by reducing it to a generalized hypergeometric equation.\\n\\nThis work is organized as follows: in section II, we shall present the Klein-Gordon equation in spherical coordinates for spin-0 particle in the presence of equal scalar and vector noncentral Kratzer plus the new ring-shaped potential and we also separate it into radial and angular parts. Section III is devoted to a brief description of the NU method.\\n\\nIn section IV, we present the exact solutions to the radial and angular parts of the Klein-Gordon equation in D-dimensions. Finally, the relevant conclusions are given in section V.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. THE KLEIN-GORDON EQUATION WITH EQUAL SCALAR AND VECTOR POTENTIALS\",\n \"text\": \"In relativistic quantum mechanics, we usually use the Klein-Gordon equation for describing a scalar particle, i.e., the spin-0 particle dynamics. The discussion of the relativistic behavior of spin-zero particles requires understanding the single particle spectrum and the 4 exact solutions to the Klein Gordon equation which are constructed by using the four-vector potential A λ (λ = 0, 1, 2, 3) and the scalar potential (S). In order to simplify the solution of the Klein-Gordon equation, the four-vector potential can be written as A λ = (A 0 , 0, 0, 0).\\n\\nThe first component of the four-vector potential is represented by a vector potential (V ), i.e., A 0 = V. In this case, the motion of a relativistic spin-0 particle in a potential is described by the Klein-Gordon equation with the potentials V and S [1]. For the case S ≥ V, there exist bound-state (real) solutions for a relativistic spin-zero particle [15] [16] [17] [18] [19] . On the other hand, for S = V, the Klein-Gordon equation reduces to a Schrödinger-like equation and thereby the bound-state solutions are easily obtained by using the well-known methods developed in nonrelativistic quantum mechanics [2] .\\n\\nThe Klein-Gordon equation describing a scalar particle (spin-0 particle) with scalar S(r, θ, ϕ) and vector V (r, θ, ϕ) potentials is given by [2,14]\\n\\nP 2 -[E R -V (r, θ, ϕ)/2] 2 + [µ + S(r, θ, ϕ)/2] 2 ψ(r, θ, ϕ) = 0, ( 1\\n\\n)\\n\\nwhere E R , P and µ are the relativistic energy, momentum operator and rest mass of the particle, respectively. The potential terms are scaled in (1) by Alhaidari et al [14] so that in the nonrelativistic limit the interaction potential becomes V.\\n\\nIn this work, we consider the equal scalar and vector potentials case, that is, S(r, θ, ϕ) = V (r, θ, ϕ) with the recently proposed general non-central potential taken in the form of the Kratzer plus ring-shaped potential [38-40]:\\n\\nV (r, θ, ϕ) = V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ , ( 2\\n\\n) V 1 (r) = - A r + B r 2 , V 2 (θ) = Cctg 2 θ, V 3 (ϕ) = 0, ( 3\\n\\n)\\n\\nwhere A = 2a 0 r 0 , B = a 0 r 2 0 and C is positive real constant with a 0 is the dissociation energy and r 0 is the equilibrium internuclear distance [33] . The potentials in Eq. ( 3 ) introduced by Cheng-Dai [38] reduce to the Kratzer potential in the limiting case of C = 0 [33]. In fact the energy spectrum for this potential can be obtained directly by considering it as special case of the general non-central seperable potentials [30] .\\n\\nIn the relativistic atomic units (h = c = 1), the D-dimensional Klein-Gordon equation in (1) becomes [41]\\n\\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 -(E R + µ) V 1 (r) + V 2 (θ) r 2 + V 3 (ϕ) r 2 sin 2 θ + E 2 R -µ 2 ψ(r, θ, ϕ) = 0. ( 4\\n\\n)\\n\\nwith ψ(r, θ, ϕ) being the spherical total wave function separated as follows\\n\\nψ njm (r, θ, ϕ) = R(r)Y m j (θ, ϕ), R(r) = r -(D-1)/2 g(r), Y m j (θ, ϕ) = H(θ)Φ(ϕ). ( 5\\n\\n)\\n\\nInserting Eqs (3) and (5) into Eq. ( 4 ) and using the method of separation of variables, the following differential equations are obtained:\\n\\n1 r D-1 d dr r D-1 dR(r) dr - j(j + D -2) r 2 + α 2 2 α 2 1 - A r + B r 2 R(r) = 0, ( 6\\n\\n) 1 sin θ d dθ sin θ d dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ + j(j + D -2) H(θ) = 0, ( 7\\n\\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 8\\n\\n)\\n\\nwhere α 2 1 = µ -E R , α 2 2 = µ + E R , m\\n\\nand j are constants and with m 2 and λ j = j(j + D -2) are the separation constants.\\n\\nFor a nonrelativistic treatment with the same potential, the Schrödinger equation in spherical coordinates is\\n\\n1 r D-1 ∂ ∂r r D-1 ∂ ∂r + 1 r 2 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin 2 θ ∂ 2 ∂ϕ 2 + 2µ E N R -V 1 (r) - V 2 (θ) r 2 - V 3 (ϕ) r 2 sin 2 θ ψ(r, θ, ϕ) = 0. ( 9\\n\\n)\\n\\nwhere µ and E N R are the reduced mass and the nonrelativistic energy, respectively. Besides, the spherical total wave function appearing in Eq. (9) has the same representation as in Eq. 6 (5) but with the transformation j → ℓ. Inserting Eq. (5) into Eq. (9) leads to the following differential equations [39,40]:\\n\\n1 r D-1 d dr r D-1 dR(r) dr - λ D r 2 -2µ E N R + A r - B r 2 R(r) = 0, ( 10\\n\\n) 1 sin θ d dθ sin θ d dθ - m 2 + 2µC cos 2 θ sin 2 θ + ℓ(ℓ + D -2) H(θ) = 0, ( 11\\n\\n) d 2 Φ(ϕ) dϕ 2 + m 2 Φ(ϕ) = 0, ( 12\\n\\n)\\n\\nwhere m 2 and λ ℓ = ℓ(ℓ + D -2) are the separation constants. Equations (6)-(8) have the same functional form as Eqs (10)-(12). Therefore, the solution of the Klein-Gordon equation can be reduced to the solution of the Schrödinger equation with the appropriate choice of parameters: j → ℓ, α 2 1 → -E N R and α 2 2 → 2µ.\\n\\nThe solution of Eq. ( 8 ) is well-known periodic and must satisfy the period boundary condition Φ(ϕ + 2π) = Φ(ϕ) which is the azimuthal angle solution:\\n\\nΦ m (ϕ) = 1 √ 2π exp(±imϕ), m = 0, 1, 2, ..... ( 13\\n\\n)\\n\\nAdditionally, Eqs (6) and (7) are radial and polar angle equations and they will be solved by using the Nikiforov-Uvarov (NU) method [29] which is given briefly in the following section.\"\n },\n {\n \"section_type\": \"METHOD\",\n \"section_title\": \"III. NIKIFOROV-UVAROV METHOD\",\n \"text\": \"The NU method is based on reducing the second-order differential equation to a generalized equation of hypergeometric type [29] . In this sense, the Schrödinger equation, after employing an appropriate coordinate transformation s = s(r), transforms to the following form:\\n\\nψ ′′ n (s) + τ (s) σ(s) ψ ′ n (s) + σ(s) σ 2 (s) ψ n (s) = 0, ( 14\\n\\n) 7\\n\\nwhere σ(s) and σ(s) are polynomials, at most of second-degree, and τ (s) is a first-degree polynomial. Using a wave function, ψ n (s), of the simple ansatz:\\n\\nψ n (s) = φ n (s)y n (s), ( 15\\n\\n)\\n\\nreduces (14) into an equation of a hypergeometric type\\n\\nσ(s)y ′′ n (s) + τ (s)y ′ n (s) + λy n (s) = 0, ( 16\\n\\n)\\n\\nwhere\\n\\nσ(s) = π(s) φ(s) φ ′ (s) , ( 17\\n\\n)\\n\\nτ (s) = τ (s) + 2π(s), τ ′ (s) < 0, ( 18\\n\\n)\\n\\nand λ is a parameter defined as\\n\\nλ = λ n = -nτ ′ (s) - n (n -1) 2 σ ′′ (s), n = 0, 1, 2, .... ( 19\\n\\n)\\n\\nThe polynomial τ (s) with the parameter s and prime factors show the differentials at first degree be negative. It is worthwhile to note that λ or λ n are obtained from a particular solution of the form y(s) = y n (s) which is a polynomial of degree n. Further, the other part y n (s) of the wave function (14) is the hypergeometric-type function whose polynomial solutions are given by Rodrigues relation\\n\\ny n (s) = B n ρ(s) d n ds n [σ n (s)ρ(s)] , ( 20\\n\\n)\\n\\nwhere B n is the normalization constant and the weight function ρ(s) must satisfy the condition [29]\\n\\nd ds w(s) = τ (s) σ(s) w(s), w(s) = σ(s)ρ(s). ( 21\\n\\n)\\n\\nThe function π and the parameter λ are defined as\\n\\nπ(s) = σ ′ (s) -τ (s) 2 ± σ ′ (s) -τ (s) 2 2 -σ(s) + kσ(s), ( 22\\n\\n) 8 λ = k + π ′ (s). ( 23\\n\\n)\\n\\nIn principle, since π(s) has to be a polynomial of degree at most one, the expression under the square root sign in (22) can be arranged to be the square of a polynomial of first degree [29] . This is possible only if its discriminant is zero. In this case, an equation for k is obtained. After solving this equation, the obtained values of k are substituted in (22) . In addition, by comparing equations (19) and (23), we obtain the energy eigenvalues.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. EXACT SOLUTIONS OF THE RADIAL AND ANGLE-DEPENDENT EQUATIONS A. Separating variables of the Klein-Gordon equation\",\n \"text\": \"We seek to solving the radial and angular parts of the Klein-Gordon equation given by Eqs (6) and (7). Equation (6) involving the radial part can be written simply in the following form [39-41]:\\n\\nd 2 g(r) dr 2 - (M -1)(M -3) 4r 2 -α 2 2 A r - B r 2 + α 2 1 α 2 2 g(r) = 0, ( 24\\n\\n)\\n\\nwhere\\n\\nM = D + 2j. ( 25\\n\\n)\\n\\nOn the other hand, Eq. ( 7 ) involving the angular part of Klein-Gordon equation retakes the simple form\\n\\nd 2 H(θ) dθ 2 + ctg(θ) dH(θ) dθ - m 2 + Cα 2 2 cos 2 θ sin 2 θ -j(j + D -2) H(θ) = 0. ( 26\\n\\n)\\n\\nThus, Eqs (24) and (26) have to be solved latter through the NU method in the following subsections. 9 B. Eigenvalues and eigenfunctions of the angle-dependent equation\\n\\nIn order to apply NU method [29, 30, 33, 35, 36, [38] [39] [40] [42] [43] [44] , we use a suitable transformation variable s = cos θ. The polar angle part of the Klein Gordon equation in (26) can be written in the following universal associated-Legendre differential equation form [38-40]\\n\\nd 2 H(s) ds 2 - 2s 1 -s 2 dH(s) ds + 1 (1 -s 2 ) 2 j(j + D -2)(1 -s 2 ) -m 2 -Cα 2 2 s 2 H(θ) = 0. ( 27\\n\\n)\\n\\nEquation (27) has already been solved for the three-dimensional Schrödinger equation through the NU method in [38] . However, the aim in this subsection is to solve with different parameters resulting from the D-space-dimensions of Klein-Gordon equation. Further, Eq.\\n\\n(27) is compared with (14) and the following identifications are obtained\\n\\nτ (s) = -2s, σ(s) = 1 -s 2 , σ(s) = -m ′2 + (1 -s 2 )ν ′ , ( 28\\n\\n)\\n\\nwhere\\n\\nν ′ = j ′ (j ′ + D -2) = j(j + D -2) + Cα 2 2 , m ′2 = m 2 + Cα 2 2 . ( 29\\n\\n)\\n\\nInserting the above expressions into equation (22), one obtains the following function:\\n\\nπ(s) = ± (ν ′ -k)s 2 + k -ν ′ + m ′2 , ( 30\\n\\n)\\n\\nFollowing the method, the polynomial π(s) is found in the following possible values\\n\\nπ(s) = m ′ s for k 1 = ν ′ -m ′2 , -m ′ s for k 1 = ν ′ -m ′2 , m ′ for k 2 = ν ′ , -m ′ for k 2 = ν ′ . ( 31\\n\\n)\\n\\nImposing the condition τ ′ (s) < 0, for equation (18), one selects\\n\\nk 1 = ν ′ -m ′2 and π(s) = -m ′ s, ( 32\\n\\n) which yields 10 τ (s) = -2(1 + m ′ )s. ( 33\\n\\n)\\n\\nUsing equations (19) and (23), the following expressions for λ are obtained, respectively,\\n\\nλ = λ n = 2 n(1 + m ′ ) + n( n -1), ( 34\\n\\n) λ = ν ′ -m ′ (1 + m ′ ). ( 35\\n\\n)\\n\\nWe compare equations (34) and (35), the new angular momentum j values are obtained as\\n\\nj = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -4Cα 2 2 -1, ( 36\\n\\n) or j ′ = - (D -2) 2 + 1 2 (D -2) 2 + (2 n + 2m ′ + 1) 2 -1. ( 37\\n\\n)\\n\\nUsing Eqs (15)-(17) and (20)-(21), the polynomial solution of y n is expressed in terms of Jacobi polynomials [39,40] which are one of the orthogonal polynomials:\\n\\nH n (θ) = N n sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ), ( 38\\n\\n) where N n = 1 2 m ′ ( n+m ′ )! (2 n+2m ′ +1)( n+2m ′ )! n! 2\\n\\nis the normalization constant determined by +1 -1 [H n (s)] 2 ds = 1 and using the orthogonality relation of Jacobi polynomials [35, 45, 46] .\\n\\nBesides n = - (1 + 2m ′ ) 2 + 1 2 (2j + 1) 2 + 4j(D -3) + 4Cα 2 2 , ( 39\\n\\n)\\n\\nwhere m ′ is defined by equation (29).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"C. Eigensolutions of the radial equation\",\n \"text\": \"The solution of the radial part of Klein-Gordon equation, Eq. ( 24 ), for the Kratzer's potential has already been solved by means of NU-method in [39] . Very recently, using the same method, the problem for the non-central potential in (2) has been solved in three 11 dimensions (3D) by Cheng and Dai [36] . However, the aim of this subsection is to solve the problem with a different radial separation function g(r) in any arbitrary dimensions. In what follows, we present the exact bound-states (real) solution of Eq. (24). Letting\\n\\nε 2 = α 2 1 α 2 2 , 4γ 2 = (M -1)(M -3) + 4Bα 2 2 , β 2 = Aα 2 2 , ( 40\\n\\n)\\n\\nand substituting these expressions in equation (24), one gets\\n\\nd 2 g(r) dr 2 + -ε 2 r 2 + β 2 r -γ 2 r 2 g(r) = 0. ( 41\\n\\n)\\n\\nTo apply the conventional NU-method, equation (41) is compared with (14), resulting in the following expressions:\\n\\nτ (r) = 0, σ(r) = r, σ(r) = -ε 2 r 2 + β 2 r -γ 2 . ( 42\\n\\n)\\n\\nSubstituting the above expressions into equation (22) gives\\n\\nπ(r) = 1 2 ± 1 2 4ε 2 r 2 + 4(k -β 2 )r + 4γ 2 + 1. ( 43\\n\\n)\\n\\nTherefore, we can determine the constant k by using the condition that the discriminant of the square root is zero, that is\\n\\nk = β 2 ± ε 4γ 2 + 1, 4γ 2 + 1 = (D + 2j -2) 2 + 4Bα 2 2 . ( 44\\n\\n)\\n\\nIn view of that, we arrive at the following four possible functions of π(r) :\\n\\nπ(r) = 1 2 + εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 -εr + 1 2 √ 4γ 2 + 1 for k 1 = β 2 + ε √ 4γ 2 + 1, 1 2 + εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1, 1 2 -εr -1 2 √ 4γ 2 + 1 for k 2 = β 2 -ε √ 4γ 2 + 1. ( 45\\n\\n)\\n\\nThe correct value of π(r) is chosen such that the function τ (r) given by Eq. (18) will have negative derivative [29] . So we can select the physical values to be\\n\\nk = β 2 -ε 4γ 2 + 1 and π(r) = 1 2 -εr - 1 2 4γ 2 + 1 , ( 46\\n\\n)\\n\\n12 which yield τ (r) = -2εr + (1 + 4γ 2 + 1), τ ′ (r) = -2ε < 0. (47) Using Eqs (19) and (23), the following expressions for λ are obtained, respectively,\\n\\nλ = λ n = 2nε, n = 0, 1, 2, ..., ( 48\\n\\n) λ = δ 2 -ε(1 + 4γ 2 + 1). ( 49\\n\\n)\\n\\nSo we can obtain the Klein Gordon energy eigenvalues from the following relation:\\n\\n1 + 2n + (D + 2j -2) 2 + 4(µ + E R )B µ -E R = A µ + E R , ( 50\\n\\n)\\n\\nand hence for the Kratzer plus the new ring-shaped potential, it becomes\\n\\n1 + 2n + (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 51\\n\\n)\\n\\nwith j defined in (36) . Although Eq. (51) is exactly solvable for E R but it looks to be little complicated. Further, it is interesting to investigate the solution for the Coulomb potential.\\n\\nTherefore, applying the following transformations: A = Ze 2 , B = 0, and j = ℓ, the central part of the potential in (3) turns into the Coulomb potential with Klein Gordon solution for the energy spectra given by\\n\\nE R = µ 1 - 2q 2 e 2 q 2 e 2 + (2n + 2ℓ + D -1) 2 , n, ℓ = 0, 1, 2, ..., ( 52\\n\\n)\\n\\nwhere q = Ze is the charge of the nucleus. Further, Eq. ( 52 ) can be expanded as a series in the nucleus charge as\\n\\nE R = µ - 2µq 2 e 2 (2n + 2ℓ + D -1) 2 + 2µq 4 e 4 (2n + 2ℓ + D -1) 4 -O(qe) 6 , ( 53\\n\\n)\\n\\nThe physical meaning of each term in the last equation was given in detail by Ref. [36] .\\n\\nBesides, the difference from the conventional nonrelativistic form is because of the choice of the vector V (r, θ, ϕ) and scalar S(r, θ, ϕ) parts of the potential in Eq. ( 1 ). 13 Overmore, if the value of j obtained by Eq.(36) is inserted into the eigenvalues of the radial part of the Klein Gordon equation with the noncentral potential given by Eq. (51), we finally find the energy eigenvalues for a bound electron in the presence of a noncentral potential by Eq. (2) as\\n\\n1 + 2n + (2j ′ + D -2) 2 + 4(a 0 r 2 0 -C)(µ + E R ) µ -E R = 2a 0 r 0 µ + E R , ( 54\\n\\n)\\n\\nwhere m ′ = m 2 + C(µ + E R ) and n is given by Eq. (39). On the other hand, the solution of the Schrödinger equation, Eq. ( 9 ), for this potential has already been obtained by using the same method in Ref. [39] and it is in the Coulombic-like form:\\n\\nE N R = - 8µa 2 0 r 2 0 2n + 1 + (2ℓ ′ + D -2) 2 + 8µ(a 0 r 2 0 -C) 2 , n = 0, 1, 2, ... ( 55\\n\\n) 2ℓ ′ + D -2 = (D -2) 2 + (2 n + 2m ′ + 1) 2 -1, ( 56\\n\\n)\\n\\nwhere m ′ = √ m 2 + 2µC. Also, applying the following appropriate transformation: 54 ) provides exactly the nonrelativistic limit given by Eq. (55).\\n\\nµ+E R → 2µ, µ -E R → -E N R , j → ℓ to Eq. (\\n\\nIn what follows, let us now turn attention to find the radial wavefunctions for this potential. Substituting the values of σ(r), π(r) and τ (r) in Eqs (42), (45) and (47) into Eqs.\\n\\n(17) and (21), we find φ(r) = r (ζ+1)\\n\\n/2 e -εr , ( 57\\n\\n) ρ(r) = r ζ e -2εr , ( 58\\n\\n)\\n\\nwhere ζ = √ 4γ 2 + 1. Then from equation (20), we obtain\\n\\ny nj (r) = B nj r -ζ e 2εr d n dr n r n+ζ e -2εr , ( 59\\n\\n)\\n\\nand the wave function g(r) can be written in the form of the generalized Laguerre polynomials as 14 g(ρ) = C nj ρ 2ε\\n\\n(1+ζ)/2\\n\\ne -ρ/2 L ζ n (ρ), ( 60\\n\\n)\\n\\nwhere for Kratzer's potential we have\\n\\nζ = (D + 2j -2) 2 + 4a 0 r 2 0 (µ + E R ), ρ = 2εr. ( 61\\n\\n)\\n\\nFinally, the radial wave functions of the Klein-Gordon equation are obtained R(ρ) = C nj ρ 2ε\\n\\n(ζ+2-D)/2\\n\\ne -ρ/2 L ζ n (ρ), ( 62\\n\\n)\\n\\nwhere C nj is the normalization constant to be determined below. Using the normalization condition, ∞ 0 R 2 (r)r D-1 dr = 1, and the orthogonality relation of the generalized Laguerre polynomials, (2n+η+1)(n+η)! n! , we have\\n\\n∞ 0 z η+1 e -z [L η n (z)] 2 dz =\\n\\nC nj = 2 µ 2 -E 2 R 1+ ζ 2 n! (2n + ζ + 1) (n + ζ)! . ( 63\\n\\n)\\n\\nFinally, we may express the normalized total wave functions as\\n\\nψ(r, θ, ϕ) = 2 µ 2 -E 2 R 1+ ζ 2 2 m ′ ( n + m ′ )! (2 n + 2m ′ + 1)( n + 2m ′ )! n!n! 2π (2n + ζ + 1) (n + ζ)! ×r (ζ+2-D) 2 exp(-µ 2 -E 2 R r)L ζ n (2 µ 2 -E 2 R r) sin m ′ (θ)P (m ′ ,m ′ ) n (cos θ) exp(±imϕ). ( 64\\n\\n)\\n\\nwhere ζ is defined in Eq. ( 61 ) and m ′ is given after the Eq. (54).\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"V. CONCLUSIONS\",\n \"text\": \"The relativistic spin-0 particle D-dimensional Klein-Gordon equation has been solved easily for its exact bound-states with equal scalar and vector ring-shaped Kratzer potential through the conventional NU method. The analytical expressions for the total energy levels and eigenfunctions of this system can be reduced to their conventional three-dimensional space form upon setting D = 3. Further, the noncentral potentials treated in [30] can be introduced as perturbation to the Kratzer's potential by adjusting the strength of the 15 coupling constant C in terms of a 0 , which is the coupling constant of the Kratzer's potential.\\n\\nAdditionally, the radial and polar angle wave functions of Klein-Gordon equation are found in terms of Laguerre and Jacobi polynomials, respectively. The method presented in this paper is general and worth extending to the solution of other interaction problems. This method is very simple and useful in solving other complicated systems analytically without given a restiction conditions on the solution of some quantum systems as the case in the other models. We have seen that for the nonrelativistic model, the exact energy spectra can be obtained either by solving the Schrödinger equation in (9) (cf. Ref. [39] or Eq. ( 55 )) or by applying appropriate transformation to the relativistic solution given by Eq. ( 54 ). Finally, we point out that these exact results obtained for this new proposed form of the potential (2) may have some interesting applications in the study of different quantum mechanical systems, atomic and molecular physics.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"ACKNOWLEDGMENTS\",\n \"text\": \"This research was partially supported by the Scientific and Technological Research Council of Turkey. S.M. Ikhdair wishes to dedicate this work to his family for their love and assistance. 16 REFERENCES [1] T.Y. Wu and W.Y. Pauchy Hwang, Relativistic Quantum Mechanics and Quantum Fields (World Scientific, Singapore, 1991). [2] W. Greiner, Relativistic Quantum Mechanics: Wave Equations, 3rd edn (springer, Berlin, 2000). [3] A.D. Alhaidari, Phys. Rev. Lett. 87 (2001) 210405; 88 (2002) 189901.\\n\\n[4] G. Chen, Mod. Phys. Lett. A 19 (2004) 2009; J.-Y. Guo, J. Meng and F.-X. Xu, Chin.\\n\\nPhys. Lett. 20 (2003) 602; A.D. Alhaidari, J. Phys. A: Math. Gen. 34 (2001) 9827; 35 (2002) 6207; M. S ¸imşek and H. Egrifes, J. Phys. A: Math. Gen. 37 (2004) 4379.\\n\\n[5] J.-Y. Guo, X.-Z. Fang and F.-X. Xu, Phys. Rev. A 66 (2002) 062105; C. Berkdemir, A.\\n\\nBerkdemir and R. Sever, J. Phys. A: Math. Gen. 39 (2006) 13455. [6] G. Chen, Acta Phys. Sinica 50 (2001) 1651; Ö. Yeşiltaş, Phys. Scr. 75 (2007) 41.\\n\\n[7] G. Chen and Z.M. Lou, Acta Phys. Sinica 52 (2003) 1071. [8] S.M. Ikhdair and R. Sever, J. Mol. Structure:THEOCHEM 806 (2007) 155; G. Chen, Z.D. Chen and Z.M. Lou, Chin. Phys. 13 (2004) 279. [9] W.C. Qiang, Chin. Phys. 12 (2003) 136.\\n\\n[10] W.C. Qiang, Chin. Phys. 13 (2004) 571. [11] G. Chen, Phys. Lett. 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Zhedabov, Phys. Lett. A 153 (1991) 177; C. Quesne, J. Phys. A 21 (1988) 3093.\\n\\n[21] M. Kibler and P. Winternitz, J. Phys. A 20 (1987) 4097.\\n\\n[22] H. Hartmann and D. Schuch, Int. J. Quantum Chem. 18 (1980) 125. [23] M. Kibler and T. Negadi, Phys. Lett. A 124 (1987) 42; A. Guha and S. Mukherjee, J.\\n\\nMath. Phys. 28 (1989) 840; G. E. Draganescu, C. Campiogotto and M. Kibler, Phys.\\n\\nLett. A 170 (1992) 339; M. Kibler and C. Campiogotto, Phys. Lett. A 181 (1993) 1; V. M. Villalba, Phys. Lett. A 193 (1994) 218. [24] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [25] P. A. M. Dirac, Proc. R. Soc. London Ser. A 133 (1931) 60. [26] B.P. Mandal, Int. J. Mod. Phys. A 15 (2000) 1225.\\n\\n[27] R. Dutt, A. Gangopadhyaya and U.P. Sukhatme, Am. J. Phys. 65 (5) (1997) 400. [28] B. Gönül and İ. Zorba, Phys. Lett. A 269 (2000) 83. [29] A.F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics 18 (Birkhauser, Bassel, 1988). [30] S.M. Ikhdair and R. Sever, to appear in the Int. J. Theor. Phys. (preprint quantph//0702186). [31] H.S. Valk, Am. J. Phys. 54 (1986) 921. [32] Q.W. Chao, Chin. Phys. 13 (5) (2004) 575.\\n\\n[33] C. Berkdemir, A. Berkdemir and J.G. Han, Chem. Phys. Lett. 417 (2006) 326. [34] A. Bastida et al, J. Chem. Phys. 93 (1990) 3408. [35] F. Yasuk, A. Durmuş and I. Boztosun, J. Math. Phys. 47 (2006) 082302. [36] C. Berkdemir, Am. J. Phys. 75 (2007) 81. [37] C.Y. Chen and S.H. Dong, Phys. Lett. A 335 (2005) 374. [38] Y.F. Cheng and T.Q. Dai, Phys. Scr. 75 (2007) 274. [39] S.M. Ikhdair and R. Sever, preprint quant-ph/0703008; quant-ph/0703042. [40] S.M. Ikhdair and R. Sever, preprint quant-ph/0703131. [41] S.M. Ikhdair and R. Sever, Z. Phys. C 56 (1992) 155; C 58 (1993) 153; D 28 (1993) 1; Hadronic J. 15 (1992) 389; Int. J. Mod. Phys. A 18 (2003) 4215; A 19 (2004) 1771; A 20 (2005) 4035; A 20 (2005) 6509; A 21 (2006) 2191; A 21 (2006) 3989; A 21 (2006) 6699; Int. J. Mod. Phys. E (in press) (preprint hep-ph/0504176); S. Ikhdair et al, Tr. J. Phys. 16 (1992) 510; 17 (1993) 474. [42] S.M. Ikhdair and R. Sever, Int. J. Theor. Phys. (DOI 10.1007/s10773-006-9317-7; J.\\n\\nMath. Chem. (DOI 10.1007/s10910-006-9115-8). [43] S.M. Ikhdair and R. Sever, Ann. Phys. (Leipzig) 16 (3) (2007) 218. [44] S.M. Ikhdair and R. Sever, to appear in the Int. J. Mod. Phys. E (preprint quant-19 ph/0611065). [45] G. Sezgo, Orthogonal Polynomials (American Mathematical Society, New York, 1939). [46] N.N. Lebedev, Special Functions and Their Applications (Prentice-Hall, Englewood Cliffs, NJ, 1965). 20\"\n }\n]","isConversational":false}}},{"rowIdx":88,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0490"},"arxiv_id":{"kind":"string","value":"0704.0490"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevD.76.111701"},"content_hash":{"kind":"string","value":"33a1d1b81936bb4d490d360820f2123e47c0f5994f09751e7382ad683a799b76"},"title":{"kind":"string","value":"Long Distance Signaling Using Axion-like Particles"},"abstract":{"kind":"string","value":"The possible existence of axion-like particles could lead to a new type of\nlong distance communication. In this work, basic antenna concepts are defined\nand a Friis-like equation is derived to facilitate long-distance link\ncalculations. An example calculation is presented showing that communication\nover distances of 1000 km or more may be possible for $m_{a}< 3.5$ meV and\n$g_{a\\gamma \\gamma} > 5 \\times 10^{- 8} {\\text{GeV}}^{- 1}$."},"authors":{"kind":"list like","value":["Daniel D. Stancil"],"string":"[\n \"Daniel D. Stancil\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["hep-ph","astro-ph","hep-th"],"string":"[\n \"hep-ph\",\n \"astro-ph\",\n \"hep-th\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"hep-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"The possible existence of axion-like particles could lead to a new type of long distance communication. In this work, basic antenna concepts are defined and a Friis-like equation is derived to facilitate long-distance link calculations. An example calculation is presented showing that communication over distances of 1000 km or more may be possible for 3.5\n\nsuggested coupling between the axion and electromagnetic fields that was much larger than thought possible based on solar axion observations by the CAST collaboration [5] .\n\nAlthough mechanisms have been proposed to reconcile the reports [6] , the PVLAS collaboration recently retracted the results [7] and an independent group has reported a negative result from a photon regeneration experiment that excludes the PVLAS result [8] .\n\nThere does not now appear to be any experimental evidence of a coupling strength inconsistent with CAST observations. However, since recent work has suggested mechanisms whereby such strong coupling may be possible, I believe it remains interesting to consider the implications of stronger-than-expected axion-photon coupling.\n\nIn particular, I would like to call attention to the observation that a new type of longdistance signaling and communication may be possible. It may be possible to construct a communication system that cannot be blocked-even communicating directly through the diameter of the earth. This would make reliable worldwide signaling possible without the use of either satellites or the ionosphere, and would enable communication to locations previously inaccessible, such as submarines at the bottom of the sea and mines deep beneath the earth. The signal would also be very difficult to intercept since the axion beam would be essentially as narrow as a laser beam used to create it, and most of the path would be underground. With advances in power and sensitivity, it may also be possible to use axion signaling in space communications. For example, using such a system, communication with points on the far side of the moon may be possible without the use of lunar satellites.\n\nCommunication systems using neutrinos have also been proposed [9], and would have many of the same characteristics as the proposed axion system. However, the generation and detection of neutrinos requires massive particle accelerators and scintillation detectors [10] . Also, full deflection over 4π steradians would not be practical, though limited beam steering could be achieved using a magnetic field to deflect the precursor pion beam. Finally, it would be difficult to consider modulation techniques more sophisticated than simple amplitude modulation. In contrast, using axion mass and coupling values not yet experimentally explored, it appears that worldwide communication would be possible with a fully steerable axion system about the size of a medium-size telescope. Further, since the signals at the input and output would be electromagnetic waves, any existing modulation technology could be used. However, as with neutrinos, the lack of strong interactions with matter presents challenges with respect to the generation and detection of axions. Sikivie proposed an experimental approach for detecting axions via their coupling to the electromagnetic field [11] . The coupling was obtained by considering the Lagrangian density\n\n2 2 1 1 1 1 4 4 2 2 a a L F F g aF F a a m a μν μν μ μν γγ μν μ = - - + ∂ ∂ - , (1)\n\nwhere\n\nF A A μν μ ν ν μ = ∂ -∂ is the electromagnetic field tensor, ( ) , A V A α =\n\nwhere V is the electric scalar potential and A is the magnetic vector potential, a is the axion field, a g γγ is the coupling constant between the electromagnetic and axion fields, and the electromagnetic dual tensor is given by 1 2 F F γδ αβ αβγδ ε = . In these equations we have\n\ntaken 1 h c = = .\n\nAs an example, consider the coupling between plane waves propagating along the z direction caused by a strong static magnetic field parallel to the polarization of the incident electromagnetic wave. If the time dependence is exp(\n\n) i t ω -\n\n, where ω is the frequency of the incident linearly polarized electromagnetic wave, the equations of motion obtained from (1) reduce to the coupled equations ( )\n\n2 2 2 2 2 0 0 2 2 , . a a a a A m a i g B A A i g B a z z γγ γγ ω ω ω ω ∂ ∂ + - =- + = ∂ ∂ (2)\n\nThus the static magnetic field B 0 couples the photon and axion fields. An apparatus for the generation and detection of axions based on this coupling is shown in Figure 1(a) . This is sometimes referred to as an \"invisible light through walls\" experiment [4, 12, 13 ].\n\nAn electromagnetic wave with amplitude (0) A enters a region of magnetic field of strength B OT extending over a distance L T . If the coupling is sufficiently weak that the change in A over the transmit conversion region and the change in a over the receive conversion region are negligible, then the conversion loss through the system will be [12] 0 / in R T\n\nP P p p = , (3)\n\nwhere in P is the optical input power, 0 P is the optical output power, T p , R p are the probabilities of photon-axion (and axion-photon) conversion in the transmitter and receiver, respectively, and the conversion probability is [11, 12, 14] 2\n\n0 1 sin /2 2 /2 a a qL p g BL k q L γγ ω ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ . (4)\n\nHere a q k k γ =indicates the phase mismatch between the photon and axion fields.\n\nThe efficiency of generating axions and regenerating photons can be greatly increased by adding electromagnetic resonators, as shown in Figure 1 (b) [4, 15] . In this light to pass through the magnetic field multiple times, increasing the conversion probability by the factor 2 / T F π , where F T is the finesse of the resonator in the axion generator (transmitter). This factor can be interpreted as the effective number of photon passes in the resonator. Axions are also emitted in the backward direction owing to the counter-propagating light in the resonator, resulting in half the particles traveling in an unwanted direction. Consequently, the probability of conversion in a given direction is increased by the factor / T F π . As also shown in Figure 1(b), a resonator on the photon regenerator (receiver) likewise increases the axion-photon conversion probability [15]. Since regenerated photons will be emitted in both directions, detectors are placed on both ends of the receiving optical resonator, enhancing the photon regeneration probability by the factor 2 / R F π , where F R is the finesse of the receiving resonator. (If the power from a single end is collected, the factor would be / R F π , as with the transmitter.) Finally, to turn this into a communication system, we add appropriate electromagnetic wave modulators and detectors as shown in Figure 1(b).\n\nThe conversion loss equation ( 3 ) is valid when the transmitter and receiver are sufficiently close together that beam diffraction can be neglected, and when the transmitter and receiver have equal cross-sectional areas. For signaling over long distances, neither assumption will be valid in general. To treat the long-distance case, we first calculate the radiated axion field, then calculate the regenerated photons resulting when the radiated field reaches the receiver.\n\nThe general solution to Eq. ( 2 ) is given by [12] 3 0\n\n( ) ( ) ( ) 4 a ik r r a V e a r i g d r A r B r r r γγ ω π ′ - ′ ′ ′ = ′ - ∫ i . ( 5\n\n)\n\nIf the observation point r is very far away from all points in the source volume V, then we obtain the far-field approximation for the axion field\n\nT A x F A ik z γ π =\n\n, where 0 A is the amplitude of the incident electromagnetic wave, and the cylinder is contained in a resonant cavity with finesse T F .\n\nUsing Eq. ( 6 ), the far-field potential is found to be\n\n( ) ( ) ( ) 1 0 0 sin / 2 ( ) 2 / 2 a ik r az T a T T a T T T a T az T k k L J k R F e a r i g A B L s r k R k k L γ ρ γγ ρ γ ω π π ⎡ ⎤⎡ ⎤ - ⎣ ⎦ ≈ ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ , (7)\n\nwhere sin , cos\n\n1 0 lim / 1/ 2 a a T a T k J k R k R ρ ρ ρ → ⎡ ⎤ = ⎣ ⎦ .\n\nConsequently, the axion field on axis is\n\n[ ] 0 0 sin / 2 ( ) 4 / 2 a ik r T T a T T T T qL F e a r i g A B L s r q L γγ ω π π ≈ . ( 8\n\n)\n\nThe time averaged transmitted power density is ( )\n\n2 2 2 1 ( ) ( ) 2 2 a T a a T Ti n k F S r k a r s p P r ω π π = = , (9)\n\nwhere (0)\n\nin T P S s γ = , and\n\n( ) 2 1 (0) 0 2 S k A γ γ ω = .\n\nIf this axion power density is incident upon a photon regenerator at distance r that is perfectly aligned with the transmitter, then the received power is\n\n0 (2 / ) ( ) R R a R P F p S r s π = . ( 10\n\n)\n\nSubstituting Eq. (9) for the power flux ( )\n\na S r gives 2 0 2 2 4 in a R T R R T T P k F F P ps ps r π π π π = . ( 11\n\n)\n\nThis expression can be understood in terms of antenna theory for electromagnetic waves.\n\nIn this context, we refer to the apparatus consisting of the resonator and the structure creating the magnetic field as an axion antenna. In analogy with conventional antenna theory, we define the directivity as 2 ( ) /( 4 )\n\na rad S r D P r π = , ( 12\n\n)\n\nwhere P rad is the total power radiated by the transmitting antenna. To obtain the total power radiated, we could integrate the power flux (9) over a sphere enclosing the antenna. However, it is easier to do the calculation in the near field using the axion field at the aperture of the transmitting antenna. Using ( ) ( / ) (0)\n\na T T T S L F p S γ π = , we have 2 ( ) (2 / ) rad a T T T T in P S L s F p P π = = . (13)\n\nSubstituting ( 13 ) for the total radiated power and (9) for the power flux, the directivity simplifies to\n\n2 2 (2 / ) (4 / )( /2) T a T a T D s s π λ π λ = = . ( 14\n\n)\n\nThe relation between the directivity and physical area ( 14 ) is ½ that found in conventional antenna theory, or equivalently, the area appears to be half the physical area. This results from the bi-directional radiation properties of the resonator. Defining an efficiency as /\n\nrad in P P η = , we also define the antenna gain as 2 (2 / ) (2 / ) T T T T T a T G D F p s η π π λ = = , (15) where (2 / ) T T T F p η π = .\n\nNext suppose that at some distant location this transmitted field is incident upon a receive antenna with length L R , radius R R , and finesse F R . From (10) and assuming the photons emitted from both ends of the receive antenna are collected, the total power collected will be 0 As with electromagnetic antennas, the ratio of effective area to gain is found to be independent of the details of the antenna, other than whether or not photons are collected from both ends of the antenna when used to receive:\n\n2 , , / / / (4 ) . e R R e T T c a s G s G n λ π = = ( 17\n\n)\n\nIf photons were collected only from one end of the receive antenna ( 1 c n = ), then Eq. ( 17 ) would be identical to conventional antenna theory.\n\nWith these definitions, the expression for the received power ( 11 ) can be interpreted as a Friis-like equation:\n\n( ) ( )( )\n\n2 2 2 0 , , , / / 4 / ( ) / in c T R a e T c T a e R P P n G G r s n s r λ π λ = = . (18) Here , c T n is the value used to compute , e T s according to Eq. (16). The ratio , , / e T c T s n is independent of the choice of , c T\n\nn , as it should be since the number of detectors that might be used on receive is independent of the transmit properties of the antenna.\n\nIt is also useful to note that if the magnetic field is uniform (i.e., wigglers, or quasi-phase matching, are not used [12] ), then there is an optimum length for the conversion region of an antenna. This occurs when ( )\n\nsin / 2 1 qL = , or qL π = . The optimum length is found to be 2 ( / ) opt a L m γ λ ω = . ( 19\n\n)\n\nIn obtaining this expression, we have used 2 /(2 ) a q m ω ≈ , which is valid for a m ω .\n\nThe diffraction-limited power pattern of the radiated axion field is determined by the aperture size in wavelengths through the Bessel function term in ( 7 ):\n\n( ) ( )\n\n2 1 4 sin /( sin ) d a a P J k R k R θ θ θ = ⎡ ⎤ ⎣ ⎦ . ( 20\n\n)\n\nThe diffraction beam width between first nulls is determined by the first zero of the Airy disc, and for small angles is given by the well-known expression\n\n1.22 / d FWFN T R γ θ λ ≈ . (21)\n\nSimilarly, the diffraction beam width at half maximum is determined by the roots of ( )\n\n) / 0.514 / d FWHM a T T R R γ θ πλ λ = ≈ . 1/ 2 d P θ = , or (1.616 /\n\nIn contrast, the conversion-limited power pattern is given by ( ) ( )\n\n2 sin cos / 2 ( ) 2 cos / 2 a c a k k L P k k L γ γ θ π θ θ ⎡ ⎤ ⎡ ⎤ - ⎣ ⎦ ⎢ ⎥ = ⎢ ⎥ - ⎣ ⎦ , (23)\n\nand depends on both the length in wavelengths and the velocity mismatch. For the optimum length L given by ( 19 ), the conversion beam widths are approximately given by 2( / ), 1.06( / )\n\nc c FWFN a FWHM a m m θ ω θ ω ≈ ≈ . ( 24\n\n)\n\nFor quantum-limited detection, the channel capacity is [16] log [17], and c is the velocity of light. Equation (25) can be combined with (11) or (18) to find the axion parameters that would permit a particular channel capacity at a given distance for a particular experimental apparatus.\n\n(1 / ) d R C N ν η ν = Δ + Δ , (25)\n\nAs an example, consider transmitters and receivers with 1064 nm (1.17 eV)\n\nγ λ = , 10 W in P = , 2 0.01 m s = , 0 10 T B = , 3 L = m, 5 3.1 10 F = × , 0.5 d η =\n\n, and a minimum information capacity of 1 bps. Figure 3 shows the inverse coupling strength curves\n\n( ) 1/ a a M m g γγ =\n\nfor distances of 1000 km and the diameter of the earth. Also shown is the curve for communication between the earth and the far side of the moon using the \"4+4\" experimental apparatus proposed in [15] . The shaded regions are excluded by the results from the BFRT collaboration [13] and Robilliard et al. [8] . The dots represent extensions of the curves if quasi-phase matching (QPM) is used by periodically reversing the magnetic field [12] . The minimum period of the reversal is taken to be twice the beam diameter for the 3 m system example, and 28.6 m for the 4+4 system. From the figure, communication over distances in excess of 1000 km should be possible for 3.5\n\na m < meV and 7 2 10 GeV M < × ( 8 1 5 10 GeV a g γγ -- Note that the receive resonator must be tuned to the same frequency as the transmit resonator to within a fraction of the resonator line width which is 161 Hz in this example. This will present a significant challenge, especially since the resonators are remote from one another. A possible approach would be to use atomic clocks to stabilize both the transmit frequency and a local reference frequency at the receiver. The receive resonator would then be locked to the reference to get as close as possible to the correct frequency, then slowly tuned until the signal is located. A feedback loop could then be closed to lock the receive resonator to the signal.\n\nIn summary, for 3.5 > × ), it may be possible to realize a new type of wireless signaling that cannot be blocked or shielded. An example calculation shows that communication between points located diametrically opposite on the earth should be possible. This could enable world-wide communication without the use of satellites or the ionosphere. However, with present knowledge, the signaling will be limited to low data rates, perhaps on the order of a few bits per second for terrestrial links. This estimate assumes 3 m long generation/regeneration regions to allow fully steerable instruments. Though not easily steerable, the apparatus in the \"4+4\" experiment proposed by Sikivie et al., [15] may enable communication to the far side of the moon for 0.3 a m < meV and 6 6 10 GeV M < × . I would like to acknowledge helpful discussions with Jim Lesh, Rich Holman, Jeff Peterson, Pierre Sikivie, and David Tanner during the development of these ideas. Electronic address: stancil@cmu.edu 17. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6 th Edition (Oxford Univ. Press, New York, 2007), pp. 171. 18. C-C. Chen, op cit., p. 106. B 0R L T L R Mod Laser Demod Data In Data Out B 0T Optical Detectors B 0T B 0R Photon input Photon output Axions penetrate barrier opaque to photons L T L R (a) Photons Axions Cavity mirrors (b) Figure 1, Stancil, Phys. Rev. D. x y z r θ -L T /2 L T /2"},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"The possible existence of axion-like particles could lead to a new type of long distance communication. In this work, basic antenna concepts are defined and a Friis-like equation is derived to facilitate long-distance link calculations. An example calculation is presented showing that communication over distances of 1000 km or more may be possible for 3.5"},{"section_type":"OTHER","section_title":"Untitled Section","text":"suggested coupling between the axion and electromagnetic fields that was much larger than thought possible based on solar axion observations by the CAST collaboration [5] .\n\nAlthough mechanisms have been proposed to reconcile the reports [6] , the PVLAS collaboration recently retracted the results [7] and an independent group has reported a negative result from a photon regeneration experiment that excludes the PVLAS result [8] .\n\nThere does not now appear to be any experimental evidence of a coupling strength inconsistent with CAST observations. However, since recent work has suggested mechanisms whereby such strong coupling may be possible, I believe it remains interesting to consider the implications of stronger-than-expected axion-photon coupling.\n\nIn particular, I would like to call attention to the observation that a new type of longdistance signaling and communication may be possible. It may be possible to construct a communication system that cannot be blocked-even communicating directly through the diameter of the earth. This would make reliable worldwide signaling possible without the use of either satellites or the ionosphere, and would enable communication to locations previously inaccessible, such as submarines at the bottom of the sea and mines deep beneath the earth. The signal would also be very difficult to intercept since the axion beam would be essentially as narrow as a laser beam used to create it, and most of the path would be underground. With advances in power and sensitivity, it may also be possible to use axion signaling in space communications. For example, using such a system, communication with points on the far side of the moon may be possible without the use of lunar satellites.\n\nCommunication systems using neutrinos have also been proposed [9], and would have many of the same characteristics as the proposed axion system. However, the generation and detection of neutrinos requires massive particle accelerators and scintillation detectors [10] . Also, full deflection over 4π steradians would not be practical, though limited beam steering could be achieved using a magnetic field to deflect the precursor pion beam. Finally, it would be difficult to consider modulation techniques more sophisticated than simple amplitude modulation. In contrast, using axion mass and coupling values not yet experimentally explored, it appears that worldwide communication would be possible with a fully steerable axion system about the size of a medium-size telescope. Further, since the signals at the input and output would be electromagnetic waves, any existing modulation technology could be used. However, as with neutrinos, the lack of strong interactions with matter presents challenges with respect to the generation and detection of axions. Sikivie proposed an experimental approach for detecting axions via their coupling to the electromagnetic field [11] . The coupling was obtained by considering the Lagrangian density\n\n2 2 1 1 1 1 4 4 2 2 a a L F F g aF F a a m a μν μν μ μν γγ μν μ = - - + ∂ ∂ - , (1)\n\nwhere\n\nF A A μν μ ν ν μ = ∂ -∂ is the electromagnetic field tensor, ( ) , A V A α =\n\nwhere V is the electric scalar potential and A is the magnetic vector potential, a is the axion field, a g γγ is the coupling constant between the electromagnetic and axion fields, and the electromagnetic dual tensor is given by 1 2 F F γδ αβ αβγδ ε = . In these equations we have\n\ntaken 1 h c = = .\n\nAs an example, consider the coupling between plane waves propagating along the z direction caused by a strong static magnetic field parallel to the polarization of the incident electromagnetic wave. If the time dependence is exp(\n\n) i t ω -\n\n, where ω is the frequency of the incident linearly polarized electromagnetic wave, the equations of motion obtained from (1) reduce to the coupled equations ( )\n\n2 2 2 2 2 0 0 2 2 , . a a a a A m a i g B A A i g B a z z γγ γγ ω ω ω ω ∂ ∂ + - =- + = ∂ ∂ (2)\n\nThus the static magnetic field B 0 couples the photon and axion fields. An apparatus for the generation and detection of axions based on this coupling is shown in Figure 1(a) . This is sometimes referred to as an \"invisible light through walls\" experiment [4, 12, 13 ].\n\nAn electromagnetic wave with amplitude (0) A enters a region of magnetic field of strength B OT extending over a distance L T . If the coupling is sufficiently weak that the change in A over the transmit conversion region and the change in a over the receive conversion region are negligible, then the conversion loss through the system will be [12] 0 / in R T\n\nP P p p = , (3)\n\nwhere in P is the optical input power, 0 P is the optical output power, T p , R p are the probabilities of photon-axion (and axion-photon) conversion in the transmitter and receiver, respectively, and the conversion probability is [11, 12, 14] 2\n\n0 1 sin /2 2 /2 a a qL p g BL k q L γγ ω ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ . (4)\n\nHere a q k k γ =indicates the phase mismatch between the photon and axion fields.\n\nThe efficiency of generating axions and regenerating photons can be greatly increased by adding electromagnetic resonators, as shown in Figure 1 (b) [4, 15] . In this light to pass through the magnetic field multiple times, increasing the conversion probability by the factor 2 / T F π , where F T is the finesse of the resonator in the axion generator (transmitter). This factor can be interpreted as the effective number of photon passes in the resonator. Axions are also emitted in the backward direction owing to the counter-propagating light in the resonator, resulting in half the particles traveling in an unwanted direction. Consequently, the probability of conversion in a given direction is increased by the factor / T F π . As also shown in Figure 1(b), a resonator on the photon regenerator (receiver) likewise increases the axion-photon conversion probability [15]. Since regenerated photons will be emitted in both directions, detectors are placed on both ends of the receiving optical resonator, enhancing the photon regeneration probability by the factor 2 / R F π , where F R is the finesse of the receiving resonator. (If the power from a single end is collected, the factor would be / R F π , as with the transmitter.) Finally, to turn this into a communication system, we add appropriate electromagnetic wave modulators and detectors as shown in Figure 1(b).\n\nThe conversion loss equation ( 3 ) is valid when the transmitter and receiver are sufficiently close together that beam diffraction can be neglected, and when the transmitter and receiver have equal cross-sectional areas. For signaling over long distances, neither assumption will be valid in general. To treat the long-distance case, we first calculate the radiated axion field, then calculate the regenerated photons resulting when the radiated field reaches the receiver.\n\nThe general solution to Eq. ( 2 ) is given by [12] 3 0\n\n( ) ( ) ( ) 4 a ik r r a V e a r i g d r A r B r r r γγ ω π ′ - ′ ′ ′ = ′ - ∫ i . ( 5\n\n)\n\nIf the observation point r is very far away from all points in the source volume V, then we obtain the far-field approximation for the axion field\n\nT A x F A ik z γ π =\n\n, where 0 A is the amplitude of the incident electromagnetic wave, and the cylinder is contained in a resonant cavity with finesse T F .\n\nUsing Eq. ( 6 ), the far-field potential is found to be\n\n( ) ( ) ( ) 1 0 0 sin / 2 ( ) 2 / 2 a ik r az T a T T a T T T a T az T k k L J k R F e a r i g A B L s r k R k k L γ ρ γγ ρ γ ω π π ⎡ ⎤⎡ ⎤ - ⎣ ⎦ ≈ ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ , (7)\n\nwhere sin , cos\n\n1 0 lim / 1/ 2 a a T a T k J k R k R ρ ρ ρ → ⎡ ⎤ = ⎣ ⎦ .\n\nConsequently, the axion field on axis is\n\n[ ] 0 0 sin / 2 ( ) 4 / 2 a ik r T T a T T T T qL F e a r i g A B L s r q L γγ ω π π ≈ . ( 8\n\n)\n\nThe time averaged transmitted power density is ( )\n\n2 2 2 1 ( ) ( ) 2 2 a T a a T Ti n k F S r k a r s p P r ω π π = = , (9)\n\nwhere (0)\n\nin T P S s γ = , and\n\n( ) 2 1 (0) 0 2 S k A γ γ ω = .\n\nIf this axion power density is incident upon a photon regenerator at distance r that is perfectly aligned with the transmitter, then the received power is\n\n0 (2 / ) ( ) R R a R P F p S r s π = . ( 10\n\n)\n\nSubstituting Eq. (9) for the power flux ( )\n\na S r gives 2 0 2 2 4 in a R T R R T T P k F F P ps ps r π π π π = . ( 11\n\n)\n\nThis expression can be understood in terms of antenna theory for electromagnetic waves.\n\nIn this context, we refer to the apparatus consisting of the resonator and the structure creating the magnetic field as an axion antenna. In analogy with conventional antenna theory, we define the directivity as 2 ( ) /( 4 )\n\na rad S r D P r π = , ( 12\n\n)\n\nwhere P rad is the total power radiated by the transmitting antenna. To obtain the total power radiated, we could integrate the power flux (9) over a sphere enclosing the antenna. However, it is easier to do the calculation in the near field using the axion field at the aperture of the transmitting antenna. Using ( ) ( / ) (0)\n\na T T T S L F p S γ π = , we have 2 ( ) (2 / ) rad a T T T T in P S L s F p P π = = . (13)\n\nSubstituting ( 13 ) for the total radiated power and (9) for the power flux, the directivity simplifies to\n\n2 2 (2 / ) (4 / )( /2) T a T a T D s s π λ π λ = = . ( 14\n\n)\n\nThe relation between the directivity and physical area ( 14 ) is ½ that found in conventional antenna theory, or equivalently, the area appears to be half the physical area. This results from the bi-directional radiation properties of the resonator. Defining an efficiency as /\n\nrad in P P η = , we also define the antenna gain as 2 (2 / ) (2 / ) T T T T T a T G D F p s η π π λ = = , (15) where (2 / ) T T T F p η π = .\n\nNext suppose that at some distant location this transmitted field is incident upon a receive antenna with length L R , radius R R , and finesse F R . From (10) and assuming the photons emitted from both ends of the receive antenna are collected, the total power collected will be 0 As with electromagnetic antennas, the ratio of effective area to gain is found to be independent of the details of the antenna, other than whether or not photons are collected from both ends of the antenna when used to receive:\n\n2 , , / / / (4 ) . e R R e T T c a s G s G n λ π = = ( 17\n\n)\n\nIf photons were collected only from one end of the receive antenna ( 1 c n = ), then Eq. ( 17 ) would be identical to conventional antenna theory.\n\nWith these definitions, the expression for the received power ( 11 ) can be interpreted as a Friis-like equation:\n\n( ) ( )( )\n\n2 2 2 0 , , , / / 4 / ( ) / in c T R a e T c T a e R P P n G G r s n s r λ π λ = = . (18) Here , c T n is the value used to compute , e T s according to Eq. (16). The ratio , , / e T c T s n is independent of the choice of , c T\n\nn , as it should be since the number of detectors that might be used on receive is independent of the transmit properties of the antenna.\n\nIt is also useful to note that if the magnetic field is uniform (i.e., wigglers, or quasi-phase matching, are not used [12] ), then there is an optimum length for the conversion region of an antenna. This occurs when ( )\n\nsin / 2 1 qL = , or qL π = . The optimum length is found to be 2 ( / ) opt a L m γ λ ω = . ( 19\n\n)\n\nIn obtaining this expression, we have used 2 /(2 ) a q m ω ≈ , which is valid for a m ω .\n\nThe diffraction-limited power pattern of the radiated axion field is determined by the aperture size in wavelengths through the Bessel function term in ( 7 ):\n\n( ) ( )\n\n2 1 4 sin /( sin ) d a a P J k R k R θ θ θ = ⎡ ⎤ ⎣ ⎦ . ( 20\n\n)\n\nThe diffraction beam width between first nulls is determined by the first zero of the Airy disc, and for small angles is given by the well-known expression\n\n1.22 / d FWFN T R γ θ λ ≈ . (21)\n\nSimilarly, the diffraction beam width at half maximum is determined by the roots of ( )\n\n) / 0.514 / d FWHM a T T R R γ θ πλ λ = ≈ . 1/ 2 d P θ = , or (1.616 /\n\nIn contrast, the conversion-limited power pattern is given by ( ) ( )\n\n2 sin cos / 2 ( ) 2 cos / 2 a c a k k L P k k L γ γ θ π θ θ ⎡ ⎤ ⎡ ⎤ - ⎣ ⎦ ⎢ ⎥ = ⎢ ⎥ - ⎣ ⎦ , (23)\n\nand depends on both the length in wavelengths and the velocity mismatch. For the optimum length L given by ( 19 ), the conversion beam widths are approximately given by 2( / ), 1.06( / )\n\nc c FWFN a FWHM a m m θ ω θ ω ≈ ≈ . ( 24\n\n)\n\nFor quantum-limited detection, the channel capacity is [16] log [17], and c is the velocity of light. Equation (25) can be combined with (11) or (18) to find the axion parameters that would permit a particular channel capacity at a given distance for a particular experimental apparatus.\n\n(1 / ) d R C N ν η ν = Δ + Δ , (25)\n\nAs an example, consider transmitters and receivers with 1064 nm (1.17 eV)\n\nγ λ = , 10 W in P = , 2 0.01 m s = , 0 10 T B = , 3 L = m, 5 3.1 10 F = × , 0.5 d η =\n\n, and a minimum information capacity of 1 bps. Figure 3 shows the inverse coupling strength curves\n\n( ) 1/ a a M m g γγ =\n\nfor distances of 1000 km and the diameter of the earth. Also shown is the curve for communication between the earth and the far side of the moon using the \"4+4\" experimental apparatus proposed in [15] . The shaded regions are excluded by the results from the BFRT collaboration [13] and Robilliard et al. [8] . The dots represent extensions of the curves if quasi-phase matching (QPM) is used by periodically reversing the magnetic field [12] . The minimum period of the reversal is taken to be twice the beam diameter for the 3 m system example, and 28.6 m for the 4+4 system. From the figure, communication over distances in excess of 1000 km should be possible for 3.5\n\na m < meV and 7 2 10 GeV M < × ( 8 1 5 10 GeV a g γγ -- Note that the receive resonator must be tuned to the same frequency as the transmit resonator to within a fraction of the resonator line width which is 161 Hz in this example. This will present a significant challenge, especially since the resonators are remote from one another. A possible approach would be to use atomic clocks to stabilize both the transmit frequency and a local reference frequency at the receiver. The receive resonator would then be locked to the reference to get as close as possible to the correct frequency, then slowly tuned until the signal is located. A feedback loop could then be closed to lock the receive resonator to the signal.\n\nIn summary, for 3.5 > × ), it may be possible to realize a new type of wireless signaling that cannot be blocked or shielded. An example calculation shows that communication between points located diametrically opposite on the earth should be possible. This could enable world-wide communication without the use of satellites or the ionosphere. However, with present knowledge, the signaling will be limited to low data rates, perhaps on the order of a few bits per second for terrestrial links. This estimate assumes 3 m long generation/regeneration regions to allow fully steerable instruments. Though not easily steerable, the apparatus in the \"4+4\" experiment proposed by Sikivie et al., [15] may enable communication to the far side of the moon for 0.3 a m < meV and 6 6 10 GeV M < × . I would like to acknowledge helpful discussions with Jim Lesh, Rich Holman, Jeff Peterson, Pierre Sikivie, and David Tanner during the development of these ideas. Electronic address: stancil@cmu.edu 17. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6 th Edition (Oxford Univ. Press, New York, 2007), pp. 171. 18. C-C. Chen, op cit., p. 106. B 0R L T L R Mod Laser Demod Data In Data Out B 0T Optical Detectors B 0T B 0R Photon input Photon output Axions penetrate barrier opaque to photons L T L R (a) Photons Axions Cavity mirrors (b) Figure 1, Stancil, Phys. Rev. D. x y z r θ -L T /2 L T /2"}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"The possible existence of axion-like particles could lead to a new type of long distance communication. In this work, basic antenna concepts are defined and a Friis-like equation is derived to facilitate long-distance link calculations. An example calculation is presented showing that communication over distances of 1000 km or more may be possible for 3.5\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Untitled Section\",\n \"text\": \"suggested coupling between the axion and electromagnetic fields that was much larger than thought possible based on solar axion observations by the CAST collaboration [5] .\\n\\nAlthough mechanisms have been proposed to reconcile the reports [6] , the PVLAS collaboration recently retracted the results [7] and an independent group has reported a negative result from a photon regeneration experiment that excludes the PVLAS result [8] .\\n\\nThere does not now appear to be any experimental evidence of a coupling strength inconsistent with CAST observations. However, since recent work has suggested mechanisms whereby such strong coupling may be possible, I believe it remains interesting to consider the implications of stronger-than-expected axion-photon coupling.\\n\\nIn particular, I would like to call attention to the observation that a new type of longdistance signaling and communication may be possible. It may be possible to construct a communication system that cannot be blocked-even communicating directly through the diameter of the earth. This would make reliable worldwide signaling possible without the use of either satellites or the ionosphere, and would enable communication to locations previously inaccessible, such as submarines at the bottom of the sea and mines deep beneath the earth. The signal would also be very difficult to intercept since the axion beam would be essentially as narrow as a laser beam used to create it, and most of the path would be underground. With advances in power and sensitivity, it may also be possible to use axion signaling in space communications. For example, using such a system, communication with points on the far side of the moon may be possible without the use of lunar satellites.\\n\\nCommunication systems using neutrinos have also been proposed [9], and would have many of the same characteristics as the proposed axion system. However, the generation and detection of neutrinos requires massive particle accelerators and scintillation detectors [10] . Also, full deflection over 4π steradians would not be practical, though limited beam steering could be achieved using a magnetic field to deflect the precursor pion beam. Finally, it would be difficult to consider modulation techniques more sophisticated than simple amplitude modulation. In contrast, using axion mass and coupling values not yet experimentally explored, it appears that worldwide communication would be possible with a fully steerable axion system about the size of a medium-size telescope. Further, since the signals at the input and output would be electromagnetic waves, any existing modulation technology could be used. However, as with neutrinos, the lack of strong interactions with matter presents challenges with respect to the generation and detection of axions. Sikivie proposed an experimental approach for detecting axions via their coupling to the electromagnetic field [11] . The coupling was obtained by considering the Lagrangian density\\n\\n2 2 1 1 1 1 4 4 2 2 a a L F F g aF F a a m a μν μν μ μν γγ μν μ = - - + ∂ ∂ - , (1)\\n\\nwhere\\n\\nF A A μν μ ν ν μ = ∂ -∂ is the electromagnetic field tensor, ( ) , A V A α =\\n\\nwhere V is the electric scalar potential and A is the magnetic vector potential, a is the axion field, a g γγ is the coupling constant between the electromagnetic and axion fields, and the electromagnetic dual tensor is given by 1 2 F F γδ αβ αβγδ ε = . In these equations we have\\n\\ntaken 1 h c = = .\\n\\nAs an example, consider the coupling between plane waves propagating along the z direction caused by a strong static magnetic field parallel to the polarization of the incident electromagnetic wave. If the time dependence is exp(\\n\\n) i t ω -\\n\\n, where ω is the frequency of the incident linearly polarized electromagnetic wave, the equations of motion obtained from (1) reduce to the coupled equations ( )\\n\\n2 2 2 2 2 0 0 2 2 , . a a a a A m a i g B A A i g B a z z γγ γγ ω ω ω ω ∂ ∂ + - =- + = ∂ ∂ (2)\\n\\nThus the static magnetic field B 0 couples the photon and axion fields. An apparatus for the generation and detection of axions based on this coupling is shown in Figure 1(a) . This is sometimes referred to as an \\\"invisible light through walls\\\" experiment [4, 12, 13 ].\\n\\nAn electromagnetic wave with amplitude (0) A enters a region of magnetic field of strength B OT extending over a distance L T . If the coupling is sufficiently weak that the change in A over the transmit conversion region and the change in a over the receive conversion region are negligible, then the conversion loss through the system will be [12] 0 / in R T\\n\\nP P p p = , (3)\\n\\nwhere in P is the optical input power, 0 P is the optical output power, T p , R p are the probabilities of photon-axion (and axion-photon) conversion in the transmitter and receiver, respectively, and the conversion probability is [11, 12, 14] 2\\n\\n0 1 sin /2 2 /2 a a qL p g BL k q L γγ ω ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ . (4)\\n\\nHere a q k k γ =indicates the phase mismatch between the photon and axion fields.\\n\\nThe efficiency of generating axions and regenerating photons can be greatly increased by adding electromagnetic resonators, as shown in Figure 1 (b) [4, 15] . In this light to pass through the magnetic field multiple times, increasing the conversion probability by the factor 2 / T F π , where F T is the finesse of the resonator in the axion generator (transmitter). This factor can be interpreted as the effective number of photon passes in the resonator. Axions are also emitted in the backward direction owing to the counter-propagating light in the resonator, resulting in half the particles traveling in an unwanted direction. Consequently, the probability of conversion in a given direction is increased by the factor / T F π . As also shown in Figure 1(b), a resonator on the photon regenerator (receiver) likewise increases the axion-photon conversion probability [15]. Since regenerated photons will be emitted in both directions, detectors are placed on both ends of the receiving optical resonator, enhancing the photon regeneration probability by the factor 2 / R F π , where F R is the finesse of the receiving resonator. (If the power from a single end is collected, the factor would be / R F π , as with the transmitter.) Finally, to turn this into a communication system, we add appropriate electromagnetic wave modulators and detectors as shown in Figure 1(b).\\n\\nThe conversion loss equation ( 3 ) is valid when the transmitter and receiver are sufficiently close together that beam diffraction can be neglected, and when the transmitter and receiver have equal cross-sectional areas. For signaling over long distances, neither assumption will be valid in general. To treat the long-distance case, we first calculate the radiated axion field, then calculate the regenerated photons resulting when the radiated field reaches the receiver.\\n\\nThe general solution to Eq. ( 2 ) is given by [12] 3 0\\n\\n( ) ( ) ( ) 4 a ik r r a V e a r i g d r A r B r r r γγ ω π ′ - ′ ′ ′ = ′ - ∫ i . ( 5\\n\\n)\\n\\nIf the observation point r is very far away from all points in the source volume V, then we obtain the far-field approximation for the axion field\\n\\nT A x F A ik z γ π =\\n\\n, where 0 A is the amplitude of the incident electromagnetic wave, and the cylinder is contained in a resonant cavity with finesse T F .\\n\\nUsing Eq. ( 6 ), the far-field potential is found to be\\n\\n( ) ( ) ( ) 1 0 0 sin / 2 ( ) 2 / 2 a ik r az T a T T a T T T a T az T k k L J k R F e a r i g A B L s r k R k k L γ ρ γγ ρ γ ω π π ⎡ ⎤⎡ ⎤ - ⎣ ⎦ ≈ ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ , (7)\\n\\nwhere sin , cos\\n\\n1 0 lim / 1/ 2 a a T a T k J k R k R ρ ρ ρ → ⎡ ⎤ = ⎣ ⎦ .\\n\\nConsequently, the axion field on axis is\\n\\n[ ] 0 0 sin / 2 ( ) 4 / 2 a ik r T T a T T T T qL F e a r i g A B L s r q L γγ ω π π ≈ . ( 8\\n\\n)\\n\\nThe time averaged transmitted power density is ( )\\n\\n2 2 2 1 ( ) ( ) 2 2 a T a a T Ti n k F S r k a r s p P r ω π π = = , (9)\\n\\nwhere (0)\\n\\nin T P S s γ = , and\\n\\n( ) 2 1 (0) 0 2 S k A γ γ ω = .\\n\\nIf this axion power density is incident upon a photon regenerator at distance r that is perfectly aligned with the transmitter, then the received power is\\n\\n0 (2 / ) ( ) R R a R P F p S r s π = . ( 10\\n\\n)\\n\\nSubstituting Eq. (9) for the power flux ( )\\n\\na S r gives 2 0 2 2 4 in a R T R R T T P k F F P ps ps r π π π π = . ( 11\\n\\n)\\n\\nThis expression can be understood in terms of antenna theory for electromagnetic waves.\\n\\nIn this context, we refer to the apparatus consisting of the resonator and the structure creating the magnetic field as an axion antenna. In analogy with conventional antenna theory, we define the directivity as 2 ( ) /( 4 )\\n\\na rad S r D P r π = , ( 12\\n\\n)\\n\\nwhere P rad is the total power radiated by the transmitting antenna. To obtain the total power radiated, we could integrate the power flux (9) over a sphere enclosing the antenna. However, it is easier to do the calculation in the near field using the axion field at the aperture of the transmitting antenna. Using ( ) ( / ) (0)\\n\\na T T T S L F p S γ π = , we have 2 ( ) (2 / ) rad a T T T T in P S L s F p P π = = . (13)\\n\\nSubstituting ( 13 ) for the total radiated power and (9) for the power flux, the directivity simplifies to\\n\\n2 2 (2 / ) (4 / )( /2) T a T a T D s s π λ π λ = = . ( 14\\n\\n)\\n\\nThe relation between the directivity and physical area ( 14 ) is ½ that found in conventional antenna theory, or equivalently, the area appears to be half the physical area. This results from the bi-directional radiation properties of the resonator. Defining an efficiency as /\\n\\nrad in P P η = , we also define the antenna gain as 2 (2 / ) (2 / ) T T T T T a T G D F p s η π π λ = = , (15) where (2 / ) T T T F p η π = .\\n\\nNext suppose that at some distant location this transmitted field is incident upon a receive antenna with length L R , radius R R , and finesse F R . From (10) and assuming the photons emitted from both ends of the receive antenna are collected, the total power collected will be 0 As with electromagnetic antennas, the ratio of effective area to gain is found to be independent of the details of the antenna, other than whether or not photons are collected from both ends of the antenna when used to receive:\\n\\n2 , , / / / (4 ) . e R R e T T c a s G s G n λ π = = ( 17\\n\\n)\\n\\nIf photons were collected only from one end of the receive antenna ( 1 c n = ), then Eq. ( 17 ) would be identical to conventional antenna theory.\\n\\nWith these definitions, the expression for the received power ( 11 ) can be interpreted as a Friis-like equation:\\n\\n( ) ( )( )\\n\\n2 2 2 0 , , , / / 4 / ( ) / in c T R a e T c T a e R P P n G G r s n s r λ π λ = = . (18) Here , c T n is the value used to compute , e T s according to Eq. (16). The ratio , , / e T c T s n is independent of the choice of , c T\\n\\nn , as it should be since the number of detectors that might be used on receive is independent of the transmit properties of the antenna.\\n\\nIt is also useful to note that if the magnetic field is uniform (i.e., wigglers, or quasi-phase matching, are not used [12] ), then there is an optimum length for the conversion region of an antenna. This occurs when ( )\\n\\nsin / 2 1 qL = , or qL π = . The optimum length is found to be 2 ( / ) opt a L m γ λ ω = . ( 19\\n\\n)\\n\\nIn obtaining this expression, we have used 2 /(2 ) a q m ω ≈ , which is valid for a m ω .\\n\\nThe diffraction-limited power pattern of the radiated axion field is determined by the aperture size in wavelengths through the Bessel function term in ( 7 ):\\n\\n( ) ( )\\n\\n2 1 4 sin /( sin ) d a a P J k R k R θ θ θ = ⎡ ⎤ ⎣ ⎦ . ( 20\\n\\n)\\n\\nThe diffraction beam width between first nulls is determined by the first zero of the Airy disc, and for small angles is given by the well-known expression\\n\\n1.22 / d FWFN T R γ θ λ ≈ . (21)\\n\\nSimilarly, the diffraction beam width at half maximum is determined by the roots of ( )\\n\\n) / 0.514 / d FWHM a T T R R γ θ πλ λ = ≈ . 1/ 2 d P θ = , or (1.616 /\\n\\nIn contrast, the conversion-limited power pattern is given by ( ) ( )\\n\\n2 sin cos / 2 ( ) 2 cos / 2 a c a k k L P k k L γ γ θ π θ θ ⎡ ⎤ ⎡ ⎤ - ⎣ ⎦ ⎢ ⎥ = ⎢ ⎥ - ⎣ ⎦ , (23)\\n\\nand depends on both the length in wavelengths and the velocity mismatch. For the optimum length L given by ( 19 ), the conversion beam widths are approximately given by 2( / ), 1.06( / )\\n\\nc c FWFN a FWHM a m m θ ω θ ω ≈ ≈ . ( 24\\n\\n)\\n\\nFor quantum-limited detection, the channel capacity is [16] log [17], and c is the velocity of light. Equation (25) can be combined with (11) or (18) to find the axion parameters that would permit a particular channel capacity at a given distance for a particular experimental apparatus.\\n\\n(1 / ) d R C N ν η ν = Δ + Δ , (25)\\n\\nAs an example, consider transmitters and receivers with 1064 nm (1.17 eV)\\n\\nγ λ = , 10 W in P = , 2 0.01 m s = , 0 10 T B = , 3 L = m, 5 3.1 10 F = × , 0.5 d η =\\n\\n, and a minimum information capacity of 1 bps. Figure 3 shows the inverse coupling strength curves\\n\\n( ) 1/ a a M m g γγ =\\n\\nfor distances of 1000 km and the diameter of the earth. Also shown is the curve for communication between the earth and the far side of the moon using the \\\"4+4\\\" experimental apparatus proposed in [15] . The shaded regions are excluded by the results from the BFRT collaboration [13] and Robilliard et al. [8] . The dots represent extensions of the curves if quasi-phase matching (QPM) is used by periodically reversing the magnetic field [12] . The minimum period of the reversal is taken to be twice the beam diameter for the 3 m system example, and 28.6 m for the 4+4 system. From the figure, communication over distances in excess of 1000 km should be possible for 3.5\\n\\na m < meV and 7 2 10 GeV M < × ( 8 1 5 10 GeV a g γγ -- Note that the receive resonator must be tuned to the same frequency as the transmit resonator to within a fraction of the resonator line width which is 161 Hz in this example. This will present a significant challenge, especially since the resonators are remote from one another. A possible approach would be to use atomic clocks to stabilize both the transmit frequency and a local reference frequency at the receiver. The receive resonator would then be locked to the reference to get as close as possible to the correct frequency, then slowly tuned until the signal is located. A feedback loop could then be closed to lock the receive resonator to the signal.\\n\\nIn summary, for 3.5 > × ), it may be possible to realize a new type of wireless signaling that cannot be blocked or shielded. An example calculation shows that communication between points located diametrically opposite on the earth should be possible. This could enable world-wide communication without the use of satellites or the ionosphere. However, with present knowledge, the signaling will be limited to low data rates, perhaps on the order of a few bits per second for terrestrial links. This estimate assumes 3 m long generation/regeneration regions to allow fully steerable instruments. Though not easily steerable, the apparatus in the \\\"4+4\\\" experiment proposed by Sikivie et al., [15] may enable communication to the far side of the moon for 0.3 a m < meV and 6 6 10 GeV M < × . I would like to acknowledge helpful discussions with Jim Lesh, Rich Holman, Jeff Peterson, Pierre Sikivie, and David Tanner during the development of these ideas. Electronic address: stancil@cmu.edu 17. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6 th Edition (Oxford Univ. Press, New York, 2007), pp. 171. 18. C-C. Chen, op cit., p. 106. B 0R L T L R Mod Laser Demod Data In Data Out B 0T Optical Detectors B 0T B 0R Photon input Photon output Axions penetrate barrier opaque to photons L T L R (a) Photons Axions Cavity mirrors (b) Figure 1, Stancil, Phys. Rev. D. x y z r θ -L T /2 L T /2\"\n }\n]","isConversational":false}}},{"rowIdx":89,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0495"},"arxiv_id":{"kind":"string","value":"0704.0495"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.3842/SIGMA.2007.075"},"content_hash":{"kind":"string","value":"1901032a0ef23b90804048b2aef0d82a6e8e98bc6f56ec72e49cd08936aac7b0"},"title":{"kind":"string","value":"The Veldkamp Space of Two-Qubits"},"abstract":{"kind":"string","value":"Given a remarkable representation of the generalized Pauli operators of\ntwo-qubits in terms of the points of the generalized quadrangle of order two,\nW(2), it is shown that specific subsets of these operators can also be\nassociated with the points and lines of the four-dimensional projective space\nover the Galois field with two elements - the so-called Veldkamp space of W(2).\nAn intriguing novelty is the recognition of (uni- and tri-centric) triads and\nspecific pentads of the Pauli operators in addition to the \"classical\" subsets\nanswering to geometric hyperplanes of W(2)."},"authors":{"kind":"list like","value":["Metod Saniga (ASTRINSTSAV)","Michel Planat (FEMTO-ST)","Petr Pracna\n (JH-Inst)","Hans Havlicek (TUW)"],"string":"[\n \"Metod Saniga (ASTRINSTSAV)\",\n \"Michel Planat (FEMTO-ST)\",\n \"Petr Pracna\\n (JH-Inst)\",\n \"Hans Havlicek (TUW)\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["quant-ph","math-ph","math.MP"],"string":"[\n \"quant-ph\",\n \"math-ph\",\n \"math.MP\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"A deeper understanding of the structure of Hilbert spaces of finite dimensions is of utmost importance for quantum information theory. Recently, we made an important step in this respect by demonstrating that the commutation algebra of the generalized Pauli operators on the 2 N -dimensional Hilbert spaces is embodied in the geometry of the symplectic polar space of rank N and order two [1, 2, 3] . The case of two-qubit operator space, N = 2, was scrutinized in very detail [1, 3] by explicitly demonstrating, in different ways, the correspondence between various subsets of the generalized Pauli operators/matrices and the fundamental subgeometries of the associated rank-two polar space -the (unique) generalized quadrangle of order two. In this paper we will reveal another interesting geometry hidden behind the Pauli operators of two-qubits, namely that of the Veldkamp space defined on this generalized quadrangle.\n\nIn this section we will briefly highlight the basics of the theory of finite generalized quadrangles [4] and introduce the concept of the Veldkamp space of a point-line incidence geometry [5] to be employed in what follows. 2 M. Saniga, M. Planat, P. Pracna and H. Havlicek A finite generalized quadrangle of order (s, t), usually denoted GQ(s, t), is an incidence structure S = (P, B, I), where P and B are disjoint (non-empty) sets of objects, called respectively points and lines, and where I is a symmetric point-line incidence relation satisfying the following axioms [4]: (i) each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point; and (iii) if x is a point and L is a line not incident with x, then there exists a unique pair (y, M ) ∈ P × B for which xIM IyIL; from these axioms it readily follows that |P | = (s + 1)(st + 1) and |B| = (t + 1)(st + 1). It is obvious that there exists a point-line duality with respect to which each of the axioms is self-dual. Interchanging points and lines in S thus yields a generalized quadrangle S D of order (t, s), called the dual of S. If s = t, S is said to have order s. The generalized quadrangle of order (s, 1) is called a grid and that of order (1, t) a dual grid. A generalized quadrangle with both s > 1 and t > 1 is called thick.\n\nGiven two points x and y of S one writes x ∼ y and says that x and y are collinear if there exists a line L of S incident with both. For any x ∈ P denote x ⊥ = {y ∈ P |y ∼ x} and note that x ∈ x ⊥ ; obviously, x ⊥ = 1 + s + st. Given an arbitrary subset A of P , the perp(-set) of A, A ⊥ , is defined as A ⊥ = {x ⊥ |x ∈ A} and A ⊥⊥ := (A ⊥ ) ⊥ . A triple of pairwise non-collinear points of S is called a triad; given any triad T , a point of T ⊥ is called its center and we say that T is acentric, centric or unicentric according as |T ⊥ | is, respectively, zero, non-zero or one. An ovoid of a generalized quadrangle S is a set of points of S such that each line of S is incident with exactly one point of the set; hence, each ovoid contains st + 1 points.\n\nThe concept of crucial importance is a geometric hyperplane H of a point-line geometry Γ(P, B), which is a proper subset of P such that each line of Γ meets H in one or all points [6] . For Γ = GQ(s, t), it is well known that H is one of the following three kinds: (i) the perp-set of a point x, x ⊥ ; (ii) a (full) subquadrangle of order (s, t ′ ), t ′ < t; and (iii) an ovoid.\n\nFinally, we need to introduce the notion of the Veldkamp space of a point-line incidence geometry Γ(P, B), V(Γ) [5] . V(Γ) is the space in which (i) a point is a geometric hyperplane of Γ and (ii) a line is the collection i = 1, 2) , where H 1 and H 2 are distinct points of V(Γ). 1 If Γ = S, from the preceding paragraph we learn that the points of V(S) are, in general, of three different types.\n\nH 1 H 2 of all geometric hyperplanes H of Γ such that H 1 H 2 = H 1 H = H 2 H or H = H i (\n\nThe smallest thick GQ is obviously the one with s = t = 2, dubbed the \"doily.\" This quadrangle has a number of interesting representations of which we mention the most important two [4]. One, frequently denoted as W 3 (2) or simply W (2), is in terms of the points of P G(3, 2) (i.e., the three-dimensional projective space over the Galois field with two elements) together with the totally isotropic lines with respect to a symplectic polarity. The other, usually denoted as Q(4, 2), is in terms of points and lines of a parabolic quadric in P G (4, 2) . By abuse of notation, any GQ isomorphic to W (2) will also be denoted by this symbol. From the preceding section we readily get that W (2) is endowed with 15 points/lines, each line contains three points and, dually, each point is on three lines; moreover, it is a self-dual object, i.e., isomorphic to its dual. W (2) features all the three kinds of hyperplanes, of the following cardinalities [5]: 15 perp-sets, x ⊥ , seven points each; 10 grids (of order (2, 1)), nine points each; and six ovoids, five points each -as depicted in Fig. 1 . The quadrangle exhibits two distinct kinds of triads, viz. unicentric and tricentric. A point of W (2) is the center of four distinct unicentric triads (Fig. 2 , left ); hence, 1 It is important to mention here that the definition of Veldkamp space given by Shult in [7] is more restrictive than that of Buekenhout and Cohen [5] adopted in this paper.\n\nThe Veldkamp Space of Two-Qubits 3 Figure 1. The three kinds of geometric hyperplanes of W (2). The points of the quadrangle are represented by small circles and its lines are illustrated by the straight segments as well as by the segments of circles; note that not every intersection of two segments counts for a point of the quadrangle. The upper panel shows the points' perp-sets (yellow bullets), the middle panel grids (red bullets) and the bottom panel ovoids (blue bullets); the use of different colouring will become clear later. Each pictureexcept that in the bottom right-hand corner -stands for five different hyperplanes, the four other being obtained from it by its successive rotations through 72 degrees around the center of the pentagon.\n\nFigure 2. Left: -The four distinct unicentric triads (grey bullets) and their common center (black bullet); note that the triads intersect pairwise in a single point and their union covers fully the center's perp-set. Right: -A grid (red bullets) and its complement as a disjoint union of two complementary tricentric triads (black and grey bullets); the two triads are also seen to comprise a dual grid (of order (1, 2)). 4 M. Saniga, M. Planat, P. Pracna and H. Havlicek Figure 3. The five different kinds of the lines of V(W (2)), each being uniquely determined by the properties of its core-set (black bullets). Note that the \"yellow\" hyperplanes (i.e., perp-sets) occur in each type, and yellow is also the colour of two homogeneous (i.e., endowed with only one kind of a hyperplane) types (2nd and 3rd row). It is also worth mentioning that the cardinality of core-sets is an odd number not exceeding five. The three hyperplanes of any line are always in such relation to each other that their union comprises all the points of W (2).\n\nThe Veldkamp Space of Two-Qubits 5 the number of such triads is 4 × 15 = 60. Tricentric triads always come in \"complementary\" pairs, one representing the centers of the other, and each such pair is the complement of a grid of W (2) (Fig. 2, right); hence, the number of such triads is 2 × 10 = 20. A unicentric triad is always a subset of an ovoid, which is never the case for a tricentric triad; the latter, in graphcombinatorial terms, representing a complete bipartite graph on six vertices. Now, we have enough background information at hand to reveal the structure of the Veldkamp space of our \"doily\", V(W (2)). 2\n\nFrom the definition given in Section 2, we easily see that V(W (2)) consists of 31 points of which 15 are represented/generated by single-point perp-sets, 10 by grids and six by ovoids. The lines of V(W (2)) feature three points each and are of five distinct types, as illustrated in Fig. 3. These types differ from each other in the cardinality and structure of \"core-sets\", i.e., the sets of points of W (2) shared by all the three hyperplanes forming a given line. As it is obvious from Fig. 3, the lines of the first three types (the first three rows of the figure) have the core-sets of the same cardinality, three, differing from each other only in the structure of these sets as being unicentric triads, tricentric triads and triples of collinear points, respectively. The lines of the fourth type have as core-sets pentads of points, each being a quadruple of points collinear with a given point of W (2), whereas core-sets of the last type's lines feature just a single point. A much more interesting issue is the composition of the lines. Just a brief look at Fig. 3 reveals that geometric hyperplanes of only one kind, namely perp-sets, are present on each line of V(W (2)); grids and ovoids occur only on two kinds of the lines. We also see that the purely homogeneous types are those whose core-sets feature collinear triples and tricentric triads, the most heterogeneous type -the one exhibiting all the three kinds of hyperplanes -being that characterized by unicentric triads. We also notice that there are no lines comprising solely grids and/or solely ovoids, nor the lines featuring only grids and ovoids, which seems to be connected with the fact that the cardinality of a core-set is an odd number. From the properties of W (2) and its triads as discussed above it readily follows that the number of the lines of type one to five is 60, 20, 15, 45 and 15, respectively, totalling 155. All these observations and facts are gathered in Table 1 . We conclude this section with the observation that V(W (2)) has the same number of points (31) and lines (155) as P G(4, 2), the four-dimensional projective space over the Galois field of two elements [8] ; this is not a coincidence, as the two spaces are, in fact, isomorphic to each other [5].\n\n4 Pauli operators of two-qubits in light of V(W (2))\n\nAs discovered in [1] (see also [3]), the fifteen generalized Pauli operators/matrices associated with the Hilbert space of two-qubits (see, e.g., [9]) can be put into a one-to-one correspondence with the fifteen points of the generalized quadrangle W (2) in such a way that their commutation algebra is completely and uniquely reproduced by the geometry of W (2) in which the concept commuting/non-commuting translates into that of collinear/non-collinear. Given this mapping, it was possible to ascribe a definitive geometrical meaning to sets of three pairwise commuting generalized Pauli operators in terms of lines of W (2) and to other three kinds of distinguished subsets of the operators having their counterparts in geometric hyperplanes of W (2) as shown in Table 2 (see [1, 3] for more details). Yet, V(W (2)) puts this bijection in a different light, in which other three subsets of the Pauli operators come into play, namely those represented by the two types of a triad and by the specific pentads occurring as the core-sets of the lines of V(W (2)) (Table 1 ). As already mentioned, the role of tricentric triads of the operators has 2 As this paper is primarily aimed at physicists rather than mathematicians, in what follows we opt for an elementary and self-contained exposition of the Veldkamp space of W (2); this explanation is based only upon some very simple properties of W (2) readily to be grasped from its depiction as \"the doily\", and does not presuppose/require any further background from the reader. 6 M. Saniga, M. Planat, P. Pracna and H. Havlicek Table 1. A succinct summary of the properties of the five different types of the lines of V(W (2)) in terms of the core-sets and the types of geometric hyperplanes featured by a generic line of a given type. The last column gives the total number of lines per the corresponding type.\n\nType of Core-Set Perp-Sets Grids Ovoids # Single Point 1 0 2 15 Collinear Triple 3 0 0 15 Unicentric Triad 1 1 1 60 Tricentric Triad 3 0 0 20 Pentad 1 2 0 45 Table 2. Three kinds of the distinguished subsets of the generalized Pauli operators of two-qubits (PO) viewed as the geometric hyperplanes in the generalized quadrangle of order two (GQ) [1, 3].\n\nPO set of five mutually set of six operators nine operators of non-commuting operators commuting with a given one a Mermin's square GQ ovoid perp-set\\{reference point} grid been recognized in disguise of complete bipartite graphs on six vertices [3]. A true novelty here is obviously unicentric triads and pentads of the generalized Pauli operators as these are all intimately connected with single-point perp-sets; given a point of W (2) (i.e., a generalized Pauli operator of two-qubits), its perp-set fully encompasses four unicentric triads (Fig. 2 , left) and three pentads (Fig. 3 , 4th row) of points/operators. This feature has also a very interesting aspect in connection with the conjecture relating the existence of mutually unbiased bases and finite projective planes raised in [10] , because with each point x of W (2) there is associated a projective plane of order two (the Fano plane) whose points are the elements of x ⊥ and whose lines are the spans {u,\n\nv} ⊥⊥ , where u, v ∈ x ⊥ with u = v [4].\n\nIdentifying the Pauli operators of a two-qubit system with the points of the generalized quadrangle of order two led to the discovery of three distinguished subsets of the operators in terms of geometric hyperplanes of the quadrangle. Here we go one level higher, and identifying these subsets with the points of the associated Veldkamp space leads to recognition of nother remarkable subsets of the Pauli operators, viz. unicentric triads and pentads. It is really intriguing to see that these are the core-sets of the two kinds of lines that both feature grids alias Mermin squares. As it is well known, Mermin squares, which reveal certain important aspects of the entanglement of the system, play a crucial role in the proof of the Kochen-Specker theorem in dimension four and our approach gives a novel geometrical meaning to this [3, 11] . At the Veldkamp space level it turns out of particular importance to study relations between eigenvectors of the above-mentioned unicentric triads and pentad of operators in order to reveal finer, hitherto unnoticed traits of the structure of Mermin squares. These seem to be intimately connected with the existence of outer automorphisms of the symmetric group on six letters, which is the full group of automorphisms of our quadrangle; as this group is the only symmetric group possessing (non-trivial) outer automorphisms, this implies that two-qubits have a rather special footing among multiple qubit systems. All these aspects deserve special attention and will therefore be dealt with in a separate paper.\n\nConcerning three-qubits, our preliminary study indicates that the corresponding finite geometry differs fundamentally from that of W (2) in the sense that it contains multi-lines, i.e., two or more lines passing through two distinct points [12] . As we do not have a full picture at hand yet, we cannot see if it admits hyperplanes and so lends itself to constructing the corresponding Veldkamp space. If the latter does exist, it is likely to differ substantially from that of twoqubits, which would imply the expected difference between entanglement properties of the two\n\nThe Veldkamp Space of Two-Qubits 7 kinds of systems; if not, this will only further strengthen the above-mentioned uniqueness of two-qubits.\n\nBy employing the concept of the Veldkamp space of the generalized quadrangle of order two, we were able to recognize other, on top of those examined in [1, 2, 3] , distinguished subsets of generalized Pauli operators of two-level quantum systems, namely unicentric triads and pentads of them. It may well be that these two kinds of subsets of the two-qubit Pauli operators hold an important key for getting deeper insights into the nature of finite geometries underlying multiple higher-level quantum systems [12, 13] , in particular when the dimension of Hilbert space is not a power of a prime [14].\n\nThis work was partially supported by the Science and Technology Assistance Agency under the contract # APVT-51-012704, the VEGA grant agency projects # 2/6070/26 and # 7012 (all from Slovak Republic), the trans-national ECO-NET project # 12651NJ \"Geometries Over Finite Rings and the Properties of Mutually Unbiased Bases\" (France), the CNRS-SAV Project # 20246 \"Projective and Related Geometries for Quantum Information\" (France/Slovakia) and by the Action Austria-Slovakia project # 58s2 \"Finite Geometries Behind Hilbert Spaces\"."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"Introduction","text":"A deeper understanding of the structure of Hilbert spaces of finite dimensions is of utmost importance for quantum information theory. Recently, we made an important step in this respect by demonstrating that the commutation algebra of the generalized Pauli operators on the 2 N -dimensional Hilbert spaces is embodied in the geometry of the symplectic polar space of rank N and order two [1, 2, 3] . The case of two-qubit operator space, N = 2, was scrutinized in very detail [1, 3] by explicitly demonstrating, in different ways, the correspondence between various subsets of the generalized Pauli operators/matrices and the fundamental subgeometries of the associated rank-two polar space -the (unique) generalized quadrangle of order two. In this paper we will reveal another interesting geometry hidden behind the Pauli operators of two-qubits, namely that of the Veldkamp space defined on this generalized quadrangle."},{"section_type":"OTHER","section_title":"Finite generalized quadrangles and Veldkamp spaces","text":"In this section we will briefly highlight the basics of the theory of finite generalized quadrangles [4] and introduce the concept of the Veldkamp space of a point-line incidence geometry [5] to be employed in what follows. 2 M. Saniga, M. Planat, P. Pracna and H. Havlicek A finite generalized quadrangle of order (s, t), usually denoted GQ(s, t), is an incidence structure S = (P, B, I), where P and B are disjoint (non-empty) sets of objects, called respectively points and lines, and where I is a symmetric point-line incidence relation satisfying the following axioms [4]: (i) each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point; and (iii) if x is a point and L is a line not incident with x, then there exists a unique pair (y, M ) ∈ P × B for which xIM IyIL; from these axioms it readily follows that |P | = (s + 1)(st + 1) and |B| = (t + 1)(st + 1). It is obvious that there exists a point-line duality with respect to which each of the axioms is self-dual. Interchanging points and lines in S thus yields a generalized quadrangle S D of order (t, s), called the dual of S. If s = t, S is said to have order s. The generalized quadrangle of order (s, 1) is called a grid and that of order (1, t) a dual grid. A generalized quadrangle with both s > 1 and t > 1 is called thick.\n\nGiven two points x and y of S one writes x ∼ y and says that x and y are collinear if there exists a line L of S incident with both. For any x ∈ P denote x ⊥ = {y ∈ P |y ∼ x} and note that x ∈ x ⊥ ; obviously, x ⊥ = 1 + s + st. Given an arbitrary subset A of P , the perp(-set) of A, A ⊥ , is defined as A ⊥ = {x ⊥ |x ∈ A} and A ⊥⊥ := (A ⊥ ) ⊥ . A triple of pairwise non-collinear points of S is called a triad; given any triad T , a point of T ⊥ is called its center and we say that T is acentric, centric or unicentric according as |T ⊥ | is, respectively, zero, non-zero or one. An ovoid of a generalized quadrangle S is a set of points of S such that each line of S is incident with exactly one point of the set; hence, each ovoid contains st + 1 points.\n\nThe concept of crucial importance is a geometric hyperplane H of a point-line geometry Γ(P, B), which is a proper subset of P such that each line of Γ meets H in one or all points [6] . For Γ = GQ(s, t), it is well known that H is one of the following three kinds: (i) the perp-set of a point x, x ⊥ ; (ii) a (full) subquadrangle of order (s, t ′ ), t ′ < t; and (iii) an ovoid.\n\nFinally, we need to introduce the notion of the Veldkamp space of a point-line incidence geometry Γ(P, B), V(Γ) [5] . V(Γ) is the space in which (i) a point is a geometric hyperplane of Γ and (ii) a line is the collection i = 1, 2) , where H 1 and H 2 are distinct points of V(Γ). 1 If Γ = S, from the preceding paragraph we learn that the points of V(S) are, in general, of three different types.\n\nH 1 H 2 of all geometric hyperplanes H of Γ such that H 1 H 2 = H 1 H = H 2 H or H = H i ("},{"section_type":"OTHER","section_title":"The smallest thick GQ and its Veldkamp space","text":"The smallest thick GQ is obviously the one with s = t = 2, dubbed the \"doily.\" This quadrangle has a number of interesting representations of which we mention the most important two [4]. One, frequently denoted as W 3 (2) or simply W (2), is in terms of the points of P G(3, 2) (i.e., the three-dimensional projective space over the Galois field with two elements) together with the totally isotropic lines with respect to a symplectic polarity. The other, usually denoted as Q(4, 2), is in terms of points and lines of a parabolic quadric in P G (4, 2) . By abuse of notation, any GQ isomorphic to W (2) will also be denoted by this symbol. From the preceding section we readily get that W (2) is endowed with 15 points/lines, each line contains three points and, dually, each point is on three lines; moreover, it is a self-dual object, i.e., isomorphic to its dual. W (2) features all the three kinds of hyperplanes, of the following cardinalities [5]: 15 perp-sets, x ⊥ , seven points each; 10 grids (of order (2, 1)), nine points each; and six ovoids, five points each -as depicted in Fig. 1 . The quadrangle exhibits two distinct kinds of triads, viz. unicentric and tricentric. A point of W (2) is the center of four distinct unicentric triads (Fig. 2 , left ); hence, 1 It is important to mention here that the definition of Veldkamp space given by Shult in [7] is more restrictive than that of Buekenhout and Cohen [5] adopted in this paper.\n\nThe Veldkamp Space of Two-Qubits 3 Figure 1. The three kinds of geometric hyperplanes of W (2). The points of the quadrangle are represented by small circles and its lines are illustrated by the straight segments as well as by the segments of circles; note that not every intersection of two segments counts for a point of the quadrangle. The upper panel shows the points' perp-sets (yellow bullets), the middle panel grids (red bullets) and the bottom panel ovoids (blue bullets); the use of different colouring will become clear later. Each pictureexcept that in the bottom right-hand corner -stands for five different hyperplanes, the four other being obtained from it by its successive rotations through 72 degrees around the center of the pentagon.\n\nFigure 2. Left: -The four distinct unicentric triads (grey bullets) and their common center (black bullet); note that the triads intersect pairwise in a single point and their union covers fully the center's perp-set. Right: -A grid (red bullets) and its complement as a disjoint union of two complementary tricentric triads (black and grey bullets); the two triads are also seen to comprise a dual grid (of order (1, 2)). 4 M. Saniga, M. Planat, P. Pracna and H. Havlicek Figure 3. The five different kinds of the lines of V(W (2)), each being uniquely determined by the properties of its core-set (black bullets). Note that the \"yellow\" hyperplanes (i.e., perp-sets) occur in each type, and yellow is also the colour of two homogeneous (i.e., endowed with only one kind of a hyperplane) types (2nd and 3rd row). It is also worth mentioning that the cardinality of core-sets is an odd number not exceeding five. The three hyperplanes of any line are always in such relation to each other that their union comprises all the points of W (2).\n\nThe Veldkamp Space of Two-Qubits 5 the number of such triads is 4 × 15 = 60. Tricentric triads always come in \"complementary\" pairs, one representing the centers of the other, and each such pair is the complement of a grid of W (2) (Fig. 2, right); hence, the number of such triads is 2 × 10 = 20. A unicentric triad is always a subset of an ovoid, which is never the case for a tricentric triad; the latter, in graphcombinatorial terms, representing a complete bipartite graph on six vertices. Now, we have enough background information at hand to reveal the structure of the Veldkamp space of our \"doily\", V(W (2)). 2\n\nFrom the definition given in Section 2, we easily see that V(W (2)) consists of 31 points of which 15 are represented/generated by single-point perp-sets, 10 by grids and six by ovoids. The lines of V(W (2)) feature three points each and are of five distinct types, as illustrated in Fig. 3. These types differ from each other in the cardinality and structure of \"core-sets\", i.e., the sets of points of W (2) shared by all the three hyperplanes forming a given line. As it is obvious from Fig. 3, the lines of the first three types (the first three rows of the figure) have the core-sets of the same cardinality, three, differing from each other only in the structure of these sets as being unicentric triads, tricentric triads and triples of collinear points, respectively. The lines of the fourth type have as core-sets pentads of points, each being a quadruple of points collinear with a given point of W (2), whereas core-sets of the last type's lines feature just a single point. A much more interesting issue is the composition of the lines. Just a brief look at Fig. 3 reveals that geometric hyperplanes of only one kind, namely perp-sets, are present on each line of V(W (2)); grids and ovoids occur only on two kinds of the lines. We also see that the purely homogeneous types are those whose core-sets feature collinear triples and tricentric triads, the most heterogeneous type -the one exhibiting all the three kinds of hyperplanes -being that characterized by unicentric triads. We also notice that there are no lines comprising solely grids and/or solely ovoids, nor the lines featuring only grids and ovoids, which seems to be connected with the fact that the cardinality of a core-set is an odd number. From the properties of W (2) and its triads as discussed above it readily follows that the number of the lines of type one to five is 60, 20, 15, 45 and 15, respectively, totalling 155. All these observations and facts are gathered in Table 1 . We conclude this section with the observation that V(W (2)) has the same number of points (31) and lines (155) as P G(4, 2), the four-dimensional projective space over the Galois field of two elements [8] ; this is not a coincidence, as the two spaces are, in fact, isomorphic to each other [5].\n\n4 Pauli operators of two-qubits in light of V(W (2))\n\nAs discovered in [1] (see also [3]), the fifteen generalized Pauli operators/matrices associated with the Hilbert space of two-qubits (see, e.g., [9]) can be put into a one-to-one correspondence with the fifteen points of the generalized quadrangle W (2) in such a way that their commutation algebra is completely and uniquely reproduced by the geometry of W (2) in which the concept commuting/non-commuting translates into that of collinear/non-collinear. Given this mapping, it was possible to ascribe a definitive geometrical meaning to sets of three pairwise commuting generalized Pauli operators in terms of lines of W (2) and to other three kinds of distinguished subsets of the operators having their counterparts in geometric hyperplanes of W (2) as shown in Table 2 (see [1, 3] for more details). Yet, V(W (2)) puts this bijection in a different light, in which other three subsets of the Pauli operators come into play, namely those represented by the two types of a triad and by the specific pentads occurring as the core-sets of the lines of V(W (2)) (Table 1 ). As already mentioned, the role of tricentric triads of the operators has 2 As this paper is primarily aimed at physicists rather than mathematicians, in what follows we opt for an elementary and self-contained exposition of the Veldkamp space of W (2); this explanation is based only upon some very simple properties of W (2) readily to be grasped from its depiction as \"the doily\", and does not presuppose/require any further background from the reader. 6 M. Saniga, M. Planat, P. Pracna and H. Havlicek Table 1. A succinct summary of the properties of the five different types of the lines of V(W (2)) in terms of the core-sets and the types of geometric hyperplanes featured by a generic line of a given type. The last column gives the total number of lines per the corresponding type.\n\nType of Core-Set Perp-Sets Grids Ovoids # Single Point 1 0 2 15 Collinear Triple 3 0 0 15 Unicentric Triad 1 1 1 60 Tricentric Triad 3 0 0 20 Pentad 1 2 0 45 Table 2. Three kinds of the distinguished subsets of the generalized Pauli operators of two-qubits (PO) viewed as the geometric hyperplanes in the generalized quadrangle of order two (GQ) [1, 3].\n\nPO set of five mutually set of six operators nine operators of non-commuting operators commuting with a given one a Mermin's square GQ ovoid perp-set\\{reference point} grid been recognized in disguise of complete bipartite graphs on six vertices [3]. A true novelty here is obviously unicentric triads and pentads of the generalized Pauli operators as these are all intimately connected with single-point perp-sets; given a point of W (2) (i.e., a generalized Pauli operator of two-qubits), its perp-set fully encompasses four unicentric triads (Fig. 2 , left) and three pentads (Fig. 3 , 4th row) of points/operators. This feature has also a very interesting aspect in connection with the conjecture relating the existence of mutually unbiased bases and finite projective planes raised in [10] , because with each point x of W (2) there is associated a projective plane of order two (the Fano plane) whose points are the elements of x ⊥ and whose lines are the spans {u,\n\nv} ⊥⊥ , where u, v ∈ x ⊥ with u = v [4].\n\nIdentifying the Pauli operators of a two-qubit system with the points of the generalized quadrangle of order two led to the discovery of three distinguished subsets of the operators in terms of geometric hyperplanes of the quadrangle. Here we go one level higher, and identifying these subsets with the points of the associated Veldkamp space leads to recognition of nother remarkable subsets of the Pauli operators, viz. unicentric triads and pentads. It is really intriguing to see that these are the core-sets of the two kinds of lines that both feature grids alias Mermin squares. As it is well known, Mermin squares, which reveal certain important aspects of the entanglement of the system, play a crucial role in the proof of the Kochen-Specker theorem in dimension four and our approach gives a novel geometrical meaning to this [3, 11] . At the Veldkamp space level it turns out of particular importance to study relations between eigenvectors of the above-mentioned unicentric triads and pentad of operators in order to reveal finer, hitherto unnoticed traits of the structure of Mermin squares. These seem to be intimately connected with the existence of outer automorphisms of the symmetric group on six letters, which is the full group of automorphisms of our quadrangle; as this group is the only symmetric group possessing (non-trivial) outer automorphisms, this implies that two-qubits have a rather special footing among multiple qubit systems. All these aspects deserve special attention and will therefore be dealt with in a separate paper.\n\nConcerning three-qubits, our preliminary study indicates that the corresponding finite geometry differs fundamentally from that of W (2) in the sense that it contains multi-lines, i.e., two or more lines passing through two distinct points [12] . As we do not have a full picture at hand yet, we cannot see if it admits hyperplanes and so lends itself to constructing the corresponding Veldkamp space. If the latter does exist, it is likely to differ substantially from that of twoqubits, which would imply the expected difference between entanglement properties of the two\n\nThe Veldkamp Space of Two-Qubits 7 kinds of systems; if not, this will only further strengthen the above-mentioned uniqueness of two-qubits."},{"section_type":"CONCLUSION","section_title":"Conclusion","text":"By employing the concept of the Veldkamp space of the generalized quadrangle of order two, we were able to recognize other, on top of those examined in [1, 2, 3] , distinguished subsets of generalized Pauli operators of two-level quantum systems, namely unicentric triads and pentads of them. It may well be that these two kinds of subsets of the two-qubit Pauli operators hold an important key for getting deeper insights into the nature of finite geometries underlying multiple higher-level quantum systems [12, 13] , in particular when the dimension of Hilbert space is not a power of a prime [14]."},{"section_type":"OTHER","section_title":"Acknowledgements","text":"This work was partially supported by the Science and Technology Assistance Agency under the contract # APVT-51-012704, the VEGA grant agency projects # 2/6070/26 and # 7012 (all from Slovak Republic), the trans-national ECO-NET project # 12651NJ \"Geometries Over Finite Rings and the Properties of Mutually Unbiased Bases\" (France), the CNRS-SAV Project # 20246 \"Projective and Related Geometries for Quantum Information\" (France/Slovakia) and by the Action Austria-Slovakia project # 58s2 \"Finite Geometries Behind Hilbert Spaces\"."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction\",\n \"text\": \"A deeper understanding of the structure of Hilbert spaces of finite dimensions is of utmost importance for quantum information theory. Recently, we made an important step in this respect by demonstrating that the commutation algebra of the generalized Pauli operators on the 2 N -dimensional Hilbert spaces is embodied in the geometry of the symplectic polar space of rank N and order two [1, 2, 3] . The case of two-qubit operator space, N = 2, was scrutinized in very detail [1, 3] by explicitly demonstrating, in different ways, the correspondence between various subsets of the generalized Pauli operators/matrices and the fundamental subgeometries of the associated rank-two polar space -the (unique) generalized quadrangle of order two. In this paper we will reveal another interesting geometry hidden behind the Pauli operators of two-qubits, namely that of the Veldkamp space defined on this generalized quadrangle.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Finite generalized quadrangles and Veldkamp spaces\",\n \"text\": \"In this section we will briefly highlight the basics of the theory of finite generalized quadrangles [4] and introduce the concept of the Veldkamp space of a point-line incidence geometry [5] to be employed in what follows. 2 M. Saniga, M. Planat, P. Pracna and H. Havlicek A finite generalized quadrangle of order (s, t), usually denoted GQ(s, t), is an incidence structure S = (P, B, I), where P and B are disjoint (non-empty) sets of objects, called respectively points and lines, and where I is a symmetric point-line incidence relation satisfying the following axioms [4]: (i) each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point; and (iii) if x is a point and L is a line not incident with x, then there exists a unique pair (y, M ) ∈ P × B for which xIM IyIL; from these axioms it readily follows that |P | = (s + 1)(st + 1) and |B| = (t + 1)(st + 1). It is obvious that there exists a point-line duality with respect to which each of the axioms is self-dual. Interchanging points and lines in S thus yields a generalized quadrangle S D of order (t, s), called the dual of S. If s = t, S is said to have order s. The generalized quadrangle of order (s, 1) is called a grid and that of order (1, t) a dual grid. A generalized quadrangle with both s > 1 and t > 1 is called thick.\\n\\nGiven two points x and y of S one writes x ∼ y and says that x and y are collinear if there exists a line L of S incident with both. For any x ∈ P denote x ⊥ = {y ∈ P |y ∼ x} and note that x ∈ x ⊥ ; obviously, x ⊥ = 1 + s + st. Given an arbitrary subset A of P , the perp(-set) of A, A ⊥ , is defined as A ⊥ = {x ⊥ |x ∈ A} and A ⊥⊥ := (A ⊥ ) ⊥ . A triple of pairwise non-collinear points of S is called a triad; given any triad T , a point of T ⊥ is called its center and we say that T is acentric, centric or unicentric according as |T ⊥ | is, respectively, zero, non-zero or one. An ovoid of a generalized quadrangle S is a set of points of S such that each line of S is incident with exactly one point of the set; hence, each ovoid contains st + 1 points.\\n\\nThe concept of crucial importance is a geometric hyperplane H of a point-line geometry Γ(P, B), which is a proper subset of P such that each line of Γ meets H in one or all points [6] . For Γ = GQ(s, t), it is well known that H is one of the following three kinds: (i) the perp-set of a point x, x ⊥ ; (ii) a (full) subquadrangle of order (s, t ′ ), t ′ < t; and (iii) an ovoid.\\n\\nFinally, we need to introduce the notion of the Veldkamp space of a point-line incidence geometry Γ(P, B), V(Γ) [5] . V(Γ) is the space in which (i) a point is a geometric hyperplane of Γ and (ii) a line is the collection i = 1, 2) , where H 1 and H 2 are distinct points of V(Γ). 1 If Γ = S, from the preceding paragraph we learn that the points of V(S) are, in general, of three different types.\\n\\nH 1 H 2 of all geometric hyperplanes H of Γ such that H 1 H 2 = H 1 H = H 2 H or H = H i (\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The smallest thick GQ and its Veldkamp space\",\n \"text\": \"The smallest thick GQ is obviously the one with s = t = 2, dubbed the \\\"doily.\\\" This quadrangle has a number of interesting representations of which we mention the most important two [4]. One, frequently denoted as W 3 (2) or simply W (2), is in terms of the points of P G(3, 2) (i.e., the three-dimensional projective space over the Galois field with two elements) together with the totally isotropic lines with respect to a symplectic polarity. The other, usually denoted as Q(4, 2), is in terms of points and lines of a parabolic quadric in P G (4, 2) . By abuse of notation, any GQ isomorphic to W (2) will also be denoted by this symbol. From the preceding section we readily get that W (2) is endowed with 15 points/lines, each line contains three points and, dually, each point is on three lines; moreover, it is a self-dual object, i.e., isomorphic to its dual. W (2) features all the three kinds of hyperplanes, of the following cardinalities [5]: 15 perp-sets, x ⊥ , seven points each; 10 grids (of order (2, 1)), nine points each; and six ovoids, five points each -as depicted in Fig. 1 . The quadrangle exhibits two distinct kinds of triads, viz. unicentric and tricentric. A point of W (2) is the center of four distinct unicentric triads (Fig. 2 , left ); hence, 1 It is important to mention here that the definition of Veldkamp space given by Shult in [7] is more restrictive than that of Buekenhout and Cohen [5] adopted in this paper.\\n\\nThe Veldkamp Space of Two-Qubits 3 Figure 1. The three kinds of geometric hyperplanes of W (2). The points of the quadrangle are represented by small circles and its lines are illustrated by the straight segments as well as by the segments of circles; note that not every intersection of two segments counts for a point of the quadrangle. The upper panel shows the points' perp-sets (yellow bullets), the middle panel grids (red bullets) and the bottom panel ovoids (blue bullets); the use of different colouring will become clear later. Each pictureexcept that in the bottom right-hand corner -stands for five different hyperplanes, the four other being obtained from it by its successive rotations through 72 degrees around the center of the pentagon.\\n\\nFigure 2. Left: -The four distinct unicentric triads (grey bullets) and their common center (black bullet); note that the triads intersect pairwise in a single point and their union covers fully the center's perp-set. Right: -A grid (red bullets) and its complement as a disjoint union of two complementary tricentric triads (black and grey bullets); the two triads are also seen to comprise a dual grid (of order (1, 2)). 4 M. Saniga, M. Planat, P. Pracna and H. Havlicek Figure 3. The five different kinds of the lines of V(W (2)), each being uniquely determined by the properties of its core-set (black bullets). Note that the \\\"yellow\\\" hyperplanes (i.e., perp-sets) occur in each type, and yellow is also the colour of two homogeneous (i.e., endowed with only one kind of a hyperplane) types (2nd and 3rd row). It is also worth mentioning that the cardinality of core-sets is an odd number not exceeding five. The three hyperplanes of any line are always in such relation to each other that their union comprises all the points of W (2).\\n\\nThe Veldkamp Space of Two-Qubits 5 the number of such triads is 4 × 15 = 60. Tricentric triads always come in \\\"complementary\\\" pairs, one representing the centers of the other, and each such pair is the complement of a grid of W (2) (Fig. 2, right); hence, the number of such triads is 2 × 10 = 20. A unicentric triad is always a subset of an ovoid, which is never the case for a tricentric triad; the latter, in graphcombinatorial terms, representing a complete bipartite graph on six vertices. Now, we have enough background information at hand to reveal the structure of the Veldkamp space of our \\\"doily\\\", V(W (2)). 2\\n\\nFrom the definition given in Section 2, we easily see that V(W (2)) consists of 31 points of which 15 are represented/generated by single-point perp-sets, 10 by grids and six by ovoids. The lines of V(W (2)) feature three points each and are of five distinct types, as illustrated in Fig. 3. These types differ from each other in the cardinality and structure of \\\"core-sets\\\", i.e., the sets of points of W (2) shared by all the three hyperplanes forming a given line. As it is obvious from Fig. 3, the lines of the first three types (the first three rows of the figure) have the core-sets of the same cardinality, three, differing from each other only in the structure of these sets as being unicentric triads, tricentric triads and triples of collinear points, respectively. The lines of the fourth type have as core-sets pentads of points, each being a quadruple of points collinear with a given point of W (2), whereas core-sets of the last type's lines feature just a single point. A much more interesting issue is the composition of the lines. Just a brief look at Fig. 3 reveals that geometric hyperplanes of only one kind, namely perp-sets, are present on each line of V(W (2)); grids and ovoids occur only on two kinds of the lines. We also see that the purely homogeneous types are those whose core-sets feature collinear triples and tricentric triads, the most heterogeneous type -the one exhibiting all the three kinds of hyperplanes -being that characterized by unicentric triads. We also notice that there are no lines comprising solely grids and/or solely ovoids, nor the lines featuring only grids and ovoids, which seems to be connected with the fact that the cardinality of a core-set is an odd number. From the properties of W (2) and its triads as discussed above it readily follows that the number of the lines of type one to five is 60, 20, 15, 45 and 15, respectively, totalling 155. All these observations and facts are gathered in Table 1 . We conclude this section with the observation that V(W (2)) has the same number of points (31) and lines (155) as P G(4, 2), the four-dimensional projective space over the Galois field of two elements [8] ; this is not a coincidence, as the two spaces are, in fact, isomorphic to each other [5].\\n\\n4 Pauli operators of two-qubits in light of V(W (2))\\n\\nAs discovered in [1] (see also [3]), the fifteen generalized Pauli operators/matrices associated with the Hilbert space of two-qubits (see, e.g., [9]) can be put into a one-to-one correspondence with the fifteen points of the generalized quadrangle W (2) in such a way that their commutation algebra is completely and uniquely reproduced by the geometry of W (2) in which the concept commuting/non-commuting translates into that of collinear/non-collinear. Given this mapping, it was possible to ascribe a definitive geometrical meaning to sets of three pairwise commuting generalized Pauli operators in terms of lines of W (2) and to other three kinds of distinguished subsets of the operators having their counterparts in geometric hyperplanes of W (2) as shown in Table 2 (see [1, 3] for more details). Yet, V(W (2)) puts this bijection in a different light, in which other three subsets of the Pauli operators come into play, namely those represented by the two types of a triad and by the specific pentads occurring as the core-sets of the lines of V(W (2)) (Table 1 ). As already mentioned, the role of tricentric triads of the operators has 2 As this paper is primarily aimed at physicists rather than mathematicians, in what follows we opt for an elementary and self-contained exposition of the Veldkamp space of W (2); this explanation is based only upon some very simple properties of W (2) readily to be grasped from its depiction as \\\"the doily\\\", and does not presuppose/require any further background from the reader. 6 M. Saniga, M. Planat, P. Pracna and H. Havlicek Table 1. A succinct summary of the properties of the five different types of the lines of V(W (2)) in terms of the core-sets and the types of geometric hyperplanes featured by a generic line of a given type. The last column gives the total number of lines per the corresponding type.\\n\\nType of Core-Set Perp-Sets Grids Ovoids # Single Point 1 0 2 15 Collinear Triple 3 0 0 15 Unicentric Triad 1 1 1 60 Tricentric Triad 3 0 0 20 Pentad 1 2 0 45 Table 2. Three kinds of the distinguished subsets of the generalized Pauli operators of two-qubits (PO) viewed as the geometric hyperplanes in the generalized quadrangle of order two (GQ) [1, 3].\\n\\nPO set of five mutually set of six operators nine operators of non-commuting operators commuting with a given one a Mermin's square GQ ovoid perp-set\\\\{reference point} grid been recognized in disguise of complete bipartite graphs on six vertices [3]. A true novelty here is obviously unicentric triads and pentads of the generalized Pauli operators as these are all intimately connected with single-point perp-sets; given a point of W (2) (i.e., a generalized Pauli operator of two-qubits), its perp-set fully encompasses four unicentric triads (Fig. 2 , left) and three pentads (Fig. 3 , 4th row) of points/operators. This feature has also a very interesting aspect in connection with the conjecture relating the existence of mutually unbiased bases and finite projective planes raised in [10] , because with each point x of W (2) there is associated a projective plane of order two (the Fano plane) whose points are the elements of x ⊥ and whose lines are the spans {u,\\n\\nv} ⊥⊥ , where u, v ∈ x ⊥ with u = v [4].\\n\\nIdentifying the Pauli operators of a two-qubit system with the points of the generalized quadrangle of order two led to the discovery of three distinguished subsets of the operators in terms of geometric hyperplanes of the quadrangle. Here we go one level higher, and identifying these subsets with the points of the associated Veldkamp space leads to recognition of nother remarkable subsets of the Pauli operators, viz. unicentric triads and pentads. It is really intriguing to see that these are the core-sets of the two kinds of lines that both feature grids alias Mermin squares. As it is well known, Mermin squares, which reveal certain important aspects of the entanglement of the system, play a crucial role in the proof of the Kochen-Specker theorem in dimension four and our approach gives a novel geometrical meaning to this [3, 11] . At the Veldkamp space level it turns out of particular importance to study relations between eigenvectors of the above-mentioned unicentric triads and pentad of operators in order to reveal finer, hitherto unnoticed traits of the structure of Mermin squares. These seem to be intimately connected with the existence of outer automorphisms of the symmetric group on six letters, which is the full group of automorphisms of our quadrangle; as this group is the only symmetric group possessing (non-trivial) outer automorphisms, this implies that two-qubits have a rather special footing among multiple qubit systems. All these aspects deserve special attention and will therefore be dealt with in a separate paper.\\n\\nConcerning three-qubits, our preliminary study indicates that the corresponding finite geometry differs fundamentally from that of W (2) in the sense that it contains multi-lines, i.e., two or more lines passing through two distinct points [12] . As we do not have a full picture at hand yet, we cannot see if it admits hyperplanes and so lends itself to constructing the corresponding Veldkamp space. If the latter does exist, it is likely to differ substantially from that of twoqubits, which would imply the expected difference between entanglement properties of the two\\n\\nThe Veldkamp Space of Two-Qubits 7 kinds of systems; if not, this will only further strengthen the above-mentioned uniqueness of two-qubits.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"Conclusion\",\n \"text\": \"By employing the concept of the Veldkamp space of the generalized quadrangle of order two, we were able to recognize other, on top of those examined in [1, 2, 3] , distinguished subsets of generalized Pauli operators of two-level quantum systems, namely unicentric triads and pentads of them. It may well be that these two kinds of subsets of the two-qubit Pauli operators hold an important key for getting deeper insights into the nature of finite geometries underlying multiple higher-level quantum systems [12, 13] , in particular when the dimension of Hilbert space is not a power of a prime [14].\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgements\",\n \"text\": \"This work was partially supported by the Science and Technology Assistance Agency under the contract # APVT-51-012704, the VEGA grant agency projects # 2/6070/26 and # 7012 (all from Slovak Republic), the trans-national ECO-NET project # 12651NJ \\\"Geometries Over Finite Rings and the Properties of Mutually Unbiased Bases\\\" (France), the CNRS-SAV Project # 20246 \\\"Projective and Related Geometries for Quantum Information\\\" (France/Slovakia) and by the Action Austria-Slovakia project # 58s2 \\\"Finite Geometries Behind Hilbert Spaces\\\".\"\n }\n]","isConversational":false}}},{"rowIdx":90,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0505"},"arxiv_id":{"kind":"string","value":"0704.0505"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevD.76.085001"},"content_hash":{"kind":"string","value":"aeda426d1071e75e895afdc97caa1fa9e76af03cbfa583d2180a72529e4ba2a9"},"title":{"kind":"string","value":"Exact Solutions of Einstein-Yang-Mills Theory with Higher-Derivative\n Coupling"},"abstract":{"kind":"string","value":"We construct a classical solution of an Einstein-Yang-Mills system with a\nfourth order term with respect to the field strength of the Yang-Mills field.\nThe solution provides a spontaneous compactification proposed by Cremmer and\nScherk; ten-dimensional space-time with a cosmological constant is compactified\nto the four-dimensional Minkowski space with a six-dimensional sphere S^6 on\nwhich an instanton solution exists. The radius of the sphere is not a modulus\nbut is determined by the gauge coupling and the four-derivative coupling\nconstants and the Newton's constant. We also construct a solution of\nten-dimensional theory without a cosmological constant compactified to AdS_4 x\nS^6."},"authors":{"kind":"list like","value":["Hironobu Kihara","Muneto Nitta"],"string":"[\n \"Hironobu Kihara\",\n \"Muneto Nitta\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["hep-th","gr-qc"],"string":"[\n \"hep-th\",\n \"gr-qc\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"We construct a classical solution of an Einstein-Yang-Mills system with a fourth order term with respect to the field strength of the Yang-Mills field. The solution provides a compactification proposed by Cremmer and Scherk; ten-dimensional space-time with a cosmological constant is compactified to the four-dimensional Minkowski space with a six-dimensional sphere S 6 on which an instanton solution exists. The radius of the sphere is not a modulus but is determined by the gauge coupling and the four-derivative coupling constants and the Newton's constant. We also construct a solution of ten-dimensional theory without a cosmological constant compactified to AdS 4 × S 6 .\n\nUnification of fundamental forces with space-time and matter often requires higherdimensional space-time rather than our four-dimensional Universe. The early Kaluza-Klein theory unifies gravity and the electro-magnetic interaction by considering five-dimensional space-time with one direction compactified into a circle S 1 [1] . This old idea has been revisited several times. After supergravity was discovered many people tried to unify all forces and matter in higher-dimensional space-time compactified on various internal manifolds [2] .\n\nString theory was proposed as the most attractive candidate of unification, but it is defined only in ten-dimensional space-time. In order to realize four-dimensional Universe one has to find a suitable six-dimensional internal space. So many candidates of such spaces were proposed; Calabi-Yau manifolds and orbifold models. Internal manifolds can be deformed with satisfying the Einstein equation, and these degrees of freedom are called the moduli.\n\nThe moduli introduce unwanted massless particles in four-dimensional world. Recently a new mechanism has been suggested to fix these moduli by turning on the Ramond-Ramond flux on the internal space [3, 4] . This flux compactification has been extensively studied in these years.\n\nWe would like to revise the compactification scenario with fixed moduli proposed by Cremmer and Scherk long time ago [5] (see also [6, 7] ) in a theory with a cosmological constant. By placing solitons on a compact internal space they showed decompactifying limit with large radius of the internal space is disfavored and the radius is fixed to a certain value determined by coupling constants. They considered the 't Hooft-Polyakov monopole [8] on S 2 and the Yang-Mills instanton [9] on S 4 , both of which can satisfy, with proper coupling constants, the first order (self-dual) equations rather than the second order equations of motion, but their solutions on higher dimensional sphere are not the case. Since string theory is defined in ten dimensions, it is natural to consider this scenario with stable BPS solitons on a six-dimensional internal space like S 6 .\n\nHigher dimensional generalization of self-dual equations was suggested by Tchrakian some years ago [10] . Eight dimensional case is known as octonionic instantons [11] . Though several works have been done for generalized self-dual equations [12, 13] , a six-dimensional case was not discussed because of the lack of conformal property. Recently we have found a new solution to the generalized self-dual equations in an SO (6) pure Yang-Mills theory with a fourth order term with respect to the field strength of the Yang-Mills field (a four-derivative term) on a six-dimensional sphere S 6 [14] .\n\nIn this letter we propose to use this solution in the context of a compactification of the Cremmer-Scherk type. In our model ten-dimensional space-time with (without) a cosmological constant is compactified to a four-dimensional Minkowski space M 4 (anti de Sitter space AdS 4 ) with a six-dimensional sphere S 6 , where dimensionality of the internal space, six, is required by the four derivative term. Unlike the case of the absence of gravity [14] the four-derivative coupling constant α can differ from the constant β in the generalized self-dual equations. When the relation α = β holds the generalized self-dual equations become the Bogomol'nyi equations and solutions are BPS. We find for both M 4 × S 6 and AdS 4 × S 6 that certain relations exist between the radius of S 6 , the gauge coupling, the four-derivative coupling α and the gravitational coupling constants. When the four-derivative coupling constant α vanishes in the case of M 4 × S 6 , these relations reduce to those of the original work by Cremmer and Scherk. The advantage of our model to the Cremmer-Scherk model is that the Yang-Mills soliton in our model satisfies the self-dual equations (the Bogomol'nyi equations for α = β) rather than usual equations of motion in the case of the Cremmer-Scherk model. This ensures the stability of configuration at least for the sector of Yang-Mills fields.\n\nLet us consider that space-time is a ten-dimensional manifold. We consider an Einstein-Yang-Mills theory. Our action contains as dynamical variables the Yang-Mills (gauge) fields\n\nA [ab] μ\n\nand a graviton field or the metric ĝμν . Indices with a hat \"ˆ\" will refer to a tendimensional space-time (X 0 , X 1 , • • • , X 9 ). Latin indices (a, b, • • • ) run from 1 to 6 and refer to an internal space. The Clifford algebra associated with the orthogonal group SO( 6 ) is useful and we represent generators of the Lie algebra so(6) as their elements. The Clifford algebra is defined by gamma matrices {Γ a } which satisfy the following anti-commutation relations, {Γ a , Γ b } = 2δ ab . These matrices can be realized as 8 × 8 matrices with complex coefficients. The generators of so( 6 ) are represented by\n\nΓ ab = 1 2 [Γ a , Γ b ].\n\nWe often abbreviate the Yang-Mills fields as\n\nA μ = 1 2 A [ab]\n\nμ Γ ab and we also use notations with differential forms. Thus the gauge fields are expressed as A = A μdX μ. In this notation, the corresponding field strength F is written as F = dA + eA ∧ A, where e is a gauge coupling. Covariant derivative\n\nD μ on an adjoint representation Y = 1 2 Y [ab] Γ ab is defined as D μY = ∂ μY + e(A μY -Y A μ),\n\nwhere Y is a scalar multiplet. The action S total consists of two parts. One is the Einstein-Hilbert action S E and the other S Y M T is a Yang-Mills action with a term which is the fourth power of the field strength F . Such a quartic term has been studied by Tchrakian [10] and so we call it the Tchrakian term. The total action is:\n\nS total = S E + S Y M T , S E = 1 16πG dvR , S Y M T = 1 16 Tr -F ∧ * F + α 2 (F ∧ F ) ∧ * (F ∧ F ) -V 0 dv . (1)\n\nHere the 10-form dv is an invariant volume form with respect to the metric ĝ and is written as dv = √ -ĝd 10 X in a local patch. The scalar curvature is denoted by R. The asterisk \" * \" denotes the Hodge dual operator. This operator defines an inner product over differential forms, and for a given form ω, ω ∧ * ω is proportional to the invariant volume form dv. 1 The parameters of this action are the Newton's gravitational constant G, the gauge coupling e, the four-derivative coupling α and the cosmological constant V 0 .\n\nWe show the explicit form of the Yang-Mills part with components of A and F ,\n\nS Y M T = -dv 1 8 F [ab] μν F μν,[ab] + α 2 8 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 16 S μν ρσ S μν ρσ + 1 2 V 0 , (2)\n\nF = 1 4 F [ab] μν dX μ ∧ dX ν Γ ab , S μν ρσ = F [ab] μν F [ab] ρσ + F [ab] μρ F [ab] σ ν + F [ab] μσ F [ab] ν ρ , (3)\n\nT [ab][cd] μν ρσ = F [ab] μν F [cd] ρσ + F [ab] μρ F [cd] σ ν + F [ab] μσ F [cd] ν ρ , T [abcd] μν ρσ = 1 6 T [ab][cd]\n\nμν ρσ + T\n\n[ac][db] μν ρσ + T [ad][bc] μν ρσ + T [cd][ab] μν ρσ + T [db][ac] μν ρσ + T [bc][ad] μν ρσ .\n\nThe Euler-Lagrange equations from these actions read the usual Einstein equation and the equations for the Yang-Mills fields:\n\nR μν - 1 2 ĝμν R = 8πGT μν , D μ √ -gF μν -2α 2 √ -gF [μν F ρσ] F ρσ = 0 . (5)\n\nHere the energy-momentum tensor T μν is obtained by the variation of the Yang-Mills part with respect to the metric:\n\nT μν = 1 2 F μ ρ,[ab] F [ab] ν ρ + α 2 T [abcd] μρσ τ T ν ρστ ,[ab][cd] + α 2 3 • 2 S μρσ τ S ν ρσ τ - 1 2 g μν χ χ = 1 4 F [ab] μν F μν,[ab] + α 2 4 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 8 S μν ρσ S μν ρσ + V 0 . (6)\n\nTo solve these equations, we make an ansatz which is the same as that of Cremmer-Scherk.\n\nOur ansatz for the metric is the following:\n\nds 2 = η µν dx µ dx ν + δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J = ĝμν dX μdX ν , y 2 = 6 a=1 (y I ) 2 , ( 7\n\n)\n\n1 The Hodge dual operator acting on a differential form on a space with Minkowski signature satisfies the following relation:\n\n(F µν dx µν ) ∧ * (F ρσ dx ρσ ) = -F µν F µν dv.\n\nwhere the coordinates X are the total space-time coordinates. The metric η µν = diag(-+ ++) is the Lorentz metric on the four-dimensional Minkowski space. Greek indices without a hat \"ˆ\", for instance µ will refer to the first four variables. Capital indices (I, J, • • • ) run from one to six and refer to the compact space. The six-dimensional space is taken as a sphere with a radius R 0 . The Riemann tensor, Ricci tensor and scalar curvature are\n\nR I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ , R = 30 R 2 0 . (8)\n\nThe rest components of the curvature tensor vanish. In this space, the Einstein equations in (5) reduce to simple equations,\n\n- 1 2 η µν 30 R 2 0 = 8πGT µν , 0 = T µI , - 1 2 20 R 2 0 g IJ = 8πGT IJ . (9)\n\nWe now make ansatzes for the gauge fields. We assume that the fields A do not depend on the four-dimensional directions, ∂ µ A = 0, and they have no four-dimensional components A µ = 0. This implies that the field strengths are two forms on the six-dimensional sphere:\n\nA = A I (y)dy I , F = 1 2\n\nF IJ dy I ∧ dy J . With these ansatzes, the four-dimensional part of the energy-momentum tensor becomes -1 2 η µν χ, and the equation reduces to 30/R 2 0 = 8πGχ. This equation requires that the χ is a constant. Suppose that the field strength fulfils the generalized self-dual condition\n\nF = iβγ 7 * 6 (F ∧ F ), ( 10\n\n)\n\nwhere β is a real parameter. Here \" * 6 \" means the Hodge dual on the six-dimensional sphere.\n\nThen the second part of the equations of motion ( 5 ) is fulfilled automatically by the relation DF = 0, where the exterior covariant derivative is defined as DF = dF +e (A ∧ F -F ∧ A).\n\nIn fact we have an explicit solution to the self-dual equation:\n\nA = 1 4eR 2 0 y a e b Γ ab , F = 1 4eR 2 0 e a ∧ e b Γ ab , β = eR 2 0 3 . ( 11\n\n)\n\nHere we identify the internal space index and the sphere index. The energy-momentum tensor of this configuration becomes\n\nζ ≡ α 2 β 2 , χ = (1 + ζ) 15 4e 2 R 4 0 + V 0 , T IJ = - 1 2 (1 -ζ) 5 4e 2 R 4 0 + V 0 g IJ . (12)\n\nWith these ansatzes we obtain algebraic equations from the Einstein equations:\n\n30 R 2 0 = 8πG 1 2 (1 + ζ) 15 2e 2 R 4 0 + V 0 , 10 R 2 0 = 8πG (1 -ζ) 5 8e 2 R 4 0 + V 0 2 , (13)\n\nFrom these we finally obtain\n\n1 πG = 1 e 2 R 2 0 (2 + 4ζ) , V 0 = 15 4e 2 R 4 0 (1 + 3ζ) . (14)\n\nWhen the four-derivative coupling vanishes, α = 0 and therefore ζ = 0, these relations reduce to those of the Cremmer and Scherk [5] . 2 When the relation α = β holds (ζ = 1) our solution saturates the Bogomol'nyi bound and becomes a BPS state. The energy density is given by an integral over S 6 as follows:\n\nE S 6 Y M T = 1 16 S 6 Tr -F ∧ * 6 F + α 2 (F ∧ F ) ∧ * 6 (F ∧ F ) = 1 16 S 6 Tr (iF ∓ αγ 7 * 6 (F ∧ F )) ∧ * 6 (iF ∓ αγ 7 * 6 (F ∧ F )) ± i 8 α S 6 Trγ 7 F ∧ F ∧ F ≥ ± i 8 α S 6 Trγ 7 F ∧ F ∧ F = ∓ 1 2 3 S 6 ǫ abcdef F [ab] ∧ F [cd] ∧ F [ef ] ≡ ±Q , (15)\n\nwhere the field strength F has only components along S 6 . If the coupling α is equal to β, the solution of eq. ( 10 ) satisfies the Bogomol'nyi equation and the energy attains the local minimum. We can also consider a system coupled with scalar fields. Suppose that scalar fields Q m transform as a representation of SO (6) . The index m labels the representation space. Let us add an action S Q of the scalar fields Q with a Higgs potential\n\nS Q = 1 2 dvD μQ m D μQ m + V (Q 2 ) , D μQ m = ∂ μQ m - 1 2 ieA [ab] μ R(Γ ab ) mm ′ Q m ′ (16)\n\nto S total . The equations of motion are modified. In general, our solution mentioned above does not satisfy the modified equations any more. However, for the scalars which fulfil the covariantly constant condition D μQ m = 0 and attain the absolute minimum V (Q) = 0, the configurations of A and g in equations ( 7 ), (11) are still solutions for the modified equations. Here the constant value of the minimum is shifted to 0. Thus we can argue the Higgs mechanism around our solutions.\n\nNext we suppose that the four-dimensional part is an anti-de Sitter space AdS 4 of radius R A . Our ansatz for the metric is the following:\n\nds 2 = η µν (x)dx µ dx ν + g IJ (y)dy I dy J = ĝμν dX μdX ν , (17)\n\ng IJ (y)dy\n\nI dy J = δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J y 2 = 6 a=1 (y I ) 2 , η µν (x)dx µ dx ν = R 2 A cos 2 θ -dτ 2 + dθ 2 + sin 2 θdΩ 2 , dΩ 2 = |dz| 2 (1 + |z| 2 /4) 2 , (18)\n\nwhere z parametrizes a whole complex plane. The metric η µν (x) is a maximally symmetric metric on the four-dimensional anti-de Sitter space. The Riemann tensor and the Ricci tensor are\n\nR µ νρσ = - 1 R 2 A δ µ ρ η νσ -δ µ σ η νρ , R µν = - 3 R 2 A η µν , R I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ . (19)\n\nThe total scalar curvature is obtained by a summation of those of two parts: R = -\n\n12 R 2 A + 30 R 2 0 .\n\nIn this space, the Einstein equations are\n\nR µν - 1 2 η µν R = 8πGT µν , R IJ - 1 2 g IJ R = 8πGT IJ . (20)\n\nThe ansatz for the gauge fields is the same as previous one and the energy momentum tensor does not change. With these ansatzes, we obtain algebraic equations from the Einstein\n\nequations as 3 R 2 A - 15 R 2 0 = -4πG (1 + ζ) 15 4e 2 R 4 0 + V 0 , 6 R 2 A - 10 R 2 0 = -4πG (1 -ζ) 5 4e 2 R 4 0 + V 0 . (21)\n\nWe are interested in a possible relation to string theory and therefore we consider the case with the vanishing cosmological constant, V 0 = 0. In this case, the radii (R A , R 0 ) are written by the couplings,\n\nR 2 0 = (5 + 7ζ) πG 4e 2 , R 2 A = 5 + 7ζ 5 + 15ζ R 2 0 . (22)\n\nThus the additional higher derivative coupling term of the Tchrakian type does not affect critically to the equations of motion. When ζ = 1 our solution becomes a solution of the Bogomol'nyi equation again.\n\nOur solutions introduced in this letter are new solutions of the system with a Tchrakian term. The origin of this term has not been clear so far but it seems rather universal in order to construct solitons with codimensions higher than four: for instance it has played a crucial role to construct a finite energy monopole (with codimension five) in a six-dimensional spacetime [13] . Though the parameter ζ(= α 2 /β 2 ) is a free parameter, we expect that the system goes to ζ = 1 because it becomes BPS. There are several discussions on the (in)stability of higher-dimensional Yang-Mills theories [15] . To compute the mass spectra of the fluctuations around our solutions is a future work. When the scalar fields Q m are non-trivially coupled, the system may allow BPS composite solitons which are made of solitons with different codimensions, as in the case of usual self-dual Yang-Mills equations coupled to Higgs fields [16] .\n\nFinally our solution of AdS 4 × S 6 may have a relation with D2-branes, and we hope that there exists some impact on AdS/CFT duality [17] ."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"We construct a classical solution of an Einstein-Yang-Mills system with a fourth order term with respect to the field strength of the Yang-Mills field. The solution provides a compactification proposed by Cremmer and Scherk; ten-dimensional space-time with a cosmological constant is compactified to the four-dimensional Minkowski space with a six-dimensional sphere S 6 on which an instanton solution exists. The radius of the sphere is not a modulus but is determined by the gauge coupling and the four-derivative coupling constants and the Newton's constant. We also construct a solution of ten-dimensional theory without a cosmological constant compactified to AdS 4 × S 6 ."},{"section_type":"OTHER","section_title":"Untitled Section","text":"Unification of fundamental forces with space-time and matter often requires higherdimensional space-time rather than our four-dimensional Universe. The early Kaluza-Klein theory unifies gravity and the electro-magnetic interaction by considering five-dimensional space-time with one direction compactified into a circle S 1 [1] . This old idea has been revisited several times. After supergravity was discovered many people tried to unify all forces and matter in higher-dimensional space-time compactified on various internal manifolds [2] .\n\nString theory was proposed as the most attractive candidate of unification, but it is defined only in ten-dimensional space-time. In order to realize four-dimensional Universe one has to find a suitable six-dimensional internal space. So many candidates of such spaces were proposed; Calabi-Yau manifolds and orbifold models. Internal manifolds can be deformed with satisfying the Einstein equation, and these degrees of freedom are called the moduli.\n\nThe moduli introduce unwanted massless particles in four-dimensional world. Recently a new mechanism has been suggested to fix these moduli by turning on the Ramond-Ramond flux on the internal space [3, 4] . This flux compactification has been extensively studied in these years.\n\nWe would like to revise the compactification scenario with fixed moduli proposed by Cremmer and Scherk long time ago [5] (see also [6, 7] ) in a theory with a cosmological constant. By placing solitons on a compact internal space they showed decompactifying limit with large radius of the internal space is disfavored and the radius is fixed to a certain value determined by coupling constants. They considered the 't Hooft-Polyakov monopole [8] on S 2 and the Yang-Mills instanton [9] on S 4 , both of which can satisfy, with proper coupling constants, the first order (self-dual) equations rather than the second order equations of motion, but their solutions on higher dimensional sphere are not the case. Since string theory is defined in ten dimensions, it is natural to consider this scenario with stable BPS solitons on a six-dimensional internal space like S 6 .\n\nHigher dimensional generalization of self-dual equations was suggested by Tchrakian some years ago [10] . Eight dimensional case is known as octonionic instantons [11] . Though several works have been done for generalized self-dual equations [12, 13] , a six-dimensional case was not discussed because of the lack of conformal property. Recently we have found a new solution to the generalized self-dual equations in an SO (6) pure Yang-Mills theory with a fourth order term with respect to the field strength of the Yang-Mills field (a four-derivative term) on a six-dimensional sphere S 6 [14] .\n\nIn this letter we propose to use this solution in the context of a compactification of the Cremmer-Scherk type. In our model ten-dimensional space-time with (without) a cosmological constant is compactified to a four-dimensional Minkowski space M 4 (anti de Sitter space AdS 4 ) with a six-dimensional sphere S 6 , where dimensionality of the internal space, six, is required by the four derivative term. Unlike the case of the absence of gravity [14] the four-derivative coupling constant α can differ from the constant β in the generalized self-dual equations. When the relation α = β holds the generalized self-dual equations become the Bogomol'nyi equations and solutions are BPS. We find for both M 4 × S 6 and AdS 4 × S 6 that certain relations exist between the radius of S 6 , the gauge coupling, the four-derivative coupling α and the gravitational coupling constants. When the four-derivative coupling constant α vanishes in the case of M 4 × S 6 , these relations reduce to those of the original work by Cremmer and Scherk. The advantage of our model to the Cremmer-Scherk model is that the Yang-Mills soliton in our model satisfies the self-dual equations (the Bogomol'nyi equations for α = β) rather than usual equations of motion in the case of the Cremmer-Scherk model. This ensures the stability of configuration at least for the sector of Yang-Mills fields.\n\nLet us consider that space-time is a ten-dimensional manifold. We consider an Einstein-Yang-Mills theory. Our action contains as dynamical variables the Yang-Mills (gauge) fields\n\nA [ab] μ\n\nand a graviton field or the metric ĝμν . Indices with a hat \"ˆ\" will refer to a tendimensional space-time (X 0 , X 1 , • • • , X 9 ). Latin indices (a, b, • • • ) run from 1 to 6 and refer to an internal space. The Clifford algebra associated with the orthogonal group SO( 6 ) is useful and we represent generators of the Lie algebra so(6) as their elements. The Clifford algebra is defined by gamma matrices {Γ a } which satisfy the following anti-commutation relations, {Γ a , Γ b } = 2δ ab . These matrices can be realized as 8 × 8 matrices with complex coefficients. The generators of so( 6 ) are represented by\n\nΓ ab = 1 2 [Γ a , Γ b ].\n\nWe often abbreviate the Yang-Mills fields as\n\nA μ = 1 2 A [ab]\n\nμ Γ ab and we also use notations with differential forms. Thus the gauge fields are expressed as A = A μdX μ. In this notation, the corresponding field strength F is written as F = dA + eA ∧ A, where e is a gauge coupling. Covariant derivative\n\nD μ on an adjoint representation Y = 1 2 Y [ab] Γ ab is defined as D μY = ∂ μY + e(A μY -Y A μ),\n\nwhere Y is a scalar multiplet. The action S total consists of two parts. One is the Einstein-Hilbert action S E and the other S Y M T is a Yang-Mills action with a term which is the fourth power of the field strength F . Such a quartic term has been studied by Tchrakian [10] and so we call it the Tchrakian term. The total action is:\n\nS total = S E + S Y M T , S E = 1 16πG dvR , S Y M T = 1 16 Tr -F ∧ * F + α 2 (F ∧ F ) ∧ * (F ∧ F ) -V 0 dv . (1)\n\nHere the 10-form dv is an invariant volume form with respect to the metric ĝ and is written as dv = √ -ĝd 10 X in a local patch. The scalar curvature is denoted by R. The asterisk \" * \" denotes the Hodge dual operator. This operator defines an inner product over differential forms, and for a given form ω, ω ∧ * ω is proportional to the invariant volume form dv. 1 The parameters of this action are the Newton's gravitational constant G, the gauge coupling e, the four-derivative coupling α and the cosmological constant V 0 .\n\nWe show the explicit form of the Yang-Mills part with components of A and F ,\n\nS Y M T = -dv 1 8 F [ab] μν F μν,[ab] + α 2 8 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 16 S μν ρσ S μν ρσ + 1 2 V 0 , (2)\n\nF = 1 4 F [ab] μν dX μ ∧ dX ν Γ ab , S μν ρσ = F [ab] μν F [ab] ρσ + F [ab] μρ F [ab] σ ν + F [ab] μσ F [ab] ν ρ , (3)\n\nT [ab][cd] μν ρσ = F [ab] μν F [cd] ρσ + F [ab] μρ F [cd] σ ν + F [ab] μσ F [cd] ν ρ , T [abcd] μν ρσ = 1 6 T [ab][cd]\n\nμν ρσ + T\n\n[ac][db] μν ρσ + T [ad][bc] μν ρσ + T [cd][ab] μν ρσ + T [db][ac] μν ρσ + T [bc][ad] μν ρσ .\n\nThe Euler-Lagrange equations from these actions read the usual Einstein equation and the equations for the Yang-Mills fields:\n\nR μν - 1 2 ĝμν R = 8πGT μν , D μ √ -gF μν -2α 2 √ -gF [μν F ρσ] F ρσ = 0 . (5)\n\nHere the energy-momentum tensor T μν is obtained by the variation of the Yang-Mills part with respect to the metric:\n\nT μν = 1 2 F μ ρ,[ab] F [ab] ν ρ + α 2 T [abcd] μρσ τ T ν ρστ ,[ab][cd] + α 2 3 • 2 S μρσ τ S ν ρσ τ - 1 2 g μν χ χ = 1 4 F [ab] μν F μν,[ab] + α 2 4 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 8 S μν ρσ S μν ρσ + V 0 . (6)\n\nTo solve these equations, we make an ansatz which is the same as that of Cremmer-Scherk.\n\nOur ansatz for the metric is the following:\n\nds 2 = η µν dx µ dx ν + δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J = ĝμν dX μdX ν , y 2 = 6 a=1 (y I ) 2 , ( 7\n\n)\n\n1 The Hodge dual operator acting on a differential form on a space with Minkowski signature satisfies the following relation:\n\n(F µν dx µν ) ∧ * (F ρσ dx ρσ ) = -F µν F µν dv.\n\nwhere the coordinates X are the total space-time coordinates. The metric η µν = diag(-+ ++) is the Lorentz metric on the four-dimensional Minkowski space. Greek indices without a hat \"ˆ\", for instance µ will refer to the first four variables. Capital indices (I, J, • • • ) run from one to six and refer to the compact space. The six-dimensional space is taken as a sphere with a radius R 0 . The Riemann tensor, Ricci tensor and scalar curvature are\n\nR I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ , R = 30 R 2 0 . (8)\n\nThe rest components of the curvature tensor vanish. In this space, the Einstein equations in (5) reduce to simple equations,\n\n- 1 2 η µν 30 R 2 0 = 8πGT µν , 0 = T µI , - 1 2 20 R 2 0 g IJ = 8πGT IJ . (9)\n\nWe now make ansatzes for the gauge fields. We assume that the fields A do not depend on the four-dimensional directions, ∂ µ A = 0, and they have no four-dimensional components A µ = 0. This implies that the field strengths are two forms on the six-dimensional sphere:\n\nA = A I (y)dy I , F = 1 2\n\nF IJ dy I ∧ dy J . With these ansatzes, the four-dimensional part of the energy-momentum tensor becomes -1 2 η µν χ, and the equation reduces to 30/R 2 0 = 8πGχ. This equation requires that the χ is a constant. Suppose that the field strength fulfils the generalized self-dual condition\n\nF = iβγ 7 * 6 (F ∧ F ), ( 10\n\n)\n\nwhere β is a real parameter. Here \" * 6 \" means the Hodge dual on the six-dimensional sphere.\n\nThen the second part of the equations of motion ( 5 ) is fulfilled automatically by the relation DF = 0, where the exterior covariant derivative is defined as DF = dF +e (A ∧ F -F ∧ A).\n\nIn fact we have an explicit solution to the self-dual equation:\n\nA = 1 4eR 2 0 y a e b Γ ab , F = 1 4eR 2 0 e a ∧ e b Γ ab , β = eR 2 0 3 . ( 11\n\n)\n\nHere we identify the internal space index and the sphere index. The energy-momentum tensor of this configuration becomes\n\nζ ≡ α 2 β 2 , χ = (1 + ζ) 15 4e 2 R 4 0 + V 0 , T IJ = - 1 2 (1 -ζ) 5 4e 2 R 4 0 + V 0 g IJ . (12)\n\nWith these ansatzes we obtain algebraic equations from the Einstein equations:\n\n30 R 2 0 = 8πG 1 2 (1 + ζ) 15 2e 2 R 4 0 + V 0 , 10 R 2 0 = 8πG (1 -ζ) 5 8e 2 R 4 0 + V 0 2 , (13)\n\nFrom these we finally obtain\n\n1 πG = 1 e 2 R 2 0 (2 + 4ζ) , V 0 = 15 4e 2 R 4 0 (1 + 3ζ) . (14)\n\nWhen the four-derivative coupling vanishes, α = 0 and therefore ζ = 0, these relations reduce to those of the Cremmer and Scherk [5] . 2 When the relation α = β holds (ζ = 1) our solution saturates the Bogomol'nyi bound and becomes a BPS state. The energy density is given by an integral over S 6 as follows:\n\nE S 6 Y M T = 1 16 S 6 Tr -F ∧ * 6 F + α 2 (F ∧ F ) ∧ * 6 (F ∧ F ) = 1 16 S 6 Tr (iF ∓ αγ 7 * 6 (F ∧ F )) ∧ * 6 (iF ∓ αγ 7 * 6 (F ∧ F )) ± i 8 α S 6 Trγ 7 F ∧ F ∧ F ≥ ± i 8 α S 6 Trγ 7 F ∧ F ∧ F = ∓ 1 2 3 S 6 ǫ abcdef F [ab] ∧ F [cd] ∧ F [ef ] ≡ ±Q , (15)\n\nwhere the field strength F has only components along S 6 . If the coupling α is equal to β, the solution of eq. ( 10 ) satisfies the Bogomol'nyi equation and the energy attains the local minimum. We can also consider a system coupled with scalar fields. Suppose that scalar fields Q m transform as a representation of SO (6) . The index m labels the representation space. Let us add an action S Q of the scalar fields Q with a Higgs potential\n\nS Q = 1 2 dvD μQ m D μQ m + V (Q 2 ) , D μQ m = ∂ μQ m - 1 2 ieA [ab] μ R(Γ ab ) mm ′ Q m ′ (16)\n\nto S total . The equations of motion are modified. In general, our solution mentioned above does not satisfy the modified equations any more. However, for the scalars which fulfil the covariantly constant condition D μQ m = 0 and attain the absolute minimum V (Q) = 0, the configurations of A and g in equations ( 7 ), (11) are still solutions for the modified equations. Here the constant value of the minimum is shifted to 0. Thus we can argue the Higgs mechanism around our solutions.\n\nNext we suppose that the four-dimensional part is an anti-de Sitter space AdS 4 of radius R A . Our ansatz for the metric is the following:\n\nds 2 = η µν (x)dx µ dx ν + g IJ (y)dy I dy J = ĝμν dX μdX ν , (17)\n\ng IJ (y)dy\n\nI dy J = δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J y 2 = 6 a=1 (y I ) 2 , η µν (x)dx µ dx ν = R 2 A cos 2 θ -dτ 2 + dθ 2 + sin 2 θdΩ 2 , dΩ 2 = |dz| 2 (1 + |z| 2 /4) 2 , (18)\n\nwhere z parametrizes a whole complex plane. The metric η µν (x) is a maximally symmetric metric on the four-dimensional anti-de Sitter space. The Riemann tensor and the Ricci tensor are\n\nR µ νρσ = - 1 R 2 A δ µ ρ η νσ -δ µ σ η νρ , R µν = - 3 R 2 A η µν , R I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ . (19)\n\nThe total scalar curvature is obtained by a summation of those of two parts: R = -\n\n12 R 2 A + 30 R 2 0 .\n\nIn this space, the Einstein equations are\n\nR µν - 1 2 η µν R = 8πGT µν , R IJ - 1 2 g IJ R = 8πGT IJ . (20)\n\nThe ansatz for the gauge fields is the same as previous one and the energy momentum tensor does not change. With these ansatzes, we obtain algebraic equations from the Einstein\n\nequations as 3 R 2 A - 15 R 2 0 = -4πG (1 + ζ) 15 4e 2 R 4 0 + V 0 , 6 R 2 A - 10 R 2 0 = -4πG (1 -ζ) 5 4e 2 R 4 0 + V 0 . (21)\n\nWe are interested in a possible relation to string theory and therefore we consider the case with the vanishing cosmological constant, V 0 = 0. In this case, the radii (R A , R 0 ) are written by the couplings,\n\nR 2 0 = (5 + 7ζ) πG 4e 2 , R 2 A = 5 + 7ζ 5 + 15ζ R 2 0 . (22)\n\nThus the additional higher derivative coupling term of the Tchrakian type does not affect critically to the equations of motion. When ζ = 1 our solution becomes a solution of the Bogomol'nyi equation again.\n\nOur solutions introduced in this letter are new solutions of the system with a Tchrakian term. The origin of this term has not been clear so far but it seems rather universal in order to construct solitons with codimensions higher than four: for instance it has played a crucial role to construct a finite energy monopole (with codimension five) in a six-dimensional spacetime [13] . Though the parameter ζ(= α 2 /β 2 ) is a free parameter, we expect that the system goes to ζ = 1 because it becomes BPS. There are several discussions on the (in)stability of higher-dimensional Yang-Mills theories [15] . To compute the mass spectra of the fluctuations around our solutions is a future work. When the scalar fields Q m are non-trivially coupled, the system may allow BPS composite solitons which are made of solitons with different codimensions, as in the case of usual self-dual Yang-Mills equations coupled to Higgs fields [16] .\n\nFinally our solution of AdS 4 × S 6 may have a relation with D2-branes, and we hope that there exists some impact on AdS/CFT duality [17] ."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"We construct a classical solution of an Einstein-Yang-Mills system with a fourth order term with respect to the field strength of the Yang-Mills field. The solution provides a compactification proposed by Cremmer and Scherk; ten-dimensional space-time with a cosmological constant is compactified to the four-dimensional Minkowski space with a six-dimensional sphere S 6 on which an instanton solution exists. The radius of the sphere is not a modulus but is determined by the gauge coupling and the four-derivative coupling constants and the Newton's constant. We also construct a solution of ten-dimensional theory without a cosmological constant compactified to AdS 4 × S 6 .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Untitled Section\",\n \"text\": \"Unification of fundamental forces with space-time and matter often requires higherdimensional space-time rather than our four-dimensional Universe. The early Kaluza-Klein theory unifies gravity and the electro-magnetic interaction by considering five-dimensional space-time with one direction compactified into a circle S 1 [1] . This old idea has been revisited several times. After supergravity was discovered many people tried to unify all forces and matter in higher-dimensional space-time compactified on various internal manifolds [2] .\\n\\nString theory was proposed as the most attractive candidate of unification, but it is defined only in ten-dimensional space-time. In order to realize four-dimensional Universe one has to find a suitable six-dimensional internal space. So many candidates of such spaces were proposed; Calabi-Yau manifolds and orbifold models. Internal manifolds can be deformed with satisfying the Einstein equation, and these degrees of freedom are called the moduli.\\n\\nThe moduli introduce unwanted massless particles in four-dimensional world. Recently a new mechanism has been suggested to fix these moduli by turning on the Ramond-Ramond flux on the internal space [3, 4] . This flux compactification has been extensively studied in these years.\\n\\nWe would like to revise the compactification scenario with fixed moduli proposed by Cremmer and Scherk long time ago [5] (see also [6, 7] ) in a theory with a cosmological constant. By placing solitons on a compact internal space they showed decompactifying limit with large radius of the internal space is disfavored and the radius is fixed to a certain value determined by coupling constants. They considered the 't Hooft-Polyakov monopole [8] on S 2 and the Yang-Mills instanton [9] on S 4 , both of which can satisfy, with proper coupling constants, the first order (self-dual) equations rather than the second order equations of motion, but their solutions on higher dimensional sphere are not the case. Since string theory is defined in ten dimensions, it is natural to consider this scenario with stable BPS solitons on a six-dimensional internal space like S 6 .\\n\\nHigher dimensional generalization of self-dual equations was suggested by Tchrakian some years ago [10] . Eight dimensional case is known as octonionic instantons [11] . Though several works have been done for generalized self-dual equations [12, 13] , a six-dimensional case was not discussed because of the lack of conformal property. Recently we have found a new solution to the generalized self-dual equations in an SO (6) pure Yang-Mills theory with a fourth order term with respect to the field strength of the Yang-Mills field (a four-derivative term) on a six-dimensional sphere S 6 [14] .\\n\\nIn this letter we propose to use this solution in the context of a compactification of the Cremmer-Scherk type. In our model ten-dimensional space-time with (without) a cosmological constant is compactified to a four-dimensional Minkowski space M 4 (anti de Sitter space AdS 4 ) with a six-dimensional sphere S 6 , where dimensionality of the internal space, six, is required by the four derivative term. Unlike the case of the absence of gravity [14] the four-derivative coupling constant α can differ from the constant β in the generalized self-dual equations. When the relation α = β holds the generalized self-dual equations become the Bogomol'nyi equations and solutions are BPS. We find for both M 4 × S 6 and AdS 4 × S 6 that certain relations exist between the radius of S 6 , the gauge coupling, the four-derivative coupling α and the gravitational coupling constants. When the four-derivative coupling constant α vanishes in the case of M 4 × S 6 , these relations reduce to those of the original work by Cremmer and Scherk. The advantage of our model to the Cremmer-Scherk model is that the Yang-Mills soliton in our model satisfies the self-dual equations (the Bogomol'nyi equations for α = β) rather than usual equations of motion in the case of the Cremmer-Scherk model. This ensures the stability of configuration at least for the sector of Yang-Mills fields.\\n\\nLet us consider that space-time is a ten-dimensional manifold. We consider an Einstein-Yang-Mills theory. Our action contains as dynamical variables the Yang-Mills (gauge) fields\\n\\nA [ab] μ\\n\\nand a graviton field or the metric ĝμν . Indices with a hat \\\"ˆ\\\" will refer to a tendimensional space-time (X 0 , X 1 , • • • , X 9 ). Latin indices (a, b, • • • ) run from 1 to 6 and refer to an internal space. The Clifford algebra associated with the orthogonal group SO( 6 ) is useful and we represent generators of the Lie algebra so(6) as their elements. The Clifford algebra is defined by gamma matrices {Γ a } which satisfy the following anti-commutation relations, {Γ a , Γ b } = 2δ ab . These matrices can be realized as 8 × 8 matrices with complex coefficients. The generators of so( 6 ) are represented by\\n\\nΓ ab = 1 2 [Γ a , Γ b ].\\n\\nWe often abbreviate the Yang-Mills fields as\\n\\nA μ = 1 2 A [ab]\\n\\nμ Γ ab and we also use notations with differential forms. Thus the gauge fields are expressed as A = A μdX μ. In this notation, the corresponding field strength F is written as F = dA + eA ∧ A, where e is a gauge coupling. Covariant derivative\\n\\nD μ on an adjoint representation Y = 1 2 Y [ab] Γ ab is defined as D μY = ∂ μY + e(A μY -Y A μ),\\n\\nwhere Y is a scalar multiplet. The action S total consists of two parts. One is the Einstein-Hilbert action S E and the other S Y M T is a Yang-Mills action with a term which is the fourth power of the field strength F . Such a quartic term has been studied by Tchrakian [10] and so we call it the Tchrakian term. The total action is:\\n\\nS total = S E + S Y M T , S E = 1 16πG dvR , S Y M T = 1 16 Tr -F ∧ * F + α 2 (F ∧ F ) ∧ * (F ∧ F ) -V 0 dv . (1)\\n\\nHere the 10-form dv is an invariant volume form with respect to the metric ĝ and is written as dv = √ -ĝd 10 X in a local patch. The scalar curvature is denoted by R. The asterisk \\\" * \\\" denotes the Hodge dual operator. This operator defines an inner product over differential forms, and for a given form ω, ω ∧ * ω is proportional to the invariant volume form dv. 1 The parameters of this action are the Newton's gravitational constant G, the gauge coupling e, the four-derivative coupling α and the cosmological constant V 0 .\\n\\nWe show the explicit form of the Yang-Mills part with components of A and F ,\\n\\nS Y M T = -dv 1 8 F [ab] μν F μν,[ab] + α 2 8 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 16 S μν ρσ S μν ρσ + 1 2 V 0 , (2)\\n\\nF = 1 4 F [ab] μν dX μ ∧ dX ν Γ ab , S μν ρσ = F [ab] μν F [ab] ρσ + F [ab] μρ F [ab] σ ν + F [ab] μσ F [ab] ν ρ , (3)\\n\\nT [ab][cd] μν ρσ = F [ab] μν F [cd] ρσ + F [ab] μρ F [cd] σ ν + F [ab] μσ F [cd] ν ρ , T [abcd] μν ρσ = 1 6 T [ab][cd]\\n\\nμν ρσ + T\\n\\n[ac][db] μν ρσ + T [ad][bc] μν ρσ + T [cd][ab] μν ρσ + T [db][ac] μν ρσ + T [bc][ad] μν ρσ .\\n\\nThe Euler-Lagrange equations from these actions read the usual Einstein equation and the equations for the Yang-Mills fields:\\n\\nR μν - 1 2 ĝμν R = 8πGT μν , D μ √ -gF μν -2α 2 √ -gF [μν F ρσ] F ρσ = 0 . (5)\\n\\nHere the energy-momentum tensor T μν is obtained by the variation of the Yang-Mills part with respect to the metric:\\n\\nT μν = 1 2 F μ ρ,[ab] F [ab] ν ρ + α 2 T [abcd] μρσ τ T ν ρστ ,[ab][cd] + α 2 3 • 2 S μρσ τ S ν ρσ τ - 1 2 g μν χ χ = 1 4 F [ab] μν F μν,[ab] + α 2 4 T [abcd] μν ρσ T μν ρσ,[ab][cd] + α 2 3 • 8 S μν ρσ S μν ρσ + V 0 . (6)\\n\\nTo solve these equations, we make an ansatz which is the same as that of Cremmer-Scherk.\\n\\nOur ansatz for the metric is the following:\\n\\nds 2 = η µν dx µ dx ν + δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J = ĝμν dX μdX ν , y 2 = 6 a=1 (y I ) 2 , ( 7\\n\\n)\\n\\n1 The Hodge dual operator acting on a differential form on a space with Minkowski signature satisfies the following relation:\\n\\n(F µν dx µν ) ∧ * (F ρσ dx ρσ ) = -F µν F µν dv.\\n\\nwhere the coordinates X are the total space-time coordinates. The metric η µν = diag(-+ ++) is the Lorentz metric on the four-dimensional Minkowski space. Greek indices without a hat \\\"ˆ\\\", for instance µ will refer to the first four variables. Capital indices (I, J, • • • ) run from one to six and refer to the compact space. The six-dimensional space is taken as a sphere with a radius R 0 . The Riemann tensor, Ricci tensor and scalar curvature are\\n\\nR I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ , R = 30 R 2 0 . (8)\\n\\nThe rest components of the curvature tensor vanish. In this space, the Einstein equations in (5) reduce to simple equations,\\n\\n- 1 2 η µν 30 R 2 0 = 8πGT µν , 0 = T µI , - 1 2 20 R 2 0 g IJ = 8πGT IJ . (9)\\n\\nWe now make ansatzes for the gauge fields. We assume that the fields A do not depend on the four-dimensional directions, ∂ µ A = 0, and they have no four-dimensional components A µ = 0. This implies that the field strengths are two forms on the six-dimensional sphere:\\n\\nA = A I (y)dy I , F = 1 2\\n\\nF IJ dy I ∧ dy J . With these ansatzes, the four-dimensional part of the energy-momentum tensor becomes -1 2 η µν χ, and the equation reduces to 30/R 2 0 = 8πGχ. This equation requires that the χ is a constant. Suppose that the field strength fulfils the generalized self-dual condition\\n\\nF = iβγ 7 * 6 (F ∧ F ), ( 10\\n\\n)\\n\\nwhere β is a real parameter. Here \\\" * 6 \\\" means the Hodge dual on the six-dimensional sphere.\\n\\nThen the second part of the equations of motion ( 5 ) is fulfilled automatically by the relation DF = 0, where the exterior covariant derivative is defined as DF = dF +e (A ∧ F -F ∧ A).\\n\\nIn fact we have an explicit solution to the self-dual equation:\\n\\nA = 1 4eR 2 0 y a e b Γ ab , F = 1 4eR 2 0 e a ∧ e b Γ ab , β = eR 2 0 3 . ( 11\\n\\n)\\n\\nHere we identify the internal space index and the sphere index. The energy-momentum tensor of this configuration becomes\\n\\nζ ≡ α 2 β 2 , χ = (1 + ζ) 15 4e 2 R 4 0 + V 0 , T IJ = - 1 2 (1 -ζ) 5 4e 2 R 4 0 + V 0 g IJ . (12)\\n\\nWith these ansatzes we obtain algebraic equations from the Einstein equations:\\n\\n30 R 2 0 = 8πG 1 2 (1 + ζ) 15 2e 2 R 4 0 + V 0 , 10 R 2 0 = 8πG (1 -ζ) 5 8e 2 R 4 0 + V 0 2 , (13)\\n\\nFrom these we finally obtain\\n\\n1 πG = 1 e 2 R 2 0 (2 + 4ζ) , V 0 = 15 4e 2 R 4 0 (1 + 3ζ) . (14)\\n\\nWhen the four-derivative coupling vanishes, α = 0 and therefore ζ = 0, these relations reduce to those of the Cremmer and Scherk [5] . 2 When the relation α = β holds (ζ = 1) our solution saturates the Bogomol'nyi bound and becomes a BPS state. The energy density is given by an integral over S 6 as follows:\\n\\nE S 6 Y M T = 1 16 S 6 Tr -F ∧ * 6 F + α 2 (F ∧ F ) ∧ * 6 (F ∧ F ) = 1 16 S 6 Tr (iF ∓ αγ 7 * 6 (F ∧ F )) ∧ * 6 (iF ∓ αγ 7 * 6 (F ∧ F )) ± i 8 α S 6 Trγ 7 F ∧ F ∧ F ≥ ± i 8 α S 6 Trγ 7 F ∧ F ∧ F = ∓ 1 2 3 S 6 ǫ abcdef F [ab] ∧ F [cd] ∧ F [ef ] ≡ ±Q , (15)\\n\\nwhere the field strength F has only components along S 6 . If the coupling α is equal to β, the solution of eq. ( 10 ) satisfies the Bogomol'nyi equation and the energy attains the local minimum. We can also consider a system coupled with scalar fields. Suppose that scalar fields Q m transform as a representation of SO (6) . The index m labels the representation space. Let us add an action S Q of the scalar fields Q with a Higgs potential\\n\\nS Q = 1 2 dvD μQ m D μQ m + V (Q 2 ) , D μQ m = ∂ μQ m - 1 2 ieA [ab] μ R(Γ ab ) mm ′ Q m ′ (16)\\n\\nto S total . The equations of motion are modified. In general, our solution mentioned above does not satisfy the modified equations any more. However, for the scalars which fulfil the covariantly constant condition D μQ m = 0 and attain the absolute minimum V (Q) = 0, the configurations of A and g in equations ( 7 ), (11) are still solutions for the modified equations. Here the constant value of the minimum is shifted to 0. Thus we can argue the Higgs mechanism around our solutions.\\n\\nNext we suppose that the four-dimensional part is an anti-de Sitter space AdS 4 of radius R A . Our ansatz for the metric is the following:\\n\\nds 2 = η µν (x)dx µ dx ν + g IJ (y)dy I dy J = ĝμν dX μdX ν , (17)\\n\\ng IJ (y)dy\\n\\nI dy J = δ IJ (1 + y 2 /4R 2 0 ) 2 dy I dy J y 2 = 6 a=1 (y I ) 2 , η µν (x)dx µ dx ν = R 2 A cos 2 θ -dτ 2 + dθ 2 + sin 2 θdΩ 2 , dΩ 2 = |dz| 2 (1 + |z| 2 /4) 2 , (18)\\n\\nwhere z parametrizes a whole complex plane. The metric η µν (x) is a maximally symmetric metric on the four-dimensional anti-de Sitter space. The Riemann tensor and the Ricci tensor are\\n\\nR µ νρσ = - 1 R 2 A δ µ ρ η νσ -δ µ σ η νρ , R µν = - 3 R 2 A η µν , R I JKL = 1 R 2 0 δ I K g JL -δ I L g JK , R IJ = 5 R 2 0 g IJ . (19)\\n\\nThe total scalar curvature is obtained by a summation of those of two parts: R = -\\n\\n12 R 2 A + 30 R 2 0 .\\n\\nIn this space, the Einstein equations are\\n\\nR µν - 1 2 η µν R = 8πGT µν , R IJ - 1 2 g IJ R = 8πGT IJ . (20)\\n\\nThe ansatz for the gauge fields is the same as previous one and the energy momentum tensor does not change. With these ansatzes, we obtain algebraic equations from the Einstein\\n\\nequations as 3 R 2 A - 15 R 2 0 = -4πG (1 + ζ) 15 4e 2 R 4 0 + V 0 , 6 R 2 A - 10 R 2 0 = -4πG (1 -ζ) 5 4e 2 R 4 0 + V 0 . (21)\\n\\nWe are interested in a possible relation to string theory and therefore we consider the case with the vanishing cosmological constant, V 0 = 0. In this case, the radii (R A , R 0 ) are written by the couplings,\\n\\nR 2 0 = (5 + 7ζ) πG 4e 2 , R 2 A = 5 + 7ζ 5 + 15ζ R 2 0 . (22)\\n\\nThus the additional higher derivative coupling term of the Tchrakian type does not affect critically to the equations of motion. When ζ = 1 our solution becomes a solution of the Bogomol'nyi equation again.\\n\\nOur solutions introduced in this letter are new solutions of the system with a Tchrakian term. The origin of this term has not been clear so far but it seems rather universal in order to construct solitons with codimensions higher than four: for instance it has played a crucial role to construct a finite energy monopole (with codimension five) in a six-dimensional spacetime [13] . Though the parameter ζ(= α 2 /β 2 ) is a free parameter, we expect that the system goes to ζ = 1 because it becomes BPS. There are several discussions on the (in)stability of higher-dimensional Yang-Mills theories [15] . To compute the mass spectra of the fluctuations around our solutions is a future work. When the scalar fields Q m are non-trivially coupled, the system may allow BPS composite solitons which are made of solitons with different codimensions, as in the case of usual self-dual Yang-Mills equations coupled to Higgs fields [16] .\\n\\nFinally our solution of AdS 4 × S 6 may have a relation with D2-branes, and we hope that there exists some impact on AdS/CFT duality [17] .\"\n }\n]","isConversational":false}}},{"rowIdx":91,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0507"},"arxiv_id":{"kind":"string","value":"0704.0507"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevD.76.124023"},"content_hash":{"kind":"string","value":"22f44bb72e223a65c9213c7d3684c85f57dc3b9bd321cac32466088966d0a47e"},"title":{"kind":"string","value":"E_6 and the bipartite entanglement of three qutrits"},"abstract":{"kind":"string","value":"Recent investigations have established an analogy between the entropy of\nfour-dimensional supersymmetric black holes in string theory and entanglement\nin quantum information theory. Examples include: (1) N=2 STU black holes and\nthe tripartite entanglement of three qubits (2-state systems), where the common\nsymmetry is [SL(2)]^3 and (2) N=8 black holes and the tripartite entanglement\nof seven qubits where the common symmetry is E_7 which contains [SL(2)]^7. Here\nwe present another example: N=8 black holes (or black strings) in five\ndimensions and the bipartite entanglement of three qutrits (3-state systems),\nwhere the common symmetry is E_6 which contains [SL(3)]^3. Both the black hole\n(or black string) entropy and the entanglement measure are provided by the\nCartan cubic E_6 invariant. Similar analogies exist for ``magic'' N=2\nsupergravity black holes in both four and five dimensions."},"authors":{"kind":"list like","value":["M. J. Duff and S. Ferrara"],"string":"[\n \"M. J. Duff and S. Ferrara\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["hep-th","quant-ph"],"string":"[\n \"hep-th\",\n \"quant-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"1 m.duff@imperial.ac.uk 2 Sergio.Ferrara@cern.ch Contents 1 D = 4 black holes and qubits 1.1 N = 2 black holes and the tripartite entanglement of three qubits . . . . . . 1.2 N = 2 black holes and the bipartite entanglement of two qubits . . . . . . . 1.3 N = 8 black holes and the tripartite entanglement of seven qubits . . . . . . 1.4 Magic supergravities in D = 4 . . . . . . . . . . . . . . . . . . . . . . . . . .\n\n2 Five-dimensional supergravity 3 D = 5 black holes and qutrits 3.1 N = 2 black holes and the bipartite entanglement of two qutrits . . . . . . . 3.2 N = 8 black holes and the bipartite entanglement of three qutrits . . . . . . 3.3 Magic supergravities in D = 5 . . . . . . . . . . . . . . . . . . . . . . . . . .\n\n4 Conclusions 5 Acknowledgements 2 1 D = 4 black holes and qubits\n\nIt sometimes happens that two very different areas of theoretical physics share the same mathematics. This may eventually lead to the realisation that they are, in fact, dual descriptions of the same physical phenomena, or it may not. Either way, it frequently leads to new insights in both areas. Recent papers [1, 2, 3, 4, 5, 6] have established an analogy between the entropy of certain four-dimensional supersymmetric black holes in string theory and entanglement measures in quantum information theory. In this paper we extend the analogy from four dimensions to five which also involves going from two-state systems (qubits) to three-state systems (qutrits). We begin by recalling the four-dimensional examples:\n\n1.1 N = 2 black holes and the tripartite entanglement of three qubits\n\nThe three qubit system (Alice, Bob, Charlie) is described by the state\n\n|Ψ = a ABC |ABC (1.1)\n\nwhere A = 0, 1, so the Hilbert space has dimension 2 3 = 8. The complex numbers a ABC transforms as a (2, 2, 2) under SL(2, C) A × SL(2, C) B × SL(2, C) C . The tripartite entanglement is measured by the 3-tangle [7, 8]\n\nτ 3 (ABC) = 4|Det a ABC |. (1.2)\n\nwhere Det a ABC is Cayley's hyperdeterminant [9] .\n\nDet a = - 1 2 ǫ A 1 A 2 ǫ B 1 B 2 ǫ A 3 A 4 ǫ B 3 B 4 ǫ C 1 C 4 ǫ C 2 C 3 a A 1 B 1 C 1 a A 2 B 2 C 2 a A 3 B 3 C 3 a A 4 B 4 C 4 (1.\n\n3) The hyperdeterminant is invariant under SL(2) A × SL(2) B × SL(2) C and under a triality that interchanges A, B and C. In the context of stringy black holes the 8 a ABC are the 4 electric and 4 magnetic charges of the N = 2 STU black hole [10] and hence take on real (integer) values. The ST U model corresponds to N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . The Bekenstein-Hawking entropy of the black hole, S, was first calculated in [11] . The connection to quantum information theory arises by noting [1] that it can also be expressed in terms of Cayley's hyperdeterminant\n\nS = π |Det a ABC |.\n\n(1.4) One can then establish a dictionary between the classification of various entangled states (separable A-B-C; bipartite entangled A-BC, B-CA, C-AB; tripartite entangled W; tripartite entangled GHZ) and the classfication of various \"small\" and \"large\" BPS and non-BPS black holes [1, 2, 3, 4, 5, 6] . For example, the GHZ state [12] |Ψ ∼ |111 + |000 (1.5) 3 with Det a ABC ≥ 0 corresponds to a large non-BPS 2-charge black hole; the W-state |Ψ ∼ |100 + |010 + |001 (1.6) with Det a ABC = 0 corresponds to a small-BPS 3-charge black hole; the GHZ-state\n\n|Ψ = -|000 + |011 + |101 + |110 (1.7)\n\ncorresponds to a large BPS 4-charge black hole.\n\n1.2 N = 2 black holes and the bipartite entanglement of two qubits\n\nAn even simpler example [2] is provided by the two qubit system (Alice and Bob) described by the state |Ψ = a AB |AB (1.8) where A = 0, 1, and the Hilbert space has dimension 2 2 = 4. The a AB transforms as a (2, 2) under SL(2, C) A × SL(2, C) B . The entanglement is measured by the 2-tangle τ 2 (AB) = C 2 (AB) (1.9) where C(AB) = 2 |det a AB | (1.10) is the concurrence. The determinant is invariant under SL(2, C) A × SL(2, C) B and under a duality that interchanges A and B. Here it is sufficient to look at N = 2 supergravity coupled to just one vector multiplet and the 4 a AB are the 2 electric and 2 magnetic charges of the axion-dilaton black hole with entropy S = π|det a AB | (1.11) For example, the Bell state |Ψ ∼ |11 + |00 (1.12) with det a AB ≥ 0 corresponds to a large non-BPS 2-charge black hole.\n\nWe recall that in the case of D = 4, N = 8 supergravity, the the 28 electric and 28 magnetic charges belong to the 56 of E 7 (7) . The black hole entropy is [15, 18] (1.13) where J 4 is Cartan's quartic E 7 invariant [13, 14] . It may be written\n\nS = π |J 4 |\n\nJ 4 = P ij Q jk P kl Q li -1 4 P ij Q ij P kl Q kl + 1 96 ǫ ijklmnop Q ij Q kl Q mn Q op + ǫ ijklmnop P ij P kl P mn P op . (1.14) 4\n\nwhere P ij and Q jk are 8 × 8 antisymmetric matrices. The qubit interpretation [4] relies on the decomposition E 7 (C) ⊃ [SL(2, C)] 7 (1.15) under which 56 → 2, 2, 1, 2, 1, 1, 1) + (1, 2, 2, 1, 2, 1, 1 1, 1, 2, 2, 1, 2, 1) +(1, 1, 1, 2, 2, 1, 2) +(2, 1, 1, 1, 2, 2, 1) + (1, 2, 1, 1, 1, 2, 2 2, 1, 2, 1, 1, 1, 2) (1.16) suggesting the tripartite entanglement of seven qubits (Alice, Bob, Charlie, Daisy, Emma, Fred and George) described by the state.\n\n(\n\n) +(\n\n) +(\n\n|Ψ = a ABD |ABD +b BCE |BCE +c CDF |CDF +d DEG |DEG +e EF A |EF A +f F GB |F GB +g GAC |GAC (1.17)\n\nwhere A = 0, 1, so the Hilbert space has dimension 7.2 3 = 56. The a, b, c, d, e, f, g transform as a 56 of E 7 (C). The entanglement may be represented by a heptagon where the vertices A,B,C,D,E,F,G represent the seven qubits and the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC represent the tripartite entanglement. See Figure 1 . Alternatively, we can use the Fano plane. See Figure 2 . The Fano plane also corresponds to the multiplication table of the octonions 3 The measure of the tripartite entanglement of the seven qubits is provided by the 3-tangle\n\nτ 3 (ABCDEF G) = 4|J 4 | (1.18) with J 4 ∼ a 4 + b 4 + c 4 + d 4 + e 4 + f 4 + g 4 +\n\n3 Not the \"split\" octonions as was incorrectly stated in the published version of [4].\n\nA B C D E F G Figure 1: The E 7 entanglement diagram. Each of the seven vertices A,B,C,D,E,F,G represents a qubit and each of the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC describes a tripartite entanglement.\n\n2[a 2 b 2 + b 2 c 2 + c 2 d 2 + d 2 e 2 + e 2 f 2 + f 2 g 2 + g 2 a 2 + a 2 c 2 + b 2 d 2 + c 2 e 2 + d 2 f 2 + e 2 g 2 + f 2 a 2 + g 2 b 2 + a 2 d 2 + b 2 e 2 + c 2 f 2 + d 2 g 2 + e 2 a 2 + f 2 b 2 + g 2 c 2 ] +8[bcdf + cdeg + def a + ef gb + f gac + gabd + abce] (1.19)\n\nwhere products like a 4 = (ABD)(ABD)(ABD)(ABD)\n\n= ǫ A 1 A 2 ǫ B 1 B 2 ǫ D 1 D 4 ǫ A 3 A 4 ǫ B 3 B 4 ǫ D 2 D 3 a A 1 B 1 D 1 a A 2 B 2 D 2 a A 3 B 3 D 3 a A 4 B 4 D 4 (1.20)\n\nexclude four individuals (here Charlie, Emma, Fred and George), products like\n\na 2 b 2 = (ABD)(ABD)(F GB)(F GB) = ǫ A 1 A 2 ǫ B 1 B 3 ǫ D 1 D 2 ǫ F 3 F 4 ǫ G 3 G 4 ǫ B 2 B 4 a A 1 B 1 D 1 a A 2 B 2 D 2 b F 3 G 3 B 3 b F 4 G 4 B 4 (1.21)\n\nexclude two individuals (here Charlie and Emma), and products like\n\nabce = (ABD)(BCE)(CDF )(EF A) = ǫ A 1 A 4 ǫ B 1 B 2 ǫ C 2 C 3 ǫ D 1 D 3 ǫ E 2 E 4 ǫ F 3 F 4 a A 1 B 1 D 1 b B 2 C 2 E 2 c C 3 D 3 F 3 e E 4 F 4 A 4 (1.22)\n\nexclude one individual (here George) 4 . Once again large non-BPS, small BPS and large BPS black holes correspond to states with J 4 > 0, J 4 = 0 and J 4 < 0, respectively.\n\n4 This corrects the corresponding equation in the published version of [4] which had the wrong index contraction.\n\nF E B G C A D\n\nFigure 2: The Fano plane has seven points, representing the seven qubits, and seven lines (the circle counts as a line) with three points on every line, representing the tripartite entanglement, and three lines through every point.\n\nThe black holes described by Cayley's hyperdeterminant are those of N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . In [4] the following four-dimensional generalizations were considered: 1) N = 2 supergravity coupled to l vector multiplets where the symmetry is SL(2, Z) × SO(l -1, 2, Z) and the black holes carry charges belonging to the (2, l + 1) representation (l + 1 electric plus l + 1 magnetic).\n\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SL(2, Z) × SO(6, m, Z) where the black holes carry charges belonging to the (2, 6 + m) representation (m + 6 electric plus m + 6 magnetic).\n\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 7(7) (Z) and the black holes carry charges belonging to the fundamental 56-dimensional representation (28 electric plus 28 magnetic).\n\nIn all three case there exist quartic invariants akin to Cayley's hyperdeterminant whose square root yields the corresponding black hole entropy. In [4] we succeeded in giving a quantum theoretic interpretation in the N = 8 case together with its truncations to N = 4 (with m = 6) and N = 2 (with l = 3, the case we already knew [1]).\n\nHowever, as suggested by Levay [5], one might also consider the \"magic\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 4 supergravity coupled to 6, 9, 15 and 27 vector multiplets with symmetries Sp(6, Z), SU(3, 3), SO * (12) and E 7(-25) , respectively. Once again, as has been shown just recently [20] , in all cases there are quartic invariants whose square root yields the corresponding black hole entropy.\n\nHere we demonstrate that the black-hole/qubit correspondence does indeed continue to 7 hold for magic supergravities. The crucial observation is that, although the black hole charges a ABC are real (integer) numbers and the entropy (1.13) is invariant under E 7 (7)(Z), the coefficients a ABC that appear in the qutrit state (1.17) are complex. So the three tangle (1.18) is invariant under E 7 (C) which contains both E 7(7) (Z) and E 7(-25) (Z) as subgroups.\n\nTo find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\n\nE 7(7) (Z) ⊃ [SL(2)(Z)] 7 (1.23) but E 7(-25) (Z) ⊃ [SL(2)(Z)] 7 (1.24)\n\nis irrelevant. All that matters is that\n\nE 7 (C) ⊃ [SL(2)(C)] 7 (1.25)\n\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic) . Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) above. When we looked at the seven qubit subsector E 7 (C) ⊃ SL(2, C) × SO(12, C), we gave an N = 4 supergravity interpretationwith symmetry SL(2, R)×SO(6, 6) [4], but we could equally have given an interpretation in terms of N = 2 supergravity coupled to 11 vector multiplets with symmetry SL(2, R) × SO(10, 2).\n\nMoreover, SO(l -1, 2) is contained in SO(l + 1, C) and SO(6, m) is contained in SO(12 + m, C) so we can give a qubit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(4n, C) which contains [SL(2, C)] 2n .\n\nIn five dimensions we might consider: 1) N = 2 supergravity coupled to l+1 vector multiplets where the symmetry is SO(1, 1, Z)× SO(l, 1, Z) and the black holes carry charges belonging to the (l + 1) representation (all electric) .\n\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SO(1, 1, Z)× SO(m, 5, Z) where the black holes carry charges belonging to the (m + 5) representation (all electric).\n\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 6(6) (Z) and the black holes carry charges belonging to the fundamental 27-dimensional representation (all electric).\n\nThe electrically charged objects are point-like and the magnetic duals are one-dimensional, or string-like, transforming according to the contrgredient representation. In all three cases above there exist cubic invariants akin to the determinant which yield the corresponding black hole or black string entropy.\n\nIn this section we briefly describe the salient properties of maximal N = 8 case, following [16] . We have 27 abelian gauge fields which transform in the fundamental representation of 8 6(6) . The first invariant of E 6(6) is the cubic invariant [13, 17, 16, 18, 19]\n\nE\n\nJ 3 = q ij Ω jl q lm Ω mn q np Ω pi (2.1)\n\nwhere q ij is the charge vector transforming as a 27 which can be represented as traceless Sp(8) matrix. The entropy of a black hole with charges q ij is then given by\n\nS = π |J 3 | (2.2)\n\nWe will see that a configuration with J 3 = 0 preserves 1/8 of the supersymmetries. If J 3 = 0 and ∂J 3 ∂q i = 0 then it preserves 1/4 of the supersymmetries, and finally if ∂J 3 ∂q i = 0 (and the charge vector q i is non-zero), the configuration preserves 1/2 of the supersymmetries. We will show this by choosing a particular basis for the charges, the general result following by U-duality.\n\nIn five dimensions the compact group H is USp(8). We choose our conventions so that USp(2) = SU(2). In the commutator of the supersymmetry generators we have a central charge matrix Z ab which can be brought to a normal form by a USp(8) transformation. In the normal form the central charge matrix can be written as\n\ne ab = s 1 + s 2 -s 3 0 0 0 0 s 1 + s 3 -s 2 0 0 0 0 s 2 + s 3 -s 1 0 0 0 0 -(s 1 + s 2 + s 3 ) × 0 1 -1 0 (2.3) we can order s i so that s 1 ≥ s 2 ≥ |s 3 |.\n\nThe cubic invariant, in this basis, becomes\n\nJ 3 = s 1 s 2 s 3 (2.4)\n\nEven though the eigenvalues s i might depend on the moduli, the invariant (2.4) only depends on the quantized values of the charges. We can write a generic charge configuration as UeU t , where e is the normal frame as above, and the invariant will then be (2.4). There are three distinct possibilities\n\nJ 3 = 0 s 1 , s 2 , s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 , s 2 = 0, s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 = 0, s 2 , s 3 = 0 (2.5)\n\nTaking the case of type II on T 5 we can choose the rotation in such a way that, for example, s 1 corresponds to solitonic five-brane charge, s 2 to fundamental string winding charge along some direction and s 3 to Kaluza-Klein momentum along the same direction. We can see that in this specific example the three possibilities in (2.5) break 1/8, 1/4 and 1/2 supersymmetries. The respective orbits are\n\nE 6 (6) F 4(4) E 6 (6) SO(5, 4) ×T 16 9 E 6 (6) SO(5, 5) ×T 16 (2.6)\n\nThis also shows that one can generically choose a basis for the charges so that all others are related by U-duality. The basis chosen here is the S-dual of the D-brane basis usually chosen for describing black holes in type II B on T 5 . All others are related by U-duality to this particular choice. Note that, in contrast to the four-dimensional case where flipping the sign of J 4 (1.14) interchanges BPS and non-BPS black holes, the sign of the J 3 (2.4) is not important since it changes under a CPT transformation. There is no non-BPS orbit in five dimensions.\n\nIn five dimensions there are also string-like configurations which are the magnetic duals of the configurations considered here. They transform in the contragredient 27 ′ representation and the solutions preserving 1/2, 1/4, 1/8 supersymmetries are characterized in an analogous way. We could also have configurations where we have both point-like and string-like ch the point-like charge is uniformly distributed along the string, it is more natural to consider this configuration as a point-like object in D = 4 by dimensional reduction.\n\nIt is useful to decompose the U-duality group into the T-duality group and the S-duality group. The decomposition reads E 6 → SO(5, 5) × SO(1, 1), leading to\n\n27 → 16 1 + 10 -2 + 1 4 (2.7)\n\nThe last term in (2.7) corresponds to the NS five-brane charge. The 16 correspond to the D-brane charges and the 10 correspond to the 5 directions of KK momentum and the 5 directions of fundamental string winding, which are the charges that explicitly appear in string perturbation theory. The cubic invariant has the decomposition (27) 3 → 10 -2 10 -2 1 4 + 16 1 16 1 10 -2 (2.8) This is saying that in order to have a non-zero area black hole we must have three NS charges (more precisely some \"perturbative\" charges and a solitonic five-brane); or we can have two D-brane charges and one NS charge. In particular, it is not possible to have a black hole with a non-zero horizon area with purely D-brane charges. Notice that the non-compact nature of the groups is crucial in this classification.\n\nSo far, all the quantum information analogies involve four-dimensional black holes and qubits. In order to find an analogy with five-dimensional black holes we invoke three state systems called qutrits.\n\nThe two qutrit system (Alice and Bob) is described by the state\n\n|Ψ = a AB |AB 10\n\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3 2 = 9. The a AB transforms as a (3, 3) under SL(3) A × SL(3) B . The bipartite entanglement is measured by the concurrence [21]\n\nC(AB) = 3 3/2 |det a AB |. (3.1)\n\nThe determinant is invariant under SL(3, C) A × SL(3, C) B and under a duality that interchanges A and B. The black hole interpretation is provided by N = 2 supergravity coupled to 8 vector multiplets with symmetry SL(3, C) where the black hole charges transform as a 9. The entropy is given by\n\nS = π|det a AB | (3.2)\n\n3.2 N = 8 black holes and the bipartite entanglement of three qutrits\n\nAs we have seen in section (2) in the case of D = 5, N = 8 supergravity, the black hole charges belong to the 27 of E 6(6) and the entropy is given by (2.2).\n\nThe qutrit interpretation now relies on the decomposition\n\nE 6 (C) ⊃ SL(3, C) A × SL(3, C) B × SL(3, C) C (3.3) under which 27 → (3, 3, 1) + (3 ′ , 1, 3) + (1, 3 ′ , 3 ′ ) (3.4)\n\nsuggesting the bipartite entanglement of three qutrits (Alice, Bob, Charlie). However, the larger symmetry requires that they undergo at most bipartite entanglement of a very specific kind, where each person has bipartite entanglement with the other two:\n\n|Ψ = a AB |AB + b B C |BC + c CA |CA (3.5)\n\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3.3 2 = 27. The three states transforms as a pair of triplets under two of the SL(3)'s and singlets under the remaining one. Individually, therefore, the bipartite entanglement of each of the three states is given by the determinant (3.1). Taken together however, we see from (3.4) that they transform as a complex 27 of E 6 (C). The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements. See Fig. 3. The N=2 truncation of section 3.1 is represented by just the line AB with endpoints A and B. Note that: 1) Any pair of states has an individual in common 2) Each individual is excluded from one out of the three states The entanglement measure will be given by the concurrence\n\nC(ABC) = 3 3/2 |J 3 | (3.6)\n\nJ 3 being the singlet in 27 × 27 × 27:\n\nJ 3 ∼ a 3 + b 3 + c 3 + 6abc (3.7) 11 A B C\n\nFigure 3: The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements.\n\nwhere the products\n\na 3 = ǫ A 1 A 2 A 3 ǫ B 1 B 2 B 3 a A 1 B 1 a A 2 B 2 a A 3 B 3 (3.8) b 3 = ǫ B 1 B 2 B 3 ǫ C 1 C 2 C 3 b B 1 C 1 b B 2 C 2 b B 3 C 3 (3.9) c 3 = ǫ C 1 C 2 C 3 ǫ A 1 A 2 A 3 c C 1 A 1 c C 2 A 2 c C 3 A 3 (3.10)\n\nexclude one individual (Charlie, Alice, and Bob respectively), and the product abc = a AB b B C c CA (3.11) excludes none.\n\nJust as in four dimensions, one might also consider the \"magic\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 5 supergravity coupled to 5, 8, 14 and 26 vector multiplets with symmetries SL(3, R), SL(3, C), SU * (6) and E 6(-26) respectively. Once again, in all cases there are cubic invariants whose square root yields the corresponding black hole entropy [20] . Here we demonstrate that the black-hole/qubit correspondence continue to hold for these D = 5 magic supergravities, as well as D = 4 . Once again, the crucial observation is that, although the black hole charges a AB are real (integer) numbers and the entropy (2.2) is invariant under E 6(6) (Z), the coefficients a AB that appear in the wave function (3.5) are complex. So the 2-tangle (3.6) is invariant under E 6 (C) which contains both E 6(6) (Z) and E 6(-26) (Z) as subgroups. To find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\n\nE 6(6) (Z) ⊃ [SL(3)(Z)] 3 (3.12) 12 but E 6(-26) (Z) ⊃ [SL(3)(Z)] 3 ( 3\n\n.13) is irrelevant. All that matters is that\n\nE 6 (C) ⊃ [SL(3)(C)] 3 (3.14)\n\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic). In fact, the example of section 3.1 corresponds to the complex case. Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) of section (2). SO(l, 1) is contained in SO(l + 1, C) and SO(m, 5) is contained in SO(5 + m, C), so we can give a qutrit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(6n, C) which contains [SL(3, C)] n .\n\nWe note that the 27-dimensional Hilbert space given in (3.4) and (3.5) is not a subspace of the 3 3 -dimensional three qutrit Hilbert space given by (3, 3, 3) , but rather a direct sum of three 3 2 -dimensional Hilbert spaces. It is, however, a subspace of the 7 3 -dimensional three 7-dit Hilbert space given by (7, 7, 7) . Consider the decomposition 3, 3, 1) + (3, 1, 3) + (1, 3, 3) 1, 1, 1) This contains the subspace that describes the bipartite entanglement of three qutrits, namely\n\nSL(7) A × SL(7) B × SL(7) C → SL(3) A × SL(3) B × SL(3) C under which (7, 7, 7) → (3 ′ , 3 ′ , 3 ′ ) + (3 ′ , 3 ′ , 3) + (3 ′ , 3, 3 ′ ) + (3, 3 ′ , 3 ′ ) + (3 ′ , 3, 3) + (3, 3 ′ , 3) + (3, 3, 3 ′ ) + (3, 3, 3) +(3 ′ , 3 ′ , 1) + (3 ′ , 1, 3 ′ ) + (1, 3 ′ , 3 ′ ) + (3 ′ , 1, 3) + (3 ′ , 3, 1) + (1, 3, 3 ′ ) +(\n\n+ (3, 1, 3 ′ ) + (3, 3 ′ , 1) + (1, 3 ′ , 3) +(3 ′ , 1, 1) + (1, 3 ′ , 1) + (1, 1, 3 ′ ) + (3, 1, 1) + (1, 3, 1) + (1, 1, 3) +(\n\n(3 ′ , 3, 1) + (3, 1, 3) + (1, 3 ′ , 3 ′ )\n\nSo the triangle entanglement we have described fits within conventional quantum information theory. Our analogy between black holes and quantum information remains, for the moment, just that. We know of no physics connecting them.\n\nNevertheless, just as the exceptional group E 7 describes the tripartite entanglement of seven qubits [4, 5] , we have seen is this paper that the exceptional group E 6 describes the bipartite entanglement of three qutrits. In the E 7 case, the quartic Cartan invariant provides both the measure of entanglement and the entropy of the four-dimensional N = 8 black hole, 13 whereas in the E 6 case, the cubic Cartan invariant provides both the measure of entanglement and the entropy of the five-dimensional N = 8 black hole.\n\nMoreover, we have seen that similar analogies exist not only for the N = 4 and N = 2 truncations, but also for the magic N = 2 supergravities in both four and five dimensions (In the four-dimensional case, this had previously been conjectured by Levay[4, 5]). Murat Gunaydin has suggested (private communication) that the appearance of octonions implies a connection to quaternionic and/or octonionic quantum mechanics. This was not apparent (at least to us) in the four-dimensional N = 8 case [4], but the appearance in the five dimensional magic N = 2 case of SL(3, R), SL(3, C), SL(3, H) and SL(3, O) is more suggestive.\n\nMJD has enjoyed useful conversations with Leron Borsten, Hajar Ebrahim, Chris Hull, Martin Plenio and Tony Sudbery. This work was supported in part by the National Science Foundation under grant number PHY-0245337 and PHY-0555605. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The work of S.F. has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104 Constituents, fundamental forces and symmetries of the universe, in association with INFN Frascati National Laboratories and by the D.O.E grant DE-FG03-91ER40662, Task C. The work of MJD is supported in part by PPARC under rolling grant PPA/G/O/2002/00474, PP/D50744X/1."},"sections":{"kind":"list like","value":[{"section_type":"OTHER","section_title":"Untitled Section","text":"1 m.duff@imperial.ac.uk 2 Sergio.Ferrara@cern.ch Contents 1 D = 4 black holes and qubits 1.1 N = 2 black holes and the tripartite entanglement of three qubits . . . . . . 1.2 N = 2 black holes and the bipartite entanglement of two qubits . . . . . . . 1.3 N = 8 black holes and the tripartite entanglement of seven qubits . . . . . . 1.4 Magic supergravities in D = 4 . . . . . . . . . . . . . . . . . . . . . . . . . .\n\n2 Five-dimensional supergravity 3 D = 5 black holes and qutrits 3.1 N = 2 black holes and the bipartite entanglement of two qutrits . . . . . . . 3.2 N = 8 black holes and the bipartite entanglement of three qutrits . . . . . . 3.3 Magic supergravities in D = 5 . . . . . . . . . . . . . . . . . . . . . . . . . .\n\n4 Conclusions 5 Acknowledgements 2 1 D = 4 black holes and qubits\n\nIt sometimes happens that two very different areas of theoretical physics share the same mathematics. This may eventually lead to the realisation that they are, in fact, dual descriptions of the same physical phenomena, or it may not. Either way, it frequently leads to new insights in both areas. Recent papers [1, 2, 3, 4, 5, 6] have established an analogy between the entropy of certain four-dimensional supersymmetric black holes in string theory and entanglement measures in quantum information theory. In this paper we extend the analogy from four dimensions to five which also involves going from two-state systems (qubits) to three-state systems (qutrits). We begin by recalling the four-dimensional examples:\n\n1.1 N = 2 black holes and the tripartite entanglement of three qubits\n\nThe three qubit system (Alice, Bob, Charlie) is described by the state\n\n|Ψ = a ABC |ABC (1.1)\n\nwhere A = 0, 1, so the Hilbert space has dimension 2 3 = 8. The complex numbers a ABC transforms as a (2, 2, 2) under SL(2, C) A × SL(2, C) B × SL(2, C) C . The tripartite entanglement is measured by the 3-tangle [7, 8]\n\nτ 3 (ABC) = 4|Det a ABC |. (1.2)\n\nwhere Det a ABC is Cayley's hyperdeterminant [9] .\n\nDet a = - 1 2 ǫ A 1 A 2 ǫ B 1 B 2 ǫ A 3 A 4 ǫ B 3 B 4 ǫ C 1 C 4 ǫ C 2 C 3 a A 1 B 1 C 1 a A 2 B 2 C 2 a A 3 B 3 C 3 a A 4 B 4 C 4 (1.\n\n3) The hyperdeterminant is invariant under SL(2) A × SL(2) B × SL(2) C and under a triality that interchanges A, B and C. In the context of stringy black holes the 8 a ABC are the 4 electric and 4 magnetic charges of the N = 2 STU black hole [10] and hence take on real (integer) values. The ST U model corresponds to N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . The Bekenstein-Hawking entropy of the black hole, S, was first calculated in [11] . The connection to quantum information theory arises by noting [1] that it can also be expressed in terms of Cayley's hyperdeterminant\n\nS = π |Det a ABC |.\n\n(1.4) One can then establish a dictionary between the classification of various entangled states (separable A-B-C; bipartite entangled A-BC, B-CA, C-AB; tripartite entangled W; tripartite entangled GHZ) and the classfication of various \"small\" and \"large\" BPS and non-BPS black holes [1, 2, 3, 4, 5, 6] . For example, the GHZ state [12] |Ψ ∼ |111 + |000 (1.5) 3 with Det a ABC ≥ 0 corresponds to a large non-BPS 2-charge black hole; the W-state |Ψ ∼ |100 + |010 + |001 (1.6) with Det a ABC = 0 corresponds to a small-BPS 3-charge black hole; the GHZ-state\n\n|Ψ = -|000 + |011 + |101 + |110 (1.7)\n\ncorresponds to a large BPS 4-charge black hole.\n\n1.2 N = 2 black holes and the bipartite entanglement of two qubits\n\nAn even simpler example [2] is provided by the two qubit system (Alice and Bob) described by the state |Ψ = a AB |AB (1.8) where A = 0, 1, and the Hilbert space has dimension 2 2 = 4. The a AB transforms as a (2, 2) under SL(2, C) A × SL(2, C) B . The entanglement is measured by the 2-tangle τ 2 (AB) = C 2 (AB) (1.9) where C(AB) = 2 |det a AB | (1.10) is the concurrence. The determinant is invariant under SL(2, C) A × SL(2, C) B and under a duality that interchanges A and B. Here it is sufficient to look at N = 2 supergravity coupled to just one vector multiplet and the 4 a AB are the 2 electric and 2 magnetic charges of the axion-dilaton black hole with entropy S = π|det a AB | (1.11) For example, the Bell state |Ψ ∼ |11 + |00 (1.12) with det a AB ≥ 0 corresponds to a large non-BPS 2-charge black hole."},{"section_type":"OTHER","section_title":"N = 8 black holes and the tripartite entanglement of seven qubits","text":"We recall that in the case of D = 4, N = 8 supergravity, the the 28 electric and 28 magnetic charges belong to the 56 of E 7 (7) . The black hole entropy is [15, 18] (1.13) where J 4 is Cartan's quartic E 7 invariant [13, 14] . It may be written\n\nS = π |J 4 |\n\nJ 4 = P ij Q jk P kl Q li -1 4 P ij Q ij P kl Q kl + 1 96 ǫ ijklmnop Q ij Q kl Q mn Q op + ǫ ijklmnop P ij P kl P mn P op . (1.14) 4\n\nwhere P ij and Q jk are 8 × 8 antisymmetric matrices. The qubit interpretation [4] relies on the decomposition E 7 (C) ⊃ [SL(2, C)] 7 (1.15) under which 56 → 2, 2, 1, 2, 1, 1, 1) + (1, 2, 2, 1, 2, 1, 1 1, 1, 2, 2, 1, 2, 1) +(1, 1, 1, 2, 2, 1, 2) +(2, 1, 1, 1, 2, 2, 1) + (1, 2, 1, 1, 1, 2, 2 2, 1, 2, 1, 1, 1, 2) (1.16) suggesting the tripartite entanglement of seven qubits (Alice, Bob, Charlie, Daisy, Emma, Fred and George) described by the state.\n\n(\n\n) +(\n\n) +(\n\n|Ψ = a ABD |ABD +b BCE |BCE +c CDF |CDF +d DEG |DEG +e EF A |EF A +f F GB |F GB +g GAC |GAC (1.17)\n\nwhere A = 0, 1, so the Hilbert space has dimension 7.2 3 = 56. The a, b, c, d, e, f, g transform as a 56 of E 7 (C). The entanglement may be represented by a heptagon where the vertices A,B,C,D,E,F,G represent the seven qubits and the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC represent the tripartite entanglement. See Figure 1 . Alternatively, we can use the Fano plane. See Figure 2 . The Fano plane also corresponds to the multiplication table of the octonions 3 The measure of the tripartite entanglement of the seven qubits is provided by the 3-tangle\n\nτ 3 (ABCDEF G) = 4|J 4 | (1.18) with J 4 ∼ a 4 + b 4 + c 4 + d 4 + e 4 + f 4 + g 4 +\n\n3 Not the \"split\" octonions as was incorrectly stated in the published version of [4].\n\nA B C D E F G Figure 1: The E 7 entanglement diagram. Each of the seven vertices A,B,C,D,E,F,G represents a qubit and each of the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC describes a tripartite entanglement.\n\n2[a 2 b 2 + b 2 c 2 + c 2 d 2 + d 2 e 2 + e 2 f 2 + f 2 g 2 + g 2 a 2 + a 2 c 2 + b 2 d 2 + c 2 e 2 + d 2 f 2 + e 2 g 2 + f 2 a 2 + g 2 b 2 + a 2 d 2 + b 2 e 2 + c 2 f 2 + d 2 g 2 + e 2 a 2 + f 2 b 2 + g 2 c 2 ] +8[bcdf + cdeg + def a + ef gb + f gac + gabd + abce] (1.19)\n\nwhere products like a 4 = (ABD)(ABD)(ABD)(ABD)\n\n= ǫ A 1 A 2 ǫ B 1 B 2 ǫ D 1 D 4 ǫ A 3 A 4 ǫ B 3 B 4 ǫ D 2 D 3 a A 1 B 1 D 1 a A 2 B 2 D 2 a A 3 B 3 D 3 a A 4 B 4 D 4 (1.20)\n\nexclude four individuals (here Charlie, Emma, Fred and George), products like\n\na 2 b 2 = (ABD)(ABD)(F GB)(F GB) = ǫ A 1 A 2 ǫ B 1 B 3 ǫ D 1 D 2 ǫ F 3 F 4 ǫ G 3 G 4 ǫ B 2 B 4 a A 1 B 1 D 1 a A 2 B 2 D 2 b F 3 G 3 B 3 b F 4 G 4 B 4 (1.21)\n\nexclude two individuals (here Charlie and Emma), and products like\n\nabce = (ABD)(BCE)(CDF )(EF A) = ǫ A 1 A 4 ǫ B 1 B 2 ǫ C 2 C 3 ǫ D 1 D 3 ǫ E 2 E 4 ǫ F 3 F 4 a A 1 B 1 D 1 b B 2 C 2 E 2 c C 3 D 3 F 3 e E 4 F 4 A 4 (1.22)\n\nexclude one individual (here George) 4 . Once again large non-BPS, small BPS and large BPS black holes correspond to states with J 4 > 0, J 4 = 0 and J 4 < 0, respectively.\n\n4 This corrects the corresponding equation in the published version of [4] which had the wrong index contraction.\n\nF E B G C A D\n\nFigure 2: The Fano plane has seven points, representing the seven qubits, and seven lines (the circle counts as a line) with three points on every line, representing the tripartite entanglement, and three lines through every point."},{"section_type":"OTHER","section_title":"Magic supergravities in D = 4","text":"The black holes described by Cayley's hyperdeterminant are those of N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . In [4] the following four-dimensional generalizations were considered: 1) N = 2 supergravity coupled to l vector multiplets where the symmetry is SL(2, Z) × SO(l -1, 2, Z) and the black holes carry charges belonging to the (2, l + 1) representation (l + 1 electric plus l + 1 magnetic).\n\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SL(2, Z) × SO(6, m, Z) where the black holes carry charges belonging to the (2, 6 + m) representation (m + 6 electric plus m + 6 magnetic).\n\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 7(7) (Z) and the black holes carry charges belonging to the fundamental 56-dimensional representation (28 electric plus 28 magnetic).\n\nIn all three case there exist quartic invariants akin to Cayley's hyperdeterminant whose square root yields the corresponding black hole entropy. In [4] we succeeded in giving a quantum theoretic interpretation in the N = 8 case together with its truncations to N = 4 (with m = 6) and N = 2 (with l = 3, the case we already knew [1]).\n\nHowever, as suggested by Levay [5], one might also consider the \"magic\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 4 supergravity coupled to 6, 9, 15 and 27 vector multiplets with symmetries Sp(6, Z), SU(3, 3), SO * (12) and E 7(-25) , respectively. Once again, as has been shown just recently [20] , in all cases there are quartic invariants whose square root yields the corresponding black hole entropy.\n\nHere we demonstrate that the black-hole/qubit correspondence does indeed continue to 7 hold for magic supergravities. The crucial observation is that, although the black hole charges a ABC are real (integer) numbers and the entropy (1.13) is invariant under E 7 (7)(Z), the coefficients a ABC that appear in the qutrit state (1.17) are complex. So the three tangle (1.18) is invariant under E 7 (C) which contains both E 7(7) (Z) and E 7(-25) (Z) as subgroups.\n\nTo find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\n\nE 7(7) (Z) ⊃ [SL(2)(Z)] 7 (1.23) but E 7(-25) (Z) ⊃ [SL(2)(Z)] 7 (1.24)\n\nis irrelevant. All that matters is that\n\nE 7 (C) ⊃ [SL(2)(C)] 7 (1.25)\n\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic) . Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) above. When we looked at the seven qubit subsector E 7 (C) ⊃ SL(2, C) × SO(12, C), we gave an N = 4 supergravity interpretationwith symmetry SL(2, R)×SO(6, 6) [4], but we could equally have given an interpretation in terms of N = 2 supergravity coupled to 11 vector multiplets with symmetry SL(2, R) × SO(10, 2).\n\nMoreover, SO(l -1, 2) is contained in SO(l + 1, C) and SO(6, m) is contained in SO(12 + m, C) so we can give a qubit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(4n, C) which contains [SL(2, C)] 2n ."},{"section_type":"OTHER","section_title":"Five-dimensional supergravity","text":"In five dimensions we might consider: 1) N = 2 supergravity coupled to l+1 vector multiplets where the symmetry is SO(1, 1, Z)× SO(l, 1, Z) and the black holes carry charges belonging to the (l + 1) representation (all electric) .\n\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SO(1, 1, Z)× SO(m, 5, Z) where the black holes carry charges belonging to the (m + 5) representation (all electric).\n\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 6(6) (Z) and the black holes carry charges belonging to the fundamental 27-dimensional representation (all electric).\n\nThe electrically charged objects are point-like and the magnetic duals are one-dimensional, or string-like, transforming according to the contrgredient representation. In all three cases above there exist cubic invariants akin to the determinant which yield the corresponding black hole or black string entropy.\n\nIn this section we briefly describe the salient properties of maximal N = 8 case, following [16] . We have 27 abelian gauge fields which transform in the fundamental representation of 8 6(6) . The first invariant of E 6(6) is the cubic invariant [13, 17, 16, 18, 19]\n\nE\n\nJ 3 = q ij Ω jl q lm Ω mn q np Ω pi (2.1)\n\nwhere q ij is the charge vector transforming as a 27 which can be represented as traceless Sp(8) matrix. The entropy of a black hole with charges q ij is then given by\n\nS = π |J 3 | (2.2)\n\nWe will see that a configuration with J 3 = 0 preserves 1/8 of the supersymmetries. If J 3 = 0 and ∂J 3 ∂q i = 0 then it preserves 1/4 of the supersymmetries, and finally if ∂J 3 ∂q i = 0 (and the charge vector q i is non-zero), the configuration preserves 1/2 of the supersymmetries. We will show this by choosing a particular basis for the charges, the general result following by U-duality.\n\nIn five dimensions the compact group H is USp(8). We choose our conventions so that USp(2) = SU(2). In the commutator of the supersymmetry generators we have a central charge matrix Z ab which can be brought to a normal form by a USp(8) transformation. In the normal form the central charge matrix can be written as\n\ne ab = s 1 + s 2 -s 3 0 0 0 0 s 1 + s 3 -s 2 0 0 0 0 s 2 + s 3 -s 1 0 0 0 0 -(s 1 + s 2 + s 3 ) × 0 1 -1 0 (2.3) we can order s i so that s 1 ≥ s 2 ≥ |s 3 |.\n\nThe cubic invariant, in this basis, becomes\n\nJ 3 = s 1 s 2 s 3 (2.4)\n\nEven though the eigenvalues s i might depend on the moduli, the invariant (2.4) only depends on the quantized values of the charges. We can write a generic charge configuration as UeU t , where e is the normal frame as above, and the invariant will then be (2.4). There are three distinct possibilities\n\nJ 3 = 0 s 1 , s 2 , s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 , s 2 = 0, s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 = 0, s 2 , s 3 = 0 (2.5)\n\nTaking the case of type II on T 5 we can choose the rotation in such a way that, for example, s 1 corresponds to solitonic five-brane charge, s 2 to fundamental string winding charge along some direction and s 3 to Kaluza-Klein momentum along the same direction. We can see that in this specific example the three possibilities in (2.5) break 1/8, 1/4 and 1/2 supersymmetries. The respective orbits are\n\nE 6 (6) F 4(4) E 6 (6) SO(5, 4) ×T 16 9 E 6 (6) SO(5, 5) ×T 16 (2.6)\n\nThis also shows that one can generically choose a basis for the charges so that all others are related by U-duality. The basis chosen here is the S-dual of the D-brane basis usually chosen for describing black holes in type II B on T 5 . All others are related by U-duality to this particular choice. Note that, in contrast to the four-dimensional case where flipping the sign of J 4 (1.14) interchanges BPS and non-BPS black holes, the sign of the J 3 (2.4) is not important since it changes under a CPT transformation. There is no non-BPS orbit in five dimensions.\n\nIn five dimensions there are also string-like configurations which are the magnetic duals of the configurations considered here. They transform in the contragredient 27 ′ representation and the solutions preserving 1/2, 1/4, 1/8 supersymmetries are characterized in an analogous way. We could also have configurations where we have both point-like and string-like ch the point-like charge is uniformly distributed along the string, it is more natural to consider this configuration as a point-like object in D = 4 by dimensional reduction.\n\nIt is useful to decompose the U-duality group into the T-duality group and the S-duality group. The decomposition reads E 6 → SO(5, 5) × SO(1, 1), leading to\n\n27 → 16 1 + 10 -2 + 1 4 (2.7)\n\nThe last term in (2.7) corresponds to the NS five-brane charge. The 16 correspond to the D-brane charges and the 10 correspond to the 5 directions of KK momentum and the 5 directions of fundamental string winding, which are the charges that explicitly appear in string perturbation theory. The cubic invariant has the decomposition (27) 3 → 10 -2 10 -2 1 4 + 16 1 16 1 10 -2 (2.8) This is saying that in order to have a non-zero area black hole we must have three NS charges (more precisely some \"perturbative\" charges and a solitonic five-brane); or we can have two D-brane charges and one NS charge. In particular, it is not possible to have a black hole with a non-zero horizon area with purely D-brane charges. Notice that the non-compact nature of the groups is crucial in this classification."},{"section_type":"OTHER","section_title":"D = 5 black holes and qutrits","text":"So far, all the quantum information analogies involve four-dimensional black holes and qubits. In order to find an analogy with five-dimensional black holes we invoke three state systems called qutrits."},{"section_type":"OTHER","section_title":"N = 2 black holes and the bipartite entanglement of two qutrits","text":"The two qutrit system (Alice and Bob) is described by the state\n\n|Ψ = a AB |AB 10\n\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3 2 = 9. The a AB transforms as a (3, 3) under SL(3) A × SL(3) B . The bipartite entanglement is measured by the concurrence [21]\n\nC(AB) = 3 3/2 |det a AB |. (3.1)\n\nThe determinant is invariant under SL(3, C) A × SL(3, C) B and under a duality that interchanges A and B. The black hole interpretation is provided by N = 2 supergravity coupled to 8 vector multiplets with symmetry SL(3, C) where the black hole charges transform as a 9. The entropy is given by\n\nS = π|det a AB | (3.2)\n\n3.2 N = 8 black holes and the bipartite entanglement of three qutrits\n\nAs we have seen in section (2) in the case of D = 5, N = 8 supergravity, the black hole charges belong to the 27 of E 6(6) and the entropy is given by (2.2).\n\nThe qutrit interpretation now relies on the decomposition\n\nE 6 (C) ⊃ SL(3, C) A × SL(3, C) B × SL(3, C) C (3.3) under which 27 → (3, 3, 1) + (3 ′ , 1, 3) + (1, 3 ′ , 3 ′ ) (3.4)\n\nsuggesting the bipartite entanglement of three qutrits (Alice, Bob, Charlie). However, the larger symmetry requires that they undergo at most bipartite entanglement of a very specific kind, where each person has bipartite entanglement with the other two:\n\n|Ψ = a AB |AB + b B C |BC + c CA |CA (3.5)\n\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3.3 2 = 27. The three states transforms as a pair of triplets under two of the SL(3)'s and singlets under the remaining one. Individually, therefore, the bipartite entanglement of each of the three states is given by the determinant (3.1). Taken together however, we see from (3.4) that they transform as a complex 27 of E 6 (C). The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements. See Fig. 3. The N=2 truncation of section 3.1 is represented by just the line AB with endpoints A and B. Note that: 1) Any pair of states has an individual in common 2) Each individual is excluded from one out of the three states The entanglement measure will be given by the concurrence\n\nC(ABC) = 3 3/2 |J 3 | (3.6)\n\nJ 3 being the singlet in 27 × 27 × 27:\n\nJ 3 ∼ a 3 + b 3 + c 3 + 6abc (3.7) 11 A B C\n\nFigure 3: The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements.\n\nwhere the products\n\na 3 = ǫ A 1 A 2 A 3 ǫ B 1 B 2 B 3 a A 1 B 1 a A 2 B 2 a A 3 B 3 (3.8) b 3 = ǫ B 1 B 2 B 3 ǫ C 1 C 2 C 3 b B 1 C 1 b B 2 C 2 b B 3 C 3 (3.9) c 3 = ǫ C 1 C 2 C 3 ǫ A 1 A 2 A 3 c C 1 A 1 c C 2 A 2 c C 3 A 3 (3.10)\n\nexclude one individual (Charlie, Alice, and Bob respectively), and the product abc = a AB b B C c CA (3.11) excludes none."},{"section_type":"OTHER","section_title":"Magic supergravities in D = 5","text":"Just as in four dimensions, one might also consider the \"magic\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 5 supergravity coupled to 5, 8, 14 and 26 vector multiplets with symmetries SL(3, R), SL(3, C), SU * (6) and E 6(-26) respectively. Once again, in all cases there are cubic invariants whose square root yields the corresponding black hole entropy [20] . Here we demonstrate that the black-hole/qubit correspondence continue to hold for these D = 5 magic supergravities, as well as D = 4 . Once again, the crucial observation is that, although the black hole charges a AB are real (integer) numbers and the entropy (2.2) is invariant under E 6(6) (Z), the coefficients a AB that appear in the wave function (3.5) are complex. So the 2-tangle (3.6) is invariant under E 6 (C) which contains both E 6(6) (Z) and E 6(-26) (Z) as subgroups. To find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\n\nE 6(6) (Z) ⊃ [SL(3)(Z)] 3 (3.12) 12 but E 6(-26) (Z) ⊃ [SL(3)(Z)] 3 ( 3\n\n.13) is irrelevant. All that matters is that\n\nE 6 (C) ⊃ [SL(3)(C)] 3 (3.14)\n\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic). In fact, the example of section 3.1 corresponds to the complex case. Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) of section (2). SO(l, 1) is contained in SO(l + 1, C) and SO(m, 5) is contained in SO(5 + m, C), so we can give a qutrit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(6n, C) which contains [SL(3, C)] n ."},{"section_type":"CONCLUSION","section_title":"Conclusions","text":"We note that the 27-dimensional Hilbert space given in (3.4) and (3.5) is not a subspace of the 3 3 -dimensional three qutrit Hilbert space given by (3, 3, 3) , but rather a direct sum of three 3 2 -dimensional Hilbert spaces. It is, however, a subspace of the 7 3 -dimensional three 7-dit Hilbert space given by (7, 7, 7) . Consider the decomposition 3, 3, 1) + (3, 1, 3) + (1, 3, 3) 1, 1, 1) This contains the subspace that describes the bipartite entanglement of three qutrits, namely\n\nSL(7) A × SL(7) B × SL(7) C → SL(3) A × SL(3) B × SL(3) C under which (7, 7, 7) → (3 ′ , 3 ′ , 3 ′ ) + (3 ′ , 3 ′ , 3) + (3 ′ , 3, 3 ′ ) + (3, 3 ′ , 3 ′ ) + (3 ′ , 3, 3) + (3, 3 ′ , 3) + (3, 3, 3 ′ ) + (3, 3, 3) +(3 ′ , 3 ′ , 1) + (3 ′ , 1, 3 ′ ) + (1, 3 ′ , 3 ′ ) + (3 ′ , 1, 3) + (3 ′ , 3, 1) + (1, 3, 3 ′ ) +(\n\n+ (3, 1, 3 ′ ) + (3, 3 ′ , 1) + (1, 3 ′ , 3) +(3 ′ , 1, 1) + (1, 3 ′ , 1) + (1, 1, 3 ′ ) + (3, 1, 1) + (1, 3, 1) + (1, 1, 3) +(\n\n(3 ′ , 3, 1) + (3, 1, 3) + (1, 3 ′ , 3 ′ )\n\nSo the triangle entanglement we have described fits within conventional quantum information theory. Our analogy between black holes and quantum information remains, for the moment, just that. We know of no physics connecting them.\n\nNevertheless, just as the exceptional group E 7 describes the tripartite entanglement of seven qubits [4, 5] , we have seen is this paper that the exceptional group E 6 describes the bipartite entanglement of three qutrits. In the E 7 case, the quartic Cartan invariant provides both the measure of entanglement and the entropy of the four-dimensional N = 8 black hole, 13 whereas in the E 6 case, the cubic Cartan invariant provides both the measure of entanglement and the entropy of the five-dimensional N = 8 black hole.\n\nMoreover, we have seen that similar analogies exist not only for the N = 4 and N = 2 truncations, but also for the magic N = 2 supergravities in both four and five dimensions (In the four-dimensional case, this had previously been conjectured by Levay[4, 5]). Murat Gunaydin has suggested (private communication) that the appearance of octonions implies a connection to quaternionic and/or octonionic quantum mechanics. This was not apparent (at least to us) in the four-dimensional N = 8 case [4], but the appearance in the five dimensional magic N = 2 case of SL(3, R), SL(3, C), SL(3, H) and SL(3, O) is more suggestive."},{"section_type":"OTHER","section_title":"Acknowledgements","text":"MJD has enjoyed useful conversations with Leron Borsten, Hajar Ebrahim, Chris Hull, Martin Plenio and Tony Sudbery. This work was supported in part by the National Science Foundation under grant number PHY-0245337 and PHY-0555605. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The work of S.F. has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104 Constituents, fundamental forces and symmetries of the universe, in association with INFN Frascati National Laboratories and by the D.O.E grant DE-FG03-91ER40662, Task C. The work of MJD is supported in part by PPARC under rolling grant PPA/G/O/2002/00474, PP/D50744X/1."}],"string":"[\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Untitled Section\",\n \"text\": \"1 m.duff@imperial.ac.uk 2 Sergio.Ferrara@cern.ch Contents 1 D = 4 black holes and qubits 1.1 N = 2 black holes and the tripartite entanglement of three qubits . . . . . . 1.2 N = 2 black holes and the bipartite entanglement of two qubits . . . . . . . 1.3 N = 8 black holes and the tripartite entanglement of seven qubits . . . . . . 1.4 Magic supergravities in D = 4 . . . . . . . . . . . . . . . . . . . . . . . . . .\\n\\n2 Five-dimensional supergravity 3 D = 5 black holes and qutrits 3.1 N = 2 black holes and the bipartite entanglement of two qutrits . . . . . . . 3.2 N = 8 black holes and the bipartite entanglement of three qutrits . . . . . . 3.3 Magic supergravities in D = 5 . . . . . . . . . . . . . . . . . . . . . . . . . .\\n\\n4 Conclusions 5 Acknowledgements 2 1 D = 4 black holes and qubits\\n\\nIt sometimes happens that two very different areas of theoretical physics share the same mathematics. This may eventually lead to the realisation that they are, in fact, dual descriptions of the same physical phenomena, or it may not. Either way, it frequently leads to new insights in both areas. Recent papers [1, 2, 3, 4, 5, 6] have established an analogy between the entropy of certain four-dimensional supersymmetric black holes in string theory and entanglement measures in quantum information theory. In this paper we extend the analogy from four dimensions to five which also involves going from two-state systems (qubits) to three-state systems (qutrits). We begin by recalling the four-dimensional examples:\\n\\n1.1 N = 2 black holes and the tripartite entanglement of three qubits\\n\\nThe three qubit system (Alice, Bob, Charlie) is described by the state\\n\\n|Ψ = a ABC |ABC (1.1)\\n\\nwhere A = 0, 1, so the Hilbert space has dimension 2 3 = 8. The complex numbers a ABC transforms as a (2, 2, 2) under SL(2, C) A × SL(2, C) B × SL(2, C) C . The tripartite entanglement is measured by the 3-tangle [7, 8]\\n\\nτ 3 (ABC) = 4|Det a ABC |. (1.2)\\n\\nwhere Det a ABC is Cayley's hyperdeterminant [9] .\\n\\nDet a = - 1 2 ǫ A 1 A 2 ǫ B 1 B 2 ǫ A 3 A 4 ǫ B 3 B 4 ǫ C 1 C 4 ǫ C 2 C 3 a A 1 B 1 C 1 a A 2 B 2 C 2 a A 3 B 3 C 3 a A 4 B 4 C 4 (1.\\n\\n3) The hyperdeterminant is invariant under SL(2) A × SL(2) B × SL(2) C and under a triality that interchanges A, B and C. In the context of stringy black holes the 8 a ABC are the 4 electric and 4 magnetic charges of the N = 2 STU black hole [10] and hence take on real (integer) values. The ST U model corresponds to N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . The Bekenstein-Hawking entropy of the black hole, S, was first calculated in [11] . The connection to quantum information theory arises by noting [1] that it can also be expressed in terms of Cayley's hyperdeterminant\\n\\nS = π |Det a ABC |.\\n\\n(1.4) One can then establish a dictionary between the classification of various entangled states (separable A-B-C; bipartite entangled A-BC, B-CA, C-AB; tripartite entangled W; tripartite entangled GHZ) and the classfication of various \\\"small\\\" and \\\"large\\\" BPS and non-BPS black holes [1, 2, 3, 4, 5, 6] . For example, the GHZ state [12] |Ψ ∼ |111 + |000 (1.5) 3 with Det a ABC ≥ 0 corresponds to a large non-BPS 2-charge black hole; the W-state |Ψ ∼ |100 + |010 + |001 (1.6) with Det a ABC = 0 corresponds to a small-BPS 3-charge black hole; the GHZ-state\\n\\n|Ψ = -|000 + |011 + |101 + |110 (1.7)\\n\\ncorresponds to a large BPS 4-charge black hole.\\n\\n1.2 N = 2 black holes and the bipartite entanglement of two qubits\\n\\nAn even simpler example [2] is provided by the two qubit system (Alice and Bob) described by the state |Ψ = a AB |AB (1.8) where A = 0, 1, and the Hilbert space has dimension 2 2 = 4. The a AB transforms as a (2, 2) under SL(2, C) A × SL(2, C) B . The entanglement is measured by the 2-tangle τ 2 (AB) = C 2 (AB) (1.9) where C(AB) = 2 |det a AB | (1.10) is the concurrence. The determinant is invariant under SL(2, C) A × SL(2, C) B and under a duality that interchanges A and B. Here it is sufficient to look at N = 2 supergravity coupled to just one vector multiplet and the 4 a AB are the 2 electric and 2 magnetic charges of the axion-dilaton black hole with entropy S = π|det a AB | (1.11) For example, the Bell state |Ψ ∼ |11 + |00 (1.12) with det a AB ≥ 0 corresponds to a large non-BPS 2-charge black hole.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"N = 8 black holes and the tripartite entanglement of seven qubits\",\n \"text\": \"We recall that in the case of D = 4, N = 8 supergravity, the the 28 electric and 28 magnetic charges belong to the 56 of E 7 (7) . The black hole entropy is [15, 18] (1.13) where J 4 is Cartan's quartic E 7 invariant [13, 14] . It may be written\\n\\nS = π |J 4 |\\n\\nJ 4 = P ij Q jk P kl Q li -1 4 P ij Q ij P kl Q kl + 1 96 ǫ ijklmnop Q ij Q kl Q mn Q op + ǫ ijklmnop P ij P kl P mn P op . (1.14) 4\\n\\nwhere P ij and Q jk are 8 × 8 antisymmetric matrices. The qubit interpretation [4] relies on the decomposition E 7 (C) ⊃ [SL(2, C)] 7 (1.15) under which 56 → 2, 2, 1, 2, 1, 1, 1) + (1, 2, 2, 1, 2, 1, 1 1, 1, 2, 2, 1, 2, 1) +(1, 1, 1, 2, 2, 1, 2) +(2, 1, 1, 1, 2, 2, 1) + (1, 2, 1, 1, 1, 2, 2 2, 1, 2, 1, 1, 1, 2) (1.16) suggesting the tripartite entanglement of seven qubits (Alice, Bob, Charlie, Daisy, Emma, Fred and George) described by the state.\\n\\n(\\n\\n) +(\\n\\n) +(\\n\\n|Ψ = a ABD |ABD +b BCE |BCE +c CDF |CDF +d DEG |DEG +e EF A |EF A +f F GB |F GB +g GAC |GAC (1.17)\\n\\nwhere A = 0, 1, so the Hilbert space has dimension 7.2 3 = 56. The a, b, c, d, e, f, g transform as a 56 of E 7 (C). The entanglement may be represented by a heptagon where the vertices A,B,C,D,E,F,G represent the seven qubits and the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC represent the tripartite entanglement. See Figure 1 . Alternatively, we can use the Fano plane. See Figure 2 . The Fano plane also corresponds to the multiplication table of the octonions 3 The measure of the tripartite entanglement of the seven qubits is provided by the 3-tangle\\n\\nτ 3 (ABCDEF G) = 4|J 4 | (1.18) with J 4 ∼ a 4 + b 4 + c 4 + d 4 + e 4 + f 4 + g 4 +\\n\\n3 Not the \\\"split\\\" octonions as was incorrectly stated in the published version of [4].\\n\\nA B C D E F G Figure 1: The E 7 entanglement diagram. Each of the seven vertices A,B,C,D,E,F,G represents a qubit and each of the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC describes a tripartite entanglement.\\n\\n2[a 2 b 2 + b 2 c 2 + c 2 d 2 + d 2 e 2 + e 2 f 2 + f 2 g 2 + g 2 a 2 + a 2 c 2 + b 2 d 2 + c 2 e 2 + d 2 f 2 + e 2 g 2 + f 2 a 2 + g 2 b 2 + a 2 d 2 + b 2 e 2 + c 2 f 2 + d 2 g 2 + e 2 a 2 + f 2 b 2 + g 2 c 2 ] +8[bcdf + cdeg + def a + ef gb + f gac + gabd + abce] (1.19)\\n\\nwhere products like a 4 = (ABD)(ABD)(ABD)(ABD)\\n\\n= ǫ A 1 A 2 ǫ B 1 B 2 ǫ D 1 D 4 ǫ A 3 A 4 ǫ B 3 B 4 ǫ D 2 D 3 a A 1 B 1 D 1 a A 2 B 2 D 2 a A 3 B 3 D 3 a A 4 B 4 D 4 (1.20)\\n\\nexclude four individuals (here Charlie, Emma, Fred and George), products like\\n\\na 2 b 2 = (ABD)(ABD)(F GB)(F GB) = ǫ A 1 A 2 ǫ B 1 B 3 ǫ D 1 D 2 ǫ F 3 F 4 ǫ G 3 G 4 ǫ B 2 B 4 a A 1 B 1 D 1 a A 2 B 2 D 2 b F 3 G 3 B 3 b F 4 G 4 B 4 (1.21)\\n\\nexclude two individuals (here Charlie and Emma), and products like\\n\\nabce = (ABD)(BCE)(CDF )(EF A) = ǫ A 1 A 4 ǫ B 1 B 2 ǫ C 2 C 3 ǫ D 1 D 3 ǫ E 2 E 4 ǫ F 3 F 4 a A 1 B 1 D 1 b B 2 C 2 E 2 c C 3 D 3 F 3 e E 4 F 4 A 4 (1.22)\\n\\nexclude one individual (here George) 4 . Once again large non-BPS, small BPS and large BPS black holes correspond to states with J 4 > 0, J 4 = 0 and J 4 < 0, respectively.\\n\\n4 This corrects the corresponding equation in the published version of [4] which had the wrong index contraction.\\n\\nF E B G C A D\\n\\nFigure 2: The Fano plane has seven points, representing the seven qubits, and seven lines (the circle counts as a line) with three points on every line, representing the tripartite entanglement, and three lines through every point.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Magic supergravities in D = 4\",\n \"text\": \"The black holes described by Cayley's hyperdeterminant are those of N = 2 supergravity coupled to three vector multiplets, where the symmetry is [SL(2, Z)] 3 . In [4] the following four-dimensional generalizations were considered: 1) N = 2 supergravity coupled to l vector multiplets where the symmetry is SL(2, Z) × SO(l -1, 2, Z) and the black holes carry charges belonging to the (2, l + 1) representation (l + 1 electric plus l + 1 magnetic).\\n\\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SL(2, Z) × SO(6, m, Z) where the black holes carry charges belonging to the (2, 6 + m) representation (m + 6 electric plus m + 6 magnetic).\\n\\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 7(7) (Z) and the black holes carry charges belonging to the fundamental 56-dimensional representation (28 electric plus 28 magnetic).\\n\\nIn all three case there exist quartic invariants akin to Cayley's hyperdeterminant whose square root yields the corresponding black hole entropy. In [4] we succeeded in giving a quantum theoretic interpretation in the N = 8 case together with its truncations to N = 4 (with m = 6) and N = 2 (with l = 3, the case we already knew [1]).\\n\\nHowever, as suggested by Levay [5], one might also consider the \\\"magic\\\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 4 supergravity coupled to 6, 9, 15 and 27 vector multiplets with symmetries Sp(6, Z), SU(3, 3), SO * (12) and E 7(-25) , respectively. Once again, as has been shown just recently [20] , in all cases there are quartic invariants whose square root yields the corresponding black hole entropy.\\n\\nHere we demonstrate that the black-hole/qubit correspondence does indeed continue to 7 hold for magic supergravities. The crucial observation is that, although the black hole charges a ABC are real (integer) numbers and the entropy (1.13) is invariant under E 7 (7)(Z), the coefficients a ABC that appear in the qutrit state (1.17) are complex. So the three tangle (1.18) is invariant under E 7 (C) which contains both E 7(7) (Z) and E 7(-25) (Z) as subgroups.\\n\\nTo find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\\n\\nE 7(7) (Z) ⊃ [SL(2)(Z)] 7 (1.23) but E 7(-25) (Z) ⊃ [SL(2)(Z)] 7 (1.24)\\n\\nis irrelevant. All that matters is that\\n\\nE 7 (C) ⊃ [SL(2)(C)] 7 (1.25)\\n\\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic) . Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) above. When we looked at the seven qubit subsector E 7 (C) ⊃ SL(2, C) × SO(12, C), we gave an N = 4 supergravity interpretationwith symmetry SL(2, R)×SO(6, 6) [4], but we could equally have given an interpretation in terms of N = 2 supergravity coupled to 11 vector multiplets with symmetry SL(2, R) × SO(10, 2).\\n\\nMoreover, SO(l -1, 2) is contained in SO(l + 1, C) and SO(6, m) is contained in SO(12 + m, C) so we can give a qubit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(4n, C) which contains [SL(2, C)] 2n .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Five-dimensional supergravity\",\n \"text\": \"In five dimensions we might consider: 1) N = 2 supergravity coupled to l+1 vector multiplets where the symmetry is SO(1, 1, Z)× SO(l, 1, Z) and the black holes carry charges belonging to the (l + 1) representation (all electric) .\\n\\n2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SO(1, 1, Z)× SO(m, 5, Z) where the black holes carry charges belonging to the (m + 5) representation (all electric).\\n\\n3) N = 8 supergravity where the symmetry is the non-compact exceptional group E 6(6) (Z) and the black holes carry charges belonging to the fundamental 27-dimensional representation (all electric).\\n\\nThe electrically charged objects are point-like and the magnetic duals are one-dimensional, or string-like, transforming according to the contrgredient representation. In all three cases above there exist cubic invariants akin to the determinant which yield the corresponding black hole or black string entropy.\\n\\nIn this section we briefly describe the salient properties of maximal N = 8 case, following [16] . We have 27 abelian gauge fields which transform in the fundamental representation of 8 6(6) . The first invariant of E 6(6) is the cubic invariant [13, 17, 16, 18, 19]\\n\\nE\\n\\nJ 3 = q ij Ω jl q lm Ω mn q np Ω pi (2.1)\\n\\nwhere q ij is the charge vector transforming as a 27 which can be represented as traceless Sp(8) matrix. The entropy of a black hole with charges q ij is then given by\\n\\nS = π |J 3 | (2.2)\\n\\nWe will see that a configuration with J 3 = 0 preserves 1/8 of the supersymmetries. If J 3 = 0 and ∂J 3 ∂q i = 0 then it preserves 1/4 of the supersymmetries, and finally if ∂J 3 ∂q i = 0 (and the charge vector q i is non-zero), the configuration preserves 1/2 of the supersymmetries. We will show this by choosing a particular basis for the charges, the general result following by U-duality.\\n\\nIn five dimensions the compact group H is USp(8). We choose our conventions so that USp(2) = SU(2). In the commutator of the supersymmetry generators we have a central charge matrix Z ab which can be brought to a normal form by a USp(8) transformation. In the normal form the central charge matrix can be written as\\n\\ne ab = s 1 + s 2 -s 3 0 0 0 0 s 1 + s 3 -s 2 0 0 0 0 s 2 + s 3 -s 1 0 0 0 0 -(s 1 + s 2 + s 3 ) × 0 1 -1 0 (2.3) we can order s i so that s 1 ≥ s 2 ≥ |s 3 |.\\n\\nThe cubic invariant, in this basis, becomes\\n\\nJ 3 = s 1 s 2 s 3 (2.4)\\n\\nEven though the eigenvalues s i might depend on the moduli, the invariant (2.4) only depends on the quantized values of the charges. We can write a generic charge configuration as UeU t , where e is the normal frame as above, and the invariant will then be (2.4). There are three distinct possibilities\\n\\nJ 3 = 0 s 1 , s 2 , s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 , s 2 = 0, s 3 = 0 J 3 = 0, ∂J 3 ∂q i = 0 s 1 = 0, s 2 , s 3 = 0 (2.5)\\n\\nTaking the case of type II on T 5 we can choose the rotation in such a way that, for example, s 1 corresponds to solitonic five-brane charge, s 2 to fundamental string winding charge along some direction and s 3 to Kaluza-Klein momentum along the same direction. We can see that in this specific example the three possibilities in (2.5) break 1/8, 1/4 and 1/2 supersymmetries. The respective orbits are\\n\\nE 6 (6) F 4(4) E 6 (6) SO(5, 4) ×T 16 9 E 6 (6) SO(5, 5) ×T 16 (2.6)\\n\\nThis also shows that one can generically choose a basis for the charges so that all others are related by U-duality. The basis chosen here is the S-dual of the D-brane basis usually chosen for describing black holes in type II B on T 5 . All others are related by U-duality to this particular choice. Note that, in contrast to the four-dimensional case where flipping the sign of J 4 (1.14) interchanges BPS and non-BPS black holes, the sign of the J 3 (2.4) is not important since it changes under a CPT transformation. There is no non-BPS orbit in five dimensions.\\n\\nIn five dimensions there are also string-like configurations which are the magnetic duals of the configurations considered here. They transform in the contragredient 27 ′ representation and the solutions preserving 1/2, 1/4, 1/8 supersymmetries are characterized in an analogous way. We could also have configurations where we have both point-like and string-like ch the point-like charge is uniformly distributed along the string, it is more natural to consider this configuration as a point-like object in D = 4 by dimensional reduction.\\n\\nIt is useful to decompose the U-duality group into the T-duality group and the S-duality group. The decomposition reads E 6 → SO(5, 5) × SO(1, 1), leading to\\n\\n27 → 16 1 + 10 -2 + 1 4 (2.7)\\n\\nThe last term in (2.7) corresponds to the NS five-brane charge. The 16 correspond to the D-brane charges and the 10 correspond to the 5 directions of KK momentum and the 5 directions of fundamental string winding, which are the charges that explicitly appear in string perturbation theory. The cubic invariant has the decomposition (27) 3 → 10 -2 10 -2 1 4 + 16 1 16 1 10 -2 (2.8) This is saying that in order to have a non-zero area black hole we must have three NS charges (more precisely some \\\"perturbative\\\" charges and a solitonic five-brane); or we can have two D-brane charges and one NS charge. In particular, it is not possible to have a black hole with a non-zero horizon area with purely D-brane charges. Notice that the non-compact nature of the groups is crucial in this classification.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"D = 5 black holes and qutrits\",\n \"text\": \"So far, all the quantum information analogies involve four-dimensional black holes and qubits. In order to find an analogy with five-dimensional black holes we invoke three state systems called qutrits.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"N = 2 black holes and the bipartite entanglement of two qutrits\",\n \"text\": \"The two qutrit system (Alice and Bob) is described by the state\\n\\n|Ψ = a AB |AB 10\\n\\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3 2 = 9. The a AB transforms as a (3, 3) under SL(3) A × SL(3) B . The bipartite entanglement is measured by the concurrence [21]\\n\\nC(AB) = 3 3/2 |det a AB |. (3.1)\\n\\nThe determinant is invariant under SL(3, C) A × SL(3, C) B and under a duality that interchanges A and B. The black hole interpretation is provided by N = 2 supergravity coupled to 8 vector multiplets with symmetry SL(3, C) where the black hole charges transform as a 9. The entropy is given by\\n\\nS = π|det a AB | (3.2)\\n\\n3.2 N = 8 black holes and the bipartite entanglement of three qutrits\\n\\nAs we have seen in section (2) in the case of D = 5, N = 8 supergravity, the black hole charges belong to the 27 of E 6(6) and the entropy is given by (2.2).\\n\\nThe qutrit interpretation now relies on the decomposition\\n\\nE 6 (C) ⊃ SL(3, C) A × SL(3, C) B × SL(3, C) C (3.3) under which 27 → (3, 3, 1) + (3 ′ , 1, 3) + (1, 3 ′ , 3 ′ ) (3.4)\\n\\nsuggesting the bipartite entanglement of three qutrits (Alice, Bob, Charlie). However, the larger symmetry requires that they undergo at most bipartite entanglement of a very specific kind, where each person has bipartite entanglement with the other two:\\n\\n|Ψ = a AB |AB + b B C |BC + c CA |CA (3.5)\\n\\nwhere A = 0, 1, 2, so the Hilbert space has dimension 3.3 2 = 27. The three states transforms as a pair of triplets under two of the SL(3)'s and singlets under the remaining one. Individually, therefore, the bipartite entanglement of each of the three states is given by the determinant (3.1). Taken together however, we see from (3.4) that they transform as a complex 27 of E 6 (C). The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements. See Fig. 3. The N=2 truncation of section 3.1 is represented by just the line AB with endpoints A and B. Note that: 1) Any pair of states has an individual in common 2) Each individual is excluded from one out of the three states The entanglement measure will be given by the concurrence\\n\\nC(ABC) = 3 3/2 |J 3 | (3.6)\\n\\nJ 3 being the singlet in 27 × 27 × 27:\\n\\nJ 3 ∼ a 3 + b 3 + c 3 + 6abc (3.7) 11 A B C\\n\\nFigure 3: The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements.\\n\\nwhere the products\\n\\na 3 = ǫ A 1 A 2 A 3 ǫ B 1 B 2 B 3 a A 1 B 1 a A 2 B 2 a A 3 B 3 (3.8) b 3 = ǫ B 1 B 2 B 3 ǫ C 1 C 2 C 3 b B 1 C 1 b B 2 C 2 b B 3 C 3 (3.9) c 3 = ǫ C 1 C 2 C 3 ǫ A 1 A 2 A 3 c C 1 A 1 c C 2 A 2 c C 3 A 3 (3.10)\\n\\nexclude one individual (Charlie, Alice, and Bob respectively), and the product abc = a AB b B C c CA (3.11) excludes none.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Magic supergravities in D = 5\",\n \"text\": \"Just as in four dimensions, one might also consider the \\\"magic\\\" supergravities [22, 23, 24] . These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) N = 2, D = 5 supergravity coupled to 5, 8, 14 and 26 vector multiplets with symmetries SL(3, R), SL(3, C), SU * (6) and E 6(-26) respectively. Once again, in all cases there are cubic invariants whose square root yields the corresponding black hole entropy [20] . Here we demonstrate that the black-hole/qubit correspondence continue to hold for these D = 5 magic supergravities, as well as D = 4 . Once again, the crucial observation is that, although the black hole charges a AB are real (integer) numbers and the entropy (2.2) is invariant under E 6(6) (Z), the coefficients a AB that appear in the wave function (3.5) are complex. So the 2-tangle (3.6) is invariant under E 6 (C) which contains both E 6(6) (Z) and E 6(-26) (Z) as subgroups. To find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact that\\n\\nE 6(6) (Z) ⊃ [SL(3)(Z)] 3 (3.12) 12 but E 6(-26) (Z) ⊃ [SL(3)(Z)] 3 ( 3\\n\\n.13) is irrelevant. All that matters is that\\n\\nE 6 (C) ⊃ [SL(3)(C)] 3 (3.14)\\n\\nThe same argument holds for the magic real, complex and quaternionic N = 2 supergravities which are, in any case truncations of N = 8 (in contrast to the octonionic). In fact, the example of section 3.1 corresponds to the complex case. Having made this observation, one may then revisit the conventional N = 2 and N = 4 cases (1) and (2) of section (2). SO(l, 1) is contained in SO(l + 1, C) and SO(m, 5) is contained in SO(5 + m, C), so we can give a qutrit interpretation to more vector multiplets for both N = 2 and N = 4, at least in the case of SO(6n, C) which contains [SL(3, C)] n .\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"Conclusions\",\n \"text\": \"We note that the 27-dimensional Hilbert space given in (3.4) and (3.5) is not a subspace of the 3 3 -dimensional three qutrit Hilbert space given by (3, 3, 3) , but rather a direct sum of three 3 2 -dimensional Hilbert spaces. It is, however, a subspace of the 7 3 -dimensional three 7-dit Hilbert space given by (7, 7, 7) . Consider the decomposition 3, 3, 1) + (3, 1, 3) + (1, 3, 3) 1, 1, 1) This contains the subspace that describes the bipartite entanglement of three qutrits, namely\\n\\nSL(7) A × SL(7) B × SL(7) C → SL(3) A × SL(3) B × SL(3) C under which (7, 7, 7) → (3 ′ , 3 ′ , 3 ′ ) + (3 ′ , 3 ′ , 3) + (3 ′ , 3, 3 ′ ) + (3, 3 ′ , 3 ′ ) + (3 ′ , 3, 3) + (3, 3 ′ , 3) + (3, 3, 3 ′ ) + (3, 3, 3) +(3 ′ , 3 ′ , 1) + (3 ′ , 1, 3 ′ ) + (1, 3 ′ , 3 ′ ) + (3 ′ , 1, 3) + (3 ′ , 3, 1) + (1, 3, 3 ′ ) +(\\n\\n+ (3, 1, 3 ′ ) + (3, 3 ′ , 1) + (1, 3 ′ , 3) +(3 ′ , 1, 1) + (1, 3 ′ , 1) + (1, 1, 3 ′ ) + (3, 1, 1) + (1, 3, 1) + (1, 1, 3) +(\\n\\n(3 ′ , 3, 1) + (3, 1, 3) + (1, 3 ′ , 3 ′ )\\n\\nSo the triangle entanglement we have described fits within conventional quantum information theory. Our analogy between black holes and quantum information remains, for the moment, just that. We know of no physics connecting them.\\n\\nNevertheless, just as the exceptional group E 7 describes the tripartite entanglement of seven qubits [4, 5] , we have seen is this paper that the exceptional group E 6 describes the bipartite entanglement of three qutrits. In the E 7 case, the quartic Cartan invariant provides both the measure of entanglement and the entropy of the four-dimensional N = 8 black hole, 13 whereas in the E 6 case, the cubic Cartan invariant provides both the measure of entanglement and the entropy of the five-dimensional N = 8 black hole.\\n\\nMoreover, we have seen that similar analogies exist not only for the N = 4 and N = 2 truncations, but also for the magic N = 2 supergravities in both four and five dimensions (In the four-dimensional case, this had previously been conjectured by Levay[4, 5]). Murat Gunaydin has suggested (private communication) that the appearance of octonions implies a connection to quaternionic and/or octonionic quantum mechanics. This was not apparent (at least to us) in the four-dimensional N = 8 case [4], but the appearance in the five dimensional magic N = 2 case of SL(3, R), SL(3, C), SL(3, H) and SL(3, O) is more suggestive.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgements\",\n \"text\": \"MJD has enjoyed useful conversations with Leron Borsten, Hajar Ebrahim, Chris Hull, Martin Plenio and Tony Sudbery. This work was supported in part by the National Science Foundation under grant number PHY-0245337 and PHY-0555605. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The work of S.F. has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104 Constituents, fundamental forces and symmetries of the universe, in association with INFN Frascati National Laboratories and by the D.O.E grant DE-FG03-91ER40662, Task C. The work of MJD is supported in part by PPARC under rolling grant PPA/G/O/2002/00474, PP/D50744X/1.\"\n }\n]","isConversational":false}}},{"rowIdx":92,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0511"},"arxiv_id":{"kind":"string","value":"0704.0511"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1007/s10946-007-0032-5"},"content_hash":{"kind":"string","value":"be892cb8194712d11f25f536c9a91df2f945e518862e27506bb42e0662579c98"},"title":{"kind":"string","value":"A unified approach to SIC-POVMs and MUBs"},"abstract":{"kind":"string","value":"A unified approach to (symmetric informationally complete) positive operator\nvalued measures and mutually unbiased bases is developed in this article. The\napproach is based on the use of operator equivalents expanded in the enveloping\nalgebra of SU(2). Emphasis is put on similarities and differences between\nSIC-POVMs and MUBs."},"authors":{"kind":"list like","value":["O. Albouy (IPNL)","M. R. Kibler (IPNL)"],"string":"[\n \"O. Albouy (IPNL)\",\n \"M. R. Kibler (IPNL)\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["quant-ph","math-ph","math.MP"],"string":"[\n \"quant-ph\",\n \"math-ph\",\n \"math.MP\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"The importance of finite-dimensional spaces for quantum mechanics is well recognized (see for instance [1]-[3]). In particular, such spaces play a major role in quantum information theory, especially for quantum cryptography and quantum state tomography [4]- [27] .\n\nAlong this vein, a symmetric informationally complete (SIC) positive operator valued measure (POVM) is a set of operators acting on a finite Hilbert space [4]-[14] (see also [3] for an infinite Hilbert space) and mutually unbiased bases (MUBs) are specific bases for such a space [15]- [27] .\n\nThe introduction of POVMs goes back to the seventies [4]- [7] . The most general quantum measurement is represented by a POVM. In the present work, we will be interested 1 in SIC-POVMs, for which the statistics of the measurement allows the reconstruction of the quantum state. Moreover, those POVMs are endowed with an extra symmetry condition (see definition in Sec. 2). The notion of MUBs (see definition in Sec. 3), implicit or explicit in the seminal works of [15] - [18] , has been the object of numerous mathematical and physical investigations during the last two decades in connection with the so-called complementary observables. Unfortunately, the question to know, for a given Hilbert space of finite dimension d, whether there exist SIC-POVMs and how many MUBs there exist has remained an open one.\n\nThe aim of this note is to develop a unified approach to SIC-POVMs and MUBs based on a complex vector space of higher dimension, viz. d 2 instead of d. We then give a specific example of this approach grounded on the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) recently used for a study of entanglement of rotationally invariant spin systems [28] and for an angular momentum study of MUBs [26, 27] .\n\nMost of the notations in this work are standard. Let us simply mention that I is the identity operator, the bar indicates complex conjugation, A † denotes the adjoint of the operator A, δ a,b stands for the Kronecker symbol for a and b, and ∆(a, b, c) is 1 or 0 according as a, b and c satisfy or not the triangular inequality.\n\nLet C d be the standard Hilbert space of dimension d endowed with its usual inner product denoted by | . As is usual, we will identify a POVM with a nonorthogonal decomposition of the identity. Thus, a discrete SIC-POVM is a set {P x : x = 1, 2, • • • , d 2 } of d 2 nonnegative operators P x acting on C d , such that:\n\n• they satisfy the trace or symmetry condition\n\nTr (P x P y ) = 1 d + 1 , x = y; ( 1\n\n)\n\nmoreover, we will assume the operators P x are normalized, thus completing this 2 condition with\n\nTr P 2 x = 1; ( 2\n\n)\n\n• they form a decomposition of the identity\n\n1 d d 2 x=1 P x = I; ( 3\n\n)\n\n• they satisfy a completeness condition: the knowledge of the probabilities p x defined by p x = Tr(P x ρ) is sufficient to reconstruct the density matrix ρ.\n\nNow, let us develop each of the operators P x on an orthonormal (with respect to the Hilbert-Schmidt product) basis {u i : i = 1, 2, • • • , d 2 } of the space of linear operators on\n\nC d P x = d 2 i=1 v i (x)u i , ( 4\n\n)\n\nwhere the operators u i satisfy Tr(u † i u j ) = δ i,j . The operators P x are thus considered as vectors\n\nv(x) = (v 1 (x), v 2 (x), • • • , v d 2 (x)) ( 5\n\n)\n\nin the Hilbert space C d 2 of dimension d 2 and the determination of the operators P x is equivalent to the determination of the components v i (x) of v(x). In this language, the trace property (1) together with the normalization condition (2) give y) is the usual Hermitian product in C d 2 . In order to compare Eq. ( 6 ) with what usually happens in the search for SIC-POVMs, we suppose from now on that the operators P x are rank-one operators. Therefore, by putting\n\nv(x) • v(y) = 1 d + 1 (dδ x,y + 1) , ( 6\n\n) where v(x) • v(y) = d 2 i=1 v i (x)v i (\n\nP x = |Φ x Φ x | ( 7\n\n) 3 with |φ x ∈ C d , the trace property (1, 2) reads | Φ x |Φ y | 2 = 1 d + 1 (dδ x,y + 1) . ( 8\n\n)\n\nFrom this point of view, to find d 2 operators P x is equivalent to finding d 2 vectors |φ x in C d satisfying Eq. ( 8 ). At the price of an increase in the number of components from d 3 (for d 2 vectors in C d ) to d 4 (for d 2 vectors in C d 2 ), we have got rid of the square modulus to result in a single scalar product (compare Eqs. (6) and (8)), what may prove to be suitable for another way to search for SIC-POVMs. Moreover, our relation (6) is independent of any hypothesis on the rank of the operators P x . In fact, there exists a lot of relations among these d 4 coefficients that decrease the effective number of coefficients to be found and give structural constraints on them. Those relations are highly sensitive to the choice of the basis {u i : i = 1, 2, • • • , d 2 } and we are going to exhibit an example of such a set of relations by choosing the basis to consist of Racah unit tensors.\n\nThe cornerstone of this approach is to identify C d with a subspace ε(j) of constant angular momentum j = (d -1)/2. Such a subspace is spanned by the set {|j, m : m = -j, -j + 1, • • • , j}, where |j, m is an eigenvector of the square and the z-component of a generalized angular momentum operator. Let u (k) be the Racah unit tensor [29]\n\nof order k (with k = 0, 1, • • • , 2j) defined by its 2k + 1 components u (k) q (where q = -k, -k + 1, • • • , k) through u (k) q = j m=-j j m ′ =-j (-1) j-m j k j -m q m ′ |j, m j, m ′ |, ( 9\n\n)\n\nwhere (• • •) denotes a 3-jm Wigner symbol. For fixed j, the (2j + 1) 2 operators u\n\n(k) q (with k = 0, 1, • • • , 2j and q = -k, -k + 1, • • • , k) act on ε(j) ∼ C d and form a basis\n\nof the Hilbert space C N of dimension N = (2j + 1) 2 , the inner product in C N being the Hilbert-Schmidt product. The formulas (involving unit tensors, 3-jm and 6-j symbols) relevant for this work are given in Appendix (see also [29] to [31] ). We must remember that those Racah operators are not normalized to unity (see relation (46)). So this will generate an extra factor when defining v i (x).\n\n4 Each operator P x can be developed as a linear combination of the operators u (k) q .\n\nHence, we have\n\nP x = 2j k=0 k q=-k c kq (x)u (k) q , ( 10\n\n)\n\nwhere the unknown expansion coefficients c kq (x) are a priori complex numbers. The determination of the operators P x is thus equivalent to the determination of the coefficients c kq (x), which are formally given by\n\nc kq (x) = (2k + 1) Φ x |u (k) q |Φ x , ( 11\n\n)\n\nas can be seen by multiplying each member of Eq. (10) by the adjoint of u (ℓ) p and then using Eq. (46) of Appendix.\n\nBy defining the vector\n\nv(x) = (v 1 (x), v 2 (x), • • • , v N (x)), N = (2j + 1) 2 ( 12\n\n) via v i (x) = 1 √ 2k + 1 c kq (x), i = k 2 + k + q + 1, ( 13\n\n)\n\nthe following properties and relations are obtained.\n\n• The first component v 1 (x) of v(x) does not depend on x since c 00 (x) = 1 √ 2j + 1 ( 14\n\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }.\n\nProof: Take the trace of Eq. (10) and use Eq. (48) of Appendix.\n\n• The components v i (x) of v(x) satisfy the complex conjugation property described by c kq (x) = (-1) q c k-q (x) ( 15\n\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\n\nProof: Use the Hermitian property of P x and Eq. (43) of Appendix.\n\n• In terms of c kq , Eq. (6) reads\n\n2j k=0 1 2k + 1 k q=-k c kq (x)c kq (y) = 1 2(j + 1) [(2j + 1)δ x,y + 1] ( 16\n\n) for all x, y ∈ {1, 2, • • • , (2j + 1) 2 },\n\nwhere the sum over q is SO(3) rotationally invariant.\n\nProof: The proof is trivial.\n\n• The coefficients c kq (x) are solutions of the nonlinear system given by\n\n1 2K + 1 c KQ (x) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ k ℓ K -q -p Q × k ℓ K j j j c kq (x)c ℓp (x) ( 17\n\n) for all x ∈ {1, 2, • • • , (2j+1) 2 }, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K+1, • • • , K}.\n\nProof: Consider P 2 x = P x and use the coupling relation (51) of Appendix involving a 3-jm and a 6-j Wigner symbols.\n\nAs a corollary of the latter property, by taking K = 0 and using Eqs. (47) and (50) of Appendix, we get again the normalization relation v(x\n\n) 2 = v(x) • v(x) = 1.\n\n• All coefficients c kq (x) are connected through the sum rule\n\n(2j+1) 2 x=1 2j k=0 k q=-k c kq (x) j k j -m q m ′ = (-1) j-m (2j + 1)δ m,m ′ , ( 18\n\n)\n\nwhich turns out to be useful for global checking purposes.\n\nProof: Take the jm-jm ′ matrix element of the resolution of the identity in terms of the operators P x /(2j + 1).\n\nA complete set of MUBs in the Hilbert space\n\nC d is a set of d(d + 1) vectors |aα ∈ C d such that | aα|bβ | 2 = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 19\n\n) 6 where a = 0, 1, • • • , d and α = 0, 1, • • • , d -1.\n\nThe indices of type a refer to the bases and, for fixed a, the index α refers to one of the d vectors of the basis corresponding to a.\n\nWe know that such a complete set exists if d is a prime or the power of a prime (e.g., see [16]-[24]).\n\nThe approach developed in Sec. 2 for SIC-POVMs can be applied to MUBs too. Let us suppose that it is possible to\n\nfind d + 1 sets S a (with a = 0, 1, • • • , d) of vectors in C d , each set S a = {|aα : α = 0, 1, • • • , d -1} containing d vectors |aα such that Eq. ( 19\n\n)\n\nbe satisfied. This amounts to finding d(d + 1) projection operators\n\nΠ aα = |aα aα| ( 20\n\n)\n\nsatisfying the trace condition\n\nTr (Π aα Π bβ ) = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 21\n\n)\n\nwhere the trace is taken on C d . Therefore, they also form a nonorthogonal decomposition of the identity\n\n1 d + 1 d a=0 d-1 α=0 Π aα = I. ( 22\n\n)\n\nAs in Sec. 2, we develop each operator Π aα on an orthonormal basis with expansion coefficients w i (aα). Thus we get vectors w(aα) in\n\nC d 2 w(aα) = (w 1 (aα), w 2 (aα), • • • , w d 2 (aα)) ( 23\n\n) such that w(aα) • w(bβ) = δ α,β δ a,b + 1 d (1 -δ a,b ) ( 24\n\n) for all a, b ∈ {0, 1, • • • , d} and α, β ∈ {0, 1, • • • , d -1}.\n\nNow we draw the same relations as for POVMs by choosing the Racah operators to be our basis in C d 2 . We assume once again that the Hilbert space C d is realized by ε(j) with j = (d -1)/2. Then, each operator Π aα can be developed on the basis of the (2j + 1) 2 7 operators u (k) q as\n\nΠ aα = 2j k=0 k q=-k d kq (aα)u (k) q , ( 25\n\n)\n\nto be compared with Eq. (10). The expansion coefficients are d kq (aα) = (2k + 1) aα|u\n\n(k) q |aα ( 26\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1,\n\n• • • , 2j} and q ∈ {-k, -k + 1, • • • , k}. For a and α fixed, the complex coefficients d kq (aα) define a vector\n\nw(aα) = (w 1 (aα), w 2 (aα), • • • , w N (aα)) , N = (2j + 1) 2 ( 27\n\n)\n\nin the Hilbert space C N , the components of which are given by\n\nw i (aα) = 1 √ 2k + 1 d kq (aα), i = k 2 + k + q + 1. ( 28\n\n)\n\nWe are thus led to the following properties and relations. The proofs are similar to those in Sec. 2.\n\n• First component w 1 (aα) of w(aα):\n\nd 00 (aα) = 1 √ 2j + 1 ( 29\n\n) for all a ∈ {0, 1, • • • , 2j + 1} and α ∈ {0, 1, • • • , 2j}.\n\n• Complex conjugation property:\n\nd kq (aα) = (-1) q d k-q (aα) ( 30\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\n\n• Rotational invariance:\n\n2j k=0 1 2k + 1 k q=-k d kq (aα)d kq (bβ) = δ α,β δ a,b + 1 2j + 1 (1 -δ a,b ) ( 31\n\n) for all a, b ∈ {0, 1, • • • , 2j + 1} and α, β ∈ {0, 1, • • • , 2j}.\n\n• Tensor product formula: 1 2K + 1 d KQ (aα) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ\n\nk ℓ K -q -p Q × k ℓ K j j j d kq (aα)d ℓp (aα) ( 32\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K + 1, • • • , K}.\n\n• Sum rule:\n\n2j+1 a=0 2j α=0 2j k=0 k q=-k d kq (aα) j k j -m q m ′ = (-1) j-m 2(2j + 1)δ m,m ′ ( 33\n\n)\n\nwhich involves all coefficients d kq (aα).\n\nAlthough the structure of the relations in Sec. 1 on the one hand and Sec. 2 on the other hand is very similar, there are deep differences between the two sets of results. The similarities are reminiscent of the fact that both MUBs and SIC-POVMs can be linked to finite affine planes [12, 13, 22, 23, 25] and to complex projective 2-designs [8, 10, 19, 24] .\n\nOn the other side, there are two arguments in favor of the differences between relations (6) and (24). First, the problem of constructing SIC-POVMs in dimension d is not equivalent to the existence of an affine plane of order d [12, 13] . Second, there is a consensus around the conjecture according to which there exists a complete set of MUBs in dimension d if and only if there exists an affine plane of order d [22].\n\nIn dimension d, to find d 2 operators P x of a SIC-POVM acting on the Hilbert space\n\nC d amounts to find d 2 vectors v(x) in the Hilbert space C N with N = d 2 satisfying v x = 1, v(x) • v(y) = 1 d + 1 for x = y ( 34\n\n)\n\n(the norm v(x) of each vector v(x) is 1 and the angle ω xy of any pair of vectors v(x) and v(y) is ω xy = cos -1 [1/(d + 1)] for x = y).\n\nIn a similar way, to find d + 1 MUBs of C d is equivalent to find d + 1 sets S a (with a = 0, 1, • • • , d) of d vectors, i.e., d(d + 1) vectors in all, w(aα) in C N with N = d 2 satisfying w(aα) • w(aβ) = δ α,β , w(aα) • w(bβ) = 1 d for a = b (35) (each set S a consists of d orthonormalized vectors and the angle ω aαbβ of any vector w(aα) of a set S a with any vector w(bβ) of a set S b is ω aαbβ = cos -1 (1/d) for a = b).\n\nAccording to a well accepted conjecture [8, 10], SIC-POVMs should exist in any\n\ndimension. The present study shows that in order to prove this conjecture it is sufficient to prove that Eq. (34) admits solutions for any value of d.\n\nThe situation is different for MUBs. In dimension d, it is known that there exist d + 1 sets of d vectors of type |aα in C d satisfying Eq. (19) when d is a prime or the power of a prime. This shows that Eq. ( 35 ) can be solved for d prime or power of a prime. For d prime, it is possible to find an explicit solution of Eq. ( 19 ). In fact, we have [26, 27] |aα = 1 √ 2j + 1 j m=-j ω (j+m)(j-m+1)a/2+(j+m)\n\nα |j, m , ( 36\n\n) ω = exp i 2π 2j + 1 , j = 1 2 (d -1) ( 37\n\n) for a, α ∈ {0, 1, • • • , 2j} while |aα = |j, m ( 38\n\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j. Then, Eq. (26) yields d kq (aα) = 2k + 1 2j + 1 j m=-j j m ′ =-j ω θ(m,m ′ ) (-1) j-m j k j -m q m ′ , ( 39\n\n) θ(m, m ′ ) = (m -m ′ ) 1 2 (1 -m -m ′ )a + α ( 40\n\n) for a, α ∈ {0, 1, • • • , 2j} while d kq (aα) = δ q,0 (2k + 1)(-1) j-m j k j -m 0 m ( 41\n\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j.\n\nIt can be shown that Eqs. (40) and (41) are in agreement with the results of Sec. 3. We thus have a solution of the equations for 10 the results of Sec. 3 when d is prime. As an open problem, it would be worthwhile to find an explicit solution for the coefficients d kq (aα) when d = 2j + 1 is any positive power of a prime. Finally, note that to prove (or disprove) the conjecture according to which a complete set of MUBs in dimension d exists only if d is a prime or the power of a prime is equivalent to prove (or disprove) that Eq. ( 35 ) has a solution only if d is a prime or the power of a prime.\n\nAPPENDIX: WIGNER-RACAH ALGEBRA OF SU(2) ⊃ U(1)\n\nWe limit ourselves to those basic formulas for the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) which are necessary to derive the results of this paper. The summations in this appendix have to be extended to the allowed values for the involved magnetic and angular momentum quantum numbers.\n\nThe definition (9) of the components u (k) q of the Racah unit tensor u (k) yields\n\nj, m|u (k) q |j, m ′ = (-1) j-m j k j -m q m ′ , ( 42\n\n)\n\nfrom which we easily obtain the Hermitian conjugation property\n\nu (k) q † = (-1) q u (k) -q . ( 43\n\n)\n\nThe 3-jm Wigner symbol in Eq. (42) satisfies the orthogonality relations\n\nmm ′ j j ′ k m m ′ q j j ′ ℓ m m ′ p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j ′ , k) ( 44\n\n) and kq (2k + 1) j j ′ k m m ′ q j j ′ k M M ′ q = δ m,M δ m ′ ,M ′ . ( 45\n\n)\n\nThe trace relation on the space ε(j)\n\nTr u (k) q † u (ℓ) p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j, k) ( 46\n\n)\n\n11 easily follows by combining Eqs. (42) and (44). Furthermore, by introducing\n\nj j ′ 0 m -m ′ 0 = δ j,j ′ δ m,m ′ (-1) j-m 1 √ 2j + 1 ( 47\n\n)\n\nin Eq. ( 44 ), we obtain the sum rule\n\nm (-1) j-m j k j -m q m = 2j + 1δ k,0 δ q,0 ∆(j, k, j), ( 48\n\n)\n\nknown in spectroscopy as the barycenter theorem.\n\nThere are several relations involving 3-jm and 6-j symbols. In particular, we have\n\nmm ′ M (-1) j-M j j -m q M j ℓ j -M p m ′ j K j -m Q m ′ = (-1) 2j-Q k ℓ K -q -p Q k ℓ K j j j , ( 49\n\n)\n\nwhere {• • •} denotes a 6-j Wigner symbol (or W Racah coefficient). Note that the introduction of k ℓ 0 j j J = δ k,ℓ (-1) j+k+J 1 (2k + 1)(2j + 1) (50) in Eq. (49) gives back Eq. ( 44 ). Equation (49) is central in the derivation of the coupling relation\n\nu (k) q u (ℓ) p = KQ (-1) 2j-Q (2K + 1) k ℓ K -q -p Q k ℓ K j j j u (K) Q . ( 51\n\n)\n\nEquation (51) makes it possible to calculate the commutator [u\n\n(k) q , u (ℓ)\n\np ] which shows that the set {u (k) q\n\n: k = 0, 1, • • • , 2j; q = -k, -k + 1, • • •\n\n, k} can be used to span the Lie algebra of the unitary group U(2j + 1). The latter result is at the root of the expansions (17) and (32).\n\nAfter the submission of the present paper for publication in Journal of Russian Laser Research, a pre-print dealing with the existence of SIC-POVMs was posted on arXiv [32] .\n\nThe main result in [32] is that SIC-POVMs exist in all dimensions. As a corollary of this result, Eq. (34) admits solutions in any dimension.\n\n12\n\nThis work was presented at the International Conference on Squeezed States and Uncertainty Relations, University of Bradford, England (ICSSUR'07). The authors wish to thank the organizer A. Vourdas and are grateful to D. M. Appleby, V. I. Man'ko and M.\n\nPlanat for interesting comments. 13 References [1] A. 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B 20, 1942 (2006) [15] J. Schwinger, Proc. Nat. Acad. Sci. USA 46, 570 (1960) [16] P. Delsarte, J. M. Goethals and J. J. Seidel, Philips Res. Repts. 30, 91 (1975) 14 [17] I. D. Ivanović, J. Phys. A: Math. Gen. 14, 3241 (1981) [18] W. K. Wootters, Ann. Phys. (N.Y.) 176, 1 (1987) [19] H. Barnum, Preprint quant-ph/0205155 (2002) [20] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury and F. Vatan, Algorithmica 34, 512 (2002) [21] A. O. Pittenger and M. H. Rubin, Linear Alg. Appl. 390, 255 (2004) [22] M. Saniga, M. Planat and H. Rosu, J. Opt. B: Quantum Semiclassical Opt. 6, L19 (2004) [23] I. Bengtsson and Å. Ericsson, Open Syst. Inf. Dyn. 12, 107 (2005) [24] A. Klappenecker and M. Rötteler, Preprint quant-ph/0502031 (2005) [25] W. K. Wootters, Found. Phys. 36, 112 (2006) [26] M. R. Kibler and M. Planat, Int. J. Mod. Phys. B 20, 1802 (2006) [27] O. Albouy and M. R. Kibler, SIGMA 3, article 076 (2007) [28] H.-P. Breuer, J. Phys. A: Math. Gen. 38, 9019 (2005) [29] G. Racah, Phys. Rev. 62, 438 (1942) [30] U. Fano and G. Racah, \"Irreducible Tensorial Sets\", New York: Academic (1959) [31] M. Kibler and G. Grenet, J. Math. Phys. 21, 422 (1980) [32] J.L. Hall and A. Rao, Preprint quant-ph/0707.3002v1 (20 July 2007) 15"},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"INTRODUCTION","text":"The importance of finite-dimensional spaces for quantum mechanics is well recognized (see for instance [1]-[3]). In particular, such spaces play a major role in quantum information theory, especially for quantum cryptography and quantum state tomography [4]- [27] .\n\nAlong this vein, a symmetric informationally complete (SIC) positive operator valued measure (POVM) is a set of operators acting on a finite Hilbert space [4]-[14] (see also [3] for an infinite Hilbert space) and mutually unbiased bases (MUBs) are specific bases for such a space [15]- [27] .\n\nThe introduction of POVMs goes back to the seventies [4]- [7] . The most general quantum measurement is represented by a POVM. In the present work, we will be interested 1 in SIC-POVMs, for which the statistics of the measurement allows the reconstruction of the quantum state. Moreover, those POVMs are endowed with an extra symmetry condition (see definition in Sec. 2). The notion of MUBs (see definition in Sec. 3), implicit or explicit in the seminal works of [15] - [18] , has been the object of numerous mathematical and physical investigations during the last two decades in connection with the so-called complementary observables. Unfortunately, the question to know, for a given Hilbert space of finite dimension d, whether there exist SIC-POVMs and how many MUBs there exist has remained an open one.\n\nThe aim of this note is to develop a unified approach to SIC-POVMs and MUBs based on a complex vector space of higher dimension, viz. d 2 instead of d. We then give a specific example of this approach grounded on the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) recently used for a study of entanglement of rotationally invariant spin systems [28] and for an angular momentum study of MUBs [26, 27] .\n\nMost of the notations in this work are standard. Let us simply mention that I is the identity operator, the bar indicates complex conjugation, A † denotes the adjoint of the operator A, δ a,b stands for the Kronecker symbol for a and b, and ∆(a, b, c) is 1 or 0 according as a, b and c satisfy or not the triangular inequality."},{"section_type":"OTHER","section_title":"SIC-POVMs","text":"Let C d be the standard Hilbert space of dimension d endowed with its usual inner product denoted by | . As is usual, we will identify a POVM with a nonorthogonal decomposition of the identity. Thus, a discrete SIC-POVM is a set {P x : x = 1, 2, • • • , d 2 } of d 2 nonnegative operators P x acting on C d , such that:\n\n• they satisfy the trace or symmetry condition\n\nTr (P x P y ) = 1 d + 1 , x = y; ( 1\n\n)\n\nmoreover, we will assume the operators P x are normalized, thus completing this 2 condition with\n\nTr P 2 x = 1; ( 2\n\n)\n\n• they form a decomposition of the identity\n\n1 d d 2 x=1 P x = I; ( 3\n\n)\n\n• they satisfy a completeness condition: the knowledge of the probabilities p x defined by p x = Tr(P x ρ) is sufficient to reconstruct the density matrix ρ.\n\nNow, let us develop each of the operators P x on an orthonormal (with respect to the Hilbert-Schmidt product) basis {u i : i = 1, 2, • • • , d 2 } of the space of linear operators on\n\nC d P x = d 2 i=1 v i (x)u i , ( 4\n\n)\n\nwhere the operators u i satisfy Tr(u † i u j ) = δ i,j . The operators P x are thus considered as vectors\n\nv(x) = (v 1 (x), v 2 (x), • • • , v d 2 (x)) ( 5\n\n)\n\nin the Hilbert space C d 2 of dimension d 2 and the determination of the operators P x is equivalent to the determination of the components v i (x) of v(x). In this language, the trace property (1) together with the normalization condition (2) give y) is the usual Hermitian product in C d 2 . In order to compare Eq. ( 6 ) with what usually happens in the search for SIC-POVMs, we suppose from now on that the operators P x are rank-one operators. Therefore, by putting\n\nv(x) • v(y) = 1 d + 1 (dδ x,y + 1) , ( 6\n\n) where v(x) • v(y) = d 2 i=1 v i (x)v i (\n\nP x = |Φ x Φ x | ( 7\n\n) 3 with |φ x ∈ C d , the trace property (1, 2) reads | Φ x |Φ y | 2 = 1 d + 1 (dδ x,y + 1) . ( 8\n\n)\n\nFrom this point of view, to find d 2 operators P x is equivalent to finding d 2 vectors |φ x in C d satisfying Eq. ( 8 ). At the price of an increase in the number of components from d 3 (for d 2 vectors in C d ) to d 4 (for d 2 vectors in C d 2 ), we have got rid of the square modulus to result in a single scalar product (compare Eqs. (6) and (8)), what may prove to be suitable for another way to search for SIC-POVMs. Moreover, our relation (6) is independent of any hypothesis on the rank of the operators P x . In fact, there exists a lot of relations among these d 4 coefficients that decrease the effective number of coefficients to be found and give structural constraints on them. Those relations are highly sensitive to the choice of the basis {u i : i = 1, 2, • • • , d 2 } and we are going to exhibit an example of such a set of relations by choosing the basis to consist of Racah unit tensors.\n\nThe cornerstone of this approach is to identify C d with a subspace ε(j) of constant angular momentum j = (d -1)/2. Such a subspace is spanned by the set {|j, m : m = -j, -j + 1, • • • , j}, where |j, m is an eigenvector of the square and the z-component of a generalized angular momentum operator. Let u (k) be the Racah unit tensor [29]\n\nof order k (with k = 0, 1, • • • , 2j) defined by its 2k + 1 components u (k) q (where q = -k, -k + 1, • • • , k) through u (k) q = j m=-j j m ′ =-j (-1) j-m j k j -m q m ′ |j, m j, m ′ |, ( 9\n\n)\n\nwhere (• • •) denotes a 3-jm Wigner symbol. For fixed j, the (2j + 1) 2 operators u\n\n(k) q (with k = 0, 1, • • • , 2j and q = -k, -k + 1, • • • , k) act on ε(j) ∼ C d and form a basis\n\nof the Hilbert space C N of dimension N = (2j + 1) 2 , the inner product in C N being the Hilbert-Schmidt product. The formulas (involving unit tensors, 3-jm and 6-j symbols) relevant for this work are given in Appendix (see also [29] to [31] ). We must remember that those Racah operators are not normalized to unity (see relation (46)). So this will generate an extra factor when defining v i (x).\n\n4 Each operator P x can be developed as a linear combination of the operators u (k) q .\n\nHence, we have\n\nP x = 2j k=0 k q=-k c kq (x)u (k) q , ( 10\n\n)\n\nwhere the unknown expansion coefficients c kq (x) are a priori complex numbers. The determination of the operators P x is thus equivalent to the determination of the coefficients c kq (x), which are formally given by\n\nc kq (x) = (2k + 1) Φ x |u (k) q |Φ x , ( 11\n\n)\n\nas can be seen by multiplying each member of Eq. (10) by the adjoint of u (ℓ) p and then using Eq. (46) of Appendix.\n\nBy defining the vector\n\nv(x) = (v 1 (x), v 2 (x), • • • , v N (x)), N = (2j + 1) 2 ( 12\n\n) via v i (x) = 1 √ 2k + 1 c kq (x), i = k 2 + k + q + 1, ( 13\n\n)\n\nthe following properties and relations are obtained.\n\n• The first component v 1 (x) of v(x) does not depend on x since c 00 (x) = 1 √ 2j + 1 ( 14\n\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }.\n\nProof: Take the trace of Eq. (10) and use Eq. (48) of Appendix.\n\n• The components v i (x) of v(x) satisfy the complex conjugation property described by c kq (x) = (-1) q c k-q (x) ( 15\n\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\n\nProof: Use the Hermitian property of P x and Eq. (43) of Appendix.\n\n• In terms of c kq , Eq. (6) reads\n\n2j k=0 1 2k + 1 k q=-k c kq (x)c kq (y) = 1 2(j + 1) [(2j + 1)δ x,y + 1] ( 16\n\n) for all x, y ∈ {1, 2, • • • , (2j + 1) 2 },\n\nwhere the sum over q is SO(3) rotationally invariant.\n\nProof: The proof is trivial.\n\n• The coefficients c kq (x) are solutions of the nonlinear system given by\n\n1 2K + 1 c KQ (x) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ k ℓ K -q -p Q × k ℓ K j j j c kq (x)c ℓp (x) ( 17\n\n) for all x ∈ {1, 2, • • • , (2j+1) 2 }, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K+1, • • • , K}.\n\nProof: Consider P 2 x = P x and use the coupling relation (51) of Appendix involving a 3-jm and a 6-j Wigner symbols.\n\nAs a corollary of the latter property, by taking K = 0 and using Eqs. (47) and (50) of Appendix, we get again the normalization relation v(x\n\n) 2 = v(x) • v(x) = 1.\n\n• All coefficients c kq (x) are connected through the sum rule\n\n(2j+1) 2 x=1 2j k=0 k q=-k c kq (x) j k j -m q m ′ = (-1) j-m (2j + 1)δ m,m ′ , ( 18\n\n)\n\nwhich turns out to be useful for global checking purposes.\n\nProof: Take the jm-jm ′ matrix element of the resolution of the identity in terms of the operators P x /(2j + 1).\n\nA complete set of MUBs in the Hilbert space\n\nC d is a set of d(d + 1) vectors |aα ∈ C d such that | aα|bβ | 2 = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 19\n\n) 6 where a = 0, 1, • • • , d and α = 0, 1, • • • , d -1.\n\nThe indices of type a refer to the bases and, for fixed a, the index α refers to one of the d vectors of the basis corresponding to a.\n\nWe know that such a complete set exists if d is a prime or the power of a prime (e.g., see [16]-[24]).\n\nThe approach developed in Sec. 2 for SIC-POVMs can be applied to MUBs too. Let us suppose that it is possible to\n\nfind d + 1 sets S a (with a = 0, 1, • • • , d) of vectors in C d , each set S a = {|aα : α = 0, 1, • • • , d -1} containing d vectors |aα such that Eq. ( 19\n\n)\n\nbe satisfied. This amounts to finding d(d + 1) projection operators\n\nΠ aα = |aα aα| ( 20\n\n)\n\nsatisfying the trace condition\n\nTr (Π aα Π bβ ) = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 21\n\n)\n\nwhere the trace is taken on C d . Therefore, they also form a nonorthogonal decomposition of the identity\n\n1 d + 1 d a=0 d-1 α=0 Π aα = I. ( 22\n\n)\n\nAs in Sec. 2, we develop each operator Π aα on an orthonormal basis with expansion coefficients w i (aα). Thus we get vectors w(aα) in\n\nC d 2 w(aα) = (w 1 (aα), w 2 (aα), • • • , w d 2 (aα)) ( 23\n\n) such that w(aα) • w(bβ) = δ α,β δ a,b + 1 d (1 -δ a,b ) ( 24\n\n) for all a, b ∈ {0, 1, • • • , d} and α, β ∈ {0, 1, • • • , d -1}.\n\nNow we draw the same relations as for POVMs by choosing the Racah operators to be our basis in C d 2 . We assume once again that the Hilbert space C d is realized by ε(j) with j = (d -1)/2. Then, each operator Π aα can be developed on the basis of the (2j + 1) 2 7 operators u (k) q as\n\nΠ aα = 2j k=0 k q=-k d kq (aα)u (k) q , ( 25\n\n)\n\nto be compared with Eq. (10). The expansion coefficients are d kq (aα) = (2k + 1) aα|u\n\n(k) q |aα ( 26\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1,\n\n• • • , 2j} and q ∈ {-k, -k + 1, • • • , k}. For a and α fixed, the complex coefficients d kq (aα) define a vector\n\nw(aα) = (w 1 (aα), w 2 (aα), • • • , w N (aα)) , N = (2j + 1) 2 ( 27\n\n)\n\nin the Hilbert space C N , the components of which are given by\n\nw i (aα) = 1 √ 2k + 1 d kq (aα), i = k 2 + k + q + 1. ( 28\n\n)\n\nWe are thus led to the following properties and relations. The proofs are similar to those in Sec. 2.\n\n• First component w 1 (aα) of w(aα):\n\nd 00 (aα) = 1 √ 2j + 1 ( 29\n\n) for all a ∈ {0, 1, • • • , 2j + 1} and α ∈ {0, 1, • • • , 2j}.\n\n• Complex conjugation property:\n\nd kq (aα) = (-1) q d k-q (aα) ( 30\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\n\n• Rotational invariance:\n\n2j k=0 1 2k + 1 k q=-k d kq (aα)d kq (bβ) = δ α,β δ a,b + 1 2j + 1 (1 -δ a,b ) ( 31\n\n) for all a, b ∈ {0, 1, • • • , 2j + 1} and α, β ∈ {0, 1, • • • , 2j}.\n\n• Tensor product formula: 1 2K + 1 d KQ (aα) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ\n\nk ℓ K -q -p Q × k ℓ K j j j d kq (aα)d ℓp (aα) ( 32\n\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K + 1, • • • , K}.\n\n• Sum rule:\n\n2j+1 a=0 2j α=0 2j k=0 k q=-k d kq (aα) j k j -m q m ′ = (-1) j-m 2(2j + 1)δ m,m ′ ( 33\n\n)\n\nwhich involves all coefficients d kq (aα)."},{"section_type":"CONCLUSION","section_title":"CONCLUSIONS","text":"Although the structure of the relations in Sec. 1 on the one hand and Sec. 2 on the other hand is very similar, there are deep differences between the two sets of results. The similarities are reminiscent of the fact that both MUBs and SIC-POVMs can be linked to finite affine planes [12, 13, 22, 23, 25] and to complex projective 2-designs [8, 10, 19, 24] .\n\nOn the other side, there are two arguments in favor of the differences between relations (6) and (24). First, the problem of constructing SIC-POVMs in dimension d is not equivalent to the existence of an affine plane of order d [12, 13] . Second, there is a consensus around the conjecture according to which there exists a complete set of MUBs in dimension d if and only if there exists an affine plane of order d [22].\n\nIn dimension d, to find d 2 operators P x of a SIC-POVM acting on the Hilbert space\n\nC d amounts to find d 2 vectors v(x) in the Hilbert space C N with N = d 2 satisfying v x = 1, v(x) • v(y) = 1 d + 1 for x = y ( 34\n\n)\n\n(the norm v(x) of each vector v(x) is 1 and the angle ω xy of any pair of vectors v(x) and v(y) is ω xy = cos -1 [1/(d + 1)] for x = y).\n\nIn a similar way, to find d + 1 MUBs of C d is equivalent to find d + 1 sets S a (with a = 0, 1, • • • , d) of d vectors, i.e., d(d + 1) vectors in all, w(aα) in C N with N = d 2 satisfying w(aα) • w(aβ) = δ α,β , w(aα) • w(bβ) = 1 d for a = b (35) (each set S a consists of d orthonormalized vectors and the angle ω aαbβ of any vector w(aα) of a set S a with any vector w(bβ) of a set S b is ω aαbβ = cos -1 (1/d) for a = b).\n\nAccording to a well accepted conjecture [8, 10], SIC-POVMs should exist in any\n\ndimension. The present study shows that in order to prove this conjecture it is sufficient to prove that Eq. (34) admits solutions for any value of d.\n\nThe situation is different for MUBs. In dimension d, it is known that there exist d + 1 sets of d vectors of type |aα in C d satisfying Eq. (19) when d is a prime or the power of a prime. This shows that Eq. ( 35 ) can be solved for d prime or power of a prime. For d prime, it is possible to find an explicit solution of Eq. ( 19 ). In fact, we have [26, 27] |aα = 1 √ 2j + 1 j m=-j ω (j+m)(j-m+1)a/2+(j+m)\n\nα |j, m , ( 36\n\n) ω = exp i 2π 2j + 1 , j = 1 2 (d -1) ( 37\n\n) for a, α ∈ {0, 1, • • • , 2j} while |aα = |j, m ( 38\n\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j. Then, Eq. (26) yields d kq (aα) = 2k + 1 2j + 1 j m=-j j m ′ =-j ω θ(m,m ′ ) (-1) j-m j k j -m q m ′ , ( 39\n\n) θ(m, m ′ ) = (m -m ′ ) 1 2 (1 -m -m ′ )a + α ( 40\n\n) for a, α ∈ {0, 1, • • • , 2j} while d kq (aα) = δ q,0 (2k + 1)(-1) j-m j k j -m 0 m ( 41\n\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j.\n\nIt can be shown that Eqs. (40) and (41) are in agreement with the results of Sec. 3. We thus have a solution of the equations for 10 the results of Sec. 3 when d is prime. As an open problem, it would be worthwhile to find an explicit solution for the coefficients d kq (aα) when d = 2j + 1 is any positive power of a prime. Finally, note that to prove (or disprove) the conjecture according to which a complete set of MUBs in dimension d exists only if d is a prime or the power of a prime is equivalent to prove (or disprove) that Eq. ( 35 ) has a solution only if d is a prime or the power of a prime.\n\nAPPENDIX: WIGNER-RACAH ALGEBRA OF SU(2) ⊃ U(1)\n\nWe limit ourselves to those basic formulas for the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) which are necessary to derive the results of this paper. The summations in this appendix have to be extended to the allowed values for the involved magnetic and angular momentum quantum numbers.\n\nThe definition (9) of the components u (k) q of the Racah unit tensor u (k) yields\n\nj, m|u (k) q |j, m ′ = (-1) j-m j k j -m q m ′ , ( 42\n\n)\n\nfrom which we easily obtain the Hermitian conjugation property\n\nu (k) q † = (-1) q u (k) -q . ( 43\n\n)\n\nThe 3-jm Wigner symbol in Eq. (42) satisfies the orthogonality relations\n\nmm ′ j j ′ k m m ′ q j j ′ ℓ m m ′ p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j ′ , k) ( 44\n\n) and kq (2k + 1) j j ′ k m m ′ q j j ′ k M M ′ q = δ m,M δ m ′ ,M ′ . ( 45\n\n)\n\nThe trace relation on the space ε(j)\n\nTr u (k) q † u (ℓ) p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j, k) ( 46\n\n)\n\n11 easily follows by combining Eqs. (42) and (44). Furthermore, by introducing\n\nj j ′ 0 m -m ′ 0 = δ j,j ′ δ m,m ′ (-1) j-m 1 √ 2j + 1 ( 47\n\n)\n\nin Eq. ( 44 ), we obtain the sum rule\n\nm (-1) j-m j k j -m q m = 2j + 1δ k,0 δ q,0 ∆(j, k, j), ( 48\n\n)\n\nknown in spectroscopy as the barycenter theorem.\n\nThere are several relations involving 3-jm and 6-j symbols. In particular, we have\n\nmm ′ M (-1) j-M j j -m q M j ℓ j -M p m ′ j K j -m Q m ′ = (-1) 2j-Q k ℓ K -q -p Q k ℓ K j j j , ( 49\n\n)\n\nwhere {• • •} denotes a 6-j Wigner symbol (or W Racah coefficient). Note that the introduction of k ℓ 0 j j J = δ k,ℓ (-1) j+k+J 1 (2k + 1)(2j + 1) (50) in Eq. (49) gives back Eq. ( 44 ). Equation (49) is central in the derivation of the coupling relation\n\nu (k) q u (ℓ) p = KQ (-1) 2j-Q (2K + 1) k ℓ K -q -p Q k ℓ K j j j u (K) Q . ( 51\n\n)\n\nEquation (51) makes it possible to calculate the commutator [u\n\n(k) q , u (ℓ)\n\np ] which shows that the set {u (k) q\n\n: k = 0, 1, • • • , 2j; q = -k, -k + 1, • • •\n\n, k} can be used to span the Lie algebra of the unitary group U(2j + 1). The latter result is at the root of the expansions (17) and (32)."},{"section_type":"OTHER","section_title":"Note added in version 3","text":"After the submission of the present paper for publication in Journal of Russian Laser Research, a pre-print dealing with the existence of SIC-POVMs was posted on arXiv [32] .\n\nThe main result in [32] is that SIC-POVMs exist in all dimensions. As a corollary of this result, Eq. (34) admits solutions in any dimension.\n\n12"},{"section_type":"OTHER","section_title":"Acknowledgements","text":"This work was presented at the International Conference on Squeezed States and Uncertainty Relations, University of Bradford, England (ICSSUR'07). The authors wish to thank the organizer A. Vourdas and are grateful to D. M. Appleby, V. I. Man'ko and M.\n\nPlanat for interesting comments. 13 References [1] A. Peres, \"Quantum Theory: Concepts and Methods\", Dordrecht: Kluwer (1995) [2] A. Vourdas, J. Phys. A: Math. Gen. 38, 8453 (2005) [3] W. M. de Muynck, \"Foundations of Quantum Mechanics, an Empiricist Approach\", Dordrecht: Kluwer (2002) [4] J. M. Jauch and C. Piron, Helv. Phys. Acta 40, 559 (1967) [5] E. B. Davies and J. T. Levis, Comm. Math. Phys. 17, 239 (1970) [6] E. B. Davies, IEEE Trans. Inform. Theory IT-24, 596 (1978) [7] K. Kraus, \"States, Effects, and Operations\", Lect. Notes Phys. 190 (1983) [8] G. Zauner, Diploma Thesis, University of Wien (1999) [9] C. M. Caves, C. A. Fuchs and R. Schack, J. Math. Phys. 43, 4537 (2002) [10] J. M. Renes, R. Blume-Kohout, A. J. Scott and C. M. Caves, J. Math. Phys. 45, 2171 (2004) [11] D. M. Appleby, J. Math. Phys. 46, 052107 (2005) [12] M. Grassl, Proc. ERATO Conf. Quant. Inf. Science (EQIS 2004) ed. J. Gruska, Tokyo (2005) [13] M. Grassl, Elec. Notes Discrete Math. 20, 151 (2005) [14] S. Weigert, Int. J. Mod. Phys. B 20, 1942 (2006) [15] J. Schwinger, Proc. Nat. Acad. Sci. USA 46, 570 (1960) [16] P. Delsarte, J. M. Goethals and J. J. Seidel, Philips Res. Repts. 30, 91 (1975) 14 [17] I. D. Ivanović, J. Phys. A: Math. Gen. 14, 3241 (1981) [18] W. K. Wootters, Ann. Phys. (N.Y.) 176, 1 (1987) [19] H. Barnum, Preprint quant-ph/0205155 (2002) [20] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury and F. Vatan, Algorithmica 34, 512 (2002) [21] A. O. Pittenger and M. H. Rubin, Linear Alg. Appl. 390, 255 (2004) [22] M. Saniga, M. Planat and H. Rosu, J. Opt. B: Quantum Semiclassical Opt. 6, L19 (2004) [23] I. Bengtsson and Å. Ericsson, Open Syst. Inf. Dyn. 12, 107 (2005) [24] A. Klappenecker and M. Rötteler, Preprint quant-ph/0502031 (2005) [25] W. K. Wootters, Found. Phys. 36, 112 (2006) [26] M. R. Kibler and M. Planat, Int. J. Mod. Phys. B 20, 1802 (2006) [27] O. Albouy and M. R. Kibler, SIGMA 3, article 076 (2007) [28] H.-P. Breuer, J. Phys. A: Math. Gen. 38, 9019 (2005) [29] G. Racah, Phys. Rev. 62, 438 (1942) [30] U. Fano and G. Racah, \"Irreducible Tensorial Sets\", New York: Academic (1959) [31] M. Kibler and G. Grenet, J. Math. Phys. 21, 422 (1980) [32] J.L. Hall and A. Rao, Preprint quant-ph/0707.3002v1 (20 July 2007) 15"}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"INTRODUCTION\",\n \"text\": \"The importance of finite-dimensional spaces for quantum mechanics is well recognized (see for instance [1]-[3]). In particular, such spaces play a major role in quantum information theory, especially for quantum cryptography and quantum state tomography [4]- [27] .\\n\\nAlong this vein, a symmetric informationally complete (SIC) positive operator valued measure (POVM) is a set of operators acting on a finite Hilbert space [4]-[14] (see also [3] for an infinite Hilbert space) and mutually unbiased bases (MUBs) are specific bases for such a space [15]- [27] .\\n\\nThe introduction of POVMs goes back to the seventies [4]- [7] . The most general quantum measurement is represented by a POVM. In the present work, we will be interested 1 in SIC-POVMs, for which the statistics of the measurement allows the reconstruction of the quantum state. Moreover, those POVMs are endowed with an extra symmetry condition (see definition in Sec. 2). The notion of MUBs (see definition in Sec. 3), implicit or explicit in the seminal works of [15] - [18] , has been the object of numerous mathematical and physical investigations during the last two decades in connection with the so-called complementary observables. Unfortunately, the question to know, for a given Hilbert space of finite dimension d, whether there exist SIC-POVMs and how many MUBs there exist has remained an open one.\\n\\nThe aim of this note is to develop a unified approach to SIC-POVMs and MUBs based on a complex vector space of higher dimension, viz. d 2 instead of d. We then give a specific example of this approach grounded on the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) recently used for a study of entanglement of rotationally invariant spin systems [28] and for an angular momentum study of MUBs [26, 27] .\\n\\nMost of the notations in this work are standard. Let us simply mention that I is the identity operator, the bar indicates complex conjugation, A † denotes the adjoint of the operator A, δ a,b stands for the Kronecker symbol for a and b, and ∆(a, b, c) is 1 or 0 according as a, b and c satisfy or not the triangular inequality.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"SIC-POVMs\",\n \"text\": \"Let C d be the standard Hilbert space of dimension d endowed with its usual inner product denoted by | . As is usual, we will identify a POVM with a nonorthogonal decomposition of the identity. Thus, a discrete SIC-POVM is a set {P x : x = 1, 2, • • • , d 2 } of d 2 nonnegative operators P x acting on C d , such that:\\n\\n• they satisfy the trace or symmetry condition\\n\\nTr (P x P y ) = 1 d + 1 , x = y; ( 1\\n\\n)\\n\\nmoreover, we will assume the operators P x are normalized, thus completing this 2 condition with\\n\\nTr P 2 x = 1; ( 2\\n\\n)\\n\\n• they form a decomposition of the identity\\n\\n1 d d 2 x=1 P x = I; ( 3\\n\\n)\\n\\n• they satisfy a completeness condition: the knowledge of the probabilities p x defined by p x = Tr(P x ρ) is sufficient to reconstruct the density matrix ρ.\\n\\nNow, let us develop each of the operators P x on an orthonormal (with respect to the Hilbert-Schmidt product) basis {u i : i = 1, 2, • • • , d 2 } of the space of linear operators on\\n\\nC d P x = d 2 i=1 v i (x)u i , ( 4\\n\\n)\\n\\nwhere the operators u i satisfy Tr(u † i u j ) = δ i,j . The operators P x are thus considered as vectors\\n\\nv(x) = (v 1 (x), v 2 (x), • • • , v d 2 (x)) ( 5\\n\\n)\\n\\nin the Hilbert space C d 2 of dimension d 2 and the determination of the operators P x is equivalent to the determination of the components v i (x) of v(x). In this language, the trace property (1) together with the normalization condition (2) give y) is the usual Hermitian product in C d 2 . In order to compare Eq. ( 6 ) with what usually happens in the search for SIC-POVMs, we suppose from now on that the operators P x are rank-one operators. Therefore, by putting\\n\\nv(x) • v(y) = 1 d + 1 (dδ x,y + 1) , ( 6\\n\\n) where v(x) • v(y) = d 2 i=1 v i (x)v i (\\n\\nP x = |Φ x Φ x | ( 7\\n\\n) 3 with |φ x ∈ C d , the trace property (1, 2) reads | Φ x |Φ y | 2 = 1 d + 1 (dδ x,y + 1) . ( 8\\n\\n)\\n\\nFrom this point of view, to find d 2 operators P x is equivalent to finding d 2 vectors |φ x in C d satisfying Eq. ( 8 ). At the price of an increase in the number of components from d 3 (for d 2 vectors in C d ) to d 4 (for d 2 vectors in C d 2 ), we have got rid of the square modulus to result in a single scalar product (compare Eqs. (6) and (8)), what may prove to be suitable for another way to search for SIC-POVMs. Moreover, our relation (6) is independent of any hypothesis on the rank of the operators P x . In fact, there exists a lot of relations among these d 4 coefficients that decrease the effective number of coefficients to be found and give structural constraints on them. Those relations are highly sensitive to the choice of the basis {u i : i = 1, 2, • • • , d 2 } and we are going to exhibit an example of such a set of relations by choosing the basis to consist of Racah unit tensors.\\n\\nThe cornerstone of this approach is to identify C d with a subspace ε(j) of constant angular momentum j = (d -1)/2. Such a subspace is spanned by the set {|j, m : m = -j, -j + 1, • • • , j}, where |j, m is an eigenvector of the square and the z-component of a generalized angular momentum operator. Let u (k) be the Racah unit tensor [29]\\n\\nof order k (with k = 0, 1, • • • , 2j) defined by its 2k + 1 components u (k) q (where q = -k, -k + 1, • • • , k) through u (k) q = j m=-j j m ′ =-j (-1) j-m j k j -m q m ′ |j, m j, m ′ |, ( 9\\n\\n)\\n\\nwhere (• • •) denotes a 3-jm Wigner symbol. For fixed j, the (2j + 1) 2 operators u\\n\\n(k) q (with k = 0, 1, • • • , 2j and q = -k, -k + 1, • • • , k) act on ε(j) ∼ C d and form a basis\\n\\nof the Hilbert space C N of dimension N = (2j + 1) 2 , the inner product in C N being the Hilbert-Schmidt product. The formulas (involving unit tensors, 3-jm and 6-j symbols) relevant for this work are given in Appendix (see also [29] to [31] ). We must remember that those Racah operators are not normalized to unity (see relation (46)). So this will generate an extra factor when defining v i (x).\\n\\n4 Each operator P x can be developed as a linear combination of the operators u (k) q .\\n\\nHence, we have\\n\\nP x = 2j k=0 k q=-k c kq (x)u (k) q , ( 10\\n\\n)\\n\\nwhere the unknown expansion coefficients c kq (x) are a priori complex numbers. The determination of the operators P x is thus equivalent to the determination of the coefficients c kq (x), which are formally given by\\n\\nc kq (x) = (2k + 1) Φ x |u (k) q |Φ x , ( 11\\n\\n)\\n\\nas can be seen by multiplying each member of Eq. (10) by the adjoint of u (ℓ) p and then using Eq. (46) of Appendix.\\n\\nBy defining the vector\\n\\nv(x) = (v 1 (x), v 2 (x), • • • , v N (x)), N = (2j + 1) 2 ( 12\\n\\n) via v i (x) = 1 √ 2k + 1 c kq (x), i = k 2 + k + q + 1, ( 13\\n\\n)\\n\\nthe following properties and relations are obtained.\\n\\n• The first component v 1 (x) of v(x) does not depend on x since c 00 (x) = 1 √ 2j + 1 ( 14\\n\\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }.\\n\\nProof: Take the trace of Eq. (10) and use Eq. (48) of Appendix.\\n\\n• The components v i (x) of v(x) satisfy the complex conjugation property described by c kq (x) = (-1) q c k-q (x) ( 15\\n\\n) for all x ∈ {1, 2, • • • , (2j + 1) 2 }, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\\n\\nProof: Use the Hermitian property of P x and Eq. (43) of Appendix.\\n\\n• In terms of c kq , Eq. (6) reads\\n\\n2j k=0 1 2k + 1 k q=-k c kq (x)c kq (y) = 1 2(j + 1) [(2j + 1)δ x,y + 1] ( 16\\n\\n) for all x, y ∈ {1, 2, • • • , (2j + 1) 2 },\\n\\nwhere the sum over q is SO(3) rotationally invariant.\\n\\nProof: The proof is trivial.\\n\\n• The coefficients c kq (x) are solutions of the nonlinear system given by\\n\\n1 2K + 1 c KQ (x) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ k ℓ K -q -p Q × k ℓ K j j j c kq (x)c ℓp (x) ( 17\\n\\n) for all x ∈ {1, 2, • • • , (2j+1) 2 }, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K+1, • • • , K}.\\n\\nProof: Consider P 2 x = P x and use the coupling relation (51) of Appendix involving a 3-jm and a 6-j Wigner symbols.\\n\\nAs a corollary of the latter property, by taking K = 0 and using Eqs. (47) and (50) of Appendix, we get again the normalization relation v(x\\n\\n) 2 = v(x) • v(x) = 1.\\n\\n• All coefficients c kq (x) are connected through the sum rule\\n\\n(2j+1) 2 x=1 2j k=0 k q=-k c kq (x) j k j -m q m ′ = (-1) j-m (2j + 1)δ m,m ′ , ( 18\\n\\n)\\n\\nwhich turns out to be useful for global checking purposes.\\n\\nProof: Take the jm-jm ′ matrix element of the resolution of the identity in terms of the operators P x /(2j + 1).\\n\\nA complete set of MUBs in the Hilbert space\\n\\nC d is a set of d(d + 1) vectors |aα ∈ C d such that | aα|bβ | 2 = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 19\\n\\n) 6 where a = 0, 1, • • • , d and α = 0, 1, • • • , d -1.\\n\\nThe indices of type a refer to the bases and, for fixed a, the index α refers to one of the d vectors of the basis corresponding to a.\\n\\nWe know that such a complete set exists if d is a prime or the power of a prime (e.g., see [16]-[24]).\\n\\nThe approach developed in Sec. 2 for SIC-POVMs can be applied to MUBs too. Let us suppose that it is possible to\\n\\nfind d + 1 sets S a (with a = 0, 1, • • • , d) of vectors in C d , each set S a = {|aα : α = 0, 1, • • • , d -1} containing d vectors |aα such that Eq. ( 19\\n\\n)\\n\\nbe satisfied. This amounts to finding d(d + 1) projection operators\\n\\nΠ aα = |aα aα| ( 20\\n\\n)\\n\\nsatisfying the trace condition\\n\\nTr (Π aα Π bβ ) = δ α,β δ a,b + 1 d (1 -δ a,b ), ( 21\\n\\n)\\n\\nwhere the trace is taken on C d . Therefore, they also form a nonorthogonal decomposition of the identity\\n\\n1 d + 1 d a=0 d-1 α=0 Π aα = I. ( 22\\n\\n)\\n\\nAs in Sec. 2, we develop each operator Π aα on an orthonormal basis with expansion coefficients w i (aα). Thus we get vectors w(aα) in\\n\\nC d 2 w(aα) = (w 1 (aα), w 2 (aα), • • • , w d 2 (aα)) ( 23\\n\\n) such that w(aα) • w(bβ) = δ α,β δ a,b + 1 d (1 -δ a,b ) ( 24\\n\\n) for all a, b ∈ {0, 1, • • • , d} and α, β ∈ {0, 1, • • • , d -1}.\\n\\nNow we draw the same relations as for POVMs by choosing the Racah operators to be our basis in C d 2 . We assume once again that the Hilbert space C d is realized by ε(j) with j = (d -1)/2. Then, each operator Π aα can be developed on the basis of the (2j + 1) 2 7 operators u (k) q as\\n\\nΠ aα = 2j k=0 k q=-k d kq (aα)u (k) q , ( 25\\n\\n)\\n\\nto be compared with Eq. (10). The expansion coefficients are d kq (aα) = (2k + 1) aα|u\\n\\n(k) q |aα ( 26\\n\\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1,\\n\\n• • • , 2j} and q ∈ {-k, -k + 1, • • • , k}. For a and α fixed, the complex coefficients d kq (aα) define a vector\\n\\nw(aα) = (w 1 (aα), w 2 (aα), • • • , w N (aα)) , N = (2j + 1) 2 ( 27\\n\\n)\\n\\nin the Hilbert space C N , the components of which are given by\\n\\nw i (aα) = 1 √ 2k + 1 d kq (aα), i = k 2 + k + q + 1. ( 28\\n\\n)\\n\\nWe are thus led to the following properties and relations. The proofs are similar to those in Sec. 2.\\n\\n• First component w 1 (aα) of w(aα):\\n\\nd 00 (aα) = 1 √ 2j + 1 ( 29\\n\\n) for all a ∈ {0, 1, • • • , 2j + 1} and α ∈ {0, 1, • • • , 2j}.\\n\\n• Complex conjugation property:\\n\\nd kq (aα) = (-1) q d k-q (aα) ( 30\\n\\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, k ∈ {0, 1, • • • , 2j} and q ∈ {-k, -k + 1, • • • , k}.\\n\\n• Rotational invariance:\\n\\n2j k=0 1 2k + 1 k q=-k d kq (aα)d kq (bβ) = δ α,β δ a,b + 1 2j + 1 (1 -δ a,b ) ( 31\\n\\n) for all a, b ∈ {0, 1, • • • , 2j + 1} and α, β ∈ {0, 1, • • • , 2j}.\\n\\n• Tensor product formula: 1 2K + 1 d KQ (aα) = (-1) 2j-Q 2j k=0 2j ℓ=0 k q=-k ℓ p=-ℓ\\n\\nk ℓ K -q -p Q × k ℓ K j j j d kq (aα)d ℓp (aα) ( 32\\n\\n) for all a ∈ {0, 1, • • • , 2j + 1}, α ∈ {0, 1, • • • , 2j}, K ∈ {0, 1, • • • , 2j} and Q ∈ {-K, -K + 1, • • • , K}.\\n\\n• Sum rule:\\n\\n2j+1 a=0 2j α=0 2j k=0 k q=-k d kq (aα) j k j -m q m ′ = (-1) j-m 2(2j + 1)δ m,m ′ ( 33\\n\\n)\\n\\nwhich involves all coefficients d kq (aα).\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"CONCLUSIONS\",\n \"text\": \"Although the structure of the relations in Sec. 1 on the one hand and Sec. 2 on the other hand is very similar, there are deep differences between the two sets of results. The similarities are reminiscent of the fact that both MUBs and SIC-POVMs can be linked to finite affine planes [12, 13, 22, 23, 25] and to complex projective 2-designs [8, 10, 19, 24] .\\n\\nOn the other side, there are two arguments in favor of the differences between relations (6) and (24). First, the problem of constructing SIC-POVMs in dimension d is not equivalent to the existence of an affine plane of order d [12, 13] . Second, there is a consensus around the conjecture according to which there exists a complete set of MUBs in dimension d if and only if there exists an affine plane of order d [22].\\n\\nIn dimension d, to find d 2 operators P x of a SIC-POVM acting on the Hilbert space\\n\\nC d amounts to find d 2 vectors v(x) in the Hilbert space C N with N = d 2 satisfying v x = 1, v(x) • v(y) = 1 d + 1 for x = y ( 34\\n\\n)\\n\\n(the norm v(x) of each vector v(x) is 1 and the angle ω xy of any pair of vectors v(x) and v(y) is ω xy = cos -1 [1/(d + 1)] for x = y).\\n\\nIn a similar way, to find d + 1 MUBs of C d is equivalent to find d + 1 sets S a (with a = 0, 1, • • • , d) of d vectors, i.e., d(d + 1) vectors in all, w(aα) in C N with N = d 2 satisfying w(aα) • w(aβ) = δ α,β , w(aα) • w(bβ) = 1 d for a = b (35) (each set S a consists of d orthonormalized vectors and the angle ω aαbβ of any vector w(aα) of a set S a with any vector w(bβ) of a set S b is ω aαbβ = cos -1 (1/d) for a = b).\\n\\nAccording to a well accepted conjecture [8, 10], SIC-POVMs should exist in any\\n\\ndimension. The present study shows that in order to prove this conjecture it is sufficient to prove that Eq. (34) admits solutions for any value of d.\\n\\nThe situation is different for MUBs. In dimension d, it is known that there exist d + 1 sets of d vectors of type |aα in C d satisfying Eq. (19) when d is a prime or the power of a prime. This shows that Eq. ( 35 ) can be solved for d prime or power of a prime. For d prime, it is possible to find an explicit solution of Eq. ( 19 ). In fact, we have [26, 27] |aα = 1 √ 2j + 1 j m=-j ω (j+m)(j-m+1)a/2+(j+m)\\n\\nα |j, m , ( 36\\n\\n) ω = exp i 2π 2j + 1 , j = 1 2 (d -1) ( 37\\n\\n) for a, α ∈ {0, 1, • • • , 2j} while |aα = |j, m ( 38\\n\\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j. Then, Eq. (26) yields d kq (aα) = 2k + 1 2j + 1 j m=-j j m ′ =-j ω θ(m,m ′ ) (-1) j-m j k j -m q m ′ , ( 39\\n\\n) θ(m, m ′ ) = (m -m ′ ) 1 2 (1 -m -m ′ )a + α ( 40\\n\\n) for a, α ∈ {0, 1, • • • , 2j} while d kq (aα) = δ q,0 (2k + 1)(-1) j-m j k j -m 0 m ( 41\\n\\n) for a = 2j + 1 and α = j + m = 0, 1, • • • , 2j.\\n\\nIt can be shown that Eqs. (40) and (41) are in agreement with the results of Sec. 3. We thus have a solution of the equations for 10 the results of Sec. 3 when d is prime. As an open problem, it would be worthwhile to find an explicit solution for the coefficients d kq (aα) when d = 2j + 1 is any positive power of a prime. Finally, note that to prove (or disprove) the conjecture according to which a complete set of MUBs in dimension d exists only if d is a prime or the power of a prime is equivalent to prove (or disprove) that Eq. ( 35 ) has a solution only if d is a prime or the power of a prime.\\n\\nAPPENDIX: WIGNER-RACAH ALGEBRA OF SU(2) ⊃ U(1)\\n\\nWe limit ourselves to those basic formulas for the Wigner-Racah algebra of the chain SU(2) ⊃ U(1) which are necessary to derive the results of this paper. The summations in this appendix have to be extended to the allowed values for the involved magnetic and angular momentum quantum numbers.\\n\\nThe definition (9) of the components u (k) q of the Racah unit tensor u (k) yields\\n\\nj, m|u (k) q |j, m ′ = (-1) j-m j k j -m q m ′ , ( 42\\n\\n)\\n\\nfrom which we easily obtain the Hermitian conjugation property\\n\\nu (k) q † = (-1) q u (k) -q . ( 43\\n\\n)\\n\\nThe 3-jm Wigner symbol in Eq. (42) satisfies the orthogonality relations\\n\\nmm ′ j j ′ k m m ′ q j j ′ ℓ m m ′ p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j ′ , k) ( 44\\n\\n) and kq (2k + 1) j j ′ k m m ′ q j j ′ k M M ′ q = δ m,M δ m ′ ,M ′ . ( 45\\n\\n)\\n\\nThe trace relation on the space ε(j)\\n\\nTr u (k) q † u (ℓ) p = 1 2k + 1 δ k,ℓ δ q,p ∆(j, j, k) ( 46\\n\\n)\\n\\n11 easily follows by combining Eqs. (42) and (44). Furthermore, by introducing\\n\\nj j ′ 0 m -m ′ 0 = δ j,j ′ δ m,m ′ (-1) j-m 1 √ 2j + 1 ( 47\\n\\n)\\n\\nin Eq. ( 44 ), we obtain the sum rule\\n\\nm (-1) j-m j k j -m q m = 2j + 1δ k,0 δ q,0 ∆(j, k, j), ( 48\\n\\n)\\n\\nknown in spectroscopy as the barycenter theorem.\\n\\nThere are several relations involving 3-jm and 6-j symbols. In particular, we have\\n\\nmm ′ M (-1) j-M j j -m q M j ℓ j -M p m ′ j K j -m Q m ′ = (-1) 2j-Q k ℓ K -q -p Q k ℓ K j j j , ( 49\\n\\n)\\n\\nwhere {• • •} denotes a 6-j Wigner symbol (or W Racah coefficient). Note that the introduction of k ℓ 0 j j J = δ k,ℓ (-1) j+k+J 1 (2k + 1)(2j + 1) (50) in Eq. (49) gives back Eq. ( 44 ). Equation (49) is central in the derivation of the coupling relation\\n\\nu (k) q u (ℓ) p = KQ (-1) 2j-Q (2K + 1) k ℓ K -q -p Q k ℓ K j j j u (K) Q . ( 51\\n\\n)\\n\\nEquation (51) makes it possible to calculate the commutator [u\\n\\n(k) q , u (ℓ)\\n\\np ] which shows that the set {u (k) q\\n\\n: k = 0, 1, • • • , 2j; q = -k, -k + 1, • • •\\n\\n, k} can be used to span the Lie algebra of the unitary group U(2j + 1). The latter result is at the root of the expansions (17) and (32).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Note added in version 3\",\n \"text\": \"After the submission of the present paper for publication in Journal of Russian Laser Research, a pre-print dealing with the existence of SIC-POVMs was posted on arXiv [32] .\\n\\nThe main result in [32] is that SIC-POVMs exist in all dimensions. As a corollary of this result, Eq. (34) admits solutions in any dimension.\\n\\n12\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgements\",\n \"text\": \"This work was presented at the International Conference on Squeezed States and Uncertainty Relations, University of Bradford, England (ICSSUR'07). The authors wish to thank the organizer A. Vourdas and are grateful to D. M. Appleby, V. I. Man'ko and M.\\n\\nPlanat for interesting comments. 13 References [1] A. Peres, \\\"Quantum Theory: Concepts and Methods\\\", Dordrecht: Kluwer (1995) [2] A. Vourdas, J. Phys. A: Math. Gen. 38, 8453 (2005) [3] W. M. de Muynck, \\\"Foundations of Quantum Mechanics, an Empiricist Approach\\\", Dordrecht: Kluwer (2002) [4] J. M. Jauch and C. Piron, Helv. Phys. Acta 40, 559 (1967) [5] E. B. Davies and J. T. Levis, Comm. Math. Phys. 17, 239 (1970) [6] E. B. Davies, IEEE Trans. Inform. Theory IT-24, 596 (1978) [7] K. Kraus, \\\"States, Effects, and Operations\\\", Lect. Notes Phys. 190 (1983) [8] G. Zauner, Diploma Thesis, University of Wien (1999) [9] C. M. Caves, C. A. Fuchs and R. Schack, J. Math. Phys. 43, 4537 (2002) [10] J. M. Renes, R. Blume-Kohout, A. J. Scott and C. M. Caves, J. Math. Phys. 45, 2171 (2004) [11] D. M. Appleby, J. Math. Phys. 46, 052107 (2005) [12] M. Grassl, Proc. ERATO Conf. Quant. Inf. Science (EQIS 2004) ed. J. Gruska, Tokyo (2005) [13] M. Grassl, Elec. Notes Discrete Math. 20, 151 (2005) [14] S. Weigert, Int. J. Mod. Phys. B 20, 1942 (2006) [15] J. Schwinger, Proc. Nat. Acad. Sci. USA 46, 570 (1960) [16] P. Delsarte, J. M. Goethals and J. J. Seidel, Philips Res. Repts. 30, 91 (1975) 14 [17] I. D. Ivanović, J. Phys. A: Math. Gen. 14, 3241 (1981) [18] W. K. Wootters, Ann. Phys. (N.Y.) 176, 1 (1987) [19] H. Barnum, Preprint quant-ph/0205155 (2002) [20] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury and F. Vatan, Algorithmica 34, 512 (2002) [21] A. O. Pittenger and M. H. Rubin, Linear Alg. Appl. 390, 255 (2004) [22] M. Saniga, M. Planat and H. Rosu, J. Opt. B: Quantum Semiclassical Opt. 6, L19 (2004) [23] I. Bengtsson and Å. Ericsson, Open Syst. Inf. Dyn. 12, 107 (2005) [24] A. Klappenecker and M. Rötteler, Preprint quant-ph/0502031 (2005) [25] W. K. Wootters, Found. Phys. 36, 112 (2006) [26] M. R. Kibler and M. Planat, Int. J. Mod. Phys. B 20, 1802 (2006) [27] O. Albouy and M. R. Kibler, SIGMA 3, article 076 (2007) [28] H.-P. Breuer, J. Phys. A: Math. Gen. 38, 9019 (2005) [29] G. Racah, Phys. Rev. 62, 438 (1942) [30] U. Fano and G. Racah, \\\"Irreducible Tensorial Sets\\\", New York: Academic (1959) [31] M. Kibler and G. Grenet, J. Math. Phys. 21, 422 (1980) [32] J.L. Hall and A. Rao, Preprint quant-ph/0707.3002v1 (20 July 2007) 15\"\n }\n]","isConversational":false}}},{"rowIdx":93,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0516"},"arxiv_id":{"kind":"string","value":"0704.0516"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":""},"content_hash":{"kind":"string","value":"4a119e65b80170869ee3687296117b16f01ab82d1c97a18278fec0312c5f9da7"},"title":{"kind":"string","value":"Effects of Imperfect Gate Operations in Shor's Prime Factorization\n Algorithm"},"abstract":{"kind":"string","value":"The effects of imperfect gate operations in implementation of Shor's prime\nfactorization algorithm are investigated. The gate imperfections may be\nclassified into three categories: the systematic error, the random error, and\nthe one with combined errors. It is found that Shor's algorithm is robust\nagainst the systematic errors but is vulnerable to the random errors. Error\nthreshold is given to the algorithm for a given number $N$ to be factorized."},"authors":{"kind":"list like","value":["Hao Guo","Gui Lu Long and Yang Sun"],"string":"[\n \"Hao Guo\",\n \"Gui Lu Long and Yang Sun\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["quant-ph"],"string":"[\n \"quant-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"Shor's factorization algorithm [1] is a very important quantum algorithm, through which one has demonstrated the power of quantum computers. It has greatly promoted the worldwide research in quantum computing over the past few years. In practice, however, quantum systems are subject to influence of environment, and in addition, quantum gate operations are often imperfect [2, 3] . Environment influence on the system can cause decoherence of quantum states, and gate imperfection leads to errors in quantum computing. Thanks to Shor's another important work, in which he showed that quantum error correlation can be corrected [4] . With quantum error correction scheme, errors arising from both decoherence and imperfection can be corrected.\n\nThere have been several works on the effects of decoherence on Shor's algorithm. Sun et al. discussed the effect of decoherence on the algorithm by modeling the environment [5] . Palma studied the effects of both decoherence and gate imperfection in ion trap quantum computers [6] . There have also been many other studies on the quantum algorithm [7, 8, 9, 10] .\n\nThe error correction scheme uses available resources. Thus it is important to study the robustness of the algorithm itself so that one can strike a balance between the amount of quantum error correction and the amount of qubits available. In this paper, we investigate the effects of gate imperfection on the efficiency of Shor's factorization algorithm. The results may guide us in practice to suppress deliberately those errors that influence the algorithm most sensitively. For those errors that do not affect the algorithm very much, we may ignore them as a good approximation. In addition, study of the robustness of algorithm to errors is important where one can not apply the quantum error correction at all, for instance, in cases that there are not enough qubits available.\n\nThe paper is organized as follows. Section II is devoted to an outline of Shor's algorithm and different error's modes. In Section III, we present the results. Finally, a short summary is given in Section IV.\n\nShor's algorithm consists of the following steps: 1) preparing a superposition of evenly distributed states |ψ = 1 √ q q-1 a=0 |a |0 ,\n\nwhere q = 2 L and N 2 ≤ q ≤ 2N 2 with N being the number to be factorized; 2) implementing y a modN and putting the results into the 2nd register\n\n|ψ 1 = 1 √ q q-1 a=0 |a |y a modN ;\n\n3) making a measument on the 2nd register; The state of the register is then\n\n|φ 2 = 1 √ A + 1 A j=0 |jr + l |z = y l = y jr+l modN\n\nwhere j ≤ q-l r = A.\n\n4) performing discrete Fourier transformation (DFT) on the first\n\nregister |φ 3 = c f (c) |c |z , where f (c) = √ r q q r -1 j=0 exp 2πi(jr + l) q = √ r q e 2πilc q q r -1 j=0 exp 2πijrc q .\n\n2 This term is nonzero only when c = k q r , with k = 0, 1, 2...r -1, which correspond to the peaks of the distribution in the measured results, and thus this term becomes f (c) = 1 √ r e 2πilc q . The Fourier transformation is important because it makes the state in the first register the same for all possible values in the 2nd register. The DFT is constructed by two basic gate operations: the single bit gate operation A j = 1 √ 2 1 1 1 -1 , which is also called the Walsh-Hadmard transformation, and the 2-bits controlled rotation\n\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e iθ jk with θ jk = π 2 k-j .\n\nThe gate sequence for implementing DFT is\n\n(A q-1 )(B q-2q-1 A q-2 ) . . . (B 0q-1 B 0q-2 . . . B 01 A 0 ). Errors can occur in both A j and B jk . A j is actually a rotation about y-axis through π 2 A j (θ) = e i h Syθ = I cos( θ 2 )-i sin( θ 2 )σ y = cos( θ 2 ) -sin( θ 2 ) sin( θ 2 ) cos( θ 2 )\n\n.\n\nIf the gate operation is not perfect, the rotation is not exactly π 2 . In this case, A j is a rotation of π 2 + 2δ A j (δ) = 1 √ 2 cos(δ) -sin(δ) -(sin(δ) + cos(δ)) sin(δ) + cos(δ) cos(δ) -sin(δ) .\n\nIf δ is very small, we have:\n\nA j (θ) = 1 √ 2 1 -δ -(1 + δ) 1 + δ+ 1 -δ .\n\nSimilarly, errors in B jk can be written as\n\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e i(θ jk +δ) .\n\nWith these errors, the DFT becomes\n\n|a → 1 √ q q-1 c=0 e i( 2π q/c +δc)a (1 + δ ′ c )|c = 1 √ q q-1 c=0 e i( 2πc q +δc)a (1 + δ ′ c )|c ,( 1\n\n)\n\nwhere δ c and δ ′ c denote the error of A j and B jk , respectively.\n\nLet us assume the following error modes: 1) systematic errors, where δ c or δ ′ c in (1) can only have systematic errors (EM 1 ); 2) random errors (EM 2 ), for which we assume that δ c or δ ′ c can only be random errors of the Gaussian or the uniform type; 3) coexistence of both systematic and random errors (EM 3 ). In the next section, we shall present the results of numerical simulations and discuss the effects of imperfect gate operation on the DFT algorithm, and thus on the Shor's algorithm.\n\nWe first discuss the influence of imperfect gate operations in the initial preparation\n\nA l-1 A l-2 ...A 0 |0...0 = 1 √ 2 (|0 + |1 + δ 1 (|0 -|1 )) ⊗ (|0 + |1 + δ 2 . = 1 √ 2 l 1 i1i2...in=0 |i 1 i 2 ...i n + 1 √ 2 l n R=1 δ n 1 i\n\nIf the errors are systematic, for instance, caused by the inaccurate calibration of the rotations, then δ 1 = δ 2 = . . . = δ n = δ. In this case, we can write the 2nd term as\n\n|ψ = 1 √ 2 l δ 1 i1i2...in=0 (2s -n)|i 1 i 2 ...i n ,\n\nwhere s stands for the number of 1's, and 2sn = s -(ns) is the difference in the number of 1's and 0's. Thus the results after the first procedure is\n\n1 √ 2 l 2 L -1 a=0 (|a + δ(2s -n)|a ) = 1 √ 2 l a=0 (1 + δ a )|a . (2)\n\nThis implies that after the procedure, the amplitude of each state is no longer equal, but have slight difference. Combining the effect in the initialization and in the DFT, we have\n\n(1 + δ a )(1 + δ c )e i( 2πc q +δ ′ c )a . = (1 + δ ′′ )e i( 2πc q +δ ′ c )a , where δ ′′ c = δ c + δ a . In the DFT, we have |ψ ⇒ √ r q q-1 c=0 q r -1 j=0 (1 + δ j )e i( 2πc q +δ ′ j )(jr+l) |c ,\n\nwhere we have rewrite δ ′′ as δ j here. Let P c denote the probability of getting the state |c after we perform a measurement, we have 3 ), we find that after the last measurement, each state can be extracted with a probability which is nonzero, and the offset l can't be eliminated. Eq. ( 3 ) is very complicated, so we will make some predigestions to discuss different error modes for convenience. Generally speaking, the influence of exponential error δ j is more remarkable than δ j , so we can omit the error δ j , thus\n\nP c = r q 2 q r -1 m=0 q r -1 k=0 (1 + δ m )(1 + δ k )e i( 2πc q +δ ′ m )(mr+l) ×e -i( 2πc q +δ ′ k )(kr+l) = r q 2 m k (1 + δ m )(1 + δ k ) cos[ 2πc q r(m -k) + (mr + l)δ ′ m -( From Eq. (\n\nDFT q |φ = √ r q q-1 c=0 q r j=0 e i( 2πc q +δ ′ j )(jr+l) |c . 3\n\nA. Case 1 If only systematic errors (EM 1 ) are considered, namely, all the δ j 's are equal, then f (c) can be given analytically f (c) = √ r q q r -1 j=0 e i( 2πc q +δ)(jr+l) = √ r q e il( 2πc q +δ) 1e i( 2πc q +δ)q\n\n1 -e i( 2πc q +δ)r ( 4\n\n)\n\nThe relative probability of finding c is\n\nP c = f (c) 2 = r q 2 sin 2 ( δq 2 ) sin 2 ( πcr q + δr 2 ) , and if c = k q r , then P c = r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) .\n\nIt can be easily seen that lim δ→0 P c = 1 r , which is just the case that no error is considered.\n\nWhen δ takes certain values, say, δ = 2 r (k -r q )π where k is an integer, then the summation in Eq. (4) is on longer valid. In our simulation, δ does not take these values. Here we consider the case where q = 2 7 = 128 and r = 4. For comparisons, we have drawn the relative probability for obtaining state c in Fig.1. for this given example. We have found the following results: (i) When δ is small, the errors do hardly influence the final result, for instance when c = k q r , then\n\nlim δ→0 P c = lim δ→0 r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) = 1 r .\n\nThe probability distribution is almost identical to those without errors.\n\n(ii) Let us increase δ gradually, from Fig. 2 , we see that a gradual change in the probability distribution takes place. (Here, we again consider the relative probabilities) When δ is increased to certain values, the positions of peaks change greatly. For instance at δ = 0.05, there appears a peak at c=127, whereas it is P c = 0 when no systematic errors are present. In general, the influence of systematic errors on the algorithm is a shift of the peak positions. This influences the final results directly.\n\nWhen both random errors and systematic errors are present, we add random errors to the simulation. To see the effect of different mode of random errors, we use two random number generators. One is the Gaussian mode and the other is the uniform mode. In this case, the error has the form δ = δ 0 + s, where δ 0 is the systematic error. s has a probability distribution with respect to c, depending on the uniform or the Gaussian distribution. When δ 0 = 0, we have only random errors which is our error mode 2. When δ 0 = 0, we have error mode 3. For the uniform distribution, s ∼ ±s max × u(0, 1) where u(0, 1) is evenly distributed in [0, 1] . s max indicates the maximum deviation from δ 0 . For Gaussian distribution, s ∼ N (0, σ 0 ). Through the figure, we see the following: (1) When only random errors are present (δ 0 = 0), the peak positions are not affected by these random errors. However, different random error modes cause similar results. The results for uniform random error mode are shown in Fig. 3 . For the uniform distribution error mode, with increasing δ max , the final probability distribution of the final results become irregular. In particular, when δ max is very large, all the patterns are destroyed and is hardly recognizable. Many unexpected small peaks appear. For the Gaussian distribution error mode, as shown in Fig. 4 , the influence of the error is more serious. This is because in Gaussian distribution, there is no cut-off of errors. Large errors can occur although their probability is small. The influence of σ 0 on the final results is also sensitive, because it determines the shape of the distribution. When σ 0 increases, the final probability distribution becomes very messy. A small change in σ 0 can cause a big change in the final results. (2) When δ 0 = 0, which corresponds to error mode 3, the effect is seen as to shift the positions of the peaks in addition to the influences of the random errors.\n\nTo summarize, we have analyzed the errors in Shor's factorization algorithm. It has been seen that the effect of the systematic errors is to shift the positions of the peaks, whereas the random errors change the shape of the probability distribution. For systematic errors, the shape of the distribution of the final results is hardly destroyed, though displaced. We can still use the result with several trial guesses to obtain the right results because the peak positions are shifted only slightly. However, the random errors are detrimental to the algorithm and should be reduced as much as possible. It is different from the case with Grover's algorithm where systematic errors are disastrous while random errors are less harmful [10].\n\n[1] P.W. Shor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, edited by S.\n\nGoldwasser (IEEE Computer Society Press, Los Alamitos, CA, 1994) p.124.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 p c FIG. 1: Relative probability for finding state c in the absence of errors.\n\n[2] A. Ekert and R. Jozsa, Rev. Mod. Phys. 68 (1996) 733. [3] W.G. Unruh, Phys. Rev A51 (1995) 992. [4] I. Chuang and R. laflamme, \"Quantum error correction by codding\" (1995) quant-ph/9511003. [5] C.P. Sun, H. Zhan and X.F. Liu, Phys. Rew. A58 (1998) 1810. [6] G.M. Palma, K.A. Suominen and A.K. Ekert, Proc. R. Soc. London, A 452 (1996) 567. [7] R.P. Feynman, Int. J. Theo. Phys., 21 (1982) 467. [8] D. Deutsch, Proc. R. Soc. Land. A 400 (1985) 97. [9] L.K. Grover, Phys. Rev, Lett, 79 (1997) 325. [10] G.L. Long, Y.S. Li, W.L. Zhang, C.C. Tu, Phys. Rev. A 61 (2000) 042305. [11] L.K. Grover, Phys. Rev. Lett, 80 (1998) 4329.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 (4) (3) c P (2) -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 c P -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c FIG. 2: The same as Fig.1. with systematic errors. In subfigures (1), (2), (3), (4), δ are 0.02, 0.03, 0.05 respectively. In sub-figure (4), the curve with solid circles(with higher peaks) is the result with δ = 0.1, and the one without solid circles(with lower peaks) denotes the result with δ = 0.33.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 (3) c P (1) -20 0 20 40 60 80 100 120 140 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (4) c P (2)\n\nFIG. 3: The same as Fig.1. with uniform random errors. In sub-figures (1), (2), (3), (4), smax are set to 0.01, 0.03, 0.05, 0.1 respectively.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 (2) P c -20 0 20 40 60 80 100 120 140 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 c P (3) -20 0 20 40 60 80 100 120 140 0.00 0.02 0.04 0.06 0.08 c P (4)\n\nFIG. 4: The same as Fig.1. with Gaussian random errors and systematic errors. In sub-figures (1), (2), and (3) τ are set to 0.01, 0.03 and 0.05 respectively, and δ0 = 0(without systematic errors). In sub-figure (4), both systematic and random Gaussian errors exist, where δ0 = 0.33, τ = 0.02."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"Shor's factorization algorithm [1] is a very important quantum algorithm, through which one has demonstrated the power of quantum computers. It has greatly promoted the worldwide research in quantum computing over the past few years. In practice, however, quantum systems are subject to influence of environment, and in addition, quantum gate operations are often imperfect [2, 3] . Environment influence on the system can cause decoherence of quantum states, and gate imperfection leads to errors in quantum computing. Thanks to Shor's another important work, in which he showed that quantum error correlation can be corrected [4] . With quantum error correction scheme, errors arising from both decoherence and imperfection can be corrected.\n\nThere have been several works on the effects of decoherence on Shor's algorithm. Sun et al. discussed the effect of decoherence on the algorithm by modeling the environment [5] . Palma studied the effects of both decoherence and gate imperfection in ion trap quantum computers [6] . There have also been many other studies on the quantum algorithm [7, 8, 9, 10] .\n\nThe error correction scheme uses available resources. Thus it is important to study the robustness of the algorithm itself so that one can strike a balance between the amount of quantum error correction and the amount of qubits available. In this paper, we investigate the effects of gate imperfection on the efficiency of Shor's factorization algorithm. The results may guide us in practice to suppress deliberately those errors that influence the algorithm most sensitively. For those errors that do not affect the algorithm very much, we may ignore them as a good approximation. In addition, study of the robustness of algorithm to errors is important where one can not apply the quantum error correction at all, for instance, in cases that there are not enough qubits available.\n\nThe paper is organized as follows. Section II is devoted to an outline of Shor's algorithm and different error's modes. In Section III, we present the results. Finally, a short summary is given in Section IV."},{"section_type":"OTHER","section_title":"II. SHOR'S ALGORITHM AND ERROR'S MODES","text":"Shor's algorithm consists of the following steps: 1) preparing a superposition of evenly distributed states |ψ = 1 √ q q-1 a=0 |a |0 ,\n\nwhere q = 2 L and N 2 ≤ q ≤ 2N 2 with N being the number to be factorized; 2) implementing y a modN and putting the results into the 2nd register\n\n|ψ 1 = 1 √ q q-1 a=0 |a |y a modN ;\n\n3) making a measument on the 2nd register; The state of the register is then\n\n|φ 2 = 1 √ A + 1 A j=0 |jr + l |z = y l = y jr+l modN\n\nwhere j ≤ q-l r = A.\n\n4) performing discrete Fourier transformation (DFT) on the first\n\nregister |φ 3 = c f (c) |c |z , where f (c) = √ r q q r -1 j=0 exp 2πi(jr + l) q = √ r q e 2πilc q q r -1 j=0 exp 2πijrc q .\n\n2 This term is nonzero only when c = k q r , with k = 0, 1, 2...r -1, which correspond to the peaks of the distribution in the measured results, and thus this term becomes f (c) = 1 √ r e 2πilc q . The Fourier transformation is important because it makes the state in the first register the same for all possible values in the 2nd register. The DFT is constructed by two basic gate operations: the single bit gate operation A j = 1 √ 2 1 1 1 -1 , which is also called the Walsh-Hadmard transformation, and the 2-bits controlled rotation\n\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e iθ jk with θ jk = π 2 k-j .\n\nThe gate sequence for implementing DFT is\n\n(A q-1 )(B q-2q-1 A q-2 ) . . . (B 0q-1 B 0q-2 . . . B 01 A 0 ). Errors can occur in both A j and B jk . A j is actually a rotation about y-axis through π 2 A j (θ) = e i h Syθ = I cos( θ 2 )-i sin( θ 2 )σ y = cos( θ 2 ) -sin( θ 2 ) sin( θ 2 ) cos( θ 2 )\n\n.\n\nIf the gate operation is not perfect, the rotation is not exactly π 2 . In this case, A j is a rotation of π 2 + 2δ A j (δ) = 1 √ 2 cos(δ) -sin(δ) -(sin(δ) + cos(δ)) sin(δ) + cos(δ) cos(δ) -sin(δ) .\n\nIf δ is very small, we have:\n\nA j (θ) = 1 √ 2 1 -δ -(1 + δ) 1 + δ+ 1 -δ .\n\nSimilarly, errors in B jk can be written as\n\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e i(θ jk +δ) .\n\nWith these errors, the DFT becomes\n\n|a → 1 √ q q-1 c=0 e i( 2π q/c +δc)a (1 + δ ′ c )|c = 1 √ q q-1 c=0 e i( 2πc q +δc)a (1 + δ ′ c )|c ,( 1\n\n)\n\nwhere δ c and δ ′ c denote the error of A j and B jk , respectively.\n\nLet us assume the following error modes: 1) systematic errors, where δ c or δ ′ c in (1) can only have systematic errors (EM 1 ); 2) random errors (EM 2 ), for which we assume that δ c or δ ′ c can only be random errors of the Gaussian or the uniform type; 3) coexistence of both systematic and random errors (EM 3 ). In the next section, we shall present the results of numerical simulations and discuss the effects of imperfect gate operation on the DFT algorithm, and thus on the Shor's algorithm."},{"section_type":"OTHER","section_title":"III. INFLUENCE OF IMPERFECT GATE OPERATIONS","text":"We first discuss the influence of imperfect gate operations in the initial preparation\n\nA l-1 A l-2 ...A 0 |0...0 = 1 √ 2 (|0 + |1 + δ 1 (|0 -|1 )) ⊗ (|0 + |1 + δ 2 . = 1 √ 2 l 1 i1i2...in=0 |i 1 i 2 ...i n + 1 √ 2 l n R=1 δ n 1 i\n\nIf the errors are systematic, for instance, caused by the inaccurate calibration of the rotations, then δ 1 = δ 2 = . . . = δ n = δ. In this case, we can write the 2nd term as\n\n|ψ = 1 √ 2 l δ 1 i1i2...in=0 (2s -n)|i 1 i 2 ...i n ,\n\nwhere s stands for the number of 1's, and 2sn = s -(ns) is the difference in the number of 1's and 0's. Thus the results after the first procedure is\n\n1 √ 2 l 2 L -1 a=0 (|a + δ(2s -n)|a ) = 1 √ 2 l a=0 (1 + δ a )|a . (2)\n\nThis implies that after the procedure, the amplitude of each state is no longer equal, but have slight difference. Combining the effect in the initialization and in the DFT, we have\n\n(1 + δ a )(1 + δ c )e i( 2πc q +δ ′ c )a . = (1 + δ ′′ )e i( 2πc q +δ ′ c )a , where δ ′′ c = δ c + δ a . In the DFT, we have |ψ ⇒ √ r q q-1 c=0 q r -1 j=0 (1 + δ j )e i( 2πc q +δ ′ j )(jr+l) |c ,\n\nwhere we have rewrite δ ′′ as δ j here. Let P c denote the probability of getting the state |c after we perform a measurement, we have 3 ), we find that after the last measurement, each state can be extracted with a probability which is nonzero, and the offset l can't be eliminated. Eq. ( 3 ) is very complicated, so we will make some predigestions to discuss different error modes for convenience. Generally speaking, the influence of exponential error δ j is more remarkable than δ j , so we can omit the error δ j , thus\n\nP c = r q 2 q r -1 m=0 q r -1 k=0 (1 + δ m )(1 + δ k )e i( 2πc q +δ ′ m )(mr+l) ×e -i( 2πc q +δ ′ k )(kr+l) = r q 2 m k (1 + δ m )(1 + δ k ) cos[ 2πc q r(m -k) + (mr + l)δ ′ m -( From Eq. (\n\nDFT q |φ = √ r q q-1 c=0 q r j=0 e i( 2πc q +δ ′ j )(jr+l) |c . 3\n\nA. Case 1 If only systematic errors (EM 1 ) are considered, namely, all the δ j 's are equal, then f (c) can be given analytically f (c) = √ r q q r -1 j=0 e i( 2πc q +δ)(jr+l) = √ r q e il( 2πc q +δ) 1e i( 2πc q +δ)q\n\n1 -e i( 2πc q +δ)r ( 4\n\n)\n\nThe relative probability of finding c is\n\nP c = f (c) 2 = r q 2 sin 2 ( δq 2 ) sin 2 ( πcr q + δr 2 ) , and if c = k q r , then P c = r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) .\n\nIt can be easily seen that lim δ→0 P c = 1 r , which is just the case that no error is considered.\n\nWhen δ takes certain values, say, δ = 2 r (k -r q )π where k is an integer, then the summation in Eq. (4) is on longer valid. In our simulation, δ does not take these values. Here we consider the case where q = 2 7 = 128 and r = 4. For comparisons, we have drawn the relative probability for obtaining state c in Fig.1. for this given example. We have found the following results: (i) When δ is small, the errors do hardly influence the final result, for instance when c = k q r , then\n\nlim δ→0 P c = lim δ→0 r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) = 1 r .\n\nThe probability distribution is almost identical to those without errors.\n\n(ii) Let us increase δ gradually, from Fig. 2 , we see that a gradual change in the probability distribution takes place. (Here, we again consider the relative probabilities) When δ is increased to certain values, the positions of peaks change greatly. For instance at δ = 0.05, there appears a peak at c=127, whereas it is P c = 0 when no systematic errors are present. In general, the influence of systematic errors on the algorithm is a shift of the peak positions. This influences the final results directly."},{"section_type":"OTHER","section_title":"B. Case 2","text":"When both random errors and systematic errors are present, we add random errors to the simulation. To see the effect of different mode of random errors, we use two random number generators. One is the Gaussian mode and the other is the uniform mode. In this case, the error has the form δ = δ 0 + s, where δ 0 is the systematic error. s has a probability distribution with respect to c, depending on the uniform or the Gaussian distribution. When δ 0 = 0, we have only random errors which is our error mode 2. When δ 0 = 0, we have error mode 3. For the uniform distribution, s ∼ ±s max × u(0, 1) where u(0, 1) is evenly distributed in [0, 1] . s max indicates the maximum deviation from δ 0 . For Gaussian distribution, s ∼ N (0, σ 0 ). Through the figure, we see the following: (1) When only random errors are present (δ 0 = 0), the peak positions are not affected by these random errors. However, different random error modes cause similar results. The results for uniform random error mode are shown in Fig. 3 . For the uniform distribution error mode, with increasing δ max , the final probability distribution of the final results become irregular. In particular, when δ max is very large, all the patterns are destroyed and is hardly recognizable. Many unexpected small peaks appear. For the Gaussian distribution error mode, as shown in Fig. 4 , the influence of the error is more serious. This is because in Gaussian distribution, there is no cut-off of errors. Large errors can occur although their probability is small. The influence of σ 0 on the final results is also sensitive, because it determines the shape of the distribution. When σ 0 increases, the final probability distribution becomes very messy. A small change in σ 0 can cause a big change in the final results. (2) When δ 0 = 0, which corresponds to error mode 3, the effect is seen as to shift the positions of the peaks in addition to the influences of the random errors."},{"section_type":"OTHER","section_title":"IV. SUMMARY","text":"To summarize, we have analyzed the errors in Shor's factorization algorithm. It has been seen that the effect of the systematic errors is to shift the positions of the peaks, whereas the random errors change the shape of the probability distribution. For systematic errors, the shape of the distribution of the final results is hardly destroyed, though displaced. We can still use the result with several trial guesses to obtain the right results because the peak positions are shifted only slightly. However, the random errors are detrimental to the algorithm and should be reduced as much as possible. It is different from the case with Grover's algorithm where systematic errors are disastrous while random errors are less harmful [10].\n\n[1] P.W. Shor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, edited by S.\n\nGoldwasser (IEEE Computer Society Press, Los Alamitos, CA, 1994) p.124.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 p c FIG. 1: Relative probability for finding state c in the absence of errors.\n\n[2] A. Ekert and R. Jozsa, Rev. Mod. Phys. 68 (1996) 733. [3] W.G. Unruh, Phys. Rev A51 (1995) 992. [4] I. Chuang and R. laflamme, \"Quantum error correction by codding\" (1995) quant-ph/9511003. [5] C.P. Sun, H. Zhan and X.F. Liu, Phys. Rew. A58 (1998) 1810. [6] G.M. Palma, K.A. Suominen and A.K. Ekert, Proc. R. Soc. London, A 452 (1996) 567. [7] R.P. Feynman, Int. J. Theo. Phys., 21 (1982) 467. [8] D. Deutsch, Proc. R. Soc. Land. A 400 (1985) 97. [9] L.K. Grover, Phys. Rev, Lett, 79 (1997) 325. [10] G.L. Long, Y.S. Li, W.L. Zhang, C.C. Tu, Phys. Rev. A 61 (2000) 042305. [11] L.K. Grover, Phys. Rev. Lett, 80 (1998) 4329.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 (4) (3) c P (2) -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 c P -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c FIG. 2: The same as Fig.1. with systematic errors. In subfigures (1), (2), (3), (4), δ are 0.02, 0.03, 0.05 respectively. In sub-figure (4), the curve with solid circles(with higher peaks) is the result with δ = 0.1, and the one without solid circles(with lower peaks) denotes the result with δ = 0.33.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 (3) c P (1) -20 0 20 40 60 80 100 120 140 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (4) c P (2)\n\nFIG. 3: The same as Fig.1. with uniform random errors. In sub-figures (1), (2), (3), (4), smax are set to 0.01, 0.03, 0.05, 0.1 respectively.\n\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 (2) P c -20 0 20 40 60 80 100 120 140 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 c P (3) -20 0 20 40 60 80 100 120 140 0.00 0.02 0.04 0.06 0.08 c P (4)\n\nFIG. 4: The same as Fig.1. with Gaussian random errors and systematic errors. In sub-figures (1), (2), and (3) τ are set to 0.01, 0.03 and 0.05 respectively, and δ0 = 0(without systematic errors). In sub-figure (4), both systematic and random Gaussian errors exist, where δ0 = 0.33, τ = 0.02."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"Shor's factorization algorithm [1] is a very important quantum algorithm, through which one has demonstrated the power of quantum computers. It has greatly promoted the worldwide research in quantum computing over the past few years. In practice, however, quantum systems are subject to influence of environment, and in addition, quantum gate operations are often imperfect [2, 3] . Environment influence on the system can cause decoherence of quantum states, and gate imperfection leads to errors in quantum computing. Thanks to Shor's another important work, in which he showed that quantum error correlation can be corrected [4] . With quantum error correction scheme, errors arising from both decoherence and imperfection can be corrected.\\n\\nThere have been several works on the effects of decoherence on Shor's algorithm. Sun et al. discussed the effect of decoherence on the algorithm by modeling the environment [5] . Palma studied the effects of both decoherence and gate imperfection in ion trap quantum computers [6] . There have also been many other studies on the quantum algorithm [7, 8, 9, 10] .\\n\\nThe error correction scheme uses available resources. Thus it is important to study the robustness of the algorithm itself so that one can strike a balance between the amount of quantum error correction and the amount of qubits available. In this paper, we investigate the effects of gate imperfection on the efficiency of Shor's factorization algorithm. The results may guide us in practice to suppress deliberately those errors that influence the algorithm most sensitively. For those errors that do not affect the algorithm very much, we may ignore them as a good approximation. In addition, study of the robustness of algorithm to errors is important where one can not apply the quantum error correction at all, for instance, in cases that there are not enough qubits available.\\n\\nThe paper is organized as follows. Section II is devoted to an outline of Shor's algorithm and different error's modes. In Section III, we present the results. Finally, a short summary is given in Section IV.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. SHOR'S ALGORITHM AND ERROR'S MODES\",\n \"text\": \"Shor's algorithm consists of the following steps: 1) preparing a superposition of evenly distributed states |ψ = 1 √ q q-1 a=0 |a |0 ,\\n\\nwhere q = 2 L and N 2 ≤ q ≤ 2N 2 with N being the number to be factorized; 2) implementing y a modN and putting the results into the 2nd register\\n\\n|ψ 1 = 1 √ q q-1 a=0 |a |y a modN ;\\n\\n3) making a measument on the 2nd register; The state of the register is then\\n\\n|φ 2 = 1 √ A + 1 A j=0 |jr + l |z = y l = y jr+l modN\\n\\nwhere j ≤ q-l r = A.\\n\\n4) performing discrete Fourier transformation (DFT) on the first\\n\\nregister |φ 3 = c f (c) |c |z , where f (c) = √ r q q r -1 j=0 exp 2πi(jr + l) q = √ r q e 2πilc q q r -1 j=0 exp 2πijrc q .\\n\\n2 This term is nonzero only when c = k q r , with k = 0, 1, 2...r -1, which correspond to the peaks of the distribution in the measured results, and thus this term becomes f (c) = 1 √ r e 2πilc q . The Fourier transformation is important because it makes the state in the first register the same for all possible values in the 2nd register. The DFT is constructed by two basic gate operations: the single bit gate operation A j = 1 √ 2 1 1 1 -1 , which is also called the Walsh-Hadmard transformation, and the 2-bits controlled rotation\\n\\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e iθ jk with θ jk = π 2 k-j .\\n\\nThe gate sequence for implementing DFT is\\n\\n(A q-1 )(B q-2q-1 A q-2 ) . . . (B 0q-1 B 0q-2 . . . B 01 A 0 ). Errors can occur in both A j and B jk . A j is actually a rotation about y-axis through π 2 A j (θ) = e i h Syθ = I cos( θ 2 )-i sin( θ 2 )σ y = cos( θ 2 ) -sin( θ 2 ) sin( θ 2 ) cos( θ 2 )\\n\\n.\\n\\nIf the gate operation is not perfect, the rotation is not exactly π 2 . In this case, A j is a rotation of π 2 + 2δ A j (δ) = 1 √ 2 cos(δ) -sin(δ) -(sin(δ) + cos(δ)) sin(δ) + cos(δ) cos(δ) -sin(δ) .\\n\\nIf δ is very small, we have:\\n\\nA j (θ) = 1 √ 2 1 -δ -(1 + δ) 1 + δ+ 1 -δ .\\n\\nSimilarly, errors in B jk can be written as\\n\\nB jk = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e i(θ jk +δ) .\\n\\nWith these errors, the DFT becomes\\n\\n|a → 1 √ q q-1 c=0 e i( 2π q/c +δc)a (1 + δ ′ c )|c = 1 √ q q-1 c=0 e i( 2πc q +δc)a (1 + δ ′ c )|c ,( 1\\n\\n)\\n\\nwhere δ c and δ ′ c denote the error of A j and B jk , respectively.\\n\\nLet us assume the following error modes: 1) systematic errors, where δ c or δ ′ c in (1) can only have systematic errors (EM 1 ); 2) random errors (EM 2 ), for which we assume that δ c or δ ′ c can only be random errors of the Gaussian or the uniform type; 3) coexistence of both systematic and random errors (EM 3 ). In the next section, we shall present the results of numerical simulations and discuss the effects of imperfect gate operation on the DFT algorithm, and thus on the Shor's algorithm.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. INFLUENCE OF IMPERFECT GATE OPERATIONS\",\n \"text\": \"We first discuss the influence of imperfect gate operations in the initial preparation\\n\\nA l-1 A l-2 ...A 0 |0...0 = 1 √ 2 (|0 + |1 + δ 1 (|0 -|1 )) ⊗ (|0 + |1 + δ 2 . = 1 √ 2 l 1 i1i2...in=0 |i 1 i 2 ...i n + 1 √ 2 l n R=1 δ n 1 i\\n\\nIf the errors are systematic, for instance, caused by the inaccurate calibration of the rotations, then δ 1 = δ 2 = . . . = δ n = δ. In this case, we can write the 2nd term as\\n\\n|ψ = 1 √ 2 l δ 1 i1i2...in=0 (2s -n)|i 1 i 2 ...i n ,\\n\\nwhere s stands for the number of 1's, and 2sn = s -(ns) is the difference in the number of 1's and 0's. Thus the results after the first procedure is\\n\\n1 √ 2 l 2 L -1 a=0 (|a + δ(2s -n)|a ) = 1 √ 2 l a=0 (1 + δ a )|a . (2)\\n\\nThis implies that after the procedure, the amplitude of each state is no longer equal, but have slight difference. Combining the effect in the initialization and in the DFT, we have\\n\\n(1 + δ a )(1 + δ c )e i( 2πc q +δ ′ c )a . = (1 + δ ′′ )e i( 2πc q +δ ′ c )a , where δ ′′ c = δ c + δ a . In the DFT, we have |ψ ⇒ √ r q q-1 c=0 q r -1 j=0 (1 + δ j )e i( 2πc q +δ ′ j )(jr+l) |c ,\\n\\nwhere we have rewrite δ ′′ as δ j here. Let P c denote the probability of getting the state |c after we perform a measurement, we have 3 ), we find that after the last measurement, each state can be extracted with a probability which is nonzero, and the offset l can't be eliminated. Eq. ( 3 ) is very complicated, so we will make some predigestions to discuss different error modes for convenience. Generally speaking, the influence of exponential error δ j is more remarkable than δ j , so we can omit the error δ j , thus\\n\\nP c = r q 2 q r -1 m=0 q r -1 k=0 (1 + δ m )(1 + δ k )e i( 2πc q +δ ′ m )(mr+l) ×e -i( 2πc q +δ ′ k )(kr+l) = r q 2 m k (1 + δ m )(1 + δ k ) cos[ 2πc q r(m -k) + (mr + l)δ ′ m -( From Eq. (\\n\\nDFT q |φ = √ r q q-1 c=0 q r j=0 e i( 2πc q +δ ′ j )(jr+l) |c . 3\\n\\nA. Case 1 If only systematic errors (EM 1 ) are considered, namely, all the δ j 's are equal, then f (c) can be given analytically f (c) = √ r q q r -1 j=0 e i( 2πc q +δ)(jr+l) = √ r q e il( 2πc q +δ) 1e i( 2πc q +δ)q\\n\\n1 -e i( 2πc q +δ)r ( 4\\n\\n)\\n\\nThe relative probability of finding c is\\n\\nP c = f (c) 2 = r q 2 sin 2 ( δq 2 ) sin 2 ( πcr q + δr 2 ) , and if c = k q r , then P c = r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) .\\n\\nIt can be easily seen that lim δ→0 P c = 1 r , which is just the case that no error is considered.\\n\\nWhen δ takes certain values, say, δ = 2 r (k -r q )π where k is an integer, then the summation in Eq. (4) is on longer valid. In our simulation, δ does not take these values. Here we consider the case where q = 2 7 = 128 and r = 4. For comparisons, we have drawn the relative probability for obtaining state c in Fig.1. for this given example. We have found the following results: (i) When δ is small, the errors do hardly influence the final result, for instance when c = k q r , then\\n\\nlim δ→0 P c = lim δ→0 r sin 2 ( δq 2 ) q 2 sin 2 ( δr 2 ) = 1 r .\\n\\nThe probability distribution is almost identical to those without errors.\\n\\n(ii) Let us increase δ gradually, from Fig. 2 , we see that a gradual change in the probability distribution takes place. (Here, we again consider the relative probabilities) When δ is increased to certain values, the positions of peaks change greatly. For instance at δ = 0.05, there appears a peak at c=127, whereas it is P c = 0 when no systematic errors are present. In general, the influence of systematic errors on the algorithm is a shift of the peak positions. This influences the final results directly.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Case 2\",\n \"text\": \"When both random errors and systematic errors are present, we add random errors to the simulation. To see the effect of different mode of random errors, we use two random number generators. One is the Gaussian mode and the other is the uniform mode. In this case, the error has the form δ = δ 0 + s, where δ 0 is the systematic error. s has a probability distribution with respect to c, depending on the uniform or the Gaussian distribution. When δ 0 = 0, we have only random errors which is our error mode 2. When δ 0 = 0, we have error mode 3. For the uniform distribution, s ∼ ±s max × u(0, 1) where u(0, 1) is evenly distributed in [0, 1] . s max indicates the maximum deviation from δ 0 . For Gaussian distribution, s ∼ N (0, σ 0 ). Through the figure, we see the following: (1) When only random errors are present (δ 0 = 0), the peak positions are not affected by these random errors. However, different random error modes cause similar results. The results for uniform random error mode are shown in Fig. 3 . For the uniform distribution error mode, with increasing δ max , the final probability distribution of the final results become irregular. In particular, when δ max is very large, all the patterns are destroyed and is hardly recognizable. Many unexpected small peaks appear. For the Gaussian distribution error mode, as shown in Fig. 4 , the influence of the error is more serious. This is because in Gaussian distribution, there is no cut-off of errors. Large errors can occur although their probability is small. The influence of σ 0 on the final results is also sensitive, because it determines the shape of the distribution. When σ 0 increases, the final probability distribution becomes very messy. A small change in σ 0 can cause a big change in the final results. (2) When δ 0 = 0, which corresponds to error mode 3, the effect is seen as to shift the positions of the peaks in addition to the influences of the random errors.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. SUMMARY\",\n \"text\": \"To summarize, we have analyzed the errors in Shor's factorization algorithm. It has been seen that the effect of the systematic errors is to shift the positions of the peaks, whereas the random errors change the shape of the probability distribution. For systematic errors, the shape of the distribution of the final results is hardly destroyed, though displaced. We can still use the result with several trial guesses to obtain the right results because the peak positions are shifted only slightly. However, the random errors are detrimental to the algorithm and should be reduced as much as possible. It is different from the case with Grover's algorithm where systematic errors are disastrous while random errors are less harmful [10].\\n\\n[1] P.W. Shor, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, edited by S.\\n\\nGoldwasser (IEEE Computer Society Press, Los Alamitos, CA, 1994) p.124.\\n\\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 p c FIG. 1: Relative probability for finding state c in the absence of errors.\\n\\n[2] A. Ekert and R. Jozsa, Rev. Mod. Phys. 68 (1996) 733. [3] W.G. Unruh, Phys. Rev A51 (1995) 992. [4] I. Chuang and R. laflamme, \\\"Quantum error correction by codding\\\" (1995) quant-ph/9511003. [5] C.P. Sun, H. Zhan and X.F. Liu, Phys. Rew. A58 (1998) 1810. [6] G.M. Palma, K.A. Suominen and A.K. Ekert, Proc. R. Soc. London, A 452 (1996) 567. [7] R.P. Feynman, Int. J. Theo. Phys., 21 (1982) 467. [8] D. Deutsch, Proc. R. Soc. Land. A 400 (1985) 97. [9] L.K. Grover, Phys. Rev, Lett, 79 (1997) 325. [10] G.L. Long, Y.S. Li, W.L. Zhang, C.C. Tu, Phys. Rev. A 61 (2000) 042305. [11] L.K. Grover, Phys. Rev. Lett, 80 (1998) 4329.\\n\\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 (4) (3) c P (2) -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 c P -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c FIG. 2: The same as Fig.1. with systematic errors. In subfigures (1), (2), (3), (4), δ are 0.02, 0.03, 0.05 respectively. In sub-figure (4), the curve with solid circles(with higher peaks) is the result with δ = 0.1, and the one without solid circles(with lower peaks) denotes the result with δ = 0.33.\\n\\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 (3) c P (1) -20 0 20 40 60 80 100 120 140 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (4) c P (2)\\n\\nFIG. 3: The same as Fig.1. with uniform random errors. In sub-figures (1), (2), (3), (4), smax are set to 0.01, 0.03, 0.05, 0.1 respectively.\\n\\n-20 0 20 40 60 80 100 120 140 0.00 0.05 0.10 0.15 (1) P c -20 0 20 40 60 80 100 120 140 0.00 0.01 0.02 0.03 0.04 0.05 (2) P c -20 0 20 40 60 80 100 120 140 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 c P (3) -20 0 20 40 60 80 100 120 140 0.00 0.02 0.04 0.06 0.08 c P (4)\\n\\nFIG. 4: The same as Fig.1. with Gaussian random errors and systematic errors. In sub-figures (1), (2), and (3) τ are set to 0.01, 0.03 and 0.05 respectively, and δ0 = 0(without systematic errors). In sub-figure (4), both systematic and random Gaussian errors exist, where δ0 = 0.33, τ = 0.02.\"\n }\n]","isConversational":false}}},{"rowIdx":94,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0520"},"arxiv_id":{"kind":"string","value":"0704.0520"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":""},"content_hash":{"kind":"string","value":"e2ed075f4ade80918e070091647f48ec5d73e5a1179cb737dc87b8224a6ec3ab"},"title":{"kind":"string","value":"A critical theory of quantum entanglement for the Hydrogen molecule"},"abstract":{"kind":"string","value":"In this paper we investigate some entanglement properties for the Hydrogen\nmolecule considered as a two interacting spin 1/2 (qubit) model. The\nentanglement related to the $H_{2}$ molecule is evaluated both using the von\nNeumann entropy and the Concurrence and it is compared with the corresponding\nquantities for the two interacting spin system. Many aspects of these functions\nare examinated employing in part analytical and, essentially, numerical\ntechniques. We have compared analogous results obtained by Huang and Kais a few\nyears ago. In this respect, some possible controversial situations are\npresented and discussed."},"authors":{"kind":"list like","value":["Tina A.C. Maiolo","Luigi Martina","Giulio Soliani"],"string":"[\n \"Tina A.C. Maiolo\",\n \"Luigi Martina\",\n \"Giulio Soliani\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["quant-ph","cond-mat.other","physics.atom-ph"],"string":"[\n \"quant-ph\",\n \"cond-mat.other\",\n \"physics.atom-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"Entanglement is a physical observable measured by the von Neumann entropy or, alternatively, by the Concurrence of the system under consideration.\n\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The electron correlation is not directly observable, since it is defined as the difference between the exact ground state energy of the many electrons Schrödinger equation and the Hartree-Fock energy.\n\nIn this paper we discuss the Hamiltonian which describes the Hydrogen molecule regarded as a two interacting spin 1 2 (qubit) model. In [1] it was argued that the entanglement (a quantum observable) can be used in analyzing the so-called correlation energy which is not directly observable. From our point of view, the Hydrogen molecule is dealt with a bipartite system governed by the Hamiltonian\n\nH H2 = - J 2 (1 + g)σ 1 ⊗ σ 1 - J 2 (1 -g)σ 2 ⊗ σ 2 -B(σ 3 ⊗ σ 3 + σ 0 ⊗ σ 3 ), ( 1\n\n) 1\n\nwhere σ i stand for the Pauli matrices (σ 0 = I). Actually, this model was considered in [1] in order to illustrate their method. However, here we will make some interpretative changes. Indeed, from our point of view, the states of an isolated atom are strongly reduced to a system with two energy levels related to the intensity of the magnetic field B. Relatively to this scale, the exchange interaction constant J is usually smaller than B, in order to represent the residual interatomic interactions. From the point of view of quantum chemistry, one may interpret the discrete spectrum as provided by the Hartree-Fock calculations, while the interaction coupling J models the residual multielectronic effects, not taken into account by the mean field approximation. For simplicity we limit ourselves to the ferromagnetic phase with J > 0. The parameter g, such that 0 ≤ g ≤ 1, describes the degree of anisotropy corresponding for g = 0 to the completely isotropic XY spin model. Conversely, g = 1 provides the anisotropic XY spin model, the so-called Ising model.\n\nWe notice that when the atoms are far apart, their interaction is quite weak. This corresponds to a vanishing value of J. In this situation the state of the system is completely factorized in the product state of the ground states of the indipendent spins. The corresponding total energy, in unit of B, is just the sum of the two fundamental levels, E 0 = -2, which we may consider as the Hartree-Fock approximated fundamental level in molecular structure calculations.\n\nWhen J = 0, the fundamental energy eigenvalue is E= -4 + g 2 λ 2 in Region I defined by 0 < λ ≤ 2 √ 1-g 2 , otherwise E = -λ (λ means the coupling constant) in Region II, which is the complement of I which respect to positive real axis. The corresponding (non normalized) eigenstates are |Ψ I = √ g 2 λ 2 +4+2 gλ , 0, 0, 1 and |Ψ II = 0, 1, 1, 0 , respectively. In both cases the state is entangled.\n\nSince we are dealing with pure states, the von Neumann entropy [2]\n\nS vN = -Tr ρ 1 log 2 ρ 1 ( 2\n\n)\n\nis chosen to be a measurement of the entanglement, where ρ 1 is the 1-particle reduced density matrix. However, for general mixed states other entanglement estimators (for instance, the Concurrence [4]) have to be used. In the considered case, one has S vN,I = -g 2 g 2 λ 2 + 4 log 1 2 -1 g 2 λ 2 + 4 λ 2 -g 2 g 2 λ 2 + 4 + 4 λ 2 + 8 g 2 λ 2 + 4 + 2 log 1 2 + 1 g 2 λ 2 + 4 1 g 2 λ 2 + 4 g 2 λ 2 + 2 g 2 λ 2 + 4 + 4 log(4) (3) S vN,II = 1. (4) 2 Scrutinizing Eq. (3) and Eq. ( 4 ) it emerges that the entropy is an increasing function of the coupling constant λ in Region I, but the state is maximally entangled in Region II independently from the anisotropy parameter g. One sees that, as it arises graphycally, for g = 1 the entanglement is a monotonic increasing function of the interaction coupling λ. Moreover for weak (< 1) coupling values it is always less than the 30%. Of course, for large coupling constants the entropy approaches 1, meaning that all levels are equiprobably visited by the considered spin.\n\nLimiting all further considerations to the case of weak interaction, we observe that at the boundary point λ b = 2 √ 1-g 2 a discontinuity occurs, signaling a crossing of the lowest eigenvalues and, in a more general context, a quantum phase transition [5] .\n\nAs it was pointed out in [6] , for quantifying the entanglement we can resort to the reduced density matrix. Furthermore, in [7] , Wootters has shown that for a pair of binary qubits one can use the concept of Concurrence C to measure the entanglement.\n\nThe Concurrence reads\n\nC(ρ) = max(0, ν 1 -ν 2 -ν 3 -ν 4 ), ( 5\n\n)\n\nwhere the ν i 's are the eigenvalues of the Hermitian matrix\n\nR = √ ρρ √ ρ 1 2 ,\n\nwhere ρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ), ρ * being the complex conjugate of ρ taken in the standard basis [7] . Some interesting results on the simple model (1) of the Hydrogen molecule can be achieved by realizing a comparative study of the von Neumann entropy and the Concurrence.\n\nTo this aim, we compute the Concurrence C I and C II , i. e.\n\nC I = gλ 1 g 2 λ 2 + 4 , C II = 1. ( 6\n\n)\n\nwhere I and II refer to Regions I and II, where 0 ≤ λ ≤ 2 1-g 2 , and E = -λ, respectively.\n\nIn Figure 1 a comparison between the Concurrence and the von Neumann entropy for two spins system as a function of the coupling λ for g = 1 is presented. Sec. 2 contains a comparison between the entanglement and the correlation energy. In Sec. 3 the Configuration Interaction method is introduced to compare entanglement and correlation energy. In Sec. 4 some differences between the Configuration Interaction approach and the two spin Ising model are presented. Finally, our main results are summarized in Sec. 5. 3 1 2 3 4 5 Λ 0.2 0.4 0.6 0.8 1 S vN Conc.\n\nFigure 1: Comparison between the Concurrence and the von Neumann entropy for the two spins system as a function of the coupling constant λ for g = 1.\n\n2 A comparison between the entanglement and the correlation energy\n\nNow we look for a comparison between the entanglement with the energy correlation, which as we have already recalled, it is understood as the difference of the fundamental energy level compared with respect to the corresponding value at vanishing coupling constant λ. For g = 1 and in unities of B it is given by\n\nE corr = |E 0 | -2 = 4 + λ 2 -2. ( 7\n\n)\n\nWe observe that the entanglement measure is always bounded, while E corr is a divergent function of λ. So it does not make much sense to look for simple relations valid on the entire λ-axes. Consequently, limiting ourselves to weak couplings, for 0 ≤ λ ≤ 1, we minimize the mean squared deviation\n\nI α = 1 0 ∆S 2 α dλ, with ∆S α = E corr -α S vN . ( 8\n\n)\n\nThus the minimizing parameter α min will be given by\n\nα min = 1 0 E corr S vN dλ 1 0 S 2 vN dλ ≈ -0.691217. ( 9\n\n)\n\nA formula analogous to (9) can be obtained by using the Concurrence as a measure of entanglement. In this case, by minimizing the mean squared deviation we have\n\nI C α ′ = 1 0 ∆C 2 α ′ dλ, with ∆C α ′ = E corr -α ′ C. ( 10\n\n)\n\nNow, in order to estimate the relative deviation of\n\nS vN with respect to E corr , let us report |∆S αmin |/S vN and |∆S αmin /E corr | as\n\nfunctions of λ at the optimal value α min . The graphs of these functions are shown in Figure 2. 4 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min S vN 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min E corr Figure 2: The relative quadratic deviation between the von Neumann entropy and the correlation energy with respect to the former and the latter, respectively, at the optimal value α min as a function of the coupling constant λ for g = 1.\n\nIn Figure 3 , the relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, at the optimal values α ′ min , is represented.\n\n0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min C 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min E corr Figure 3: The relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, respectively, at the optimal value α ′ min as a function of the coupling constant λ for g = 1.\n\nFrom these graphs, one can argue that the agreement between the two functions S vN and E corr is only qualitatively good, in fact, for very small λ, it is not good at all. However, in an intermediate range of values, i. e., 0.6 ≤ λ ≤ 1 the two functions are almost proportional within the 10%. Analogously, the same is true between energy and Concurrence. Even, the agreement becomes worst comparing the relative deviation of the Concurrence with respect to the correlation energy, since the range in which the relative deviations become smaller than 10% are narrower. Then, the question is whether the above results are i) sufficient to justify the conjecture advanced in [1] , i.e., entanglement can be considered as an estimation of correlation energy; ii) if such a relation has a 5 more concrete physical meaning, in particular whether the minimizing parameter α min and the vanishing point of ∆S αmin does possess any physical meaning (or α ′ min and the vanishing point of ∆C α ′ min ). Notice that in the case of the comparison for the Concurrence simpler analytical expressions appear. For instance one finds ∆C α ′ min = 0.383249 λ\n\n√ λ 2 +4 - √ λ 2 + 4 + 2 2 . Remark 2\n\nWe note that in an interval of values around α min , the deviation function (8) possesses a minimum in the interval of interest 0 ≤ λ ≤ 1, otherwise the minimum is achieved at larger value of λ, or the function is monotonically increasing (see Figure 4 ).\n\n0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 S Α 0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 d S Α dΛ\n\nFigure 4: The deviation ∆S α and its derivative with respect to λ are computed for values of -1.29(red) ≤ α ≤ -0.091(violet), for steps of 0.06. The curve drawn thicker corresponds to α min This behavior suggests to consider the function ∆S αmin as a sort of \"free energy\" , where α min mimics the \"temperature\" specific of the system. If, for some reason, we allow λ to change, then we expect that spontaneously the interaction coupling adjusts itself to the minimum of ∆S αmin . Similar considerations can be made looking at the graphs drawn for the function ∆C α ′ min and its derivative with respect to λ (see Figure 5 ). The function ∆S αmin or, alternatively, the minimum of ∆C α ′ min can be obtained algebraically. Such a minimum is at the value of the coupling constant λ SvN min ≈ 0.485 and λ C min ≈ 0.371, respectively. The authors in [1] studied numerically the von Neumann entropy and the correlation function for a Hydrogen molecule, using an old result by Herring and Flicker [8], going back to an oldest idea by Heitler and London [9], which consists in substituting the molecular binding with a position dependent exchange coupling:\n\nJ(r) ≈ 1.641 r 5 2 e -2 r Ry, ( 11\n\n)\n\nwhere r is given in Bohr radius, see Figure 6 . The maximum value taken by this function is at the point r max = 1.25. Assuming B = 0.5 Ry, i.e. 1 2 of the fundamental level of the Hydrogen atom, the maximum value λ ′ max = J(r max )/B ≈ 6 0.2 0.4 0.6 0.8 1 Λ -0.2 -0.1 0.1 0.2 0.3 C Α' 0.2 0.4 0.6 0.8 1 Λ -0.4 -0.2 0.2 0.4 d C Α' dΛ 5: The deviation ∆C α ′ and its derivative with respect to λ are computed for values of -0.98(red) ≤ α ′ ≤ 0.22(violet), for steps of 0.06. The curve drawn thicker corresponds to α ′ min 0.5 1 1.5 2 2.5 3 r 0.05 0.1 0.15 0.2 J r Ry\n\nFigure 6: The effective interaction Hydrogen-Hydrogen atom 0.470628 < λ SvN min , i.e. the value of the effective interaction value is less than the minimum for the deviation function ∆S αmin . Then, the equilibrium balance between entanglement (as von Neumann entropy) and correlation energy predicts a length of the molecule equal to r max (see the first panel of Figure 7 ). On the other hand, if we consider the energy gap 2B = 3/4 Ry, i.e. the energy step to the first excited state, one obtains the new value λ ′′ max ≈ 0.628, which goes beyond λ min , even if it is always less than 1. Now, the deviation function ∆S αmin has two minima as seen in the second panel of Figure 7 , one of which is at r ′′ -≈ 0.76 , the other one being at r ′′ + ≈ 1.91. These results should be compared with the experimental equilibrium length of the Hydrogen molecule, which is r exp ≈ 2.0. We point out that although the spin-model described by the Hamiltonian (1) is characterized by features which are essentially rough, however we are induced to answer positively to the quest for a physical meaning of the deviation function ∆S αmin . Indeed, the results elucidated in Figure 7 encourage, on one part, improvement of the computation of r in order to make more accurate the comparison with the experimental value r exp . 7 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .5 Ry 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .375 Ry r'' r''\n\nFigure 7: The von Neumann entropy for the 2-spin model for B = .5 Ry (left panel) and for B = .375 Ry (right panel) and the position depending interaction given by ( 11 ).\n\nThe first question to answer is whether this draft works also for the Concurrence. A statement about it is not obvious, since the von Neumann entropy is a nonlinear function of the Concurrence in the 2-qubits case. However, from Figure 8 one can see that the minimized deviation of the Concurrence takes one minimum for relatively large intensity of the magnetic field ( say B ≥ 0.6 Ry), while for weak fields two minima appear, corresponding to the situation depicted nearby. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 r Bohr B Ry 0.2 0.4 0.6 0.8 1 0 1 2 3 4 B(Ry) r(Bohr)\n\nFigure 8: Two contour plots of the minimized deviation of the Concurrence as a function of the magnetic field B (Ry) and of the internuclear distance r, as given by (11) . The range of values divided by the contour lines is [-0.038, 0, 04] for the left panel and [-0.03705, -0, 03000] for the right one that approximatively corresponding to the black area in the left panel.\n\nIn correspondence of the same values considered above, for B = 0.5 Ry the function ∆C α ′ min (r) has two minima at r = 0.79 and r = 1.88, while for B = 0.375 Ry they are located at r = 0.60 and r = 2.25. So one sees that the resulting equilibrium configurations are not much very close to the experimental one. The equilibrium configuration more closest to the experimental one is the minimum occurring at r = 1.88 (B = 1 2 Ry) for the function ∆C α ′ min (r).\n\n8 One sees that one of the resulting equilibrium configurations is only roughly close to the experimental one.\n\nIn other words, to conclude monitoring numerically B the equilibrium configuration more closest to experimental one in the minimum occurring at r = 1.88 for B = 1 2 Ry and at r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r).\n\nIn this Section we represent the results produced in [1], where the electron entanglement in the Hydrogen molecule, calculated by the von Neumann entropy of the reduced density matrix ρ 1 , is obtained starting by the excitation coefficients of the wave function expanded by a configuration interaction method:\n\nS ρ CISD 1 = -T r ρ CISD 1 log 2 ρ CISD 1 = = - m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 log 2 m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 + -|c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 log 2 |c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 , ( 12\n\n)\n\nwhere c 1 is the coefficient for a single excitation, and c 1,2 is the double excitation (in Appendix A of [10] more details are shown).\n\nIn this framework, entanglement (S) and correlation energy (E corr ), as functions of nucleus -nucleus separation are those in Figure 9 0 0.01 0.02 0.03 0.04 0.05 E c [ a. u. ] E c 0 1 2 3 4 5 R ( Å ) 0 0.2 0.4 0.6 0.8 1 Entanglement (S) S ( ρ 1 CISD ) H 2 Figure 9: Comparison between the entanglement, calculated by the von Neumann entropy of the reduced density matrix, and the electron correlation energy in the Hydrogen molecule.\n\nBy the results given by this model, we want to discuss and to suggest some answers to the questions i) and ii) presented in Remark 1. Even if, in order 9 to represent correlation energy and entanglement, we use two different scales, in Figure 9 we can see that entanglement has a small value in the united atom limit after it is growing for small distances till it arrives at a maximum value then it decrease till it assumes zero value at the separated atom limit and it is exactly the progress of the correlation curve.\n\nIn order to compare the entropy S with the electron correlation energy E corr , we rescale S with the parameter α min calculated with some procedure illustrated in Eq. ( 8 ) and Eq. (9) replacing the integration variable λ with R; in this way we extract\n\nα = E corr S vN dR S vN dR ≈ 0.009. ( 13\n\n)\n\nThe corresponding ∆S αmin = E corr -αS allows us to answer to the question ii); in fact, as it is shown in Figure 10 , the vanishing point of ∆S αmin is, according to the two -spin Ising model, nearby R ≈ 2 Å that corresponds to the equilibrium configuration of the Hydrogen molecule.\n\n0 1 2 3 4 5 R(Å) -0.01 0 0.01 0.02 0.03 0.04 Ec-αS\n\nFigure 10: ∆S αmin for the H 2 molecule as a function of nucleus-nucleus distance.\n\nThe model proposed in Sec. 1 provides us with a measurement of entanglement: indeed, Eq. (3) describes the von Neumann entropy as a function of coupling constant λ, for small λ. By using Eq. ( 7 ), we can express λ in terms of correlation energy and substituting it in Eq. ( 3 ) we can obtain the variation of S vN in terms of E corr .\n\nS vN = - E corr Log Ecorr 2(Ecorr +2) + (E corr + 4)Log Ecorr +4 2(Ecorr +2) (E corr + 2)Log4 . ( 14\n\n)\n\n10 In order to calculated the coefficient of proportionality among S vN and E corr we make an expansion of S vN for E corr → 0 (or equivalently for λ → 0) at the first order, obtaining a straight line characterized by an angular coefficient given by m SvN (Ecorr) = ( 1 4 )(1 + 1 Log2 ). Since this behavior is uncorrect to represent the logatithmic singularity of S vN in the origin, we make an expansion of Eq. ( 14 ), preserving the logarithmic deviation, and we obtain an expression of the form\n\nS vN = AE corr + BE corr Log(E corr ), ( 15\n\n)\n\nwhere A = 1/2 and B = -1/(4Log2).\n\n0.02 0.04 0.06 0.08 0.1 0.025 0.05 0.075 0.1 0.125 0.15 Ecorr S Linear AE+BELogE Figure 11: A comparison among the behavior of Eq. ( 14 ) and its linear approximation and the logarithmic one, for the Ising model.\n\nIn order to compare the behavior of S vN in Eq. ( 14 ), we have organized the numerical data, calculated with the method proposed in [1] , by making a correspondence between each value of E corr and its respective value of S vN , obtaining the plot in Figure 12 0 0.01 0.02 0.03 0.04 0.05 E c 0 0.2 0.4 0.6 0.8 1 S Figure 12: A correspondence of E corr and S vN by the numerical procedure suggested by [1] Of particular significance is the fact that, in the range where S is monotonically increasing, the correlation energy has its maximum, consequently S seems to be not a function. Moreover, it is important to note that E corr begins to decrease for R > 1 Å, region where the states become mixed, i. e. ,T rρ = T rρ 2 ; as depicted in Figure 13. 11 0 1 2 3 4 R(Å ) 0.4 0.6 0.8 1 1.2 1.4 Trρ 2\n\nFigure 13: The increasing of the degree of mixing in the two electron state: in black we depict the trace of ρ, in red the trace of ρ 2 .\n\nProbably, for this reason, the procedure adopted in [1] seems to be not correct: the density matrix, in fact, is calculated starting by the excitation coefficient of a wave function obtained developping with the Configuration Interaction Single Double method a pure two electrons state.\n\nHowever, even if we consider only the first branch of the plot in Figure 12 , i.e. , the numerical values of S vN corresponding with increasing values of E corr , and we fit the values around E corr → 0 with a F = AE corr + BE corr Log(E corr ) we draw out numerical values of the coefficient different from the ones used in Eq. (15) . This result is shown in Figure 14. 0.01 0.02 0.03 0.04 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 S Ecorr A=17.1 B=3.3 Figure 14: A fit of S vN as a function of E corr , around the origin, with a function of the form F = AE corr + BE corr Log(E corr ) whose coefficients A and B assume the numerical values in Figure.\n\nIn particular the arithmetic sign of the coefficient B in the two models are opposite and this implies the opposite concavity of the curve.\n\nThis fact, clearly demonstrates a not satisfactory agreement between the Ising model and the one proposed in [1] .\n\nWe have explored the role of entanglement in the model of two qubits describing the Hydrogen molecule (1), considered as a bipartite system. In our discussion we have limited to the ferromagnetic case governed by the interaction coupling parameter J > 0.\n\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The entanglement can be measured by using the von Neumann entropy or, alternatively, the notion of Concurrence [7] . To compute the entanglement it is convenient to consider two Regions, say I and II, which provide two different reduced density matrices. The entropy turns out to be an increasing function of the coupling constant λ in Region I, but the state under consideration is maximally entangled in Region II indipendently from the anisotropy parameter g.\n\nAn interesting result is that for large coupling constants the entropy approach 1, meaning that all levels are equiprobably visited by the considered spin.\n\nFor weak interactions, at the boundary point λ b = 2 √ 1-g 2 the von Neumann entropy admits a discontinuity, indicating a crossing of the lowest eigenvalues and, in a more general constext, a quantum phase transition [5] .\n\nIn Sec. 2 a comparison between the entanglement and the correlation energy is performed.\n\nTo quantifying the entanglement we resort to the reduced density matrix. The entanglement can also be measured by exploiting the concept of Concurrence.\n\nThe entanglement measure is always bounded, while the energy correlation,\n\nE corr = |E 0 | -2 =\n\n√ 4 + λ 2 -2, is a divergent function of λ. This fact tells us that to look for simple relations valid on the whole λ-axes has no sense.\n\nThus, by limiting ourselves to weak couplings, we have minimized the mean square deviation given by Eq. ( 8 ). This procedure leads to the value α min ≈ -0.691217 for the minimizing parameter (see Eq. (9)). Sec. 1 contains a comparison between the von Neumann entropy and the Concurrence. Such a comparison is illustrated in Figure 1 , for two spin system as a function of the coupling λ for g = 1. Some important points are commented in Remark 1 and Remark 2 .\n\nIn Figure 4 the deviation ∆S α and its derivatives with respect to λ are computed and α min is evaluated for α ranging in the interval -1.29 ≤ α ≤ -0.091.\n\nIn Figure 5 the minimized Concurrence deviation ∆C α (i) α ′\n\nfor the four eigenstates of the 2-spin model is shown.\n\nWe point out the existence of a perfect symmetry among the Concurrence deviations for pairs of eigenstates of opposite eigenvalues.\n\nFormula (11), due to Heitler-London [9], is reported, where the position dependent exchange coupling J(r) is expressed in term of the length r of the 13 nucleus-nucleus separation in the Hydrogen molecule.\n\nTo conclude, the magnetic field B has been monitored such that the equilibrium configuration more closest to the experimental one, r ≈ 2.00, is the minimum occurring at r = 1.88 for B = 1 2 Ry and r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r). We observe also that in the intermediate range of values, i. e., for 0.6 ≤ λ ≤ 1, the two functions SvN and the correlation energy are almost proportional within the 10%.\n\nHowever, when we organized the pairs of points (E corr , S vN ) calculated by following the procedure described by [1], it is clear that the von Neumann entropy cannot be considered a function of correlation energy. The principle cause is that the function E corr presents a maximum in the region where S vN is monotonically increasing.\n\nThe reversing behavior of correlation energy occurs in correspondence with an increase of the mixing degree of the two electrons state. The function E corr in terms of the nucleus -nucleus distance R, increases till the state is pure, on the contrary, when T r(ρ 2 ) becomes discordant from T r(ρ), the function E corr decreases.\n\nThis fact suggests us that the numerical model based on the calculation of S vN starting by the excitation coefficients c i , isn't completley correct because the density matrix is obtained as a product of two electron pure states. However, even if we consider only a branch of the plot in Figure 12 , the function obtained by the two spin Ising model, i. e., Eq. ( 14 ), is unsuitable for fitting these numerical data.\n\nOn the basis of our results, essentially grounded on numerical considerations, in the near feature we would explore more complicated systems of molecules, such as for example the ethylene or other hydrocarbons, and compare these studies with the goals obtained for the Hydrogen molecule.\n\nThe authors acknowledge the Italian Ministry of Scientific Researches (MIUR) for partial support of the present work under the project SINTESI 2004/06 and the INFN for partial support under the project Iniziativa Specifica LE41."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"Introduction and the model","text":"Entanglement is a physical observable measured by the von Neumann entropy or, alternatively, by the Concurrence of the system under consideration.\n\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The electron correlation is not directly observable, since it is defined as the difference between the exact ground state energy of the many electrons Schrödinger equation and the Hartree-Fock energy.\n\nIn this paper we discuss the Hamiltonian which describes the Hydrogen molecule regarded as a two interacting spin 1 2 (qubit) model. In [1] it was argued that the entanglement (a quantum observable) can be used in analyzing the so-called correlation energy which is not directly observable. From our point of view, the Hydrogen molecule is dealt with a bipartite system governed by the Hamiltonian\n\nH H2 = - J 2 (1 + g)σ 1 ⊗ σ 1 - J 2 (1 -g)σ 2 ⊗ σ 2 -B(σ 3 ⊗ σ 3 + σ 0 ⊗ σ 3 ), ( 1\n\n) 1\n\nwhere σ i stand for the Pauli matrices (σ 0 = I). Actually, this model was considered in [1] in order to illustrate their method. However, here we will make some interpretative changes. Indeed, from our point of view, the states of an isolated atom are strongly reduced to a system with two energy levels related to the intensity of the magnetic field B. Relatively to this scale, the exchange interaction constant J is usually smaller than B, in order to represent the residual interatomic interactions. From the point of view of quantum chemistry, one may interpret the discrete spectrum as provided by the Hartree-Fock calculations, while the interaction coupling J models the residual multielectronic effects, not taken into account by the mean field approximation. For simplicity we limit ourselves to the ferromagnetic phase with J > 0. The parameter g, such that 0 ≤ g ≤ 1, describes the degree of anisotropy corresponding for g = 0 to the completely isotropic XY spin model. Conversely, g = 1 provides the anisotropic XY spin model, the so-called Ising model.\n\nWe notice that when the atoms are far apart, their interaction is quite weak. This corresponds to a vanishing value of J. In this situation the state of the system is completely factorized in the product state of the ground states of the indipendent spins. The corresponding total energy, in unit of B, is just the sum of the two fundamental levels, E 0 = -2, which we may consider as the Hartree-Fock approximated fundamental level in molecular structure calculations.\n\nWhen J = 0, the fundamental energy eigenvalue is E= -4 + g 2 λ 2 in Region I defined by 0 < λ ≤ 2 √ 1-g 2 , otherwise E = -λ (λ means the coupling constant) in Region II, which is the complement of I which respect to positive real axis. The corresponding (non normalized) eigenstates are |Ψ I = √ g 2 λ 2 +4+2 gλ , 0, 0, 1 and |Ψ II = 0, 1, 1, 0 , respectively. In both cases the state is entangled.\n\nSince we are dealing with pure states, the von Neumann entropy [2]\n\nS vN = -Tr ρ 1 log 2 ρ 1 ( 2\n\n)\n\nis chosen to be a measurement of the entanglement, where ρ 1 is the 1-particle reduced density matrix. However, for general mixed states other entanglement estimators (for instance, the Concurrence [4]) have to be used. In the considered case, one has S vN,I = -g 2 g 2 λ 2 + 4 log 1 2 -1 g 2 λ 2 + 4 λ 2 -g 2 g 2 λ 2 + 4 + 4 λ 2 + 8 g 2 λ 2 + 4 + 2 log 1 2 + 1 g 2 λ 2 + 4 1 g 2 λ 2 + 4 g 2 λ 2 + 2 g 2 λ 2 + 4 + 4 log(4) (3) S vN,II = 1. (4) 2 Scrutinizing Eq. (3) and Eq. ( 4 ) it emerges that the entropy is an increasing function of the coupling constant λ in Region I, but the state is maximally entangled in Region II independently from the anisotropy parameter g. One sees that, as it arises graphycally, for g = 1 the entanglement is a monotonic increasing function of the interaction coupling λ. Moreover for weak (< 1) coupling values it is always less than the 30%. Of course, for large coupling constants the entropy approaches 1, meaning that all levels are equiprobably visited by the considered spin.\n\nLimiting all further considerations to the case of weak interaction, we observe that at the boundary point λ b = 2 √ 1-g 2 a discontinuity occurs, signaling a crossing of the lowest eigenvalues and, in a more general context, a quantum phase transition [5] .\n\nAs it was pointed out in [6] , for quantifying the entanglement we can resort to the reduced density matrix. Furthermore, in [7] , Wootters has shown that for a pair of binary qubits one can use the concept of Concurrence C to measure the entanglement.\n\nThe Concurrence reads\n\nC(ρ) = max(0, ν 1 -ν 2 -ν 3 -ν 4 ), ( 5\n\n)\n\nwhere the ν i 's are the eigenvalues of the Hermitian matrix\n\nR = √ ρρ √ ρ 1 2 ,\n\nwhere ρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ), ρ * being the complex conjugate of ρ taken in the standard basis [7] . Some interesting results on the simple model (1) of the Hydrogen molecule can be achieved by realizing a comparative study of the von Neumann entropy and the Concurrence.\n\nTo this aim, we compute the Concurrence C I and C II , i. e.\n\nC I = gλ 1 g 2 λ 2 + 4 , C II = 1. ( 6\n\n)\n\nwhere I and II refer to Regions I and II, where 0 ≤ λ ≤ 2 1-g 2 , and E = -λ, respectively.\n\nIn Figure 1 a comparison between the Concurrence and the von Neumann entropy for two spins system as a function of the coupling λ for g = 1 is presented. Sec. 2 contains a comparison between the entanglement and the correlation energy. In Sec. 3 the Configuration Interaction method is introduced to compare entanglement and correlation energy. In Sec. 4 some differences between the Configuration Interaction approach and the two spin Ising model are presented. Finally, our main results are summarized in Sec. 5. 3 1 2 3 4 5 Λ 0.2 0.4 0.6 0.8 1 S vN Conc.\n\nFigure 1: Comparison between the Concurrence and the von Neumann entropy for the two spins system as a function of the coupling constant λ for g = 1.\n\n2 A comparison between the entanglement and the correlation energy\n\nNow we look for a comparison between the entanglement with the energy correlation, which as we have already recalled, it is understood as the difference of the fundamental energy level compared with respect to the corresponding value at vanishing coupling constant λ. For g = 1 and in unities of B it is given by\n\nE corr = |E 0 | -2 = 4 + λ 2 -2. ( 7\n\n)\n\nWe observe that the entanglement measure is always bounded, while E corr is a divergent function of λ. So it does not make much sense to look for simple relations valid on the entire λ-axes. Consequently, limiting ourselves to weak couplings, for 0 ≤ λ ≤ 1, we minimize the mean squared deviation\n\nI α = 1 0 ∆S 2 α dλ, with ∆S α = E corr -α S vN . ( 8\n\n)\n\nThus the minimizing parameter α min will be given by\n\nα min = 1 0 E corr S vN dλ 1 0 S 2 vN dλ ≈ -0.691217. ( 9\n\n)\n\nA formula analogous to (9) can be obtained by using the Concurrence as a measure of entanglement. In this case, by minimizing the mean squared deviation we have\n\nI C α ′ = 1 0 ∆C 2 α ′ dλ, with ∆C α ′ = E corr -α ′ C. ( 10\n\n)\n\nNow, in order to estimate the relative deviation of\n\nS vN with respect to E corr , let us report |∆S αmin |/S vN and |∆S αmin /E corr | as\n\nfunctions of λ at the optimal value α min . The graphs of these functions are shown in Figure 2. 4 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min S vN 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min E corr Figure 2: The relative quadratic deviation between the von Neumann entropy and the correlation energy with respect to the former and the latter, respectively, at the optimal value α min as a function of the coupling constant λ for g = 1.\n\nIn Figure 3 , the relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, at the optimal values α ′ min , is represented.\n\n0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min C 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min E corr Figure 3: The relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, respectively, at the optimal value α ′ min as a function of the coupling constant λ for g = 1."},{"section_type":"OTHER","section_title":"Remark 1","text":"From these graphs, one can argue that the agreement between the two functions S vN and E corr is only qualitatively good, in fact, for very small λ, it is not good at all. However, in an intermediate range of values, i. e., 0.6 ≤ λ ≤ 1 the two functions are almost proportional within the 10%. Analogously, the same is true between energy and Concurrence. Even, the agreement becomes worst comparing the relative deviation of the Concurrence with respect to the correlation energy, since the range in which the relative deviations become smaller than 10% are narrower. Then, the question is whether the above results are i) sufficient to justify the conjecture advanced in [1] , i.e., entanglement can be considered as an estimation of correlation energy; ii) if such a relation has a 5 more concrete physical meaning, in particular whether the minimizing parameter α min and the vanishing point of ∆S αmin does possess any physical meaning (or α ′ min and the vanishing point of ∆C α ′ min ). Notice that in the case of the comparison for the Concurrence simpler analytical expressions appear. For instance one finds ∆C α ′ min = 0.383249 λ\n\n√ λ 2 +4 - √ λ 2 + 4 + 2 2 . Remark 2\n\nWe note that in an interval of values around α min , the deviation function (8) possesses a minimum in the interval of interest 0 ≤ λ ≤ 1, otherwise the minimum is achieved at larger value of λ, or the function is monotonically increasing (see Figure 4 ).\n\n0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 S Α 0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 d S Α dΛ\n\nFigure 4: The deviation ∆S α and its derivative with respect to λ are computed for values of -1.29(red) ≤ α ≤ -0.091(violet), for steps of 0.06. The curve drawn thicker corresponds to α min This behavior suggests to consider the function ∆S αmin as a sort of \"free energy\" , where α min mimics the \"temperature\" specific of the system. If, for some reason, we allow λ to change, then we expect that spontaneously the interaction coupling adjusts itself to the minimum of ∆S αmin . Similar considerations can be made looking at the graphs drawn for the function ∆C α ′ min and its derivative with respect to λ (see Figure 5 ). The function ∆S αmin or, alternatively, the minimum of ∆C α ′ min can be obtained algebraically. Such a minimum is at the value of the coupling constant λ SvN min ≈ 0.485 and λ C min ≈ 0.371, respectively. The authors in [1] studied numerically the von Neumann entropy and the correlation function for a Hydrogen molecule, using an old result by Herring and Flicker [8], going back to an oldest idea by Heitler and London [9], which consists in substituting the molecular binding with a position dependent exchange coupling:\n\nJ(r) ≈ 1.641 r 5 2 e -2 r Ry, ( 11\n\n)\n\nwhere r is given in Bohr radius, see Figure 6 . The maximum value taken by this function is at the point r max = 1.25. Assuming B = 0.5 Ry, i.e. 1 2 of the fundamental level of the Hydrogen atom, the maximum value λ ′ max = J(r max )/B ≈ 6 0.2 0.4 0.6 0.8 1 Λ -0.2 -0.1 0.1 0.2 0.3 C Α' 0.2 0.4 0.6 0.8 1 Λ -0.4 -0.2 0.2 0.4 d C Α' dΛ 5: The deviation ∆C α ′ and its derivative with respect to λ are computed for values of -0.98(red) ≤ α ′ ≤ 0.22(violet), for steps of 0.06. The curve drawn thicker corresponds to α ′ min 0.5 1 1.5 2 2.5 3 r 0.05 0.1 0.15 0.2 J r Ry\n\nFigure 6: The effective interaction Hydrogen-Hydrogen atom 0.470628 < λ SvN min , i.e. the value of the effective interaction value is less than the minimum for the deviation function ∆S αmin . Then, the equilibrium balance between entanglement (as von Neumann entropy) and correlation energy predicts a length of the molecule equal to r max (see the first panel of Figure 7 ). On the other hand, if we consider the energy gap 2B = 3/4 Ry, i.e. the energy step to the first excited state, one obtains the new value λ ′′ max ≈ 0.628, which goes beyond λ min , even if it is always less than 1. Now, the deviation function ∆S αmin has two minima as seen in the second panel of Figure 7 , one of which is at r ′′ -≈ 0.76 , the other one being at r ′′ + ≈ 1.91. These results should be compared with the experimental equilibrium length of the Hydrogen molecule, which is r exp ≈ 2.0. We point out that although the spin-model described by the Hamiltonian (1) is characterized by features which are essentially rough, however we are induced to answer positively to the quest for a physical meaning of the deviation function ∆S αmin . Indeed, the results elucidated in Figure 7 encourage, on one part, improvement of the computation of r in order to make more accurate the comparison with the experimental value r exp . 7 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .5 Ry 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .375 Ry r'' r''\n\nFigure 7: The von Neumann entropy for the 2-spin model for B = .5 Ry (left panel) and for B = .375 Ry (right panel) and the position depending interaction given by ( 11 ).\n\nThe first question to answer is whether this draft works also for the Concurrence. A statement about it is not obvious, since the von Neumann entropy is a nonlinear function of the Concurrence in the 2-qubits case. However, from Figure 8 one can see that the minimized deviation of the Concurrence takes one minimum for relatively large intensity of the magnetic field ( say B ≥ 0.6 Ry), while for weak fields two minima appear, corresponding to the situation depicted nearby. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 r Bohr B Ry 0.2 0.4 0.6 0.8 1 0 1 2 3 4 B(Ry) r(Bohr)\n\nFigure 8: Two contour plots of the minimized deviation of the Concurrence as a function of the magnetic field B (Ry) and of the internuclear distance r, as given by (11) . The range of values divided by the contour lines is [-0.038, 0, 04] for the left panel and [-0.03705, -0, 03000] for the right one that approximatively corresponding to the black area in the left panel.\n\nIn correspondence of the same values considered above, for B = 0.5 Ry the function ∆C α ′ min (r) has two minima at r = 0.79 and r = 1.88, while for B = 0.375 Ry they are located at r = 0.60 and r = 2.25. So one sees that the resulting equilibrium configurations are not much very close to the experimental one. The equilibrium configuration more closest to the experimental one is the minimum occurring at r = 1.88 (B = 1 2 Ry) for the function ∆C α ′ min (r).\n\n8 One sees that one of the resulting equilibrium configurations is only roughly close to the experimental one.\n\nIn other words, to conclude monitoring numerically B the equilibrium configuration more closest to experimental one in the minimum occurring at r = 1.88 for B = 1 2 Ry and at r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r)."},{"section_type":"OTHER","section_title":"A quantum chemical framework to compare entanglement and correlation energy","text":"In this Section we represent the results produced in [1], where the electron entanglement in the Hydrogen molecule, calculated by the von Neumann entropy of the reduced density matrix ρ 1 , is obtained starting by the excitation coefficients of the wave function expanded by a configuration interaction method:\n\nS ρ CISD 1 = -T r ρ CISD 1 log 2 ρ CISD 1 = = - m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 log 2 m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 + -|c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 log 2 |c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 , ( 12\n\n)\n\nwhere c 1 is the coefficient for a single excitation, and c 1,2 is the double excitation (in Appendix A of [10] more details are shown).\n\nIn this framework, entanglement (S) and correlation energy (E corr ), as functions of nucleus -nucleus separation are those in Figure 9 0 0.01 0.02 0.03 0.04 0.05 E c [ a. u. ] E c 0 1 2 3 4 5 R ( Å ) 0 0.2 0.4 0.6 0.8 1 Entanglement (S) S ( ρ 1 CISD ) H 2 Figure 9: Comparison between the entanglement, calculated by the von Neumann entropy of the reduced density matrix, and the electron correlation energy in the Hydrogen molecule.\n\nBy the results given by this model, we want to discuss and to suggest some answers to the questions i) and ii) presented in Remark 1. Even if, in order 9 to represent correlation energy and entanglement, we use two different scales, in Figure 9 we can see that entanglement has a small value in the united atom limit after it is growing for small distances till it arrives at a maximum value then it decrease till it assumes zero value at the separated atom limit and it is exactly the progress of the correlation curve.\n\nIn order to compare the entropy S with the electron correlation energy E corr , we rescale S with the parameter α min calculated with some procedure illustrated in Eq. ( 8 ) and Eq. (9) replacing the integration variable λ with R; in this way we extract\n\nα = E corr S vN dR S vN dR ≈ 0.009. ( 13\n\n)\n\nThe corresponding ∆S αmin = E corr -αS allows us to answer to the question ii); in fact, as it is shown in Figure 10 , the vanishing point of ∆S αmin is, according to the two -spin Ising model, nearby R ≈ 2 Å that corresponds to the equilibrium configuration of the Hydrogen molecule.\n\n0 1 2 3 4 5 R(Å) -0.01 0 0.01 0.02 0.03 0.04 Ec-αS\n\nFigure 10: ∆S αmin for the H 2 molecule as a function of nucleus-nucleus distance."},{"section_type":"METHOD","section_title":"Differences between the Configuration Interaction approach and the two-spin Ising model","text":"The model proposed in Sec. 1 provides us with a measurement of entanglement: indeed, Eq. (3) describes the von Neumann entropy as a function of coupling constant λ, for small λ. By using Eq. ( 7 ), we can express λ in terms of correlation energy and substituting it in Eq. ( 3 ) we can obtain the variation of S vN in terms of E corr .\n\nS vN = - E corr Log Ecorr 2(Ecorr +2) + (E corr + 4)Log Ecorr +4 2(Ecorr +2) (E corr + 2)Log4 . ( 14\n\n)\n\n10 In order to calculated the coefficient of proportionality among S vN and E corr we make an expansion of S vN for E corr → 0 (or equivalently for λ → 0) at the first order, obtaining a straight line characterized by an angular coefficient given by m SvN (Ecorr) = ( 1 4 )(1 + 1 Log2 ). Since this behavior is uncorrect to represent the logatithmic singularity of S vN in the origin, we make an expansion of Eq. ( 14 ), preserving the logarithmic deviation, and we obtain an expression of the form\n\nS vN = AE corr + BE corr Log(E corr ), ( 15\n\n)\n\nwhere A = 1/2 and B = -1/(4Log2).\n\n0.02 0.04 0.06 0.08 0.1 0.025 0.05 0.075 0.1 0.125 0.15 Ecorr S Linear AE+BELogE Figure 11: A comparison among the behavior of Eq. ( 14 ) and its linear approximation and the logarithmic one, for the Ising model.\n\nIn order to compare the behavior of S vN in Eq. ( 14 ), we have organized the numerical data, calculated with the method proposed in [1] , by making a correspondence between each value of E corr and its respective value of S vN , obtaining the plot in Figure 12 0 0.01 0.02 0.03 0.04 0.05 E c 0 0.2 0.4 0.6 0.8 1 S Figure 12: A correspondence of E corr and S vN by the numerical procedure suggested by [1] Of particular significance is the fact that, in the range where S is monotonically increasing, the correlation energy has its maximum, consequently S seems to be not a function. Moreover, it is important to note that E corr begins to decrease for R > 1 Å, region where the states become mixed, i. e. ,T rρ = T rρ 2 ; as depicted in Figure 13. 11 0 1 2 3 4 R(Å ) 0.4 0.6 0.8 1 1.2 1.4 Trρ 2\n\nFigure 13: The increasing of the degree of mixing in the two electron state: in black we depict the trace of ρ, in red the trace of ρ 2 .\n\nProbably, for this reason, the procedure adopted in [1] seems to be not correct: the density matrix, in fact, is calculated starting by the excitation coefficient of a wave function obtained developping with the Configuration Interaction Single Double method a pure two electrons state.\n\nHowever, even if we consider only the first branch of the plot in Figure 12 , i.e. , the numerical values of S vN corresponding with increasing values of E corr , and we fit the values around E corr → 0 with a F = AE corr + BE corr Log(E corr ) we draw out numerical values of the coefficient different from the ones used in Eq. (15) . This result is shown in Figure 14. 0.01 0.02 0.03 0.04 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 S Ecorr A=17.1 B=3.3 Figure 14: A fit of S vN as a function of E corr , around the origin, with a function of the form F = AE corr + BE corr Log(E corr ) whose coefficients A and B assume the numerical values in Figure.\n\nIn particular the arithmetic sign of the coefficient B in the two models are opposite and this implies the opposite concavity of the curve.\n\nThis fact, clearly demonstrates a not satisfactory agreement between the Ising model and the one proposed in [1] ."},{"section_type":"OTHER","section_title":"5 Concluding remarks","text":"We have explored the role of entanglement in the model of two qubits describing the Hydrogen molecule (1), considered as a bipartite system. In our discussion we have limited to the ferromagnetic case governed by the interaction coupling parameter J > 0.\n\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The entanglement can be measured by using the von Neumann entropy or, alternatively, the notion of Concurrence [7] . To compute the entanglement it is convenient to consider two Regions, say I and II, which provide two different reduced density matrices. The entropy turns out to be an increasing function of the coupling constant λ in Region I, but the state under consideration is maximally entangled in Region II indipendently from the anisotropy parameter g.\n\nAn interesting result is that for large coupling constants the entropy approach 1, meaning that all levels are equiprobably visited by the considered spin.\n\nFor weak interactions, at the boundary point λ b = 2 √ 1-g 2 the von Neumann entropy admits a discontinuity, indicating a crossing of the lowest eigenvalues and, in a more general constext, a quantum phase transition [5] .\n\nIn Sec. 2 a comparison between the entanglement and the correlation energy is performed.\n\nTo quantifying the entanglement we resort to the reduced density matrix. The entanglement can also be measured by exploiting the concept of Concurrence.\n\nThe entanglement measure is always bounded, while the energy correlation,\n\nE corr = |E 0 | -2 =\n\n√ 4 + λ 2 -2, is a divergent function of λ. This fact tells us that to look for simple relations valid on the whole λ-axes has no sense.\n\nThus, by limiting ourselves to weak couplings, we have minimized the mean square deviation given by Eq. ( 8 ). This procedure leads to the value α min ≈ -0.691217 for the minimizing parameter (see Eq. (9)). Sec. 1 contains a comparison between the von Neumann entropy and the Concurrence. Such a comparison is illustrated in Figure 1 , for two spin system as a function of the coupling λ for g = 1. Some important points are commented in Remark 1 and Remark 2 .\n\nIn Figure 4 the deviation ∆S α and its derivatives with respect to λ are computed and α min is evaluated for α ranging in the interval -1.29 ≤ α ≤ -0.091.\n\nIn Figure 5 the minimized Concurrence deviation ∆C α (i) α ′\n\nfor the four eigenstates of the 2-spin model is shown.\n\nWe point out the existence of a perfect symmetry among the Concurrence deviations for pairs of eigenstates of opposite eigenvalues.\n\nFormula (11), due to Heitler-London [9], is reported, where the position dependent exchange coupling J(r) is expressed in term of the length r of the 13 nucleus-nucleus separation in the Hydrogen molecule.\n\nTo conclude, the magnetic field B has been monitored such that the equilibrium configuration more closest to the experimental one, r ≈ 2.00, is the minimum occurring at r = 1.88 for B = 1 2 Ry and r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r). We observe also that in the intermediate range of values, i. e., for 0.6 ≤ λ ≤ 1, the two functions SvN and the correlation energy are almost proportional within the 10%.\n\nHowever, when we organized the pairs of points (E corr , S vN ) calculated by following the procedure described by [1], it is clear that the von Neumann entropy cannot be considered a function of correlation energy. The principle cause is that the function E corr presents a maximum in the region where S vN is monotonically increasing.\n\nThe reversing behavior of correlation energy occurs in correspondence with an increase of the mixing degree of the two electrons state. The function E corr in terms of the nucleus -nucleus distance R, increases till the state is pure, on the contrary, when T r(ρ 2 ) becomes discordant from T r(ρ), the function E corr decreases.\n\nThis fact suggests us that the numerical model based on the calculation of S vN starting by the excitation coefficients c i , isn't completley correct because the density matrix is obtained as a product of two electron pure states. However, even if we consider only a branch of the plot in Figure 12 , the function obtained by the two spin Ising model, i. e., Eq. ( 14 ), is unsuitable for fitting these numerical data.\n\nOn the basis of our results, essentially grounded on numerical considerations, in the near feature we would explore more complicated systems of molecules, such as for example the ethylene or other hydrocarbons, and compare these studies with the goals obtained for the Hydrogen molecule."},{"section_type":"OTHER","section_title":"Acknowledgments","text":"The authors acknowledge the Italian Ministry of Scientific Researches (MIUR) for partial support of the present work under the project SINTESI 2004/06 and the INFN for partial support under the project Iniziativa Specifica LE41."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction and the model\",\n \"text\": \"Entanglement is a physical observable measured by the von Neumann entropy or, alternatively, by the Concurrence of the system under consideration.\\n\\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The electron correlation is not directly observable, since it is defined as the difference between the exact ground state energy of the many electrons Schrödinger equation and the Hartree-Fock energy.\\n\\nIn this paper we discuss the Hamiltonian which describes the Hydrogen molecule regarded as a two interacting spin 1 2 (qubit) model. In [1] it was argued that the entanglement (a quantum observable) can be used in analyzing the so-called correlation energy which is not directly observable. From our point of view, the Hydrogen molecule is dealt with a bipartite system governed by the Hamiltonian\\n\\nH H2 = - J 2 (1 + g)σ 1 ⊗ σ 1 - J 2 (1 -g)σ 2 ⊗ σ 2 -B(σ 3 ⊗ σ 3 + σ 0 ⊗ σ 3 ), ( 1\\n\\n) 1\\n\\nwhere σ i stand for the Pauli matrices (σ 0 = I). Actually, this model was considered in [1] in order to illustrate their method. However, here we will make some interpretative changes. Indeed, from our point of view, the states of an isolated atom are strongly reduced to a system with two energy levels related to the intensity of the magnetic field B. Relatively to this scale, the exchange interaction constant J is usually smaller than B, in order to represent the residual interatomic interactions. From the point of view of quantum chemistry, one may interpret the discrete spectrum as provided by the Hartree-Fock calculations, while the interaction coupling J models the residual multielectronic effects, not taken into account by the mean field approximation. For simplicity we limit ourselves to the ferromagnetic phase with J > 0. The parameter g, such that 0 ≤ g ≤ 1, describes the degree of anisotropy corresponding for g = 0 to the completely isotropic XY spin model. Conversely, g = 1 provides the anisotropic XY spin model, the so-called Ising model.\\n\\nWe notice that when the atoms are far apart, their interaction is quite weak. This corresponds to a vanishing value of J. In this situation the state of the system is completely factorized in the product state of the ground states of the indipendent spins. The corresponding total energy, in unit of B, is just the sum of the two fundamental levels, E 0 = -2, which we may consider as the Hartree-Fock approximated fundamental level in molecular structure calculations.\\n\\nWhen J = 0, the fundamental energy eigenvalue is E= -4 + g 2 λ 2 in Region I defined by 0 < λ ≤ 2 √ 1-g 2 , otherwise E = -λ (λ means the coupling constant) in Region II, which is the complement of I which respect to positive real axis. The corresponding (non normalized) eigenstates are |Ψ I = √ g 2 λ 2 +4+2 gλ , 0, 0, 1 and |Ψ II = 0, 1, 1, 0 , respectively. In both cases the state is entangled.\\n\\nSince we are dealing with pure states, the von Neumann entropy [2]\\n\\nS vN = -Tr ρ 1 log 2 ρ 1 ( 2\\n\\n)\\n\\nis chosen to be a measurement of the entanglement, where ρ 1 is the 1-particle reduced density matrix. However, for general mixed states other entanglement estimators (for instance, the Concurrence [4]) have to be used. In the considered case, one has S vN,I = -g 2 g 2 λ 2 + 4 log 1 2 -1 g 2 λ 2 + 4 λ 2 -g 2 g 2 λ 2 + 4 + 4 λ 2 + 8 g 2 λ 2 + 4 + 2 log 1 2 + 1 g 2 λ 2 + 4 1 g 2 λ 2 + 4 g 2 λ 2 + 2 g 2 λ 2 + 4 + 4 log(4) (3) S vN,II = 1. (4) 2 Scrutinizing Eq. (3) and Eq. ( 4 ) it emerges that the entropy is an increasing function of the coupling constant λ in Region I, but the state is maximally entangled in Region II independently from the anisotropy parameter g. One sees that, as it arises graphycally, for g = 1 the entanglement is a monotonic increasing function of the interaction coupling λ. Moreover for weak (< 1) coupling values it is always less than the 30%. Of course, for large coupling constants the entropy approaches 1, meaning that all levels are equiprobably visited by the considered spin.\\n\\nLimiting all further considerations to the case of weak interaction, we observe that at the boundary point λ b = 2 √ 1-g 2 a discontinuity occurs, signaling a crossing of the lowest eigenvalues and, in a more general context, a quantum phase transition [5] .\\n\\nAs it was pointed out in [6] , for quantifying the entanglement we can resort to the reduced density matrix. Furthermore, in [7] , Wootters has shown that for a pair of binary qubits one can use the concept of Concurrence C to measure the entanglement.\\n\\nThe Concurrence reads\\n\\nC(ρ) = max(0, ν 1 -ν 2 -ν 3 -ν 4 ), ( 5\\n\\n)\\n\\nwhere the ν i 's are the eigenvalues of the Hermitian matrix\\n\\nR = √ ρρ √ ρ 1 2 ,\\n\\nwhere ρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ), ρ * being the complex conjugate of ρ taken in the standard basis [7] . Some interesting results on the simple model (1) of the Hydrogen molecule can be achieved by realizing a comparative study of the von Neumann entropy and the Concurrence.\\n\\nTo this aim, we compute the Concurrence C I and C II , i. e.\\n\\nC I = gλ 1 g 2 λ 2 + 4 , C II = 1. ( 6\\n\\n)\\n\\nwhere I and II refer to Regions I and II, where 0 ≤ λ ≤ 2 1-g 2 , and E = -λ, respectively.\\n\\nIn Figure 1 a comparison between the Concurrence and the von Neumann entropy for two spins system as a function of the coupling λ for g = 1 is presented. Sec. 2 contains a comparison between the entanglement and the correlation energy. In Sec. 3 the Configuration Interaction method is introduced to compare entanglement and correlation energy. In Sec. 4 some differences between the Configuration Interaction approach and the two spin Ising model are presented. Finally, our main results are summarized in Sec. 5. 3 1 2 3 4 5 Λ 0.2 0.4 0.6 0.8 1 S vN Conc.\\n\\nFigure 1: Comparison between the Concurrence and the von Neumann entropy for the two spins system as a function of the coupling constant λ for g = 1.\\n\\n2 A comparison between the entanglement and the correlation energy\\n\\nNow we look for a comparison between the entanglement with the energy correlation, which as we have already recalled, it is understood as the difference of the fundamental energy level compared with respect to the corresponding value at vanishing coupling constant λ. For g = 1 and in unities of B it is given by\\n\\nE corr = |E 0 | -2 = 4 + λ 2 -2. ( 7\\n\\n)\\n\\nWe observe that the entanglement measure is always bounded, while E corr is a divergent function of λ. So it does not make much sense to look for simple relations valid on the entire λ-axes. Consequently, limiting ourselves to weak couplings, for 0 ≤ λ ≤ 1, we minimize the mean squared deviation\\n\\nI α = 1 0 ∆S 2 α dλ, with ∆S α = E corr -α S vN . ( 8\\n\\n)\\n\\nThus the minimizing parameter α min will be given by\\n\\nα min = 1 0 E corr S vN dλ 1 0 S 2 vN dλ ≈ -0.691217. ( 9\\n\\n)\\n\\nA formula analogous to (9) can be obtained by using the Concurrence as a measure of entanglement. In this case, by minimizing the mean squared deviation we have\\n\\nI C α ′ = 1 0 ∆C 2 α ′ dλ, with ∆C α ′ = E corr -α ′ C. ( 10\\n\\n)\\n\\nNow, in order to estimate the relative deviation of\\n\\nS vN with respect to E corr , let us report |∆S αmin |/S vN and |∆S αmin /E corr | as\\n\\nfunctions of λ at the optimal value α min . The graphs of these functions are shown in Figure 2. 4 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min S vN 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 S min E corr Figure 2: The relative quadratic deviation between the von Neumann entropy and the correlation energy with respect to the former and the latter, respectively, at the optimal value α min as a function of the coupling constant λ for g = 1.\\n\\nIn Figure 3 , the relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, at the optimal values α ′ min , is represented.\\n\\n0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min C 0.2 0.4 0.6 0.8 1 Λ 0.2 0.4 0.6 0.8 1 C min E corr Figure 3: The relative quadratic deviation between the Concurrence and the correlation energy with respect to the former and the latter, respectively, at the optimal value α ′ min as a function of the coupling constant λ for g = 1.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Remark 1\",\n \"text\": \"From these graphs, one can argue that the agreement between the two functions S vN and E corr is only qualitatively good, in fact, for very small λ, it is not good at all. However, in an intermediate range of values, i. e., 0.6 ≤ λ ≤ 1 the two functions are almost proportional within the 10%. Analogously, the same is true between energy and Concurrence. Even, the agreement becomes worst comparing the relative deviation of the Concurrence with respect to the correlation energy, since the range in which the relative deviations become smaller than 10% are narrower. Then, the question is whether the above results are i) sufficient to justify the conjecture advanced in [1] , i.e., entanglement can be considered as an estimation of correlation energy; ii) if such a relation has a 5 more concrete physical meaning, in particular whether the minimizing parameter α min and the vanishing point of ∆S αmin does possess any physical meaning (or α ′ min and the vanishing point of ∆C α ′ min ). Notice that in the case of the comparison for the Concurrence simpler analytical expressions appear. For instance one finds ∆C α ′ min = 0.383249 λ\\n\\n√ λ 2 +4 - √ λ 2 + 4 + 2 2 . Remark 2\\n\\nWe note that in an interval of values around α min , the deviation function (8) possesses a minimum in the interval of interest 0 ≤ λ ≤ 1, otherwise the minimum is achieved at larger value of λ, or the function is monotonically increasing (see Figure 4 ).\\n\\n0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 S Α 0.2 0.4 0.6 0.8 1 Λ -0.1 -0.05 0.05 0.1 d S Α dΛ\\n\\nFigure 4: The deviation ∆S α and its derivative with respect to λ are computed for values of -1.29(red) ≤ α ≤ -0.091(violet), for steps of 0.06. The curve drawn thicker corresponds to α min This behavior suggests to consider the function ∆S αmin as a sort of \\\"free energy\\\" , where α min mimics the \\\"temperature\\\" specific of the system. If, for some reason, we allow λ to change, then we expect that spontaneously the interaction coupling adjusts itself to the minimum of ∆S αmin . Similar considerations can be made looking at the graphs drawn for the function ∆C α ′ min and its derivative with respect to λ (see Figure 5 ). The function ∆S αmin or, alternatively, the minimum of ∆C α ′ min can be obtained algebraically. Such a minimum is at the value of the coupling constant λ SvN min ≈ 0.485 and λ C min ≈ 0.371, respectively. The authors in [1] studied numerically the von Neumann entropy and the correlation function for a Hydrogen molecule, using an old result by Herring and Flicker [8], going back to an oldest idea by Heitler and London [9], which consists in substituting the molecular binding with a position dependent exchange coupling:\\n\\nJ(r) ≈ 1.641 r 5 2 e -2 r Ry, ( 11\\n\\n)\\n\\nwhere r is given in Bohr radius, see Figure 6 . The maximum value taken by this function is at the point r max = 1.25. Assuming B = 0.5 Ry, i.e. 1 2 of the fundamental level of the Hydrogen atom, the maximum value λ ′ max = J(r max )/B ≈ 6 0.2 0.4 0.6 0.8 1 Λ -0.2 -0.1 0.1 0.2 0.3 C Α' 0.2 0.4 0.6 0.8 1 Λ -0.4 -0.2 0.2 0.4 d C Α' dΛ 5: The deviation ∆C α ′ and its derivative with respect to λ are computed for values of -0.98(red) ≤ α ′ ≤ 0.22(violet), for steps of 0.06. The curve drawn thicker corresponds to α ′ min 0.5 1 1.5 2 2.5 3 r 0.05 0.1 0.15 0.2 J r Ry\\n\\nFigure 6: The effective interaction Hydrogen-Hydrogen atom 0.470628 < λ SvN min , i.e. the value of the effective interaction value is less than the minimum for the deviation function ∆S αmin . Then, the equilibrium balance between entanglement (as von Neumann entropy) and correlation energy predicts a length of the molecule equal to r max (see the first panel of Figure 7 ). On the other hand, if we consider the energy gap 2B = 3/4 Ry, i.e. the energy step to the first excited state, one obtains the new value λ ′′ max ≈ 0.628, which goes beyond λ min , even if it is always less than 1. Now, the deviation function ∆S αmin has two minima as seen in the second panel of Figure 7 , one of which is at r ′′ -≈ 0.76 , the other one being at r ′′ + ≈ 1.91. These results should be compared with the experimental equilibrium length of the Hydrogen molecule, which is r exp ≈ 2.0. We point out that although the spin-model described by the Hamiltonian (1) is characterized by features which are essentially rough, however we are induced to answer positively to the quest for a physical meaning of the deviation function ∆S αmin . Indeed, the results elucidated in Figure 7 encourage, on one part, improvement of the computation of r in order to make more accurate the comparison with the experimental value r exp . 7 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .5 Ry 0.5 1 1.5 2 2.5 3 r -0.015 -0.0125 -0.01 -0.0075 -0.005 -0.0025 S min r; B .375 Ry r'' r''\\n\\nFigure 7: The von Neumann entropy for the 2-spin model for B = .5 Ry (left panel) and for B = .375 Ry (right panel) and the position depending interaction given by ( 11 ).\\n\\nThe first question to answer is whether this draft works also for the Concurrence. A statement about it is not obvious, since the von Neumann entropy is a nonlinear function of the Concurrence in the 2-qubits case. However, from Figure 8 one can see that the minimized deviation of the Concurrence takes one minimum for relatively large intensity of the magnetic field ( say B ≥ 0.6 Ry), while for weak fields two minima appear, corresponding to the situation depicted nearby. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 r Bohr B Ry 0.2 0.4 0.6 0.8 1 0 1 2 3 4 B(Ry) r(Bohr)\\n\\nFigure 8: Two contour plots of the minimized deviation of the Concurrence as a function of the magnetic field B (Ry) and of the internuclear distance r, as given by (11) . The range of values divided by the contour lines is [-0.038, 0, 04] for the left panel and [-0.03705, -0, 03000] for the right one that approximatively corresponding to the black area in the left panel.\\n\\nIn correspondence of the same values considered above, for B = 0.5 Ry the function ∆C α ′ min (r) has two minima at r = 0.79 and r = 1.88, while for B = 0.375 Ry they are located at r = 0.60 and r = 2.25. So one sees that the resulting equilibrium configurations are not much very close to the experimental one. The equilibrium configuration more closest to the experimental one is the minimum occurring at r = 1.88 (B = 1 2 Ry) for the function ∆C α ′ min (r).\\n\\n8 One sees that one of the resulting equilibrium configurations is only roughly close to the experimental one.\\n\\nIn other words, to conclude monitoring numerically B the equilibrium configuration more closest to experimental one in the minimum occurring at r = 1.88 for B = 1 2 Ry and at r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A quantum chemical framework to compare entanglement and correlation energy\",\n \"text\": \"In this Section we represent the results produced in [1], where the electron entanglement in the Hydrogen molecule, calculated by the von Neumann entropy of the reduced density matrix ρ 1 , is obtained starting by the excitation coefficients of the wave function expanded by a configuration interaction method:\\n\\nS ρ CISD 1 = -T r ρ CISD 1 log 2 ρ CISD 1 = = - m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 log 2 m-1 i |c 2i+1 1 | 2 + m-1 i=1 |c 2i+1,2i+2 1,2 | 2 + -|c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 log 2 |c 0 | 2 + m-1 i=1 |c 2i+2 2 | 2 , ( 12\\n\\n)\\n\\nwhere c 1 is the coefficient for a single excitation, and c 1,2 is the double excitation (in Appendix A of [10] more details are shown).\\n\\nIn this framework, entanglement (S) and correlation energy (E corr ), as functions of nucleus -nucleus separation are those in Figure 9 0 0.01 0.02 0.03 0.04 0.05 E c [ a. u. ] E c 0 1 2 3 4 5 R ( Å ) 0 0.2 0.4 0.6 0.8 1 Entanglement (S) S ( ρ 1 CISD ) H 2 Figure 9: Comparison between the entanglement, calculated by the von Neumann entropy of the reduced density matrix, and the electron correlation energy in the Hydrogen molecule.\\n\\nBy the results given by this model, we want to discuss and to suggest some answers to the questions i) and ii) presented in Remark 1. Even if, in order 9 to represent correlation energy and entanglement, we use two different scales, in Figure 9 we can see that entanglement has a small value in the united atom limit after it is growing for small distances till it arrives at a maximum value then it decrease till it assumes zero value at the separated atom limit and it is exactly the progress of the correlation curve.\\n\\nIn order to compare the entropy S with the electron correlation energy E corr , we rescale S with the parameter α min calculated with some procedure illustrated in Eq. ( 8 ) and Eq. (9) replacing the integration variable λ with R; in this way we extract\\n\\nα = E corr S vN dR S vN dR ≈ 0.009. ( 13\\n\\n)\\n\\nThe corresponding ∆S αmin = E corr -αS allows us to answer to the question ii); in fact, as it is shown in Figure 10 , the vanishing point of ∆S αmin is, according to the two -spin Ising model, nearby R ≈ 2 Å that corresponds to the equilibrium configuration of the Hydrogen molecule.\\n\\n0 1 2 3 4 5 R(Å) -0.01 0 0.01 0.02 0.03 0.04 Ec-αS\\n\\nFigure 10: ∆S αmin for the H 2 molecule as a function of nucleus-nucleus distance.\"\n },\n {\n \"section_type\": \"METHOD\",\n \"section_title\": \"Differences between the Configuration Interaction approach and the two-spin Ising model\",\n \"text\": \"The model proposed in Sec. 1 provides us with a measurement of entanglement: indeed, Eq. (3) describes the von Neumann entropy as a function of coupling constant λ, for small λ. By using Eq. ( 7 ), we can express λ in terms of correlation energy and substituting it in Eq. ( 3 ) we can obtain the variation of S vN in terms of E corr .\\n\\nS vN = - E corr Log Ecorr 2(Ecorr +2) + (E corr + 4)Log Ecorr +4 2(Ecorr +2) (E corr + 2)Log4 . ( 14\\n\\n)\\n\\n10 In order to calculated the coefficient of proportionality among S vN and E corr we make an expansion of S vN for E corr → 0 (or equivalently for λ → 0) at the first order, obtaining a straight line characterized by an angular coefficient given by m SvN (Ecorr) = ( 1 4 )(1 + 1 Log2 ). Since this behavior is uncorrect to represent the logatithmic singularity of S vN in the origin, we make an expansion of Eq. ( 14 ), preserving the logarithmic deviation, and we obtain an expression of the form\\n\\nS vN = AE corr + BE corr Log(E corr ), ( 15\\n\\n)\\n\\nwhere A = 1/2 and B = -1/(4Log2).\\n\\n0.02 0.04 0.06 0.08 0.1 0.025 0.05 0.075 0.1 0.125 0.15 Ecorr S Linear AE+BELogE Figure 11: A comparison among the behavior of Eq. ( 14 ) and its linear approximation and the logarithmic one, for the Ising model.\\n\\nIn order to compare the behavior of S vN in Eq. ( 14 ), we have organized the numerical data, calculated with the method proposed in [1] , by making a correspondence between each value of E corr and its respective value of S vN , obtaining the plot in Figure 12 0 0.01 0.02 0.03 0.04 0.05 E c 0 0.2 0.4 0.6 0.8 1 S Figure 12: A correspondence of E corr and S vN by the numerical procedure suggested by [1] Of particular significance is the fact that, in the range where S is monotonically increasing, the correlation energy has its maximum, consequently S seems to be not a function. Moreover, it is important to note that E corr begins to decrease for R > 1 Å, region where the states become mixed, i. e. ,T rρ = T rρ 2 ; as depicted in Figure 13. 11 0 1 2 3 4 R(Å ) 0.4 0.6 0.8 1 1.2 1.4 Trρ 2\\n\\nFigure 13: The increasing of the degree of mixing in the two electron state: in black we depict the trace of ρ, in red the trace of ρ 2 .\\n\\nProbably, for this reason, the procedure adopted in [1] seems to be not correct: the density matrix, in fact, is calculated starting by the excitation coefficient of a wave function obtained developping with the Configuration Interaction Single Double method a pure two electrons state.\\n\\nHowever, even if we consider only the first branch of the plot in Figure 12 , i.e. , the numerical values of S vN corresponding with increasing values of E corr , and we fit the values around E corr → 0 with a F = AE corr + BE corr Log(E corr ) we draw out numerical values of the coefficient different from the ones used in Eq. (15) . This result is shown in Figure 14. 0.01 0.02 0.03 0.04 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 S Ecorr A=17.1 B=3.3 Figure 14: A fit of S vN as a function of E corr , around the origin, with a function of the form F = AE corr + BE corr Log(E corr ) whose coefficients A and B assume the numerical values in Figure.\\n\\nIn particular the arithmetic sign of the coefficient B in the two models are opposite and this implies the opposite concavity of the curve.\\n\\nThis fact, clearly demonstrates a not satisfactory agreement between the Ising model and the one proposed in [1] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"5 Concluding remarks\",\n \"text\": \"We have explored the role of entanglement in the model of two qubits describing the Hydrogen molecule (1), considered as a bipartite system. In our discussion we have limited to the ferromagnetic case governed by the interaction coupling parameter J > 0.\\n\\nThe concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The entanglement can be measured by using the von Neumann entropy or, alternatively, the notion of Concurrence [7] . To compute the entanglement it is convenient to consider two Regions, say I and II, which provide two different reduced density matrices. The entropy turns out to be an increasing function of the coupling constant λ in Region I, but the state under consideration is maximally entangled in Region II indipendently from the anisotropy parameter g.\\n\\nAn interesting result is that for large coupling constants the entropy approach 1, meaning that all levels are equiprobably visited by the considered spin.\\n\\nFor weak interactions, at the boundary point λ b = 2 √ 1-g 2 the von Neumann entropy admits a discontinuity, indicating a crossing of the lowest eigenvalues and, in a more general constext, a quantum phase transition [5] .\\n\\nIn Sec. 2 a comparison between the entanglement and the correlation energy is performed.\\n\\nTo quantifying the entanglement we resort to the reduced density matrix. The entanglement can also be measured by exploiting the concept of Concurrence.\\n\\nThe entanglement measure is always bounded, while the energy correlation,\\n\\nE corr = |E 0 | -2 =\\n\\n√ 4 + λ 2 -2, is a divergent function of λ. This fact tells us that to look for simple relations valid on the whole λ-axes has no sense.\\n\\nThus, by limiting ourselves to weak couplings, we have minimized the mean square deviation given by Eq. ( 8 ). This procedure leads to the value α min ≈ -0.691217 for the minimizing parameter (see Eq. (9)). Sec. 1 contains a comparison between the von Neumann entropy and the Concurrence. Such a comparison is illustrated in Figure 1 , for two spin system as a function of the coupling λ for g = 1. Some important points are commented in Remark 1 and Remark 2 .\\n\\nIn Figure 4 the deviation ∆S α and its derivatives with respect to λ are computed and α min is evaluated for α ranging in the interval -1.29 ≤ α ≤ -0.091.\\n\\nIn Figure 5 the minimized Concurrence deviation ∆C α (i) α ′\\n\\nfor the four eigenstates of the 2-spin model is shown.\\n\\nWe point out the existence of a perfect symmetry among the Concurrence deviations for pairs of eigenstates of opposite eigenvalues.\\n\\nFormula (11), due to Heitler-London [9], is reported, where the position dependent exchange coupling J(r) is expressed in term of the length r of the 13 nucleus-nucleus separation in the Hydrogen molecule.\\n\\nTo conclude, the magnetic field B has been monitored such that the equilibrium configuration more closest to the experimental one, r ≈ 2.00, is the minimum occurring at r = 1.88 for B = 1 2 Ry and r = 2.25 for B = 0.375 Ry for the function ∆C α ′ min (r). We observe also that in the intermediate range of values, i. e., for 0.6 ≤ λ ≤ 1, the two functions SvN and the correlation energy are almost proportional within the 10%.\\n\\nHowever, when we organized the pairs of points (E corr , S vN ) calculated by following the procedure described by [1], it is clear that the von Neumann entropy cannot be considered a function of correlation energy. The principle cause is that the function E corr presents a maximum in the region where S vN is monotonically increasing.\\n\\nThe reversing behavior of correlation energy occurs in correspondence with an increase of the mixing degree of the two electrons state. The function E corr in terms of the nucleus -nucleus distance R, increases till the state is pure, on the contrary, when T r(ρ 2 ) becomes discordant from T r(ρ), the function E corr decreases.\\n\\nThis fact suggests us that the numerical model based on the calculation of S vN starting by the excitation coefficients c i , isn't completley correct because the density matrix is obtained as a product of two electron pure states. However, even if we consider only a branch of the plot in Figure 12 , the function obtained by the two spin Ising model, i. e., Eq. ( 14 ), is unsuitable for fitting these numerical data.\\n\\nOn the basis of our results, essentially grounded on numerical considerations, in the near feature we would explore more complicated systems of molecules, such as for example the ethylene or other hydrocarbons, and compare these studies with the goals obtained for the Hydrogen molecule.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgments\",\n \"text\": \"The authors acknowledge the Italian Ministry of Scientific Researches (MIUR) for partial support of the present work under the project SINTESI 2004/06 and the INFN for partial support under the project Iniziativa Specifica LE41.\"\n }\n]","isConversational":false}}},{"rowIdx":95,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0523"},"arxiv_id":{"kind":"string","value":"0704.0523"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevA.76.042103"},"content_hash":{"kind":"string","value":"1ca28034a1181cbdbddecf9d042ad8b097ea11a943736a666791eb33df4d9ea3"},"title":{"kind":"string","value":"Quantum superpositions and entanglement of thermal states at high\n temperatures and their applications to quantum information processing"},"abstract":{"kind":"string","value":"We study characteristics of superpositions and entanglement of thermal states\nat high temperatures and discuss their applications to quantum information\nprocessing. We introduce thermal-state qubits and thermal-Bell states, which\nare a generalization of pure-state qubits and Bell states to thermal mixtures.\nA scheme is then presented to discriminate between the four thermal-Bell states\nwithout photon number resolving detection but with Kerr nonlinear interactions\nand two single-photon detectors. This enables one to perform quantum\nteleportation and gate operations for quantum computation with thermal-state\nqubits."},"authors":{"kind":"list like","value":["Hyunseok Jeong and Timothy C. Ralph"],"string":"[\n \"Hyunseok Jeong and Timothy C. Ralph\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["quant-ph"],"string":"[\n \"quant-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"We study characteristics of superpositions and entanglement of thermal states at high temperatures and discuss their applications to quantum information processing. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure-state qubits and Bell states to thermal mixtures. A scheme is then presented to discriminate between the four thermal-Bell states without photon number resolving detection but with Kerr nonlinear interactions and two singlephoton detectors. This enables one to perform quantum teleportation and gate operations for quantum computation with thermal-state qubits.\n\nIn many problems considered within the framework of quantum physics, physical systems are treated as pure states that can be represented by state vectors, or equivalently, by wave functions. Even though such an approach is simple and useful to address certain problems, it could often be quite different from real conditions of physical systems. This may be particularly true when one deals with macroscopic physical systems in terms of quantum physics. A macroscopic object is a complex open system which cannot avoid continuous interactions with the environment. Such a physical system is generally in a significantly mixed state and cannot be represented by a state vector. In general, mixed states are subtle objects whose properties are significantly more difficult to characterize than pure states.\n\nSchrödinger's famous cat paradox is a typical example where a massive classical object was assumed to be a pure state. It describes a counter-intuitive feature of quantum physics which dramatically appears when the principle of quantum superposition is applied to macroscopic objects. In the original paradox and its various explanations, the initial cat isolated in the steel chamber is considered a pure state that can be represented by a state vector such as |alive (or a wave function such as ψ alive ). The cat isolated from the environment is then assumed to interact with a microscopic superposition state, (|g +|e )/ √ 2, where |g and |e are the ground and excited states of a two-level atom. The cat will be dead if the atom is found in the excited state, |e , while it will remain alive if otherwise. Thus in Schrödinger's gedanken experiment the cat is entangled with the atom as (|g |alive + |e |dead ) √ 2, where the alive and dead statuses of the cat are described by the state vectors |alive and |dead . If one measures out the atomic system on the superposed basis, (|g ± |e )/ √ 2, the cat will be in a superposition of alive and dead states such as (|alive ± |dead )/ √ 2. It is often argued that such superposed states and entangled states can theoretically exist but are virtually impossible to observe because one cannot perfectly isolate a macroscopic object such as the cat from its environment [4] .\n\nHowever, this explanation is not fully satisfactory because the cat, a macroscopic object, is a complex open system which cannot be represented by a state vector. One may argue that the cat could be assumed to be in an unknown pure state such that the cat was certainly alive but the exact state of the cat was unknown. However, the interactions between the cat and its environment can cause the cat to become entangled with the environment [5] . In such a case, even though one can perfectly isolate the cat in the steel chamber from the enviroment, the cat will remain entangled with the environment due to its pre-interactions with the environment. Therefore, strictly speaking, even to assume a cat as an unknown pure state in the steel chamber is not legitimate. Thus a key point here is that it is unsatisfactory to describe the cat by a pure state such as |alive and |dead . We may need a more realistic assumption that the \"cat\" in Schrödinger's paradox was in a significantly mixed classical state. An intriguing question is then whether the quantum properties of the resulting state would still remain or diminish under such an assumption.\n\nRecently, such an analogy of Schrödinger's cat paradox, where the state corresponding to the virtual cat is a significantly mixed thermal state, was investigated [6] . A thermal state with a high temperature is considered a classical state in quantum optics. As the temperature of the thermal state increases, the degree of mixedness, which can be quantified by linear entropy, rapidly approaches the maximum value. When the temperature approaches infinity, the thermal state does not show any quantum properties. As a comparison, coherent states with large amplitudes are known as the most classical pure states [7] , and their superposition is often regarded as a superposition of classical states [8] . However, coherent states are still pure states which may not well represent truly classical systems, and they display some nonclassical features [9] . In Ref. [6] , it was shown that prominent quantum properties can actually be transferred from a microscopic superposition to a significantly mixed thermal state (i.e. a thermal state of which the degree of mixedness is close to the maximum value) at a high temperature through an experimentally feasible process. This result clarifies that unavoidable initial mixedness of the cat does not preclude strong quantum phenomena.\n\nOne of the results in Ref. [6] is that quantum entanglement can be produced between thermal states with nearly the maximum Bell-inequality violation when the temperatures of both modes goes to infinity. In previous related results, Bose et al. showed that entanglement can arise when two systems interact if one of the system are pure even when the other system is extremely mixed [10] . There is an interesting previous example shown by Filip et al. for the maximum violation of Bell's inequality when one of the modes is an extremely mixed thermal state [11] . Very recently, Ferreira et al. showed that entanglement can be generated at any finite temperature between high Q cavity mode field and a movable mirror thermal state [12] . However, in these example [10, 11, 12] only one of the modes is considered a large thermal state [10, 11, 12] and entanglement vanishes in the infinite temperature limit [10, 12] , which is obviously in contrast to the result presented in Ref. [6] . Entanglement for both of the modes at the thermal limit of the infinitely high temperature has not been found before. Remarkably, the violation of Bell's inequality in our examples reaches up to Cirel'son's bound [13] even in this infinite-temperature limit for both modes. As Vedral [14] and Ferreira et al. [12] pointed out it is believed that high temperatures reduce entanglement and all entanglement vanishes if the temperature is high enough, which is obviously not the case in Ref. [6] .\n\nThe purpose of this paper is twofold. Firstly, we review and further investigate various properties of superpositions and entanglement of thermal states at high temperatures [6] . In particular, we investigate two classes of highly mixed symmetric states in the phase space. Both the classes of these states do not show typical interference patterns in the phase space while they manifest strong singular behaviors. Interestingly, the first class of states has neither squeezing properties nor negative values in their Wigner functions, however, they are found to be highly nonclassical states. The second class of states has the maximum negativity in the Wigner function. Further, we discuss the possibility of quantum information processing with thermal-state qubits. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure Bell states. We show that four thermal-Bell states can be well discriminated by nonlinear interactions without photon number resolving measurements. Quantum teleportation and gate operations for thermal-state qubits can be realized using the Bell measurement scheme.\n\nThis paper is organized as follows. In Sec. II, we review the generation process of superpositions of thermal states and study their characteristics. In Sec. III, we study en-tanglement of thermal states, i.e., Bell inequality violations. In Sec. IV, we discuss the possibility of quantum information processing using thermal states. We first define the thermal-state qubit and the Bell-basis states using thermal-state entanglement. We then show that the four Bell states can be well discriminated by homodyne detection and two Kerr nonlinearities. It follows that quantum teleportation and quantum gate operations can be realized with thermal-state qubits. We conclude with final remarks in Sec. V.\n\nLet us first consider a two-mode harmonic oscillator system. A displaced thermal state can be defined as\n\nρ th (V, d) = d 2 αP th (V, d)|α α| ( 1\n\n)\n\nwhere |α is a coherent state of amplitude α and\n\nP th α (V, d) = 2 π(V -1) exp[- 2|α -d| 2 V -1 ] (2)\n\nwith variance V and displacement d in the phase space.\n\nThe thermal temperature τ increases as V increases as e ν/τ = (V + 1)/(V -1), where is Planck's constant and ν is the frequency [15] . Suppose that a microscopic superposition state\n\n|ψ a = 1 √ 2 (|0 a + |1 a ), (3)\n\nwhere |0 and |1 are the ground and first excited states of the harmonic oscillator, interacts with a thermal state ρ th b (V, d) and the interaction Hamiltonian is\n\nH K = λâ † âb † b (4)\n\nwhich corresponds to the cross Kerr nonlinear interaction. The resulting state is then\n\nρ ent ab = 1 2 d 2 αP th (V, d) |0 0| ⊗ |α α| + |1 0| ⊗ |αe iϕ α| + |0 1| ⊗ |α αe iϕ | + |1 1| ⊗ |αe iϕ αe iϕ | ( 5\n\n)\n\nand ϕ is determined by the strength of the nonlinearity λ and the interaction time. The Wigner representation of ρ ent ab is\n\nW ent ab (α, β) = 1 π e -2|α| 2 W th (β; d) + 2αV c (β; d) + 2[αV c (β; d)] * + (4|α| 2 -1)W th (β; de iϕ ) (6)\n\nwhere α and β are complex numbers parametrizing the phase spaces of the microscopic and macroscopic systems respectively and\n\nW th (α; d) = 2 πV exp[- 2|α -d| 2 V ], (7)\n\nV c (α; d) = 2 πJK exp[- 2 K (1 -e iϕ )d 2 - 1 J (α - 2e iϕ d K )(α * - 2d K )], (8)\n\nK = 2 + (V -1)(1 -e iϕ\n\n), J = (sin ϕ/2 + iV cos ϕ/2)/(2V sin ϕ/2 + 2i cos ϕ/2), and d has been assumed real without loss of generality. If one traces ρ ent ab over mode a, the remaining state will be simply in a classical mixture of two thermal states and its Wigner function will be positive everywhere. However, if one measures out the \"microscopic part\" on the superposed basis, i.e., (|0 a ± |1 a )/ √ 2, the \"macroscopic part\" for mode b may not lose its nonclassical characteristics. Such a measurement on the the superposed basis will reduce the remaining state to\n\nρ sup(±) = N ± s d 2 αP th (V, d) |α α| ± |αe iϕ α| ± |α αe iϕ | + |αe iϕ αe iϕ | , (9)\n\nwhere N ± s are the normalization factors, and its Wigner function is\n\nW sup(±) (α) = N ± s {W th (α; d) ± V c (α; d) ± {V c (α; d)} * + W th (α; de iϕ )}. ( 10\n\n)\n\nThe ± signs in Eqs. ( 8 ) and (9) correspond to the two possible results from the measurement of the microscopic system. The state in Eq. ( 10 ) is a superposition of two thermal states. A feasible experimental setup to generate superpositions of thermal states is atom-field interactions in cavities, where a π/2 pulse can be used to prepare the atom in a superposed state. This type of experiment has already been performed to produce a superposition of coherent states [16] . In our cases, simply thermal states can be used instead of coherent states. Another possible setup is an all-optical scheme with free-traveling fields and a cross-Kerr medium, where a standard singlephoton qubit could be used as the microscopic superposition. Recently, there have been theoretical and experimental efforts to produce and observe giant Kerr nonlinearities using electromagnetically induced transparency [17] . Furthermore, it was shown that a weak Kerr nonlinearity can still be useful if a initially strong field is employed in this type of experiment [18] . We shall further explain this with examples in Sec. III.\n\nThe negativity of the Wigner function is known as an indicator of non-classicality of quantum states. In order to observe negativity of the Wigner function in a real experiment, its absolute minimum negativity should be large enough. The minimum negativity of the Wigner function in Eq. ( 6 ) for V = 1 is -0.144 for d = 0 and -0.246 for d → ∞. Now suppose the initial state can be considered a classical thermal state by letting V ≫ 1. One might expect that the negativity would be washed out as the initial state becomes mixed, but this is not the case. The minimum negativity actually increases as V gets larger. If V → ∞, the minimum negativity of the Wigner function ( 6 ) is -0.246 regardless of d: no matter how mixed the initial thermal state was, the minimum negativity of Wigner function is found to be a large value. The point in the phase space which gives the minimum negativity when V ≫ 1 or d ≫ 0 is (-1 2 , 0) and has negativity\n\nW neg ≡ W ent ab (- 1 2 , 0) = 2(-2 + 1 V exp[-2d 2 V ]) π 2 √ e . (11)\n\nIt can be shown that\n\nW neg approaches -4/(π 2 √ e) ≈ -0.246 when either d → ∞ or V → ∞.\n\nThis effect is obviously due to the interaction between the microscopic superposition and the macroscopic thermal state. If the initial microscopic state is not superposed, e.g., |ψ a = |1 a , the resulting state will be a simple direct product, (|1 1|) a ⊗ ρ th b (V, -d). Whilst for V = 1 this state will exhibit negativity, this is washed out and tends to zero as V → ∞. Needless to say, if it was |0 a instead of |1 a , the resulting Wigner function will be a direct product of two Gaussian states whose Wigner fucntion can never be negative. The superpositon state (3) plays the crucial role in making the minimum negativity of the resulting Wigner function always saturate to a certain negative value no matter how mixed and classical the initial state of the other mode becomes. The Wigner functions of the single-mode states, W sup(±) (α), in Eq. (10) show large negative values. The minimum negativity of the Wigner function W sup(-) (α) is W sup(-) (0) = 2/π regardless of the values of V and d. On the other hand, the minimum negativity of the Wigner function W sup(+) (α) approaches 2/π for d → ∞ and disappears when d = 0.\n\nWhen ϕ = π, the state (9) becomes (12) where\n\nρ ± = N (ρ th (V, d)±σ(V, d)±σ(V, -d)+ρ th (V, -d)),\n\nσ(V, d) = d 2 αP th (V, d)| -α α| and N = 2 1 ± exp[-2d 2 V ] V 2 . ( 13\n\n)\n\nIf the initial state for mode b is a pure coherent state, i.e., V = 1, the measurement on the superposed basis for mode a will produce a superposition of two pure coherent states as\n\n| Ψ ± = 1 √ 1 ± e -2|α| 2 (|α ± | -α ), (14)\n\nwhere α = d. The probability P ± to obtain the state ρ ± is obtained as [19 ] where\n\nP ± = ψ ± |Tr b [ρ ent ab ]|ψ ± = 1 2 (1 ± exp[-2d 2 V ] V ), ( 15\n\n)\n\n|ψ ± = (|0 ±|1 )/ √ 2.\n\nThe probability approaches P ± = 1/2 when either d or V becomes large.\n\nAs an analogy of Schrödinger's cat paradox, the variance V corresponds to the size the initial \"cat\", and the distance d between the two thermal component states corresponds to distinguishability between the \"alive cat\" and the \"dead cat\". Suppose that both V and d are very large for the initial thermal state. The two thermal states ρ th (V, ±d) become macroscopically distinguishable when d ≫ √ V , and our example may become a more realistic analogy of the cat paradox in this limit. Both the states ρ ± in this case show probability distributions with two Gaussian peaks and interference fringes [6] . Figure 1 presents the probability distributions of x (≡ Re[α]) and p (≡ Im[α]) for ρ -(a) when V = 100 and d = 100 and (b) when V = 1000 and d = 300. The probability distribution of x (p) for ρ ± can be obtained by integrating the Wigner function of ρ ± over p (x). The two Gaussian peaks along the x axis and interference fringes along the p axis shown in Fig. 1 are a typical signature of a quantum superposition between macroscopically distinguishable states. The visibility v of the interference fringes is defined as [15]\n\nv = I max -I min I max + I min , (16)\n\nwhere I = dxW sup(-) (α) and the maximum should be taken over p. It can be simply shown that the visibility v is always 1 regardless of the value of V . Note that d should increase proportionally to √ V to maintain the condition of classical distingushability between the two component thermal states ρ th (V, ±d). The interference fringes with high visibility are incompatible with classical physics and evidence of quantum coherence. The fringe spacing (the distance between the fringes) does not depend on V but only on d, i.e., a pure superposition of coherent states shows the same fringe spacing for a given d. We emphasize that the states shown in Fig. 1 are \"superpositions\" of severely mixed thermal states.\n\nAn experimental realization of a nonlinear effect corresponding to ϕ = π is very demanding particularly in the presence of decoherence. Here we point out that the method using a weak nonlinear effect (ϕ ≪ π) combined with a strong field (d ≫ 1) [18] can be useful to generate a thermal-state superposition with prominent interference patterns. In Fig. 2 , we have used experimentally accessible values, V = 5, d = 2000 and ϕ = π/1000, but the fringe visibility is still 1. In this case, decoherence during the nonlinear interaction would be significantly reduced because of the decrease of the interaction time [18] . Note also that, if required, the state in Fig. 2 can be moved to the center of the phase space, for example, using a biased beam splitter (BS) and a strong coherent field [18] .\n\nLet us assume that d = 0, i.e., the initial state is the thermal state, ρ th (V, 0), at the origin of the phase space. In this case, the thermal-state superpositions, ρ ± , are produced with probabilities, P ± = (1/2){1 ± (1/V )}, respectively. Figure 3 shows the Wigner functions of ρ + dependent on the interaction time between the macro-scopic thermal state and the microscopic superposition in a cross Kerr medium. The state is always symmetric in the phase space regardless of the interaction time as shown in Fig. 3 . In this figure, the initial state is a thermal state of V = 100 (Fig. 3(a) ). In a relatively short time (θ = π/32 and θ = π/16), the state shows some interference patterns. When θ = π, the evolved state looks very localized around the origin as shown in Fig 3 . The generated state at θ = π does not show negativity of the Wigner function nor squeezing properties. On the other hand, a well defined P function does not exist for this state.\n\nIn the case of ρ -, with the same assumption d = 0, the Wigner function at ϕ = π has the minimum negativity (-2/π) at the origin regardless of V . As a result of the interaction with the microscopic superposition, a deep hole to the negative direction below zero has been formed around the origin for ρ -as shown in Fig. 4 .\n\nEntanglement between macroscopic objects and its Bell-type inequality tests are an important issue. In this section, we shall show that entanglement can be generated between high-temperature thermal states even when the temperature of each mode goes to infinity. 2 200 1 400 600 800 1000 V 2.1 2.2 2.3 B (a) 50 100 150 d 2.2 2 0 2.4 2.6 2.8 B (b) FIG. 5: (a) The optimized violation, B ≡ |B + |max, of Bell-\n\nCHSH inequality for the \"thermal-state entenglement\", ρ+, of V = 1000 (solid curve) and V = 100 (dashed curve). The Bell-violation of a pure entangled coherent state, i.e., V = 1, has been plotted for comparison (dotted curve). The Bell-violation B approaches its maximum bound, 2 √ 2, when d ≫ √ V regardless of the level of the mixedness V . (b) The optimized Bell-violation B against d for the different type of thermal-state entanglement generated using a 50:50 beam splitter from ρ + . V = 1000 (solid curve), V = 100 (dashed curve) and V = 1 (dotted curve).\n\nIf the microscopic superposition interacts with two thermal states, ρ th b (V, d) and ρ th c (V, d), and the micro-scopic particle is measured out on the superposed basis, the resulting state will be\n\nρ tm(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (17)\n\nwhere\n\nN t = 2 1 ± exp[-4d 2 V ] V 2 . ( 18\n\n)\n\nSuch two-mode thermal-state entanglement can be generated using two cavities and an atomic state detector [20] .\n\nExtending the two cavities to N cavities, entanglement of N -mode thermal states can also be generated. Such a state is an analogy of the N -mode pure GHZ state [21] but each mode is extremely mixed. Here we shall consider the Bell-CHSH inequality [22, 23] with photon number parity measurements [20, 24] . The parity measurements can be performed in a high-Q cavity using a far-off-resonant interaction between a two-level atom and the field [25] . The Bell-CHSH inequality can be represented in terms of the Winger function as [24 ]\n\n|B (±) | = π 2 4 |W tm(±) (α, β) + W tm(±) (α, β ′ ) + W tm(±) (α ′ , β) -W tm(±) (α ′ , β ′ )| ≤ 2, (19)\n\nwhere W tm(±) (α, β) is the Wigner function of ρ tm(±) in Eq. ( 17 ). As shown in Fig. 5 , the Bell-violation approaches the maximum bound for a bipartite measurement, 2 √ 2 [13] , when d ≫ √\n\nV regardless of the level of the mixedness V , i.e., the temperatures of the thermal states. Note that it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(a) . This implies that entanglement of nearly 1 ebit has been produced between the two significantly mixed thermal states for d ≫ √ V , and such \"thermal-state entanglement\" cannot be described by a local theory.\n\nA different type of macroscopic entanglement can be generated by applying the beam splitter operation\n\nexp[θ/2(e iφ â † s âd -e -iφ â † d âs )], (20)\n\non the \"thermal-state superpositions\" in Eq. ( 9 ). The state after passing through a 50:50 beam splitter can be represented as\n\nN d 2 αP th α (V, d) | α √ 2 , - α √ 2 ± | - α √ 2 , α √ 2 α √ 2 , - α √ 2 | ± - α √ 2 , α √ 2 | , (21\n\n) 200 400 600 800 1000 V 2.1 2 0 2.2 2.3 B FIG. 6: The optimized Bell-violation B against V for the slightly different type of thermal-state entanglement generated using a 50:50 beam splitter using ρ + when d = 0. 3.13 3.14 3.15 3.16 ϕ 1.6 2.2 2 2.4 2.6 2.8 B (a) 3.13 3.14 3.15 3.16 ϕ 1.6 2 2.2 2.4 2.6 2.8 B (b) where N is defined in Eq. ( 13 ). When d is large, this state violates the Bell-CHSH inequality to the maximum bound 2 √ 2 regardless of the level of mixedness V as shown in Fig. 5(b ). Again, it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(b ). Furthermore, these states severely violate Bell's inequality even when d = 0 as V increases as shown in Fig. 6 . We have found that the optimized Bell violation of these states approaches 2.32449 for V → ∞. Interestingly, this value is exactly the same as the optimized Bell-CHSH violation for a pure two-mode squeezed state in the infinite squeezing limit [26] . Note that multilmode entangled states can be generated using multiple beam splitters.\n\nIt should be noted that the Bell violations are more sensitive to the interaction time when either V or d is larger. Figure 7 clearly shows this tendency. Therefore, in order to observe the Bell violations using the mixed state of V (and d) large, the interaction time in the Kerr medium should be more accurate.\n\nIn this section, we discuss the possibility of quantum information processing with thermal-state qubits and thermal-state entanglement.\n\nWe introduce a thermal-state qubit\n\nρ ψ = |a| 2 ρ th (V, d) ± ab * σ(V, d) ± a * bσ(V, -d) + |b| 2 ρ th (V, -d), (22)\n\nwhere a and b are arbitrary complex numbers. The basis states, ρ th (V, d) and ρ th (V, -d), can be well discriminated by a homodyne measurement when d is larger than V . The thermal state qubit (22) can be re-written as\n\nρ ψ = d 2 αP th α (V, d) a|α + b| -α a * α| + b * -α| , (23)\n\nwhich can be understood as a generalization of the coherent state qubit, a|d + b|d , where |d is a coherent state of amplitude d. The thermal-state qubit (23) becomes identical to the coherent-state qubit when V = 1.\n\nWe also define four thermal-Bell states as\n\nρ Φ(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (24) ρ Ψ(±) = N t ρ th (V, d) ⊗ ρ th (V, -d) ± σ(V, d) ⊗ σ(V, -d) ± σ(V, -d) ⊗ σ(V, d) + ρ th (V, -d) ⊗ ρ th (V, d) (25)\n\nwhere N t was defined in Eq. ( 18 ). The thermal-Bell states can be written as For quantum information processing applications, it is an important task to discriminate between the four Bell states. Here we discuss two possible ways to discriminate between the thermal-Bell states (25) . We shall only briefly describe the first scheme using photon number resolving measurements and focus on the second scheme using nonlinear interactions. The first method is to simply use a 50-50 beam splitter and two photon number resolving detectors as shown in Fig. 8(a) . This scheme is basically the same as the Bellstate measurement scheme with pure entangled coherent states [27, 28] . Let us suppose that the amplitude, d, is large enough, i.e., d ≫ √ V . If the incident state was ρ Φ(+) or ρ Φ(-) , most of the photons are detected on detector A in in Fig. 8(a) . Meanwhile, most of the photons are detected on detector B when the incident state was ρ Ψ(+) or ρ Ψ(-) . The average photon numbers between the \"many-photon case\" and the \"few-photon case\" are compared in Fig. 9 . Furthermore, the states ρ Ψ(+) and ρ Φ(+) contain only even numbers of photons while ρ Ψ(-) and ρ Φ(-) contain only odd numbers of photons. Therefore, all the four Bell states can be well discriminated by analyzing numbers of photons detected at detectors A and B. For example, if detector A detects many photons while detector B detects few and the total photon number detected by the two detectors are even, this means that state ρ Φ(+) was measured by the thermal-Bell measurement. The nonzero failure probability can be made arbitrarily small by increasing d.\n\nρ Φ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, β ± | -α, -β α, β| ± -α, -β| , ( 26\n\n)\n\nρ Ψ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, -β ± | -α, β α, -β| ± -α, β| . (27)\n\nHowever, the average photon numbers of the thermal-Bell states are high when V ≫ 1 and d ≫ 1. In this case, it would be unrealistic to use photon number resolving detectors. It would be an interesting question whether\n\n0.5 1 1.5 2 2.5 3 d 10 0 0 20 30 N (a) 0.5 1 1.5 2 2.5 3 d 0 0 10 20 30 N (b)\n\nFIG. 9: The average photon number N for the \"many-photon case\" (solid line) and the \"few-photon case\" (dashed line) for V = 10 against d (a) when the input state is either ρ Φ(+) or ρ Ψ(+) and (b) when the input state is either ρ Φ(-) or ρ Ψ(-) .\n\nthese four thermal-Bell states can be distinguished by classical measurements, such as homodyne detection, instead of photon number resolving detection. Our alternative scheme employs cross-Kerr nonlinearities and single photon detectors as shown in Fig. 8(b ). Let us first suppose that the input field was ρ Φ(+) . The incident twomode state passes through a 50-50 beam splitter, BS1. The state after passing through the 50:50 beam splitter, BS1, is\n\nρ B = N t d 2 αd 2 βP th α (V, d)P th β (V, d) |η, -ξ η, -ξ| + |η, -ξ -η, ξ| + | -η, ξ η, -ξ| + | -η, ξ -η, ξ| (28)\n\nwhere η = (α+β)/ √ 2 and ξ = (α-β)/ √ 2. Two dual-rail single photon qubits, |ψ + ee ′ and |ψ + f f ′ , where\n\n|ψ + = 1 √ 2 (|0 |1 + |1 |0 ), (29)\n\nare prepared using two single photons and 50:50 beam splitters, BS2 and BS3, as shown in Fig. 8(b ). Then, traveling fields at modes c and d interacts with those of modes e and f , respectively, in cross-Kerr nonlinear media. We suppose that the interaction time is t = π/λ, and the resulting state is then\n\nρ B ′ = U ce U df ρ B cd ρ q ee ′ ρ q f f ′ U † ce U † df (30)\n\nwhere U ce = exp[iπH K ce /λ ] and ρ q = |ψ q ψ q |. An explicit form of Eq. ( 30 ) can then be simply obtained using the identity\n\nU ce |α c |0 e = |α c |0 e , U ce |α c |1 e = | -α c |1 e ( 31\n\n)\n\nwhere |α is a coherent state. However, we omit such an explicit expression in this paper for it is too lengthy. After the nonlinear interactions, the qubit parts, modes e, e ′ , f and f ′ , should be measured with the measurement basis\n\n{| + + , | + -, | -+ , | --} (32)\n\nwhere\n\n| + + = |ψ + ee ′ |ψ + f f ′ , | + -= |ψ + ee ′ |ψ -f f ′ , | -+ = |ψ -ee ′ |ψ + f f ′ , | --= |ψ -ee ′ |ψ -f f ′ , and |ψ -= (|0 |1 -|1 |0 )/ √ 2.\n\nThis measurement can be performed using two 50:50 beam splitters, BS4 and BS5, and four detectors, A1, A2, B1 and B2, as shown in Fig. 8(b ). If detector A1 and B1 click, i.e., the measurement result is | + + , the resulting state at modes c and d is\n\nρ ++ = N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η + | -η )( η| + -η|) c ⊗ (|ξ + | -ξ )( ξ| + -ξ|) d (33)\n\nNote that state ρ ++ is not normalized, which implies that the probability of obtaining the corresponding measurement result is not unity. The probability of obtaining this result is\n\nP ++ = (V + 1)(V + e -4d 2 V ) 2(V 2 + e -4d 2 V ) . (34)\n\nWhen the result is either | +or | -+ , the result is\n\nψ 2 |ρ B ′ |ψ 2 = ψ 3 |ρ B ′ |ψ 3 = 0, (35)\n\nwhich obviously means that the probability of the obtaining this result is zero. When the result is | --, i.e., detector A2 and B2 click,\n\nρ --= N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η -| -η )( η| --η|) c ⊗ (|ξ -| -ξ )( ξ| --ξ|) d , (36)\n\nwhich is not normalized. The probability of obtaining this result is\n\nP --= (V -1)(V -e -4d 2 V ) 2(V 2 + e -4d 2 V ) , (37)\n\nand it can be simply verified that P ++ +P --= 1. Therefore, only the measurement results | + + and | -can be obtained in the case of the input state ρ Φ(+) . This is exactly the same for the case of ρ Ψ(+) . In the same way, it can be shown that if either the input state was ρ Φ(-) or ρ Ψ(-) , only the measurement results | +and | -+ can be obtained. In other words, the parity of the toincoming state is perfectly well discriminated by the measurements on single-photon qubits.\n\nSubsequently, a homodyne measurement is performed for mode c by homodyne detector C as shown in Fig. 8(b) . We assume that ideal homodyne measurements are performed, i.e., when a homodyne measurement is performed the state is projected onto eigenstate |x of operator X with eigenvalue x, where\n\nX = 1 √ 2 (a + a † ). ( 38\n\n)\n\nLet us first consider the case when the measurement result for the single photon qubits is | + + . In this case, the remaining state is ρ ++ in Eq. ( 33 ). The probability distribution P ++ Φ (+) for the homodyne measurement at detector C is\n\nP ++ Φ (+) = x|Tr d [ρ ++ ]|x = V 1 2 (e -V x 2 + e -x 2 V ) π 1 2 (V + 1)\n\n. (39) Note that the superscript, ++, denotes that the qubit measurement result was | + + , and the subscript, Φ (+) , denotes that the input state was ρ Φ (+) . These notations will be used also for the other cases in this section. The same analysis can be performed for the other possible measurement outcome | --:\n\nP -- Φ (+) = x|Tr d [ρ --]|x = V 1 2 (e -V x 2 -e -x 2 V ) π 1 2 (V -1)\n\n. (40)\n\nIn the same way, for another input state, ρ Φ(-) , it is straightforward to show:\n\nP +- Φ (-) = P ++ Φ (+) , P -+ Φ (-) = P -- Φ (+) , (41)\n\nand P ++ Φ (-) = P -- Φ (-) = 0. On the other hand, if the input state was ρ Ψ(+) , the probability distributions P ++ Ψ (+) and P -- Ψ (+) at detector C are It is straightforward to show for the other input state ρ Ψ(-) :\n\nP ++ Ψ (+) = x|Tr c [ρ ++ ]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V + 2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) + 1 , (42)\n\nP -- Ψ (+) = x|Tr c [ρ --]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V -2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) -1 . ( 43\n\nP +- Ψ (-) = P -- Ψ (+) , P -+ Ψ (-) = P ++ Ψ (+) . (44)\n\nThe probability distributions P ++ Φ (±) and P ++ Ψ (±) are plotted in Fig. 10 . Figure 10 shows that when the input state was ρ Φ(+) or ρ Φ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Φ (+) and P -- Φ (+) , is located around the origin. However, when the input state was ρ Ψ(+) or ρ Ψ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Ψ (+) and P -- Ψ (+) , is located far from the origin. Therefore, two of the Bell states, ρ Φ(+) or ρ Φ(-) , can be well distinguished from the other two by the homodyne detector C for the case of the measurement outcome | + + . Finally, by combining the homodyne measurement result and the qubit measurement result, all four Bell states can be effectively distinguished. For example, let us assume that the measurement outcome of the single photon detectors was | + + and the homodyne detection outcome was around the origin, i.e., x ≈ 0. Then, one can say that state ρ Ψ(-) has been measured for the result of the thermal-Bell measurement.\n\nAs implied in Fig. 10 , the overlaps between the probability distributions around the origin, P ++ Φ (+) and P -- Φ (+) , and the other distributions, P ++ Ψ (+) and P -- Ψ (+) , are extremely small for a sufficiently large d. In other words, the distinguishability by the homodyne detection rapidly approaches 1 as d increases. As an example, we can calculate the distinguishability between the states ρ Ψ(+) and ρ Φ(+) by the homodyne measurement by detector C. The distinguishability by homodyne detection is Note also that the second scheme using homodyne detection is robust to detection inefficiency compared with the first scheme using photon number resolving measurements. In the first scheme, even if a detector misses only one photon, it will result in a completely wrong measurement outcome. In the second scheme, however, the measurement outcome will not be affected in that way. If a single photon detector misses a photon, it will be immediately recognized. Such a case can simply be discarded so that it will only degrade the success probability of the Bell measurement. The homodyne detection inefficiency will not significantly affect the result when the distributions around the origin and the distributions far from the origin are well separated, i.e., when d ≫ √ V , as shown in Fig. 10 . On the other hand, loss in the Kerr medium will have a detrimental affect.\n\nP s = 1 2 |x|<d dxP ++ c (x) + |x|≥d dxP ++ d (x) (45\n\nQuantum teleportation of a thermal-state qubit can be performed using one of the Bell states as the quantum channel. Let us assume that Alice needs to teleport a thermal-state qubit, ρ ψ , to Bob using a thermal-state entanglement, ρ Ψ(-) , shared by the two parties. The total state can be represented as\n\nAlice first needs to perform the thermal-Bell measurement described in the previous subsection. To complete the teleportation process, Bob should perform an appropriate unitary transformation on his part of the quantum channel according to the measurement result sent from Alice via a classical channel. It is straightforward to show that the required transformations are exactly the same to those for the coherent-state qubit [27] . When the measurement outcome is ρ Ψ(-) , Bob obtains a perfect replica of the original unknown qubit without any operation.\n\nWhen the measurement outcome is ρ Φ(-) , Bob should perform |α ↔ |α on his qubit in Eq. ( 23 ). Such a phase shift by π can be done using a phase shifter whose action is described by P (ϕ) = e iϕa † a , where a and a † are the annihilation and creation operators. When the outcome is ρ Ψ(+) , the transformation should be performed as |α → |α and |α → -|α . It is known that the displacement operator is a good approximation of this transformation for d ≫ 1 [29] . This transformation can also be achieved by teleporting the state again locally and repeating until the required phase shift is obtained [30] . When the outcome is ρ Φ(+) , σ x and σ z should be successively applied.\n\nIn this paper, we have studied characteristics of superpositions and entanglement of thermal states at high temperatures and discussed their applications to quantum information processing. The superpositions and entanglement of thermal states show various nonclassical properties such as interference patterns, negativity of the Wigner functions, and violations of the Bell-CHSH inequality. The Bell violations are more sensitive to the interaction time during the generation process when the thermal temperature (i.e. mixedness) of the thermalstate entanglement is larger. Therefore, in order to observe the Bell violations using the mixed state at a high temperature, the interaction time in the Kerr medium should be accurate. We have pointed out that certain superpositions of high-temperature thermal states, symmetric in the phase space, can also be generated. Some of these states have neither squeezing properties nor negative values in their Wigner functions but they are found to be highly nonclassical.\n\nWe have introduced the thermal-state qubit and thermal-Bell states for applications to quantum information processing. We have presented two possible methods for the Bell-state measurement. The Bell-state measurement enables one to perform quantum teleportation and gate operations for quantum computation with thermalstate qubits. The first scheme uses two photon number resolving detectors and a 50-50 beam splitter to discriminate the thermal-Bell states. Using the second scheme, it is possible to effectively discriminate the thermal-Bell states without photon number resolving detection. The required resources for the second scheme are two Kerr nonlinear interactions, two single photon detectors, two 50:50 beam splitters and one homodyne detector. The second scheme is more robust to inefficiency of the detectors: the inefficiency of the single photon detectors only degrades the success probability of the Bell measurement."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"We study characteristics of superpositions and entanglement of thermal states at high temperatures and discuss their applications to quantum information processing. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure-state qubits and Bell states to thermal mixtures. A scheme is then presented to discriminate between the four thermal-Bell states without photon number resolving detection but with Kerr nonlinear interactions and two singlephoton detectors. This enables one to perform quantum teleportation and gate operations for quantum computation with thermal-state qubits."},{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"In many problems considered within the framework of quantum physics, physical systems are treated as pure states that can be represented by state vectors, or equivalently, by wave functions. Even though such an approach is simple and useful to address certain problems, it could often be quite different from real conditions of physical systems. This may be particularly true when one deals with macroscopic physical systems in terms of quantum physics. A macroscopic object is a complex open system which cannot avoid continuous interactions with the environment. Such a physical system is generally in a significantly mixed state and cannot be represented by a state vector. In general, mixed states are subtle objects whose properties are significantly more difficult to characterize than pure states.\n\nSchrödinger's famous cat paradox is a typical example where a massive classical object was assumed to be a pure state. It describes a counter-intuitive feature of quantum physics which dramatically appears when the principle of quantum superposition is applied to macroscopic objects. In the original paradox and its various explanations, the initial cat isolated in the steel chamber is considered a pure state that can be represented by a state vector such as |alive (or a wave function such as ψ alive ). The cat isolated from the environment is then assumed to interact with a microscopic superposition state, (|g +|e )/ √ 2, where |g and |e are the ground and excited states of a two-level atom. The cat will be dead if the atom is found in the excited state, |e , while it will remain alive if otherwise. Thus in Schrödinger's gedanken experiment the cat is entangled with the atom as (|g |alive + |e |dead ) √ 2, where the alive and dead statuses of the cat are described by the state vectors |alive and |dead . If one measures out the atomic system on the superposed basis, (|g ± |e )/ √ 2, the cat will be in a superposition of alive and dead states such as (|alive ± |dead )/ √ 2. It is often argued that such superposed states and entangled states can theoretically exist but are virtually impossible to observe because one cannot perfectly isolate a macroscopic object such as the cat from its environment [4] .\n\nHowever, this explanation is not fully satisfactory because the cat, a macroscopic object, is a complex open system which cannot be represented by a state vector. One may argue that the cat could be assumed to be in an unknown pure state such that the cat was certainly alive but the exact state of the cat was unknown. However, the interactions between the cat and its environment can cause the cat to become entangled with the environment [5] . In such a case, even though one can perfectly isolate the cat in the steel chamber from the enviroment, the cat will remain entangled with the environment due to its pre-interactions with the environment. Therefore, strictly speaking, even to assume a cat as an unknown pure state in the steel chamber is not legitimate. Thus a key point here is that it is unsatisfactory to describe the cat by a pure state such as |alive and |dead . We may need a more realistic assumption that the \"cat\" in Schrödinger's paradox was in a significantly mixed classical state. An intriguing question is then whether the quantum properties of the resulting state would still remain or diminish under such an assumption.\n\nRecently, such an analogy of Schrödinger's cat paradox, where the state corresponding to the virtual cat is a significantly mixed thermal state, was investigated [6] . A thermal state with a high temperature is considered a classical state in quantum optics. As the temperature of the thermal state increases, the degree of mixedness, which can be quantified by linear entropy, rapidly approaches the maximum value. When the temperature approaches infinity, the thermal state does not show any quantum properties. As a comparison, coherent states with large amplitudes are known as the most classical pure states [7] , and their superposition is often regarded as a superposition of classical states [8] . However, coherent states are still pure states which may not well represent truly classical systems, and they display some nonclassical features [9] . In Ref. [6] , it was shown that prominent quantum properties can actually be transferred from a microscopic superposition to a significantly mixed thermal state (i.e. a thermal state of which the degree of mixedness is close to the maximum value) at a high temperature through an experimentally feasible process. This result clarifies that unavoidable initial mixedness of the cat does not preclude strong quantum phenomena.\n\nOne of the results in Ref. [6] is that quantum entanglement can be produced between thermal states with nearly the maximum Bell-inequality violation when the temperatures of both modes goes to infinity. In previous related results, Bose et al. showed that entanglement can arise when two systems interact if one of the system are pure even when the other system is extremely mixed [10] . There is an interesting previous example shown by Filip et al. for the maximum violation of Bell's inequality when one of the modes is an extremely mixed thermal state [11] . Very recently, Ferreira et al. showed that entanglement can be generated at any finite temperature between high Q cavity mode field and a movable mirror thermal state [12] . However, in these example [10, 11, 12] only one of the modes is considered a large thermal state [10, 11, 12] and entanglement vanishes in the infinite temperature limit [10, 12] , which is obviously in contrast to the result presented in Ref. [6] . Entanglement for both of the modes at the thermal limit of the infinitely high temperature has not been found before. Remarkably, the violation of Bell's inequality in our examples reaches up to Cirel'son's bound [13] even in this infinite-temperature limit for both modes. As Vedral [14] and Ferreira et al. [12] pointed out it is believed that high temperatures reduce entanglement and all entanglement vanishes if the temperature is high enough, which is obviously not the case in Ref. [6] .\n\nThe purpose of this paper is twofold. Firstly, we review and further investigate various properties of superpositions and entanglement of thermal states at high temperatures [6] . In particular, we investigate two classes of highly mixed symmetric states in the phase space. Both the classes of these states do not show typical interference patterns in the phase space while they manifest strong singular behaviors. Interestingly, the first class of states has neither squeezing properties nor negative values in their Wigner functions, however, they are found to be highly nonclassical states. The second class of states has the maximum negativity in the Wigner function. Further, we discuss the possibility of quantum information processing with thermal-state qubits. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure Bell states. We show that four thermal-Bell states can be well discriminated by nonlinear interactions without photon number resolving measurements. Quantum teleportation and gate operations for thermal-state qubits can be realized using the Bell measurement scheme.\n\nThis paper is organized as follows. In Sec. II, we review the generation process of superpositions of thermal states and study their characteristics. In Sec. III, we study en-tanglement of thermal states, i.e., Bell inequality violations. In Sec. IV, we discuss the possibility of quantum information processing using thermal states. We first define the thermal-state qubit and the Bell-basis states using thermal-state entanglement. We then show that the four Bell states can be well discriminated by homodyne detection and two Kerr nonlinearities. It follows that quantum teleportation and quantum gate operations can be realized with thermal-state qubits. We conclude with final remarks in Sec. V."},{"section_type":"OTHER","section_title":"II. SUPERPOSITIONS OF THERMAL STATES A. Generation of thermal-state superpositions","text":"Let us first consider a two-mode harmonic oscillator system. A displaced thermal state can be defined as\n\nρ th (V, d) = d 2 αP th (V, d)|α α| ( 1\n\n)\n\nwhere |α is a coherent state of amplitude α and\n\nP th α (V, d) = 2 π(V -1) exp[- 2|α -d| 2 V -1 ] (2)\n\nwith variance V and displacement d in the phase space.\n\nThe thermal temperature τ increases as V increases as e ν/τ = (V + 1)/(V -1), where is Planck's constant and ν is the frequency [15] . Suppose that a microscopic superposition state\n\n|ψ a = 1 √ 2 (|0 a + |1 a ), (3)\n\nwhere |0 and |1 are the ground and first excited states of the harmonic oscillator, interacts with a thermal state ρ th b (V, d) and the interaction Hamiltonian is\n\nH K = λâ † âb † b (4)\n\nwhich corresponds to the cross Kerr nonlinear interaction. The resulting state is then\n\nρ ent ab = 1 2 d 2 αP th (V, d) |0 0| ⊗ |α α| + |1 0| ⊗ |αe iϕ α| + |0 1| ⊗ |α αe iϕ | + |1 1| ⊗ |αe iϕ αe iϕ | ( 5\n\n)\n\nand ϕ is determined by the strength of the nonlinearity λ and the interaction time. The Wigner representation of ρ ent ab is\n\nW ent ab (α, β) = 1 π e -2|α| 2 W th (β; d) + 2αV c (β; d) + 2[αV c (β; d)] * + (4|α| 2 -1)W th (β; de iϕ ) (6)\n\nwhere α and β are complex numbers parametrizing the phase spaces of the microscopic and macroscopic systems respectively and\n\nW th (α; d) = 2 πV exp[- 2|α -d| 2 V ], (7)\n\nV c (α; d) = 2 πJK exp[- 2 K (1 -e iϕ )d 2 - 1 J (α - 2e iϕ d K )(α * - 2d K )], (8)\n\nK = 2 + (V -1)(1 -e iϕ\n\n), J = (sin ϕ/2 + iV cos ϕ/2)/(2V sin ϕ/2 + 2i cos ϕ/2), and d has been assumed real without loss of generality. If one traces ρ ent ab over mode a, the remaining state will be simply in a classical mixture of two thermal states and its Wigner function will be positive everywhere. However, if one measures out the \"microscopic part\" on the superposed basis, i.e., (|0 a ± |1 a )/ √ 2, the \"macroscopic part\" for mode b may not lose its nonclassical characteristics. Such a measurement on the the superposed basis will reduce the remaining state to\n\nρ sup(±) = N ± s d 2 αP th (V, d) |α α| ± |αe iϕ α| ± |α αe iϕ | + |αe iϕ αe iϕ | , (9)\n\nwhere N ± s are the normalization factors, and its Wigner function is\n\nW sup(±) (α) = N ± s {W th (α; d) ± V c (α; d) ± {V c (α; d)} * + W th (α; de iϕ )}. ( 10\n\n)\n\nThe ± signs in Eqs. ( 8 ) and (9) correspond to the two possible results from the measurement of the microscopic system. The state in Eq. ( 10 ) is a superposition of two thermal states. A feasible experimental setup to generate superpositions of thermal states is atom-field interactions in cavities, where a π/2 pulse can be used to prepare the atom in a superposed state. This type of experiment has already been performed to produce a superposition of coherent states [16] . In our cases, simply thermal states can be used instead of coherent states. Another possible setup is an all-optical scheme with free-traveling fields and a cross-Kerr medium, where a standard singlephoton qubit could be used as the microscopic superposition. Recently, there have been theoretical and experimental efforts to produce and observe giant Kerr nonlinearities using electromagnetically induced transparency [17] . Furthermore, it was shown that a weak Kerr nonlinearity can still be useful if a initially strong field is employed in this type of experiment [18] . We shall further explain this with examples in Sec. III."},{"section_type":"OTHER","section_title":"B. Negativity of the Wigner function","text":"The negativity of the Wigner function is known as an indicator of non-classicality of quantum states. In order to observe negativity of the Wigner function in a real experiment, its absolute minimum negativity should be large enough. The minimum negativity of the Wigner function in Eq. ( 6 ) for V = 1 is -0.144 for d = 0 and -0.246 for d → ∞. Now suppose the initial state can be considered a classical thermal state by letting V ≫ 1. One might expect that the negativity would be washed out as the initial state becomes mixed, but this is not the case. The minimum negativity actually increases as V gets larger. If V → ∞, the minimum negativity of the Wigner function ( 6 ) is -0.246 regardless of d: no matter how mixed the initial thermal state was, the minimum negativity of Wigner function is found to be a large value. The point in the phase space which gives the minimum negativity when V ≫ 1 or d ≫ 0 is (-1 2 , 0) and has negativity\n\nW neg ≡ W ent ab (- 1 2 , 0) = 2(-2 + 1 V exp[-2d 2 V ]) π 2 √ e . (11)\n\nIt can be shown that\n\nW neg approaches -4/(π 2 √ e) ≈ -0.246 when either d → ∞ or V → ∞.\n\nThis effect is obviously due to the interaction between the microscopic superposition and the macroscopic thermal state. If the initial microscopic state is not superposed, e.g., |ψ a = |1 a , the resulting state will be a simple direct product, (|1 1|) a ⊗ ρ th b (V, -d). Whilst for V = 1 this state will exhibit negativity, this is washed out and tends to zero as V → ∞. Needless to say, if it was |0 a instead of |1 a , the resulting Wigner function will be a direct product of two Gaussian states whose Wigner fucntion can never be negative. The superpositon state (3) plays the crucial role in making the minimum negativity of the resulting Wigner function always saturate to a certain negative value no matter how mixed and classical the initial state of the other mode becomes. The Wigner functions of the single-mode states, W sup(±) (α), in Eq. (10) show large negative values. The minimum negativity of the Wigner function W sup(-) (α) is W sup(-) (0) = 2/π regardless of the values of V and d. On the other hand, the minimum negativity of the Wigner function W sup(+) (α) approaches 2/π for d → ∞ and disappears when d = 0."},{"section_type":"OTHER","section_title":"C. Quantum interference in the phase space","text":"When ϕ = π, the state (9) becomes (12) where\n\nρ ± = N (ρ th (V, d)±σ(V, d)±σ(V, -d)+ρ th (V, -d)),\n\nσ(V, d) = d 2 αP th (V, d)| -α α| and N = 2 1 ± exp[-2d 2 V ] V 2 . ( 13\n\n)\n\nIf the initial state for mode b is a pure coherent state, i.e., V = 1, the measurement on the superposed basis for mode a will produce a superposition of two pure coherent states as\n\n| Ψ ± = 1 √ 1 ± e -2|α| 2 (|α ± | -α ), (14)\n\nwhere α = d. The probability P ± to obtain the state ρ ± is obtained as [19 ] where\n\nP ± = ψ ± |Tr b [ρ ent ab ]|ψ ± = 1 2 (1 ± exp[-2d 2 V ] V ), ( 15\n\n)\n\n|ψ ± = (|0 ±|1 )/ √ 2.\n\nThe probability approaches P ± = 1/2 when either d or V becomes large.\n\nAs an analogy of Schrödinger's cat paradox, the variance V corresponds to the size the initial \"cat\", and the distance d between the two thermal component states corresponds to distinguishability between the \"alive cat\" and the \"dead cat\". Suppose that both V and d are very large for the initial thermal state. The two thermal states ρ th (V, ±d) become macroscopically distinguishable when d ≫ √ V , and our example may become a more realistic analogy of the cat paradox in this limit. Both the states ρ ± in this case show probability distributions with two Gaussian peaks and interference fringes [6] . Figure 1 presents the probability distributions of x (≡ Re[α]) and p (≡ Im[α]) for ρ -(a) when V = 100 and d = 100 and (b) when V = 1000 and d = 300. The probability distribution of x (p) for ρ ± can be obtained by integrating the Wigner function of ρ ± over p (x). The two Gaussian peaks along the x axis and interference fringes along the p axis shown in Fig. 1 are a typical signature of a quantum superposition between macroscopically distinguishable states. The visibility v of the interference fringes is defined as [15]\n\nv = I max -I min I max + I min , (16)\n\nwhere I = dxW sup(-) (α) and the maximum should be taken over p. It can be simply shown that the visibility v is always 1 regardless of the value of V . Note that d should increase proportionally to √ V to maintain the condition of classical distingushability between the two component thermal states ρ th (V, ±d). The interference fringes with high visibility are incompatible with classical physics and evidence of quantum coherence. The fringe spacing (the distance between the fringes) does not depend on V but only on d, i.e., a pure superposition of coherent states shows the same fringe spacing for a given d. We emphasize that the states shown in Fig. 1 are \"superpositions\" of severely mixed thermal states.\n\nAn experimental realization of a nonlinear effect corresponding to ϕ = π is very demanding particularly in the presence of decoherence. Here we point out that the method using a weak nonlinear effect (ϕ ≪ π) combined with a strong field (d ≫ 1) [18] can be useful to generate a thermal-state superposition with prominent interference patterns. In Fig. 2 , we have used experimentally accessible values, V = 5, d = 2000 and ϕ = π/1000, but the fringe visibility is still 1. In this case, decoherence during the nonlinear interaction would be significantly reduced because of the decrease of the interaction time [18] . Note also that, if required, the state in Fig. 2 can be moved to the center of the phase space, for example, using a biased beam splitter (BS) and a strong coherent field [18] ."},{"section_type":"OTHER","section_title":"D. Symmetric macroscopic quantum states","text":"Let us assume that d = 0, i.e., the initial state is the thermal state, ρ th (V, 0), at the origin of the phase space. In this case, the thermal-state superpositions, ρ ± , are produced with probabilities, P ± = (1/2){1 ± (1/V )}, respectively. Figure 3 shows the Wigner functions of ρ + dependent on the interaction time between the macro-scopic thermal state and the microscopic superposition in a cross Kerr medium. The state is always symmetric in the phase space regardless of the interaction time as shown in Fig. 3 . In this figure, the initial state is a thermal state of V = 100 (Fig. 3(a) ). In a relatively short time (θ = π/32 and θ = π/16), the state shows some interference patterns. When θ = π, the evolved state looks very localized around the origin as shown in Fig 3 . The generated state at θ = π does not show negativity of the Wigner function nor squeezing properties. On the other hand, a well defined P function does not exist for this state.\n\nIn the case of ρ -, with the same assumption d = 0, the Wigner function at ϕ = π has the minimum negativity (-2/π) at the origin regardless of V . As a result of the interaction with the microscopic superposition, a deep hole to the negative direction below zero has been formed around the origin for ρ -as shown in Fig. 4 ."},{"section_type":"OTHER","section_title":"III. ENTANGLEMENT BETWEEN THERMAL STATES","text":"Entanglement between macroscopic objects and its Bell-type inequality tests are an important issue. In this section, we shall show that entanglement can be generated between high-temperature thermal states even when the temperature of each mode goes to infinity. 2 200 1 400 600 800 1000 V 2.1 2.2 2.3 B (a) 50 100 150 d 2.2 2 0 2.4 2.6 2.8 B (b) FIG. 5: (a) The optimized violation, B ≡ |B + |max, of Bell-\n\nCHSH inequality for the \"thermal-state entenglement\", ρ+, of V = 1000 (solid curve) and V = 100 (dashed curve). The Bell-violation of a pure entangled coherent state, i.e., V = 1, has been plotted for comparison (dotted curve). The Bell-violation B approaches its maximum bound, 2 √ 2, when d ≫ √ V regardless of the level of the mixedness V . (b) The optimized Bell-violation B against d for the different type of thermal-state entanglement generated using a 50:50 beam splitter from ρ + . V = 1000 (solid curve), V = 100 (dashed curve) and V = 1 (dotted curve)."},{"section_type":"OTHER","section_title":"A. Entanglement using two initial thermal states","text":"If the microscopic superposition interacts with two thermal states, ρ th b (V, d) and ρ th c (V, d), and the micro-scopic particle is measured out on the superposed basis, the resulting state will be\n\nρ tm(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (17)\n\nwhere\n\nN t = 2 1 ± exp[-4d 2 V ] V 2 . ( 18\n\n)\n\nSuch two-mode thermal-state entanglement can be generated using two cavities and an atomic state detector [20] .\n\nExtending the two cavities to N cavities, entanglement of N -mode thermal states can also be generated. Such a state is an analogy of the N -mode pure GHZ state [21] but each mode is extremely mixed. Here we shall consider the Bell-CHSH inequality [22, 23] with photon number parity measurements [20, 24] . The parity measurements can be performed in a high-Q cavity using a far-off-resonant interaction between a two-level atom and the field [25] . The Bell-CHSH inequality can be represented in terms of the Winger function as [24 ]\n\n|B (±) | = π 2 4 |W tm(±) (α, β) + W tm(±) (α, β ′ ) + W tm(±) (α ′ , β) -W tm(±) (α ′ , β ′ )| ≤ 2, (19)\n\nwhere W tm(±) (α, β) is the Wigner function of ρ tm(±) in Eq. ( 17 ). As shown in Fig. 5 , the Bell-violation approaches the maximum bound for a bipartite measurement, 2 √ 2 [13] , when d ≫ √\n\nV regardless of the level of the mixedness V , i.e., the temperatures of the thermal states. Note that it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(a) . This implies that entanglement of nearly 1 ebit has been produced between the two significantly mixed thermal states for d ≫ √ V , and such \"thermal-state entanglement\" cannot be described by a local theory."},{"section_type":"OTHER","section_title":"B. Entanglement using a beam splitter","text":"A different type of macroscopic entanglement can be generated by applying the beam splitter operation\n\nexp[θ/2(e iφ â † s âd -e -iφ â † d âs )], (20)\n\non the \"thermal-state superpositions\" in Eq. ( 9 ). The state after passing through a 50:50 beam splitter can be represented as\n\nN d 2 αP th α (V, d) | α √ 2 , - α √ 2 ± | - α √ 2 , α √ 2 α √ 2 , - α √ 2 | ± - α √ 2 , α √ 2 | , (21\n\n) 200 400 600 800 1000 V 2.1 2 0 2.2 2.3 B FIG. 6: The optimized Bell-violation B against V for the slightly different type of thermal-state entanglement generated using a 50:50 beam splitter using ρ + when d = 0. 3.13 3.14 3.15 3.16 ϕ 1.6 2.2 2 2.4 2.6 2.8 B (a) 3.13 3.14 3.15 3.16 ϕ 1.6 2 2.2 2.4 2.6 2.8 B (b) where N is defined in Eq. ( 13 ). When d is large, this state violates the Bell-CHSH inequality to the maximum bound 2 √ 2 regardless of the level of mixedness V as shown in Fig. 5(b ). Again, it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(b ). Furthermore, these states severely violate Bell's inequality even when d = 0 as V increases as shown in Fig. 6 . We have found that the optimized Bell violation of these states approaches 2.32449 for V → ∞. Interestingly, this value is exactly the same as the optimized Bell-CHSH violation for a pure two-mode squeezed state in the infinite squeezing limit [26] . Note that multilmode entangled states can be generated using multiple beam splitters.\n\nIt should be noted that the Bell violations are more sensitive to the interaction time when either V or d is larger. Figure 7 clearly shows this tendency. Therefore, in order to observe the Bell violations using the mixed state of V (and d) large, the interaction time in the Kerr medium should be more accurate."},{"section_type":"OTHER","section_title":"IV. QUANTUM INFORMATION PROCESSING WITH THERMAL-STATE QUBITS","text":"In this section, we discuss the possibility of quantum information processing with thermal-state qubits and thermal-state entanglement."},{"section_type":"OTHER","section_title":"A. Qubits and Bell-state measurements","text":"We introduce a thermal-state qubit\n\nρ ψ = |a| 2 ρ th (V, d) ± ab * σ(V, d) ± a * bσ(V, -d) + |b| 2 ρ th (V, -d), (22)\n\nwhere a and b are arbitrary complex numbers. The basis states, ρ th (V, d) and ρ th (V, -d), can be well discriminated by a homodyne measurement when d is larger than V . The thermal state qubit (22) can be re-written as\n\nρ ψ = d 2 αP th α (V, d) a|α + b| -α a * α| + b * -α| , (23)\n\nwhich can be understood as a generalization of the coherent state qubit, a|d + b|d , where |d is a coherent state of amplitude d. The thermal-state qubit (23) becomes identical to the coherent-state qubit when V = 1.\n\nWe also define four thermal-Bell states as\n\nρ Φ(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (24) ρ Ψ(±) = N t ρ th (V, d) ⊗ ρ th (V, -d) ± σ(V, d) ⊗ σ(V, -d) ± σ(V, -d) ⊗ σ(V, d) + ρ th (V, -d) ⊗ ρ th (V, d) (25)\n\nwhere N t was defined in Eq. ( 18 ). The thermal-Bell states can be written as For quantum information processing applications, it is an important task to discriminate between the four Bell states. Here we discuss two possible ways to discriminate between the thermal-Bell states (25) . We shall only briefly describe the first scheme using photon number resolving measurements and focus on the second scheme using nonlinear interactions. The first method is to simply use a 50-50 beam splitter and two photon number resolving detectors as shown in Fig. 8(a) . This scheme is basically the same as the Bellstate measurement scheme with pure entangled coherent states [27, 28] . Let us suppose that the amplitude, d, is large enough, i.e., d ≫ √ V . If the incident state was ρ Φ(+) or ρ Φ(-) , most of the photons are detected on detector A in in Fig. 8(a) . Meanwhile, most of the photons are detected on detector B when the incident state was ρ Ψ(+) or ρ Ψ(-) . The average photon numbers between the \"many-photon case\" and the \"few-photon case\" are compared in Fig. 9 . Furthermore, the states ρ Ψ(+) and ρ Φ(+) contain only even numbers of photons while ρ Ψ(-) and ρ Φ(-) contain only odd numbers of photons. Therefore, all the four Bell states can be well discriminated by analyzing numbers of photons detected at detectors A and B. For example, if detector A detects many photons while detector B detects few and the total photon number detected by the two detectors are even, this means that state ρ Φ(+) was measured by the thermal-Bell measurement. The nonzero failure probability can be made arbitrarily small by increasing d.\n\nρ Φ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, β ± | -α, -β α, β| ± -α, -β| , ( 26\n\n)\n\nρ Ψ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, -β ± | -α, β α, -β| ± -α, β| . (27)\n\nHowever, the average photon numbers of the thermal-Bell states are high when V ≫ 1 and d ≫ 1. In this case, it would be unrealistic to use photon number resolving detectors. It would be an interesting question whether\n\n0.5 1 1.5 2 2.5 3 d 10 0 0 20 30 N (a) 0.5 1 1.5 2 2.5 3 d 0 0 10 20 30 N (b)\n\nFIG. 9: The average photon number N for the \"many-photon case\" (solid line) and the \"few-photon case\" (dashed line) for V = 10 against d (a) when the input state is either ρ Φ(+) or ρ Ψ(+) and (b) when the input state is either ρ Φ(-) or ρ Ψ(-) .\n\nthese four thermal-Bell states can be distinguished by classical measurements, such as homodyne detection, instead of photon number resolving detection. Our alternative scheme employs cross-Kerr nonlinearities and single photon detectors as shown in Fig. 8(b ). Let us first suppose that the input field was ρ Φ(+) . The incident twomode state passes through a 50-50 beam splitter, BS1. The state after passing through the 50:50 beam splitter, BS1, is\n\nρ B = N t d 2 αd 2 βP th α (V, d)P th β (V, d) |η, -ξ η, -ξ| + |η, -ξ -η, ξ| + | -η, ξ η, -ξ| + | -η, ξ -η, ξ| (28)\n\nwhere η = (α+β)/ √ 2 and ξ = (α-β)/ √ 2. Two dual-rail single photon qubits, |ψ + ee ′ and |ψ + f f ′ , where\n\n|ψ + = 1 √ 2 (|0 |1 + |1 |0 ), (29)\n\nare prepared using two single photons and 50:50 beam splitters, BS2 and BS3, as shown in Fig. 8(b ). Then, traveling fields at modes c and d interacts with those of modes e and f , respectively, in cross-Kerr nonlinear media. We suppose that the interaction time is t = π/λ, and the resulting state is then\n\nρ B ′ = U ce U df ρ B cd ρ q ee ′ ρ q f f ′ U † ce U † df (30)\n\nwhere U ce = exp[iπH K ce /λ ] and ρ q = |ψ q ψ q |. An explicit form of Eq. ( 30 ) can then be simply obtained using the identity\n\nU ce |α c |0 e = |α c |0 e , U ce |α c |1 e = | -α c |1 e ( 31\n\n)\n\nwhere |α is a coherent state. However, we omit such an explicit expression in this paper for it is too lengthy. After the nonlinear interactions, the qubit parts, modes e, e ′ , f and f ′ , should be measured with the measurement basis\n\n{| + + , | + -, | -+ , | --} (32)\n\nwhere\n\n| + + = |ψ + ee ′ |ψ + f f ′ , | + -= |ψ + ee ′ |ψ -f f ′ , | -+ = |ψ -ee ′ |ψ + f f ′ , | --= |ψ -ee ′ |ψ -f f ′ , and |ψ -= (|0 |1 -|1 |0 )/ √ 2.\n\nThis measurement can be performed using two 50:50 beam splitters, BS4 and BS5, and four detectors, A1, A2, B1 and B2, as shown in Fig. 8(b ). If detector A1 and B1 click, i.e., the measurement result is | + + , the resulting state at modes c and d is\n\nρ ++ = N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η + | -η )( η| + -η|) c ⊗ (|ξ + | -ξ )( ξ| + -ξ|) d (33)\n\nNote that state ρ ++ is not normalized, which implies that the probability of obtaining the corresponding measurement result is not unity. The probability of obtaining this result is\n\nP ++ = (V + 1)(V + e -4d 2 V ) 2(V 2 + e -4d 2 V ) . (34)\n\nWhen the result is either | +or | -+ , the result is\n\nψ 2 |ρ B ′ |ψ 2 = ψ 3 |ρ B ′ |ψ 3 = 0, (35)\n\nwhich obviously means that the probability of the obtaining this result is zero. When the result is | --, i.e., detector A2 and B2 click,\n\nρ --= N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η -| -η )( η| --η|) c ⊗ (|ξ -| -ξ )( ξ| --ξ|) d , (36)\n\nwhich is not normalized. The probability of obtaining this result is\n\nP --= (V -1)(V -e -4d 2 V ) 2(V 2 + e -4d 2 V ) , (37)\n\nand it can be simply verified that P ++ +P --= 1. Therefore, only the measurement results | + + and | -can be obtained in the case of the input state ρ Φ(+) . This is exactly the same for the case of ρ Ψ(+) . In the same way, it can be shown that if either the input state was ρ Φ(-) or ρ Ψ(-) , only the measurement results | +and | -+ can be obtained. In other words, the parity of the toincoming state is perfectly well discriminated by the measurements on single-photon qubits.\n\nSubsequently, a homodyne measurement is performed for mode c by homodyne detector C as shown in Fig. 8(b) . We assume that ideal homodyne measurements are performed, i.e., when a homodyne measurement is performed the state is projected onto eigenstate |x of operator X with eigenvalue x, where\n\nX = 1 √ 2 (a + a † ). ( 38\n\n)\n\nLet us first consider the case when the measurement result for the single photon qubits is | + + . In this case, the remaining state is ρ ++ in Eq. ( 33 ). The probability distribution P ++ Φ (+) for the homodyne measurement at detector C is\n\nP ++ Φ (+) = x|Tr d [ρ ++ ]|x = V 1 2 (e -V x 2 + e -x 2 V ) π 1 2 (V + 1)\n\n. (39) Note that the superscript, ++, denotes that the qubit measurement result was | + + , and the subscript, Φ (+) , denotes that the input state was ρ Φ (+) . These notations will be used also for the other cases in this section. The same analysis can be performed for the other possible measurement outcome | --:\n\nP -- Φ (+) = x|Tr d [ρ --]|x = V 1 2 (e -V x 2 -e -x 2 V ) π 1 2 (V -1)\n\n. (40)\n\nIn the same way, for another input state, ρ Φ(-) , it is straightforward to show:\n\nP +- Φ (-) = P ++ Φ (+) , P -+ Φ (-) = P -- Φ (+) , (41)\n\nand P ++ Φ (-) = P -- Φ (-) = 0. On the other hand, if the input state was ρ Ψ(+) , the probability distributions P ++ Ψ (+) and P -- Ψ (+) at detector C are It is straightforward to show for the other input state ρ Ψ(-) :\n\nP ++ Ψ (+) = x|Tr c [ρ ++ ]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V + 2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) + 1 , (42)\n\nP -- Ψ (+) = x|Tr c [ρ --]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V -2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) -1 . ( 43\n\nP +- Ψ (-) = P -- Ψ (+) , P -+ Ψ (-) = P ++ Ψ (+) . (44)\n\nThe probability distributions P ++ Φ (±) and P ++ Ψ (±) are plotted in Fig. 10 . Figure 10 shows that when the input state was ρ Φ(+) or ρ Φ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Φ (+) and P -- Φ (+) , is located around the origin. However, when the input state was ρ Ψ(+) or ρ Ψ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Ψ (+) and P -- Ψ (+) , is located far from the origin. Therefore, two of the Bell states, ρ Φ(+) or ρ Φ(-) , can be well distinguished from the other two by the homodyne detector C for the case of the measurement outcome | + + . Finally, by combining the homodyne measurement result and the qubit measurement result, all four Bell states can be effectively distinguished. For example, let us assume that the measurement outcome of the single photon detectors was | + + and the homodyne detection outcome was around the origin, i.e., x ≈ 0. Then, one can say that state ρ Ψ(-) has been measured for the result of the thermal-Bell measurement.\n\nAs implied in Fig. 10 , the overlaps between the probability distributions around the origin, P ++ Φ (+) and P -- Φ (+) , and the other distributions, P ++ Ψ (+) and P -- Ψ (+) , are extremely small for a sufficiently large d. In other words, the distinguishability by the homodyne detection rapidly approaches 1 as d increases. As an example, we can calculate the distinguishability between the states ρ Ψ(+) and ρ Φ(+) by the homodyne measurement by detector C. The distinguishability by homodyne detection is Note also that the second scheme using homodyne detection is robust to detection inefficiency compared with the first scheme using photon number resolving measurements. In the first scheme, even if a detector misses only one photon, it will result in a completely wrong measurement outcome. In the second scheme, however, the measurement outcome will not be affected in that way. If a single photon detector misses a photon, it will be immediately recognized. Such a case can simply be discarded so that it will only degrade the success probability of the Bell measurement. The homodyne detection inefficiency will not significantly affect the result when the distributions around the origin and the distributions far from the origin are well separated, i.e., when d ≫ √ V , as shown in Fig. 10 . On the other hand, loss in the Kerr medium will have a detrimental affect.\n\nP s = 1 2 |x|<d dxP ++ c (x) + |x|≥d dxP ++ d (x) (45"},{"section_type":"OTHER","section_title":"B. Quantum teleportation and computation","text":"Quantum teleportation of a thermal-state qubit can be performed using one of the Bell states as the quantum channel. Let us assume that Alice needs to teleport a thermal-state qubit, ρ ψ , to Bob using a thermal-state entanglement, ρ Ψ(-) , shared by the two parties. The total state can be represented as\n\nAlice first needs to perform the thermal-Bell measurement described in the previous subsection. To complete the teleportation process, Bob should perform an appropriate unitary transformation on his part of the quantum channel according to the measurement result sent from Alice via a classical channel. It is straightforward to show that the required transformations are exactly the same to those for the coherent-state qubit [27] . When the measurement outcome is ρ Ψ(-) , Bob obtains a perfect replica of the original unknown qubit without any operation.\n\nWhen the measurement outcome is ρ Φ(-) , Bob should perform |α ↔ |α on his qubit in Eq. ( 23 ). Such a phase shift by π can be done using a phase shifter whose action is described by P (ϕ) = e iϕa † a , where a and a † are the annihilation and creation operators. When the outcome is ρ Ψ(+) , the transformation should be performed as |α → |α and |α → -|α . It is known that the displacement operator is a good approximation of this transformation for d ≫ 1 [29] . This transformation can also be achieved by teleporting the state again locally and repeating until the required phase shift is obtained [30] . When the outcome is ρ Φ(+) , σ x and σ z should be successively applied."},{"section_type":"CONCLUSION","section_title":"V. CONCLUSION","text":"In this paper, we have studied characteristics of superpositions and entanglement of thermal states at high temperatures and discussed their applications to quantum information processing. The superpositions and entanglement of thermal states show various nonclassical properties such as interference patterns, negativity of the Wigner functions, and violations of the Bell-CHSH inequality. The Bell violations are more sensitive to the interaction time during the generation process when the thermal temperature (i.e. mixedness) of the thermalstate entanglement is larger. Therefore, in order to observe the Bell violations using the mixed state at a high temperature, the interaction time in the Kerr medium should be accurate. We have pointed out that certain superpositions of high-temperature thermal states, symmetric in the phase space, can also be generated. Some of these states have neither squeezing properties nor negative values in their Wigner functions but they are found to be highly nonclassical.\n\nWe have introduced the thermal-state qubit and thermal-Bell states for applications to quantum information processing. We have presented two possible methods for the Bell-state measurement. The Bell-state measurement enables one to perform quantum teleportation and gate operations for quantum computation with thermalstate qubits. The first scheme uses two photon number resolving detectors and a 50-50 beam splitter to discriminate the thermal-Bell states. Using the second scheme, it is possible to effectively discriminate the thermal-Bell states without photon number resolving detection. The required resources for the second scheme are two Kerr nonlinear interactions, two single photon detectors, two 50:50 beam splitters and one homodyne detector. The second scheme is more robust to inefficiency of the detectors: the inefficiency of the single photon detectors only degrades the success probability of the Bell measurement."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"We study characteristics of superpositions and entanglement of thermal states at high temperatures and discuss their applications to quantum information processing. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure-state qubits and Bell states to thermal mixtures. A scheme is then presented to discriminate between the four thermal-Bell states without photon number resolving detection but with Kerr nonlinear interactions and two singlephoton detectors. This enables one to perform quantum teleportation and gate operations for quantum computation with thermal-state qubits.\"\n },\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"In many problems considered within the framework of quantum physics, physical systems are treated as pure states that can be represented by state vectors, or equivalently, by wave functions. Even though such an approach is simple and useful to address certain problems, it could often be quite different from real conditions of physical systems. This may be particularly true when one deals with macroscopic physical systems in terms of quantum physics. A macroscopic object is a complex open system which cannot avoid continuous interactions with the environment. Such a physical system is generally in a significantly mixed state and cannot be represented by a state vector. In general, mixed states are subtle objects whose properties are significantly more difficult to characterize than pure states.\\n\\nSchrödinger's famous cat paradox is a typical example where a massive classical object was assumed to be a pure state. It describes a counter-intuitive feature of quantum physics which dramatically appears when the principle of quantum superposition is applied to macroscopic objects. In the original paradox and its various explanations, the initial cat isolated in the steel chamber is considered a pure state that can be represented by a state vector such as |alive (or a wave function such as ψ alive ). The cat isolated from the environment is then assumed to interact with a microscopic superposition state, (|g +|e )/ √ 2, where |g and |e are the ground and excited states of a two-level atom. The cat will be dead if the atom is found in the excited state, |e , while it will remain alive if otherwise. Thus in Schrödinger's gedanken experiment the cat is entangled with the atom as (|g |alive + |e |dead ) √ 2, where the alive and dead statuses of the cat are described by the state vectors |alive and |dead . If one measures out the atomic system on the superposed basis, (|g ± |e )/ √ 2, the cat will be in a superposition of alive and dead states such as (|alive ± |dead )/ √ 2. It is often argued that such superposed states and entangled states can theoretically exist but are virtually impossible to observe because one cannot perfectly isolate a macroscopic object such as the cat from its environment [4] .\\n\\nHowever, this explanation is not fully satisfactory because the cat, a macroscopic object, is a complex open system which cannot be represented by a state vector. One may argue that the cat could be assumed to be in an unknown pure state such that the cat was certainly alive but the exact state of the cat was unknown. However, the interactions between the cat and its environment can cause the cat to become entangled with the environment [5] . In such a case, even though one can perfectly isolate the cat in the steel chamber from the enviroment, the cat will remain entangled with the environment due to its pre-interactions with the environment. Therefore, strictly speaking, even to assume a cat as an unknown pure state in the steel chamber is not legitimate. Thus a key point here is that it is unsatisfactory to describe the cat by a pure state such as |alive and |dead . We may need a more realistic assumption that the \\\"cat\\\" in Schrödinger's paradox was in a significantly mixed classical state. An intriguing question is then whether the quantum properties of the resulting state would still remain or diminish under such an assumption.\\n\\nRecently, such an analogy of Schrödinger's cat paradox, where the state corresponding to the virtual cat is a significantly mixed thermal state, was investigated [6] . A thermal state with a high temperature is considered a classical state in quantum optics. As the temperature of the thermal state increases, the degree of mixedness, which can be quantified by linear entropy, rapidly approaches the maximum value. When the temperature approaches infinity, the thermal state does not show any quantum properties. As a comparison, coherent states with large amplitudes are known as the most classical pure states [7] , and their superposition is often regarded as a superposition of classical states [8] . However, coherent states are still pure states which may not well represent truly classical systems, and they display some nonclassical features [9] . In Ref. [6] , it was shown that prominent quantum properties can actually be transferred from a microscopic superposition to a significantly mixed thermal state (i.e. a thermal state of which the degree of mixedness is close to the maximum value) at a high temperature through an experimentally feasible process. This result clarifies that unavoidable initial mixedness of the cat does not preclude strong quantum phenomena.\\n\\nOne of the results in Ref. [6] is that quantum entanglement can be produced between thermal states with nearly the maximum Bell-inequality violation when the temperatures of both modes goes to infinity. In previous related results, Bose et al. showed that entanglement can arise when two systems interact if one of the system are pure even when the other system is extremely mixed [10] . There is an interesting previous example shown by Filip et al. for the maximum violation of Bell's inequality when one of the modes is an extremely mixed thermal state [11] . Very recently, Ferreira et al. showed that entanglement can be generated at any finite temperature between high Q cavity mode field and a movable mirror thermal state [12] . However, in these example [10, 11, 12] only one of the modes is considered a large thermal state [10, 11, 12] and entanglement vanishes in the infinite temperature limit [10, 12] , which is obviously in contrast to the result presented in Ref. [6] . Entanglement for both of the modes at the thermal limit of the infinitely high temperature has not been found before. Remarkably, the violation of Bell's inequality in our examples reaches up to Cirel'son's bound [13] even in this infinite-temperature limit for both modes. As Vedral [14] and Ferreira et al. [12] pointed out it is believed that high temperatures reduce entanglement and all entanglement vanishes if the temperature is high enough, which is obviously not the case in Ref. [6] .\\n\\nThe purpose of this paper is twofold. Firstly, we review and further investigate various properties of superpositions and entanglement of thermal states at high temperatures [6] . In particular, we investigate two classes of highly mixed symmetric states in the phase space. Both the classes of these states do not show typical interference patterns in the phase space while they manifest strong singular behaviors. Interestingly, the first class of states has neither squeezing properties nor negative values in their Wigner functions, however, they are found to be highly nonclassical states. The second class of states has the maximum negativity in the Wigner function. Further, we discuss the possibility of quantum information processing with thermal-state qubits. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure Bell states. We show that four thermal-Bell states can be well discriminated by nonlinear interactions without photon number resolving measurements. Quantum teleportation and gate operations for thermal-state qubits can be realized using the Bell measurement scheme.\\n\\nThis paper is organized as follows. In Sec. II, we review the generation process of superpositions of thermal states and study their characteristics. In Sec. III, we study en-tanglement of thermal states, i.e., Bell inequality violations. In Sec. IV, we discuss the possibility of quantum information processing using thermal states. We first define the thermal-state qubit and the Bell-basis states using thermal-state entanglement. We then show that the four Bell states can be well discriminated by homodyne detection and two Kerr nonlinearities. It follows that quantum teleportation and quantum gate operations can be realized with thermal-state qubits. We conclude with final remarks in Sec. V.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. SUPERPOSITIONS OF THERMAL STATES A. Generation of thermal-state superpositions\",\n \"text\": \"Let us first consider a two-mode harmonic oscillator system. A displaced thermal state can be defined as\\n\\nρ th (V, d) = d 2 αP th (V, d)|α α| ( 1\\n\\n)\\n\\nwhere |α is a coherent state of amplitude α and\\n\\nP th α (V, d) = 2 π(V -1) exp[- 2|α -d| 2 V -1 ] (2)\\n\\nwith variance V and displacement d in the phase space.\\n\\nThe thermal temperature τ increases as V increases as e ν/τ = (V + 1)/(V -1), where is Planck's constant and ν is the frequency [15] . Suppose that a microscopic superposition state\\n\\n|ψ a = 1 √ 2 (|0 a + |1 a ), (3)\\n\\nwhere |0 and |1 are the ground and first excited states of the harmonic oscillator, interacts with a thermal state ρ th b (V, d) and the interaction Hamiltonian is\\n\\nH K = λâ † âb † b (4)\\n\\nwhich corresponds to the cross Kerr nonlinear interaction. The resulting state is then\\n\\nρ ent ab = 1 2 d 2 αP th (V, d) |0 0| ⊗ |α α| + |1 0| ⊗ |αe iϕ α| + |0 1| ⊗ |α αe iϕ | + |1 1| ⊗ |αe iϕ αe iϕ | ( 5\\n\\n)\\n\\nand ϕ is determined by the strength of the nonlinearity λ and the interaction time. The Wigner representation of ρ ent ab is\\n\\nW ent ab (α, β) = 1 π e -2|α| 2 W th (β; d) + 2αV c (β; d) + 2[αV c (β; d)] * + (4|α| 2 -1)W th (β; de iϕ ) (6)\\n\\nwhere α and β are complex numbers parametrizing the phase spaces of the microscopic and macroscopic systems respectively and\\n\\nW th (α; d) = 2 πV exp[- 2|α -d| 2 V ], (7)\\n\\nV c (α; d) = 2 πJK exp[- 2 K (1 -e iϕ )d 2 - 1 J (α - 2e iϕ d K )(α * - 2d K )], (8)\\n\\nK = 2 + (V -1)(1 -e iϕ\\n\\n), J = (sin ϕ/2 + iV cos ϕ/2)/(2V sin ϕ/2 + 2i cos ϕ/2), and d has been assumed real without loss of generality. If one traces ρ ent ab over mode a, the remaining state will be simply in a classical mixture of two thermal states and its Wigner function will be positive everywhere. However, if one measures out the \\\"microscopic part\\\" on the superposed basis, i.e., (|0 a ± |1 a )/ √ 2, the \\\"macroscopic part\\\" for mode b may not lose its nonclassical characteristics. Such a measurement on the the superposed basis will reduce the remaining state to\\n\\nρ sup(±) = N ± s d 2 αP th (V, d) |α α| ± |αe iϕ α| ± |α αe iϕ | + |αe iϕ αe iϕ | , (9)\\n\\nwhere N ± s are the normalization factors, and its Wigner function is\\n\\nW sup(±) (α) = N ± s {W th (α; d) ± V c (α; d) ± {V c (α; d)} * + W th (α; de iϕ )}. ( 10\\n\\n)\\n\\nThe ± signs in Eqs. ( 8 ) and (9) correspond to the two possible results from the measurement of the microscopic system. The state in Eq. ( 10 ) is a superposition of two thermal states. A feasible experimental setup to generate superpositions of thermal states is atom-field interactions in cavities, where a π/2 pulse can be used to prepare the atom in a superposed state. This type of experiment has already been performed to produce a superposition of coherent states [16] . In our cases, simply thermal states can be used instead of coherent states. Another possible setup is an all-optical scheme with free-traveling fields and a cross-Kerr medium, where a standard singlephoton qubit could be used as the microscopic superposition. Recently, there have been theoretical and experimental efforts to produce and observe giant Kerr nonlinearities using electromagnetically induced transparency [17] . Furthermore, it was shown that a weak Kerr nonlinearity can still be useful if a initially strong field is employed in this type of experiment [18] . We shall further explain this with examples in Sec. III.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Negativity of the Wigner function\",\n \"text\": \"The negativity of the Wigner function is known as an indicator of non-classicality of quantum states. In order to observe negativity of the Wigner function in a real experiment, its absolute minimum negativity should be large enough. The minimum negativity of the Wigner function in Eq. ( 6 ) for V = 1 is -0.144 for d = 0 and -0.246 for d → ∞. Now suppose the initial state can be considered a classical thermal state by letting V ≫ 1. One might expect that the negativity would be washed out as the initial state becomes mixed, but this is not the case. The minimum negativity actually increases as V gets larger. If V → ∞, the minimum negativity of the Wigner function ( 6 ) is -0.246 regardless of d: no matter how mixed the initial thermal state was, the minimum negativity of Wigner function is found to be a large value. The point in the phase space which gives the minimum negativity when V ≫ 1 or d ≫ 0 is (-1 2 , 0) and has negativity\\n\\nW neg ≡ W ent ab (- 1 2 , 0) = 2(-2 + 1 V exp[-2d 2 V ]) π 2 √ e . (11)\\n\\nIt can be shown that\\n\\nW neg approaches -4/(π 2 √ e) ≈ -0.246 when either d → ∞ or V → ∞.\\n\\nThis effect is obviously due to the interaction between the microscopic superposition and the macroscopic thermal state. If the initial microscopic state is not superposed, e.g., |ψ a = |1 a , the resulting state will be a simple direct product, (|1 1|) a ⊗ ρ th b (V, -d). Whilst for V = 1 this state will exhibit negativity, this is washed out and tends to zero as V → ∞. Needless to say, if it was |0 a instead of |1 a , the resulting Wigner function will be a direct product of two Gaussian states whose Wigner fucntion can never be negative. The superpositon state (3) plays the crucial role in making the minimum negativity of the resulting Wigner function always saturate to a certain negative value no matter how mixed and classical the initial state of the other mode becomes. The Wigner functions of the single-mode states, W sup(±) (α), in Eq. (10) show large negative values. The minimum negativity of the Wigner function W sup(-) (α) is W sup(-) (0) = 2/π regardless of the values of V and d. On the other hand, the minimum negativity of the Wigner function W sup(+) (α) approaches 2/π for d → ∞ and disappears when d = 0.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"C. Quantum interference in the phase space\",\n \"text\": \"When ϕ = π, the state (9) becomes (12) where\\n\\nρ ± = N (ρ th (V, d)±σ(V, d)±σ(V, -d)+ρ th (V, -d)),\\n\\nσ(V, d) = d 2 αP th (V, d)| -α α| and N = 2 1 ± exp[-2d 2 V ] V 2 . ( 13\\n\\n)\\n\\nIf the initial state for mode b is a pure coherent state, i.e., V = 1, the measurement on the superposed basis for mode a will produce a superposition of two pure coherent states as\\n\\n| Ψ ± = 1 √ 1 ± e -2|α| 2 (|α ± | -α ), (14)\\n\\nwhere α = d. The probability P ± to obtain the state ρ ± is obtained as [19 ] where\\n\\nP ± = ψ ± |Tr b [ρ ent ab ]|ψ ± = 1 2 (1 ± exp[-2d 2 V ] V ), ( 15\\n\\n)\\n\\n|ψ ± = (|0 ±|1 )/ √ 2.\\n\\nThe probability approaches P ± = 1/2 when either d or V becomes large.\\n\\nAs an analogy of Schrödinger's cat paradox, the variance V corresponds to the size the initial \\\"cat\\\", and the distance d between the two thermal component states corresponds to distinguishability between the \\\"alive cat\\\" and the \\\"dead cat\\\". Suppose that both V and d are very large for the initial thermal state. The two thermal states ρ th (V, ±d) become macroscopically distinguishable when d ≫ √ V , and our example may become a more realistic analogy of the cat paradox in this limit. Both the states ρ ± in this case show probability distributions with two Gaussian peaks and interference fringes [6] . Figure 1 presents the probability distributions of x (≡ Re[α]) and p (≡ Im[α]) for ρ -(a) when V = 100 and d = 100 and (b) when V = 1000 and d = 300. The probability distribution of x (p) for ρ ± can be obtained by integrating the Wigner function of ρ ± over p (x). The two Gaussian peaks along the x axis and interference fringes along the p axis shown in Fig. 1 are a typical signature of a quantum superposition between macroscopically distinguishable states. The visibility v of the interference fringes is defined as [15]\\n\\nv = I max -I min I max + I min , (16)\\n\\nwhere I = dxW sup(-) (α) and the maximum should be taken over p. It can be simply shown that the visibility v is always 1 regardless of the value of V . Note that d should increase proportionally to √ V to maintain the condition of classical distingushability between the two component thermal states ρ th (V, ±d). The interference fringes with high visibility are incompatible with classical physics and evidence of quantum coherence. The fringe spacing (the distance between the fringes) does not depend on V but only on d, i.e., a pure superposition of coherent states shows the same fringe spacing for a given d. We emphasize that the states shown in Fig. 1 are \\\"superpositions\\\" of severely mixed thermal states.\\n\\nAn experimental realization of a nonlinear effect corresponding to ϕ = π is very demanding particularly in the presence of decoherence. Here we point out that the method using a weak nonlinear effect (ϕ ≪ π) combined with a strong field (d ≫ 1) [18] can be useful to generate a thermal-state superposition with prominent interference patterns. In Fig. 2 , we have used experimentally accessible values, V = 5, d = 2000 and ϕ = π/1000, but the fringe visibility is still 1. In this case, decoherence during the nonlinear interaction would be significantly reduced because of the decrease of the interaction time [18] . Note also that, if required, the state in Fig. 2 can be moved to the center of the phase space, for example, using a biased beam splitter (BS) and a strong coherent field [18] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"D. Symmetric macroscopic quantum states\",\n \"text\": \"Let us assume that d = 0, i.e., the initial state is the thermal state, ρ th (V, 0), at the origin of the phase space. In this case, the thermal-state superpositions, ρ ± , are produced with probabilities, P ± = (1/2){1 ± (1/V )}, respectively. Figure 3 shows the Wigner functions of ρ + dependent on the interaction time between the macro-scopic thermal state and the microscopic superposition in a cross Kerr medium. The state is always symmetric in the phase space regardless of the interaction time as shown in Fig. 3 . In this figure, the initial state is a thermal state of V = 100 (Fig. 3(a) ). In a relatively short time (θ = π/32 and θ = π/16), the state shows some interference patterns. When θ = π, the evolved state looks very localized around the origin as shown in Fig 3 . The generated state at θ = π does not show negativity of the Wigner function nor squeezing properties. On the other hand, a well defined P function does not exist for this state.\\n\\nIn the case of ρ -, with the same assumption d = 0, the Wigner function at ϕ = π has the minimum negativity (-2/π) at the origin regardless of V . As a result of the interaction with the microscopic superposition, a deep hole to the negative direction below zero has been formed around the origin for ρ -as shown in Fig. 4 .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. ENTANGLEMENT BETWEEN THERMAL STATES\",\n \"text\": \"Entanglement between macroscopic objects and its Bell-type inequality tests are an important issue. In this section, we shall show that entanglement can be generated between high-temperature thermal states even when the temperature of each mode goes to infinity. 2 200 1 400 600 800 1000 V 2.1 2.2 2.3 B (a) 50 100 150 d 2.2 2 0 2.4 2.6 2.8 B (b) FIG. 5: (a) The optimized violation, B ≡ |B + |max, of Bell-\\n\\nCHSH inequality for the \\\"thermal-state entenglement\\\", ρ+, of V = 1000 (solid curve) and V = 100 (dashed curve). The Bell-violation of a pure entangled coherent state, i.e., V = 1, has been plotted for comparison (dotted curve). The Bell-violation B approaches its maximum bound, 2 √ 2, when d ≫ √ V regardless of the level of the mixedness V . (b) The optimized Bell-violation B against d for the different type of thermal-state entanglement generated using a 50:50 beam splitter from ρ + . V = 1000 (solid curve), V = 100 (dashed curve) and V = 1 (dotted curve).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A. Entanglement using two initial thermal states\",\n \"text\": \"If the microscopic superposition interacts with two thermal states, ρ th b (V, d) and ρ th c (V, d), and the micro-scopic particle is measured out on the superposed basis, the resulting state will be\\n\\nρ tm(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (17)\\n\\nwhere\\n\\nN t = 2 1 ± exp[-4d 2 V ] V 2 . ( 18\\n\\n)\\n\\nSuch two-mode thermal-state entanglement can be generated using two cavities and an atomic state detector [20] .\\n\\nExtending the two cavities to N cavities, entanglement of N -mode thermal states can also be generated. Such a state is an analogy of the N -mode pure GHZ state [21] but each mode is extremely mixed. Here we shall consider the Bell-CHSH inequality [22, 23] with photon number parity measurements [20, 24] . The parity measurements can be performed in a high-Q cavity using a far-off-resonant interaction between a two-level atom and the field [25] . The Bell-CHSH inequality can be represented in terms of the Winger function as [24 ]\\n\\n|B (±) | = π 2 4 |W tm(±) (α, β) + W tm(±) (α, β ′ ) + W tm(±) (α ′ , β) -W tm(±) (α ′ , β ′ )| ≤ 2, (19)\\n\\nwhere W tm(±) (α, β) is the Wigner function of ρ tm(±) in Eq. ( 17 ). As shown in Fig. 5 , the Bell-violation approaches the maximum bound for a bipartite measurement, 2 √ 2 [13] , when d ≫ √\\n\\nV regardless of the level of the mixedness V , i.e., the temperatures of the thermal states. Note that it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(a) . This implies that entanglement of nearly 1 ebit has been produced between the two significantly mixed thermal states for d ≫ √ V , and such \\\"thermal-state entanglement\\\" cannot be described by a local theory.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Entanglement using a beam splitter\",\n \"text\": \"A different type of macroscopic entanglement can be generated by applying the beam splitter operation\\n\\nexp[θ/2(e iφ â † s âd -e -iφ â † d âs )], (20)\\n\\non the \\\"thermal-state superpositions\\\" in Eq. ( 9 ). The state after passing through a 50:50 beam splitter can be represented as\\n\\nN d 2 αP th α (V, d) | α √ 2 , - α √ 2 ± | - α √ 2 , α √ 2 α √ 2 , - α √ 2 | ± - α √ 2 , α √ 2 | , (21\\n\\n) 200 400 600 800 1000 V 2.1 2 0 2.2 2.3 B FIG. 6: The optimized Bell-violation B against V for the slightly different type of thermal-state entanglement generated using a 50:50 beam splitter using ρ + when d = 0. 3.13 3.14 3.15 3.16 ϕ 1.6 2.2 2 2.4 2.6 2.8 B (a) 3.13 3.14 3.15 3.16 ϕ 1.6 2 2.2 2.4 2.6 2.8 B (b) where N is defined in Eq. ( 13 ). When d is large, this state violates the Bell-CHSH inequality to the maximum bound 2 √ 2 regardless of the level of mixedness V as shown in Fig. 5(b ). Again, it is true for both of ρ + and ρ -even though only the case of ρ + has been plotted in Fig. 5(b ). Furthermore, these states severely violate Bell's inequality even when d = 0 as V increases as shown in Fig. 6 . We have found that the optimized Bell violation of these states approaches 2.32449 for V → ∞. Interestingly, this value is exactly the same as the optimized Bell-CHSH violation for a pure two-mode squeezed state in the infinite squeezing limit [26] . Note that multilmode entangled states can be generated using multiple beam splitters.\\n\\nIt should be noted that the Bell violations are more sensitive to the interaction time when either V or d is larger. Figure 7 clearly shows this tendency. Therefore, in order to observe the Bell violations using the mixed state of V (and d) large, the interaction time in the Kerr medium should be more accurate.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. QUANTUM INFORMATION PROCESSING WITH THERMAL-STATE QUBITS\",\n \"text\": \"In this section, we discuss the possibility of quantum information processing with thermal-state qubits and thermal-state entanglement.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A. Qubits and Bell-state measurements\",\n \"text\": \"We introduce a thermal-state qubit\\n\\nρ ψ = |a| 2 ρ th (V, d) ± ab * σ(V, d) ± a * bσ(V, -d) + |b| 2 ρ th (V, -d), (22)\\n\\nwhere a and b are arbitrary complex numbers. The basis states, ρ th (V, d) and ρ th (V, -d), can be well discriminated by a homodyne measurement when d is larger than V . The thermal state qubit (22) can be re-written as\\n\\nρ ψ = d 2 αP th α (V, d) a|α + b| -α a * α| + b * -α| , (23)\\n\\nwhich can be understood as a generalization of the coherent state qubit, a|d + b|d , where |d is a coherent state of amplitude d. The thermal-state qubit (23) becomes identical to the coherent-state qubit when V = 1.\\n\\nWe also define four thermal-Bell states as\\n\\nρ Φ(±) = N t ρ th (V, d) ⊗ ρ th (V, d) ± σ(V, d) ⊗ σ(V, d) ± σ(V, -d) ⊗ σ(V, -d) + ρ th (V, -d) ⊗ ρ th (V, -d) (24) ρ Ψ(±) = N t ρ th (V, d) ⊗ ρ th (V, -d) ± σ(V, d) ⊗ σ(V, -d) ± σ(V, -d) ⊗ σ(V, d) + ρ th (V, -d) ⊗ ρ th (V, d) (25)\\n\\nwhere N t was defined in Eq. ( 18 ). The thermal-Bell states can be written as For quantum information processing applications, it is an important task to discriminate between the four Bell states. Here we discuss two possible ways to discriminate between the thermal-Bell states (25) . We shall only briefly describe the first scheme using photon number resolving measurements and focus on the second scheme using nonlinear interactions. The first method is to simply use a 50-50 beam splitter and two photon number resolving detectors as shown in Fig. 8(a) . This scheme is basically the same as the Bellstate measurement scheme with pure entangled coherent states [27, 28] . Let us suppose that the amplitude, d, is large enough, i.e., d ≫ √ V . If the incident state was ρ Φ(+) or ρ Φ(-) , most of the photons are detected on detector A in in Fig. 8(a) . Meanwhile, most of the photons are detected on detector B when the incident state was ρ Ψ(+) or ρ Ψ(-) . The average photon numbers between the \\\"many-photon case\\\" and the \\\"few-photon case\\\" are compared in Fig. 9 . Furthermore, the states ρ Ψ(+) and ρ Φ(+) contain only even numbers of photons while ρ Ψ(-) and ρ Φ(-) contain only odd numbers of photons. Therefore, all the four Bell states can be well discriminated by analyzing numbers of photons detected at detectors A and B. For example, if detector A detects many photons while detector B detects few and the total photon number detected by the two detectors are even, this means that state ρ Φ(+) was measured by the thermal-Bell measurement. The nonzero failure probability can be made arbitrarily small by increasing d.\\n\\nρ Φ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, β ± | -α, -β α, β| ± -α, -β| , ( 26\\n\\n)\\n\\nρ Ψ(±) = N t dα 2 dβ 2 P th α (V, d)P th β (V, d) |α, -β ± | -α, β α, -β| ± -α, β| . (27)\\n\\nHowever, the average photon numbers of the thermal-Bell states are high when V ≫ 1 and d ≫ 1. In this case, it would be unrealistic to use photon number resolving detectors. It would be an interesting question whether\\n\\n0.5 1 1.5 2 2.5 3 d 10 0 0 20 30 N (a) 0.5 1 1.5 2 2.5 3 d 0 0 10 20 30 N (b)\\n\\nFIG. 9: The average photon number N for the \\\"many-photon case\\\" (solid line) and the \\\"few-photon case\\\" (dashed line) for V = 10 against d (a) when the input state is either ρ Φ(+) or ρ Ψ(+) and (b) when the input state is either ρ Φ(-) or ρ Ψ(-) .\\n\\nthese four thermal-Bell states can be distinguished by classical measurements, such as homodyne detection, instead of photon number resolving detection. Our alternative scheme employs cross-Kerr nonlinearities and single photon detectors as shown in Fig. 8(b ). Let us first suppose that the input field was ρ Φ(+) . The incident twomode state passes through a 50-50 beam splitter, BS1. The state after passing through the 50:50 beam splitter, BS1, is\\n\\nρ B = N t d 2 αd 2 βP th α (V, d)P th β (V, d) |η, -ξ η, -ξ| + |η, -ξ -η, ξ| + | -η, ξ η, -ξ| + | -η, ξ -η, ξ| (28)\\n\\nwhere η = (α+β)/ √ 2 and ξ = (α-β)/ √ 2. Two dual-rail single photon qubits, |ψ + ee ′ and |ψ + f f ′ , where\\n\\n|ψ + = 1 √ 2 (|0 |1 + |1 |0 ), (29)\\n\\nare prepared using two single photons and 50:50 beam splitters, BS2 and BS3, as shown in Fig. 8(b ). Then, traveling fields at modes c and d interacts with those of modes e and f , respectively, in cross-Kerr nonlinear media. We suppose that the interaction time is t = π/λ, and the resulting state is then\\n\\nρ B ′ = U ce U df ρ B cd ρ q ee ′ ρ q f f ′ U † ce U † df (30)\\n\\nwhere U ce = exp[iπH K ce /λ ] and ρ q = |ψ q ψ q |. An explicit form of Eq. ( 30 ) can then be simply obtained using the identity\\n\\nU ce |α c |0 e = |α c |0 e , U ce |α c |1 e = | -α c |1 e ( 31\\n\\n)\\n\\nwhere |α is a coherent state. However, we omit such an explicit expression in this paper for it is too lengthy. After the nonlinear interactions, the qubit parts, modes e, e ′ , f and f ′ , should be measured with the measurement basis\\n\\n{| + + , | + -, | -+ , | --} (32)\\n\\nwhere\\n\\n| + + = |ψ + ee ′ |ψ + f f ′ , | + -= |ψ + ee ′ |ψ -f f ′ , | -+ = |ψ -ee ′ |ψ + f f ′ , | --= |ψ -ee ′ |ψ -f f ′ , and |ψ -= (|0 |1 -|1 |0 )/ √ 2.\\n\\nThis measurement can be performed using two 50:50 beam splitters, BS4 and BS5, and four detectors, A1, A2, B1 and B2, as shown in Fig. 8(b ). If detector A1 and B1 click, i.e., the measurement result is | + + , the resulting state at modes c and d is\\n\\nρ ++ = N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η + | -η )( η| + -η|) c ⊗ (|ξ + | -ξ )( ξ| + -ξ|) d (33)\\n\\nNote that state ρ ++ is not normalized, which implies that the probability of obtaining the corresponding measurement result is not unity. The probability of obtaining this result is\\n\\nP ++ = (V + 1)(V + e -4d 2 V ) 2(V 2 + e -4d 2 V ) . (34)\\n\\nWhen the result is either | +or | -+ , the result is\\n\\nψ 2 |ρ B ′ |ψ 2 = ψ 3 |ρ B ′ |ψ 3 = 0, (35)\\n\\nwhich obviously means that the probability of the obtaining this result is zero. When the result is | --, i.e., detector A2 and B2 click,\\n\\nρ --= N t 4 d 2 αd 2 βP th α (V, d)P th β (V, d) (|η -| -η )( η| --η|) c ⊗ (|ξ -| -ξ )( ξ| --ξ|) d , (36)\\n\\nwhich is not normalized. The probability of obtaining this result is\\n\\nP --= (V -1)(V -e -4d 2 V ) 2(V 2 + e -4d 2 V ) , (37)\\n\\nand it can be simply verified that P ++ +P --= 1. Therefore, only the measurement results | + + and | -can be obtained in the case of the input state ρ Φ(+) . This is exactly the same for the case of ρ Ψ(+) . In the same way, it can be shown that if either the input state was ρ Φ(-) or ρ Ψ(-) , only the measurement results | +and | -+ can be obtained. In other words, the parity of the toincoming state is perfectly well discriminated by the measurements on single-photon qubits.\\n\\nSubsequently, a homodyne measurement is performed for mode c by homodyne detector C as shown in Fig. 8(b) . We assume that ideal homodyne measurements are performed, i.e., when a homodyne measurement is performed the state is projected onto eigenstate |x of operator X with eigenvalue x, where\\n\\nX = 1 √ 2 (a + a † ). ( 38\\n\\n)\\n\\nLet us first consider the case when the measurement result for the single photon qubits is | + + . In this case, the remaining state is ρ ++ in Eq. ( 33 ). The probability distribution P ++ Φ (+) for the homodyne measurement at detector C is\\n\\nP ++ Φ (+) = x|Tr d [ρ ++ ]|x = V 1 2 (e -V x 2 + e -x 2 V ) π 1 2 (V + 1)\\n\\n. (39) Note that the superscript, ++, denotes that the qubit measurement result was | + + , and the subscript, Φ (+) , denotes that the input state was ρ Φ (+) . These notations will be used also for the other cases in this section. The same analysis can be performed for the other possible measurement outcome | --:\\n\\nP -- Φ (+) = x|Tr d [ρ --]|x = V 1 2 (e -V x 2 -e -x 2 V ) π 1 2 (V -1)\\n\\n. (40)\\n\\nIn the same way, for another input state, ρ Φ(-) , it is straightforward to show:\\n\\nP +- Φ (-) = P ++ Φ (+) , P -+ Φ (-) = P -- Φ (+) , (41)\\n\\nand P ++ Φ (-) = P -- Φ (-) = 0. On the other hand, if the input state was ρ Ψ(+) , the probability distributions P ++ Ψ (+) and P -- Ψ (+) at detector C are It is straightforward to show for the other input state ρ Ψ(-) :\\n\\nP ++ Ψ (+) = x|Tr c [ρ ++ ]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V + 2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) + 1 , (42)\\n\\nP -- Ψ (+) = x|Tr c [ρ --]|x = V 1 2 e -x{4d+(2+V 2 )x} V e (1+V 2 )x 2 V -2e 2x(2d+x) V + e x(8d+x+V 2 x) V 2π 1 2 (e 4d 2 V V ) -1 . ( 43\\n\\nP +- Ψ (-) = P -- Ψ (+) , P -+ Ψ (-) = P ++ Ψ (+) . (44)\\n\\nThe probability distributions P ++ Φ (±) and P ++ Ψ (±) are plotted in Fig. 10 . Figure 10 shows that when the input state was ρ Φ(+) or ρ Φ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Φ (+) and P -- Φ (+) , is located around the origin. However, when the input state was ρ Ψ(+) or ρ Ψ(-) , the homodyne measurement outcome by detector C, characterized by P ++ Ψ (+) and P -- Ψ (+) , is located far from the origin. Therefore, two of the Bell states, ρ Φ(+) or ρ Φ(-) , can be well distinguished from the other two by the homodyne detector C for the case of the measurement outcome | + + . Finally, by combining the homodyne measurement result and the qubit measurement result, all four Bell states can be effectively distinguished. For example, let us assume that the measurement outcome of the single photon detectors was | + + and the homodyne detection outcome was around the origin, i.e., x ≈ 0. Then, one can say that state ρ Ψ(-) has been measured for the result of the thermal-Bell measurement.\\n\\nAs implied in Fig. 10 , the overlaps between the probability distributions around the origin, P ++ Φ (+) and P -- Φ (+) , and the other distributions, P ++ Ψ (+) and P -- Ψ (+) , are extremely small for a sufficiently large d. In other words, the distinguishability by the homodyne detection rapidly approaches 1 as d increases. As an example, we can calculate the distinguishability between the states ρ Ψ(+) and ρ Φ(+) by the homodyne measurement by detector C. The distinguishability by homodyne detection is Note also that the second scheme using homodyne detection is robust to detection inefficiency compared with the first scheme using photon number resolving measurements. In the first scheme, even if a detector misses only one photon, it will result in a completely wrong measurement outcome. In the second scheme, however, the measurement outcome will not be affected in that way. If a single photon detector misses a photon, it will be immediately recognized. Such a case can simply be discarded so that it will only degrade the success probability of the Bell measurement. The homodyne detection inefficiency will not significantly affect the result when the distributions around the origin and the distributions far from the origin are well separated, i.e., when d ≫ √ V , as shown in Fig. 10 . On the other hand, loss in the Kerr medium will have a detrimental affect.\\n\\nP s = 1 2 |x|<d dxP ++ c (x) + |x|≥d dxP ++ d (x) (45\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Quantum teleportation and computation\",\n \"text\": \"Quantum teleportation of a thermal-state qubit can be performed using one of the Bell states as the quantum channel. Let us assume that Alice needs to teleport a thermal-state qubit, ρ ψ , to Bob using a thermal-state entanglement, ρ Ψ(-) , shared by the two parties. The total state can be represented as\\n\\nAlice first needs to perform the thermal-Bell measurement described in the previous subsection. To complete the teleportation process, Bob should perform an appropriate unitary transformation on his part of the quantum channel according to the measurement result sent from Alice via a classical channel. It is straightforward to show that the required transformations are exactly the same to those for the coherent-state qubit [27] . When the measurement outcome is ρ Ψ(-) , Bob obtains a perfect replica of the original unknown qubit without any operation.\\n\\nWhen the measurement outcome is ρ Φ(-) , Bob should perform |α ↔ |α on his qubit in Eq. ( 23 ). Such a phase shift by π can be done using a phase shifter whose action is described by P (ϕ) = e iϕa † a , where a and a † are the annihilation and creation operators. When the outcome is ρ Ψ(+) , the transformation should be performed as |α → |α and |α → -|α . It is known that the displacement operator is a good approximation of this transformation for d ≫ 1 [29] . This transformation can also be achieved by teleporting the state again locally and repeating until the required phase shift is obtained [30] . When the outcome is ρ Φ(+) , σ x and σ z should be successively applied.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"V. CONCLUSION\",\n \"text\": \"In this paper, we have studied characteristics of superpositions and entanglement of thermal states at high temperatures and discussed their applications to quantum information processing. The superpositions and entanglement of thermal states show various nonclassical properties such as interference patterns, negativity of the Wigner functions, and violations of the Bell-CHSH inequality. The Bell violations are more sensitive to the interaction time during the generation process when the thermal temperature (i.e. mixedness) of the thermalstate entanglement is larger. Therefore, in order to observe the Bell violations using the mixed state at a high temperature, the interaction time in the Kerr medium should be accurate. We have pointed out that certain superpositions of high-temperature thermal states, symmetric in the phase space, can also be generated. Some of these states have neither squeezing properties nor negative values in their Wigner functions but they are found to be highly nonclassical.\\n\\nWe have introduced the thermal-state qubit and thermal-Bell states for applications to quantum information processing. We have presented two possible methods for the Bell-state measurement. The Bell-state measurement enables one to perform quantum teleportation and gate operations for quantum computation with thermalstate qubits. The first scheme uses two photon number resolving detectors and a 50-50 beam splitter to discriminate the thermal-Bell states. Using the second scheme, it is possible to effectively discriminate the thermal-Bell states without photon number resolving detection. The required resources for the second scheme are two Kerr nonlinear interactions, two single photon detectors, two 50:50 beam splitters and one homodyne detector. The second scheme is more robust to inefficiency of the detectors: the inefficiency of the single photon detectors only degrades the success probability of the Bell measurement.\"\n }\n]","isConversational":false}}},{"rowIdx":96,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0526"},"arxiv_id":{"kind":"string","value":"0704.0526"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":""},"content_hash":{"kind":"string","value":"a1a26ce33595a638cf1c21c15b33b4f6450ff8e123099f4f85a016d59c12c65c"},"title":{"kind":"string","value":"Fractional WKB Approximation"},"abstract":{"kind":"string","value":"Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is\ninvestigated in this paper using the fractional calculus. In the fractional\ncase the wave function is constructed such that the phase factor is the same as\nthe Hamilton's principle function \"S\". To demonstrate our proposed approach two\nexamples are investigated in details."},"authors":{"kind":"list like","value":["Eqab M. Rabei","Ibrahim M. A. Altarazi","Sami I. Muslih","Dumitru Baleanu"],"string":"[\n \"Eqab M. Rabei\",\n \"Ibrahim M. A. Altarazi\",\n \"Sami I. Muslih\",\n \"Dumitru Baleanu\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["math-ph","hep-th","math.MP"],"string":"[\n \"math-ph\",\n \"hep-th\",\n \"math.MP\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"math-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case the wave function is constructed such that the phase factor is the same as the Hamilton's principle function \"S\". To demonstrate our proposed approach two examples are investigated in details.\n\nFractional calculus is a branch of mathematics that deal with a generalization of well-known operations of differentiations and integrations to arbitrary non-integer order, which can be real non-integer or even imaginary number.\n\nNowadays physicists have used this powerful tool to deal with some problems which were not solvable in the classical sense. Therefore, the fractional calculus became one of the most powerful and widely useful tools in describing and explaining some physical complex systems.\n\nRecently, the Euler-Lagrange equations has been presented for unconstrained and constrained fractional variational problems [1 and other references] . This technique enable us to solve some problems including describing the behavior of non-conservative systems developed by Riewe [2] , where he used the fractional derivative to construct the Lagrangian and Hamiltonian for non-conservative systems.\n\nFrom these reasons in [3] was developed a general formula for the potential of any arbitrary force conservative or not conservative, which leads directly to the consideration of dissipative effect in Lagrangian and Hamiltonian formulation. Also, the canonical quantization of non-conservative systems has been carried out in [4] .\n\nStarting from a Lagrangian containing a fractional derivative, the fractional Hamiltonian is achieved in [5] . In addition, the passage from Hamiltonian containing fractional derivatives to the fractional Hamilton-Jaccobi is achieved by Rabei et.al [6] . The equations of motion are obtained in a similar manner to the usual mechanics.\n\nAll these outstanding results using the fractional derivative make us concentrate on another branch of quntam physics. WKB approximation [7, 8, 9, 10, 14] . In this paper we are mainly interested to construct the solution of Schroödinger equation in an exponential form (Griffith 1995) starting from fractional Hamilton-Jaccobi equation and how it leads naturally to this semi-classical approximation namely fractional WKB.\n\nThe purpose of this paper is to find the solution of Schrödinger equation for some systems that have a fractional behavior in their Lagrangians and obey the WKB approximation assumptions.\n\nThe plan of this paper is as follows: In section II the derivation of generalized Hamilton-Jaccobi partial differential equation which given in [6] is briefly reviewed. In section III the fractional WKB approximation is derived. In Section IV some examples with the fractional WKB technique is reported. Section V is dedicated to conclusions.\n\nThe left and right Reimann-Loville fractional derivative are defined as follows [3] The left Riemann-Liouville fractional derivative is given by\n\n( ) τ τ τ α α α d f x dx d n x f D x a n n x a ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - Γ = ) ( ) ( 1 ) ( 1 (1)\n\nThe right Riemann-Liouville fractional derivative has the form\n\n( ) τ τ τ α β β d f x dx d n x f D b x n n b x ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - - Γ = ) ( ) ( 1 ) ( 1 (2)\n\nHere α, β are the order of derivation such that n-1≤α <n, n-1≤β<n, and they are not zero.\n\nIf α is an integer, these derivatives are defined in usual sense as\n\n) ( ) ( x f dx d x f D x a α α ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = (3-a) ) ( ) ( x f dx d x f D b x β β ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = (3-b)\n\nHamilton formalism with fractional derivative was proposed in [5]\n\nnamely ) , , , ( ) , , , ( t q D q D q L q D p q D p t p p q H b t t a b t t a β α β β α α β α - + = , ( 4\n\n) where L represents the fractional Lagrangian obtained by replacing the classical derivatives with the corresponding fractional ones [5] .\n\nHamilton's equations of motion are obtained as follows [5]\n\nt L t H ∂ ∂ - = ∂ ∂ ; ; q D p H t a α α = ∂ ∂ ; q D p H b t β β = ∂ ∂ α α β β p D p D q H b t t a + = ∂ ∂ (5)\n\nIn [6] based on the sequential derivatives the fractional Hamilton-Jacobi partial differential equation is obtained. The Hamilton-Jacobi function in configuration space is written in a similar manner to the usual mechanics by using the Reimann-Loville fractional derivative. In [6] the following generating function is used, where α and β are bigger or equal to 1. Thus, the new Hamiltonian is expressed as\n\nS t P P q D q D F F b t t a = = - - ) , , , , ( 1 1 2 β α β α ) , , , ( ' ) , , , ( t Q D Q D Q L Q D P Q D P t P P Q K b t t a b t t a β α β β α α β α - + = (7)\n\nIt is concluded that, the following relation relates the two Hamiltonians\n\ndt dF K Q D P Q D P H q D p q D p b t t a b t t a + - + = - + β β α α β β α α (8)\n\nAccording to reference [6] the function F is proposed as ) , , , , (\n\n1 1 t P P q D q D S F b t t a β α β α - - = Q D P Q D P B t t a 1 1 - - - - β β α α , (9)\n\nThe function S is called Hamilton's principle function.\n\nTherefore, requiring that the transformed Hamiltonian K shall be zero the Hamilton-Jacobi equation is satisfied. In other words Q, P α , P β are constants.\n\n0 = ∂ ∂ + t S H (10)\n\nSince Q, P α , P β are constants, The Hamilton's principle function is written as (11) where\n\n) , , , , ( 2 1 1 1 t E E q D q D S S b t t a - - = β α\n\n1 E P = α 2 E P = β\n\nIf the Hamiltonian is explicitly independent of time, then S can be written as follows (12) ) , , ( ) , ( ) , (\n\n2 1 2 1 2 1 1 1 t E E f E q D W E q D W S b t t a + + = - - β α\n\nW represents the Hamilton's characteristic function; therefore, the following equations of motion are obtained in [6] as:\n\nq D W P t a 1 1 - ∂ ∂ = α α q D W P b t 1 1 - ∂ ∂ = β β (13) 1 1 1 1 λ α = ∂ ∂ = - E W Q D t a 2 2 2 1 λ β = ∂ ∂ = - E W Q D b t (14)\n\nHere λ 1 ,λ 2 are constants.\n\nThe outstanding result regarding the meaning of the state function ψ and its relationship to Hamilton's principle function S enables us to write the exponential solution of Schrödinger equation [13] .\n\n⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = h ) , ( exp ) , ( t q S i t q ψ (15)\n\nThe phase of state function obeys the same mathematical equation, as does Hamilton's principle function S. The physical significance of S in classical mechanics is that it represents the generator of trajectories [12] for fractional systems; the fractional Hamilton's principle function is become the phase of the state function ψ. One can write the solution of Schrödinger equation under the postulated constrains by the WKB approximation and using the fractional Hamilton's principle function eq (12) . Thus we propose the fractional state function as:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t q D q D S i t q D q D b t t a b t t a , , exp ) , , ( 1 1 1 1 β α β α ψ h (16)\n\nFrom the quantization using WKB approximation [7, 8, 9, 10, 14] a general solution of Schrödinger equation is obtained using the expansion for S and then using the transformation to the N-dimensional system as:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ∏ = h t q S i q i N i i o i , exp ) ( 1 ψ ψ (17)\n\nwhere ( )\n\n) ( 1 i i io q p q = ψ (18)\n\nIn our case, S behaves like a 2-dimensional problem with two distinct momenta. Thus,\n\n(19) q D q t a 1 1 - ≡ α α P P 1 ≡ (20) q D q b t 1 2 - ≡ β β P P 2 ≡\n\nAnd the momenta are defined as operators. Therefore, we can propose the wave function ψ of the fractional system in the following form\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t E E q D q D S i P P t q D q D b t t a b t t a , , , , exp 1 ) , , ( 2 1 1 1 1 1 β α β α β α ψ h (21)\n\nand the momenta operators in the form\n\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 , ˆβ β α α h h (22)\n\nWe conclude that (21) is the solution of Schrödinger equation for any given fractional systems. If α and β both are equal to unity, then we will return to the usual classical solution of Schrödinger equation, also we can notice how the probability is inversely proportional to the momentum . ) (\n\n1 2 q p ≅ ψ IV. Examples\n\nAs a first model let us consider the following fractional Lagrangian,\n\n( ) ( ) 2 1 2 0 2 1 2 1 q D q D L t t β α + = (23)\n\nThe fractional Hamilton-Jacobi equation for this fractional Lagrangian can be calculated as:\n\n( ) ( ) . 0 2 1 2 1 2 2 = ∂ ∂ + + t S P P β α (24) where q D L P t α α 0 ∂ ∂ = ; q D L P t β β 1 ∂ ∂ =\n\nMaking use of equation ( 13 ), the fractional Hamilton-Jacobi equation (24) becomes:\n\n0 2 1 2 1 2 1 1 2 2 1 0 1 = ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - t S q D W q D W t t β α (25) Taking into account t S H ∂ ∂ - = (26)\n\nIf we apply (26) on a wave function it gives:\n\n) ( 2 1 E E E t S + - ≡ - = ∂ ∂ (27)\n\nBy using the fact that E is the total energy of the system and taking into account (27) we obtain 0 2\n\n1 2 1 2 2 1 1 2 1 2 1 0 1 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - E q D W E q D W t t β α (28)\n\nThus, both sides of (28) should be zero, and we obtain\n\nq D E W q D E W t t 1 1 2 2 1 0 1 1 2 , 2 - - = = β α\n\n(29) By using ( 12 ) and ( 21 ) we obtain It's the same as the classical solution. Also, when applying the energy operator it gives the energy eigenvalues:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + = - - - - t E q D E q D E i P P t q D q D t t t t 1 1 2 1 0 1 1 1 1 0 2 2 exp 1 ) , , (\n\n( ) ( )ψ ψ ψ β α 2 2 2 1 2 1 P P H + = ( ) ψ 2 1 E E + = (34)\n\nas in the classical case.\n\nAs a second example let us consider the following fractional Lagrangian ( ) ( )\n\n2 1 0 2 1 2 0 2 1 2 1 2 1 q q D q D q D q D L t t t t + + + + = β α β α (35)\n\nThe corresponding fractional Hamilton is calculated as follows\n\n( ) ( ) 2 2 2 2 1 1 2 1 1 2 1 q P P H - - + - = β α (36)\n\nThus, the fractional Hamilton-Jacobi equation becomes ( ) ( )\n\n0 2 1 1 2 1 1 2 1 2 2 2 = ∂ ∂ + - - + - t S q P P β α (37)\n\nThe fractional Hamilton's principle function is calculated as,\n\n) ( )\n\nt E E q D E q D E q S t t ) ( 1 2 1 2 2 1 1 1 2 1 0 1 2 + - + + + + = - - β α (38)\n\nAs a result the wave function can be written in the form ( )\n\n( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + + + + = - - - - t E q D E q D E q i P P t q D q D t t t t 1 1 2 1 0 1 2 1 1 1 0 1 2 1 2 exp 1 ) , , ( β α β α β α ψ h (39)\n\nTo identify the influence of the operators let us test the effect of the momenta\n\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 ˆβ β α α h h (40)\n\nUsing the characteristic equations, it can be shown that\n\n1 2 , 1 2 ˆ2 1 2 + = + + = E P E q P β α (41) The result shown in (41 ) is the same classical solution. When applying the energy operator it will give the energy eigenvalues ( ) ( ) ψ ψ ψ ψ β α 2 2 2 2 1 1 2 1 1 2 1 q P P H --+ -= (42) ( ) ( ) ψ ψ ψ ψ β α β α 2 2 2 2 1 2 1 q P P P P -+ + -+ = ( ) ( ) ψ } 2 1 1 2 2 2 2 2 2 2 2 2 2 1 { 2 2 1 2 2 1 2 2 q E E q E E q E q -+ + + + -+ + + + + = Then we get\n\nψ ψ E H = (43)\n\nwhich is exactly the total energy as the case for the classical systems.\n\nWe use the generating function \"S\" of the Hamilton-Jaccobi equation in its fractional form to be the phase factor of the wave function describing some potentials valid for the assumptions suggested by the WKB approximation The proof of our results arises from the new proposed concepts of the momentum and energy operators, that they give the same eigenvalues producing the ordinary results achieved by the classical approach.\n\nGiving the same eigenvalues that means this form of fractional operator also eigen, valid, and useful in effecting on a state functions."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case the wave function is constructed such that the phase factor is the same as the Hamilton's principle function \"S\". To demonstrate our proposed approach two examples are investigated in details."},{"section_type":"BACKGROUND","section_title":"I. Introduction","text":"Fractional calculus is a branch of mathematics that deal with a generalization of well-known operations of differentiations and integrations to arbitrary non-integer order, which can be real non-integer or even imaginary number.\n\nNowadays physicists have used this powerful tool to deal with some problems which were not solvable in the classical sense. Therefore, the fractional calculus became one of the most powerful and widely useful tools in describing and explaining some physical complex systems.\n\nRecently, the Euler-Lagrange equations has been presented for unconstrained and constrained fractional variational problems [1 and other references] . This technique enable us to solve some problems including describing the behavior of non-conservative systems developed by Riewe [2] , where he used the fractional derivative to construct the Lagrangian and Hamiltonian for non-conservative systems.\n\nFrom these reasons in [3] was developed a general formula for the potential of any arbitrary force conservative or not conservative, which leads directly to the consideration of dissipative effect in Lagrangian and Hamiltonian formulation. Also, the canonical quantization of non-conservative systems has been carried out in [4] .\n\nStarting from a Lagrangian containing a fractional derivative, the fractional Hamiltonian is achieved in [5] . In addition, the passage from Hamiltonian containing fractional derivatives to the fractional Hamilton-Jaccobi is achieved by Rabei et.al [6] . The equations of motion are obtained in a similar manner to the usual mechanics.\n\nAll these outstanding results using the fractional derivative make us concentrate on another branch of quntam physics. WKB approximation [7, 8, 9, 10, 14] . In this paper we are mainly interested to construct the solution of Schroödinger equation in an exponential form (Griffith 1995) starting from fractional Hamilton-Jaccobi equation and how it leads naturally to this semi-classical approximation namely fractional WKB.\n\nThe purpose of this paper is to find the solution of Schrödinger equation for some systems that have a fractional behavior in their Lagrangians and obey the WKB approximation assumptions.\n\nThe plan of this paper is as follows: In section II the derivation of generalized Hamilton-Jaccobi partial differential equation which given in [6] is briefly reviewed. In section III the fractional WKB approximation is derived. In Section IV some examples with the fractional WKB technique is reported. Section V is dedicated to conclusions."},{"section_type":"OTHER","section_title":"II. Basic Tools","text":"The left and right Reimann-Loville fractional derivative are defined as follows [3] The left Riemann-Liouville fractional derivative is given by\n\n( ) τ τ τ α α α d f x dx d n x f D x a n n x a ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - Γ = ) ( ) ( 1 ) ( 1 (1)\n\nThe right Riemann-Liouville fractional derivative has the form\n\n( ) τ τ τ α β β d f x dx d n x f D b x n n b x ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - - Γ = ) ( ) ( 1 ) ( 1 (2)\n\nHere α, β are the order of derivation such that n-1≤α <n, n-1≤β<n, and they are not zero.\n\nIf α is an integer, these derivatives are defined in usual sense as\n\n) ( ) ( x f dx d x f D x a α α ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = (3-a) ) ( ) ( x f dx d x f D b x β β ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = (3-b)\n\nHamilton formalism with fractional derivative was proposed in [5]\n\nnamely ) , , , ( ) , , , ( t q D q D q L q D p q D p t p p q H b t t a b t t a β α β β α α β α - + = , ( 4\n\n) where L represents the fractional Lagrangian obtained by replacing the classical derivatives with the corresponding fractional ones [5] .\n\nHamilton's equations of motion are obtained as follows [5]\n\nt L t H ∂ ∂ - = ∂ ∂ ; ; q D p H t a α α = ∂ ∂ ; q D p H b t β β = ∂ ∂ α α β β p D p D q H b t t a + = ∂ ∂ (5)\n\nIn [6] based on the sequential derivatives the fractional Hamilton-Jacobi partial differential equation is obtained. The Hamilton-Jacobi function in configuration space is written in a similar manner to the usual mechanics by using the Reimann-Loville fractional derivative. In [6] the following generating function is used, where α and β are bigger or equal to 1. Thus, the new Hamiltonian is expressed as\n\nS t P P q D q D F F b t t a = = - - ) , , , , ( 1 1 2 β α β α ) , , , ( ' ) , , , ( t Q D Q D Q L Q D P Q D P t P P Q K b t t a b t t a β α β β α α β α - + = (7)\n\nIt is concluded that, the following relation relates the two Hamiltonians\n\ndt dF K Q D P Q D P H q D p q D p b t t a b t t a + - + = - + β β α α β β α α (8)\n\nAccording to reference [6] the function F is proposed as ) , , , , (\n\n1 1 t P P q D q D S F b t t a β α β α - - = Q D P Q D P B t t a 1 1 - - - - β β α α , (9)\n\nThe function S is called Hamilton's principle function.\n\nTherefore, requiring that the transformed Hamiltonian K shall be zero the Hamilton-Jacobi equation is satisfied. In other words Q, P α , P β are constants.\n\n0 = ∂ ∂ + t S H (10)\n\nSince Q, P α , P β are constants, The Hamilton's principle function is written as (11) where\n\n) , , , , ( 2 1 1 1 t E E q D q D S S b t t a - - = β α\n\n1 E P = α 2 E P = β\n\nIf the Hamiltonian is explicitly independent of time, then S can be written as follows (12) ) , , ( ) , ( ) , (\n\n2 1 2 1 2 1 1 1 t E E f E q D W E q D W S b t t a + + = - - β α\n\nW represents the Hamilton's characteristic function; therefore, the following equations of motion are obtained in [6] as:\n\nq D W P t a 1 1 - ∂ ∂ = α α q D W P b t 1 1 - ∂ ∂ = β β (13) 1 1 1 1 λ α = ∂ ∂ = - E W Q D t a 2 2 2 1 λ β = ∂ ∂ = - E W Q D b t (14)\n\nHere λ 1 ,λ 2 are constants."},{"section_type":"OTHER","section_title":"III. Fractional WKB approximation","text":"The outstanding result regarding the meaning of the state function ψ and its relationship to Hamilton's principle function S enables us to write the exponential solution of Schrödinger equation [13] .\n\n⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = h ) , ( exp ) , ( t q S i t q ψ (15)\n\nThe phase of state function obeys the same mathematical equation, as does Hamilton's principle function S. The physical significance of S in classical mechanics is that it represents the generator of trajectories [12] for fractional systems; the fractional Hamilton's principle function is become the phase of the state function ψ. One can write the solution of Schrödinger equation under the postulated constrains by the WKB approximation and using the fractional Hamilton's principle function eq (12) . Thus we propose the fractional state function as:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t q D q D S i t q D q D b t t a b t t a , , exp ) , , ( 1 1 1 1 β α β α ψ h (16)\n\nFrom the quantization using WKB approximation [7, 8, 9, 10, 14] a general solution of Schrödinger equation is obtained using the expansion for S and then using the transformation to the N-dimensional system as:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ∏ = h t q S i q i N i i o i , exp ) ( 1 ψ ψ (17)\n\nwhere ( )\n\n) ( 1 i i io q p q = ψ (18)\n\nIn our case, S behaves like a 2-dimensional problem with two distinct momenta. Thus,\n\n(19) q D q t a 1 1 - ≡ α α P P 1 ≡ (20) q D q b t 1 2 - ≡ β β P P 2 ≡\n\nAnd the momenta are defined as operators. Therefore, we can propose the wave function ψ of the fractional system in the following form\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t E E q D q D S i P P t q D q D b t t a b t t a , , , , exp 1 ) , , ( 2 1 1 1 1 1 β α β α β α ψ h (21)\n\nand the momenta operators in the form\n\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 , ˆβ β α α h h (22)\n\nWe conclude that (21) is the solution of Schrödinger equation for any given fractional systems. If α and β both are equal to unity, then we will return to the usual classical solution of Schrödinger equation, also we can notice how the probability is inversely proportional to the momentum . ) (\n\n1 2 q p ≅ ψ IV. Examples"},{"section_type":"OTHER","section_title":"IV. a) Example 1:","text":"As a first model let us consider the following fractional Lagrangian,\n\n( ) ( ) 2 1 2 0 2 1 2 1 q D q D L t t β α + = (23)\n\nThe fractional Hamilton-Jacobi equation for this fractional Lagrangian can be calculated as:\n\n( ) ( ) . 0 2 1 2 1 2 2 = ∂ ∂ + + t S P P β α (24) where q D L P t α α 0 ∂ ∂ = ; q D L P t β β 1 ∂ ∂ =\n\nMaking use of equation ( 13 ), the fractional Hamilton-Jacobi equation (24) becomes:\n\n0 2 1 2 1 2 1 1 2 2 1 0 1 = ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - t S q D W q D W t t β α (25) Taking into account t S H ∂ ∂ - = (26)\n\nIf we apply (26) on a wave function it gives:\n\n) ( 2 1 E E E t S + - ≡ - = ∂ ∂ (27)\n\nBy using the fact that E is the total energy of the system and taking into account (27) we obtain 0 2\n\n1 2 1 2 2 1 1 2 1 2 1 0 1 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - E q D W E q D W t t β α (28)\n\nThus, both sides of (28) should be zero, and we obtain\n\nq D E W q D E W t t 1 1 2 2 1 0 1 1 2 , 2 - - = = β α\n\n(29) By using ( 12 ) and ( 21 ) we obtain It's the same as the classical solution. Also, when applying the energy operator it gives the energy eigenvalues:\n\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + = - - - - t E q D E q D E i P P t q D q D t t t t 1 1 2 1 0 1 1 1 1 0 2 2 exp 1 ) , , (\n\n( ) ( )ψ ψ ψ β α 2 2 2 1 2 1 P P H + = ( ) ψ 2 1 E E + = (34)\n\nas in the classical case."},{"section_type":"OTHER","section_title":"IVb.)Example 2:","text":"As a second example let us consider the following fractional Lagrangian ( ) ( )\n\n2 1 0 2 1 2 0 2 1 2 1 2 1 q q D q D q D q D L t t t t + + + + = β α β α (35)\n\nThe corresponding fractional Hamilton is calculated as follows\n\n( ) ( ) 2 2 2 2 1 1 2 1 1 2 1 q P P H - - + - = β α (36)\n\nThus, the fractional Hamilton-Jacobi equation becomes ( ) ( )\n\n0 2 1 1 2 1 1 2 1 2 2 2 = ∂ ∂ + - - + - t S q P P β α (37)\n\nThe fractional Hamilton's principle function is calculated as,\n\n) ( )\n\nt E E q D E q D E q S t t ) ( 1 2 1 2 2 1 1 1 2 1 0 1 2 + - + + + + = - - β α (38)\n\nAs a result the wave function can be written in the form ( )\n\n( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + + + + = - - - - t E q D E q D E q i P P t q D q D t t t t 1 1 2 1 0 1 2 1 1 1 0 1 2 1 2 exp 1 ) , , ( β α β α β α ψ h (39)\n\nTo identify the influence of the operators let us test the effect of the momenta\n\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 ˆβ β α α h h (40)\n\nUsing the characteristic equations, it can be shown that\n\n1 2 , 1 2 ˆ2 1 2 + = + + = E P E q P β α (41) The result shown in (41 ) is the same classical solution. When applying the energy operator it will give the energy eigenvalues ( ) ( ) ψ ψ ψ ψ β α 2 2 2 2 1 1 2 1 1 2 1 q P P H --+ -= (42) ( ) ( ) ψ ψ ψ ψ β α β α 2 2 2 2 1 2 1 q P P P P -+ + -+ = ( ) ( ) ψ } 2 1 1 2 2 2 2 2 2 2 2 2 2 1 { 2 2 1 2 2 1 2 2 q E E q E E q E q -+ + + + -+ + + + + = Then we get\n\nψ ψ E H = (43)\n\nwhich is exactly the total energy as the case for the classical systems."},{"section_type":"CONCLUSION","section_title":"V. Conclusions","text":"We use the generating function \"S\" of the Hamilton-Jaccobi equation in its fractional form to be the phase factor of the wave function describing some potentials valid for the assumptions suggested by the WKB approximation The proof of our results arises from the new proposed concepts of the momentum and energy operators, that they give the same eigenvalues producing the ordinary results achieved by the classical approach.\n\nGiving the same eigenvalues that means this form of fractional operator also eigen, valid, and useful in effecting on a state functions."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case the wave function is constructed such that the phase factor is the same as the Hamilton's principle function \\\"S\\\". To demonstrate our proposed approach two examples are investigated in details.\"\n },\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. Introduction\",\n \"text\": \"Fractional calculus is a branch of mathematics that deal with a generalization of well-known operations of differentiations and integrations to arbitrary non-integer order, which can be real non-integer or even imaginary number.\\n\\nNowadays physicists have used this powerful tool to deal with some problems which were not solvable in the classical sense. Therefore, the fractional calculus became one of the most powerful and widely useful tools in describing and explaining some physical complex systems.\\n\\nRecently, the Euler-Lagrange equations has been presented for unconstrained and constrained fractional variational problems [1 and other references] . This technique enable us to solve some problems including describing the behavior of non-conservative systems developed by Riewe [2] , where he used the fractional derivative to construct the Lagrangian and Hamiltonian for non-conservative systems.\\n\\nFrom these reasons in [3] was developed a general formula for the potential of any arbitrary force conservative or not conservative, which leads directly to the consideration of dissipative effect in Lagrangian and Hamiltonian formulation. Also, the canonical quantization of non-conservative systems has been carried out in [4] .\\n\\nStarting from a Lagrangian containing a fractional derivative, the fractional Hamiltonian is achieved in [5] . In addition, the passage from Hamiltonian containing fractional derivatives to the fractional Hamilton-Jaccobi is achieved by Rabei et.al [6] . The equations of motion are obtained in a similar manner to the usual mechanics.\\n\\nAll these outstanding results using the fractional derivative make us concentrate on another branch of quntam physics. WKB approximation [7, 8, 9, 10, 14] . In this paper we are mainly interested to construct the solution of Schroödinger equation in an exponential form (Griffith 1995) starting from fractional Hamilton-Jaccobi equation and how it leads naturally to this semi-classical approximation namely fractional WKB.\\n\\nThe purpose of this paper is to find the solution of Schrödinger equation for some systems that have a fractional behavior in their Lagrangians and obey the WKB approximation assumptions.\\n\\nThe plan of this paper is as follows: In section II the derivation of generalized Hamilton-Jaccobi partial differential equation which given in [6] is briefly reviewed. In section III the fractional WKB approximation is derived. In Section IV some examples with the fractional WKB technique is reported. Section V is dedicated to conclusions.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. Basic Tools\",\n \"text\": \"The left and right Reimann-Loville fractional derivative are defined as follows [3] The left Riemann-Liouville fractional derivative is given by\\n\\n( ) τ τ τ α α α d f x dx d n x f D x a n n x a ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - Γ = ) ( ) ( 1 ) ( 1 (1)\\n\\nThe right Riemann-Liouville fractional derivative has the form\\n\\n( ) τ τ τ α β β d f x dx d n x f D b x n n b x ∫ - - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - - Γ = ) ( ) ( 1 ) ( 1 (2)\\n\\nHere α, β are the order of derivation such that n-1≤α <n, n-1≤β<n, and they are not zero.\\n\\nIf α is an integer, these derivatives are defined in usual sense as\\n\\n) ( ) ( x f dx d x f D x a α α ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = (3-a) ) ( ) ( x f dx d x f D b x β β ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = (3-b)\\n\\nHamilton formalism with fractional derivative was proposed in [5]\\n\\nnamely ) , , , ( ) , , , ( t q D q D q L q D p q D p t p p q H b t t a b t t a β α β β α α β α - + = , ( 4\\n\\n) where L represents the fractional Lagrangian obtained by replacing the classical derivatives with the corresponding fractional ones [5] .\\n\\nHamilton's equations of motion are obtained as follows [5]\\n\\nt L t H ∂ ∂ - = ∂ ∂ ; ; q D p H t a α α = ∂ ∂ ; q D p H b t β β = ∂ ∂ α α β β p D p D q H b t t a + = ∂ ∂ (5)\\n\\nIn [6] based on the sequential derivatives the fractional Hamilton-Jacobi partial differential equation is obtained. The Hamilton-Jacobi function in configuration space is written in a similar manner to the usual mechanics by using the Reimann-Loville fractional derivative. In [6] the following generating function is used, where α and β are bigger or equal to 1. Thus, the new Hamiltonian is expressed as\\n\\nS t P P q D q D F F b t t a = = - - ) , , , , ( 1 1 2 β α β α ) , , , ( ' ) , , , ( t Q D Q D Q L Q D P Q D P t P P Q K b t t a b t t a β α β β α α β α - + = (7)\\n\\nIt is concluded that, the following relation relates the two Hamiltonians\\n\\ndt dF K Q D P Q D P H q D p q D p b t t a b t t a + - + = - + β β α α β β α α (8)\\n\\nAccording to reference [6] the function F is proposed as ) , , , , (\\n\\n1 1 t P P q D q D S F b t t a β α β α - - = Q D P Q D P B t t a 1 1 - - - - β β α α , (9)\\n\\nThe function S is called Hamilton's principle function.\\n\\nTherefore, requiring that the transformed Hamiltonian K shall be zero the Hamilton-Jacobi equation is satisfied. In other words Q, P α , P β are constants.\\n\\n0 = ∂ ∂ + t S H (10)\\n\\nSince Q, P α , P β are constants, The Hamilton's principle function is written as (11) where\\n\\n) , , , , ( 2 1 1 1 t E E q D q D S S b t t a - - = β α\\n\\n1 E P = α 2 E P = β\\n\\nIf the Hamiltonian is explicitly independent of time, then S can be written as follows (12) ) , , ( ) , ( ) , (\\n\\n2 1 2 1 2 1 1 1 t E E f E q D W E q D W S b t t a + + = - - β α\\n\\nW represents the Hamilton's characteristic function; therefore, the following equations of motion are obtained in [6] as:\\n\\nq D W P t a 1 1 - ∂ ∂ = α α q D W P b t 1 1 - ∂ ∂ = β β (13) 1 1 1 1 λ α = ∂ ∂ = - E W Q D t a 2 2 2 1 λ β = ∂ ∂ = - E W Q D b t (14)\\n\\nHere λ 1 ,λ 2 are constants.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. Fractional WKB approximation\",\n \"text\": \"The outstanding result regarding the meaning of the state function ψ and its relationship to Hamilton's principle function S enables us to write the exponential solution of Schrödinger equation [13] .\\n\\n⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = h ) , ( exp ) , ( t q S i t q ψ (15)\\n\\nThe phase of state function obeys the same mathematical equation, as does Hamilton's principle function S. The physical significance of S in classical mechanics is that it represents the generator of trajectories [12] for fractional systems; the fractional Hamilton's principle function is become the phase of the state function ψ. One can write the solution of Schrödinger equation under the postulated constrains by the WKB approximation and using the fractional Hamilton's principle function eq (12) . Thus we propose the fractional state function as:\\n\\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t q D q D S i t q D q D b t t a b t t a , , exp ) , , ( 1 1 1 1 β α β α ψ h (16)\\n\\nFrom the quantization using WKB approximation [7, 8, 9, 10, 14] a general solution of Schrödinger equation is obtained using the expansion for S and then using the transformation to the N-dimensional system as:\\n\\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ∏ = h t q S i q i N i i o i , exp ) ( 1 ψ ψ (17)\\n\\nwhere ( )\\n\\n) ( 1 i i io q p q = ψ (18)\\n\\nIn our case, S behaves like a 2-dimensional problem with two distinct momenta. Thus,\\n\\n(19) q D q t a 1 1 - ≡ α α P P 1 ≡ (20) q D q b t 1 2 - ≡ β β P P 2 ≡\\n\\nAnd the momenta are defined as operators. Therefore, we can propose the wave function ψ of the fractional system in the following form\\n\\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = - - - - t E E q D q D S i P P t q D q D b t t a b t t a , , , , exp 1 ) , , ( 2 1 1 1 1 1 β α β α β α ψ h (21)\\n\\nand the momenta operators in the form\\n\\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 , ˆβ β α α h h (22)\\n\\nWe conclude that (21) is the solution of Schrödinger equation for any given fractional systems. If α and β both are equal to unity, then we will return to the usual classical solution of Schrödinger equation, also we can notice how the probability is inversely proportional to the momentum . ) (\\n\\n1 2 q p ≅ ψ IV. Examples\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. a) Example 1:\",\n \"text\": \"As a first model let us consider the following fractional Lagrangian,\\n\\n( ) ( ) 2 1 2 0 2 1 2 1 q D q D L t t β α + = (23)\\n\\nThe fractional Hamilton-Jacobi equation for this fractional Lagrangian can be calculated as:\\n\\n( ) ( ) . 0 2 1 2 1 2 2 = ∂ ∂ + + t S P P β α (24) where q D L P t α α 0 ∂ ∂ = ; q D L P t β β 1 ∂ ∂ =\\n\\nMaking use of equation ( 13 ), the fractional Hamilton-Jacobi equation (24) becomes:\\n\\n0 2 1 2 1 2 1 1 2 2 1 0 1 = ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - t S q D W q D W t t β α (25) Taking into account t S H ∂ ∂ - = (26)\\n\\nIf we apply (26) on a wave function it gives:\\n\\n) ( 2 1 E E E t S + - ≡ - = ∂ ∂ (27)\\n\\nBy using the fact that E is the total energy of the system and taking into account (27) we obtain 0 2\\n\\n1 2 1 2 2 1 1 2 1 2 1 0 1 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - - E q D W E q D W t t β α (28)\\n\\nThus, both sides of (28) should be zero, and we obtain\\n\\nq D E W q D E W t t 1 1 2 2 1 0 1 1 2 , 2 - - = = β α\\n\\n(29) By using ( 12 ) and ( 21 ) we obtain It's the same as the classical solution. Also, when applying the energy operator it gives the energy eigenvalues:\\n\\n( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + = - - - - t E q D E q D E i P P t q D q D t t t t 1 1 2 1 0 1 1 1 1 0 2 2 exp 1 ) , , (\\n\\n( ) ( )ψ ψ ψ β α 2 2 2 1 2 1 P P H + = ( ) ψ 2 1 E E + = (34)\\n\\nas in the classical case.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IVb.)Example 2:\",\n \"text\": \"As a second example let us consider the following fractional Lagrangian ( ) ( )\\n\\n2 1 0 2 1 2 0 2 1 2 1 2 1 q q D q D q D q D L t t t t + + + + = β α β α (35)\\n\\nThe corresponding fractional Hamilton is calculated as follows\\n\\n( ) ( ) 2 2 2 2 1 1 2 1 1 2 1 q P P H - - + - = β α (36)\\n\\nThus, the fractional Hamilton-Jacobi equation becomes ( ) ( )\\n\\n0 2 1 1 2 1 1 2 1 2 2 2 = ∂ ∂ + - - + - t S q P P β α (37)\\n\\nThe fractional Hamilton's principle function is calculated as,\\n\\n) ( )\\n\\nt E E q D E q D E q S t t ) ( 1 2 1 2 2 1 1 1 2 1 0 1 2 + - + + + + = - - β α (38)\\n\\nAs a result the wave function can be written in the form ( )\\n\\n( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + + + + = - - - - t E q D E q D E q i P P t q D q D t t t t 1 1 2 1 0 1 2 1 1 1 0 1 2 1 2 exp 1 ) , , ( β α β α β α ψ h (39)\\n\\nTo identify the influence of the operators let us test the effect of the momenta\\n\\n⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = - - q D i P q D i P b t t a 1 1 ˆβ β α α h h (40)\\n\\nUsing the characteristic equations, it can be shown that\\n\\n1 2 , 1 2 ˆ2 1 2 + = + + = E P E q P β α (41) The result shown in (41 ) is the same classical solution. When applying the energy operator it will give the energy eigenvalues ( ) ( ) ψ ψ ψ ψ β α 2 2 2 2 1 1 2 1 1 2 1 q P P H --+ -= (42) ( ) ( ) ψ ψ ψ ψ β α β α 2 2 2 2 1 2 1 q P P P P -+ + -+ = ( ) ( ) ψ } 2 1 1 2 2 2 2 2 2 2 2 2 2 1 { 2 2 1 2 2 1 2 2 q E E q E E q E q -+ + + + -+ + + + + = Then we get\\n\\nψ ψ E H = (43)\\n\\nwhich is exactly the total energy as the case for the classical systems.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"V. Conclusions\",\n \"text\": \"We use the generating function \\\"S\\\" of the Hamilton-Jaccobi equation in its fractional form to be the phase factor of the wave function describing some potentials valid for the assumptions suggested by the WKB approximation The proof of our results arises from the new proposed concepts of the momentum and energy operators, that they give the same eigenvalues producing the ordinary results achieved by the classical approach.\\n\\nGiving the same eigenvalues that means this form of fractional operator also eigen, valid, and useful in effecting on a state functions.\"\n }\n]","isConversational":false}}},{"rowIdx":97,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0527"},"arxiv_id":{"kind":"string","value":"0704.0527"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevD.75.125023"},"content_hash":{"kind":"string","value":"edd179bddd702d53493d85c4e4f40b62a69e7e5d63dd63196e931a133fab844c"},"title":{"kind":"string","value":"Towards Skyrmion Stars: Large Baryon Configurations in the\n Einstein-Skyrme Model"},"abstract":{"kind":"string","value":"We investigate the large baryon number sector of the Einstein-Skyrme model as\na possible model for baryon stars. Gravitating hedgehog skyrmions have been\ninvestigated previously and the existence of stable solitonic stars excluded\ndue to energy considerations. However, in this paper we demonstrate that by\ngenerating gravitating skyrmions using rational maps, we can achieve\nmulti-baryon bound states whilst recovering spherical symmetry in the limit\nwhere B becomes large."},"authors":{"kind":"list like","value":["B.M.A.G. Piette and G.I. Probert"],"string":"[\n \"B.M.A.G. Piette and G.I. Probert\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["hep-th"],"string":"[\n \"hep-th\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"The Skyrme model, in its initial form, was proposed and developed by T.H.R. Skyrme in a series of papers as a non-linear field theory of pions [2] , [3] . Skyrme's initial idea was to think of baryons (in particular the nucleons) as secondary structures arising from a more fundamental mesonic fluid. The key property of the model was that the baryons arose as solitons in a topological manner and thus possessed a conserved topological charge identified with the baryon number.\n\nThe lowest energy stable solutions of the model are termed Skyrmions and can be thought of as baryonic solitons. The Skyrme model has been very successful in modelling the structures of various nuclei and has been shown by Witten et al. [4] to possess the general features of a low energy effective field theory for QCD. Some studies of the Skyrme model coupled to gravity have previously been undertaken [1], [5] , [6] , mainly with the motivation of a comparison of its features with those of other nonlinear field theories coupled to gravity. Of particular note is the Einstein-Yang-Mills theory, in which gravitationally bound configurations of non-abelian gauge fields are produced.\n\nOther reasons for studying the Einstein-Skyrme model are cosmological and astrophysical ones. Various authors have studied black hole formation in the model, with the conclusion that the so-called no-hair conjecture may not hold [7] , [8] .\n\nThe purpose of this paper is to study large baryon number Skyrmions or configurations of Skyrmions in the Einstein-Skyrme model. In particular, we wish to investigate if stable solitonic stars could exist within the model and to compare their properties to those of neutron stars. Preliminary studies of Skyrmion stars have predicted instability to single particle decay [1] . However this was done using the hedgehog ansatz for baryon number larger than 1 which is known to lead to unstable solutions even for the usual Skyrme model. Since then, it has been shown that the Skyrme model has stable shell-like solutions [9] which can be well approximated by the so called rational map ansatz [10] .\n\nIn this paper we use the rational map ansatz and its extension to multiple shells to construct configurations in the Gravitating Skyrme model that have a very large number of baryon. We show that those configurations, contrary to the hedgehog ansatz are bound even for very large baryon numbers.\n\nTo construct configurations that have a baryon number comparable to that of neutron star, we have to introduce a further approximation, which we call the ramp ansatz. We show 2 that this anstaz introduces further errors of only a few percent and we use it to compute very large Skyrmion configurations.\n\nThe paper is organised as follow: first we outline the Einstein-Skyrme model and discuss the main features of the results on static gravitating SU (2) hedgehogs obtained by Bizon and Chmaj [1] . We then use the rational map ansatz to construct shell like gravitating multibaryon configurations and show that for a fixed value of the coupling constant, the configurations exist only when the baryon number is below a certain critical value. Finally we introduce a ramp profile approximation to construct solutions with extremely high baryon numbers. We show how accurate it is and use it to construct Skyrmion stars configuration.\n\nThe action for gravitating Skyrmions is formed from the standard Skyrme action for the matter field and the Einstein-Hilbert action for the gravitational field.\n\nS = Z M √ -g \" L Sk - R 16πG « d 4 x. ( 1\n\n)\n\nHere L Sk is the Lagrangian density for the Skyrme model defined on the manifold M :\n\nL Sk = F 2 π 16\n\nT r(∇µU ∇ µ U -1 ) + 1 32e 2 T r[(∇µU )U -1 , (∇νU )U -1 ] 2 , (\n\n) 2\n\nwhere U belongs to SU (2). As we eventually wish to study baryon stars, we take a spherically symmetric metric, such as associated with the line element ds 2 = -A 2 (r) \" 1 -2m(r) r « dt 2 + \" 1 -2m(r) r\n\n« -1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), ( 3\n\n)\n\nwhere A(r) and m(r) are two profile functions that must be determined by solving the Einstein equations for the model. Our choice of ansatz is motivated by the fact that although in some cases we will be studying non-spherical Skyrmion configurations, the regime we are primarily interested in (i.e. Skyrmions of extremely high baryon number) will be shown to admit quasi spherical solutions. Also, for realistic values of the couplings, the gravitational interaction is small compared to the Skyrme interaction and thus the use of a spherical metric even with non-spherical configurations, is not a great problem.\n\nFrom (3), it can be shown that the Ricci scalar is\n\nR = -2 Ar 2 `-A ′′ r 2 -2A ′ r + 2A ′′ rm + A ′ m + 3A ′ rm ′ + Arm ′′ + 2Am ′ ´( 4\n\n)\n\nwhich, after integrating various terms by parts and noting that asymptotic flatness requires both A(r) and m(r) to take a constant value at spatial infinity, reduces the gravitational part of the action to Sgr = Z A(r) \" -m ′ (r) G « dr + m(∞) G . (5) For what follows, it will be convenient to scale to dimensionless variables by defining x = eFπr and µ(x) = eFπm(r)/2, resulting in one dimensionless coupling parameter for the model, α = πF 2 π G. We note that taking Fπ = 186M ev and G = 6.72 × 10 -45 M ev -2 , then the physical value of the coupling is α = 7.3 × 10 -40 .\n\nAs the Skyrme field is an SU (2) valued scalar field, at any given time one can think of it as a map from R 3 to the SU (2) manifold. Finite energy considerations impose that the field at spatial infinity should map to the same point on SU (2), say the identity. Thus, one can simply think of the Skyrme field as a map between three-spheres. All such maps fall into disjoint homotopy classes characterised by their winding number. This winding number is a conserved topological charge because no continuous deformation of the field and thus no time evolution, can allow transitions between homotopy classes. It is this topological charge that is interpreted as the baryon number.\n\nGravitating Skyrmions were first studied by Bizon and Chmaj[1] who analysed the properties of static spherically symmetric gravitating SU (2) skyrmions. Taking the Hedgehog Ansatz for the Skyrme field\n\nU = exp(i -→ σ .rF (r)) ( 6\n\n)\n\nsubject to the boundary conditions\n\nF (r = 0) = Bπ ( 7\n\n) F (r = ∞) = 0 ( 8\n\n)\n\nwhere B is the Baryon number associated with the Skyrmion configuration, they derived the Euler-Lagrange equation for the profiles F (r), (A(r) and m(r) and found that the model admit two branches of global solitonic solutions at each given baryon number, which annihilate at a critical value of the coupling parameter. Above αcrit no further solutions were found. In particular the value of the critical coupling decreased quite considerably with increasing baryon number as αcrit ≈ 0.040378/B 2 . It appears that the existence of a 4 critical coupling does not signal the collapse of a Skyrmion to form a black hole. In fact the metric factor S(x) = (1 -2µ(x) x ) is non-zero at αcrit; there simply ceases to be any stationary points of the action above the critical coupling.\n\nThe major problem with the ansatz (7) is that it leads to unstable solutions, i.e. for any given value of α, MADM (B = N ) > N MADM (B = 1). This is actually the case for the pure Skyrme model as well where the hedgehog anstaz (7) with B > 1 does not correspond to the lowest energy solution for the model. The solutions of the pure Skyrme model when B > 1 are known not to be spherically symmetric [11] but are stable i.e. E(B = N ) < N * E(B = 1).\n\nIt was actually shown by Houghton et al [10] , [12] that the multi-baryon solutions of the pure Skyrme model can be well approximated by the so called rational maps ansatz which is a generalisation of the hedgehog ansatz. While not radially symmetric, the ansatz separates its radial and angular dependence through a profile function and a rational map respectively.\n\nIn the following sections we will generalise the construction of Houghton et al to approximate the solution of the Einstein-Skyrme model.\n\nThe rational map ansatz introduced by Houghton et al.[10] works by decomposing the field into angular and radial parts. Using the polar coordinates in R 3 and defining the stereographic coordinates z = tan(θ/2) exp iφ the ansatz reads [10]\n\nU = exp (i σ • nRF (r, t)) ( 9\n\n) where nR = 1 1 + |R| 2 `2ℜ(R), 2ℑ(R), 1 -|R| 2 ´( 10\n\n)\n\nis a unit vector where R is a rational function of z.\n\nIt can be shown that the baryon number for Skyrmions constructed in this way, is equal to the degree of the rational map providing we take the boundary conditions\n\nF (r = 0) = π F (r = ∞) = 0. ( 11\n\n)\n\nSubstituting the ansatz (9) into the action for the model and scaling to dimensionless\n\n5\n\nvariables as earlier, we obtain the following reduced Hamiltonian\n\nH = 16πFπ e »Z ∞ 0 A(x) \" 1 2 S(x)F (x) ′2 x 2 + Bsin 2 F (x)(1 + S(x)F (x) ′2 ) + Isin 4 F (x) 2x 2 - µ ′ (x) α « dx + µ(∞) α - ( 12\n\n)\n\nwhere\n\nS(x) = 1 - 2µ(x) x ( 13\n\n)\n\nFrom which one obtains the following field equations\n\nµ(x) ′ = α \" 1 2 S(x)x 2 F (x) ′2 + B sin 2 F (x) + S(x)BF (x) ′2 sin 2 F (x) + I sin 4 F (x) 2x 2 « ( 14\n\n) F (x) ′′ = 1 S(x)V (x) » sin 2F (x) \" B + S(x)BF (x) ′2 + I sin 2 F (x) x 2 « - αS(x)F (x) ′3 V (x) 2 x -S(x) ′ F (x) ′ V (x) -S(x)F (x) ′ V (x) ′ ˜( 15\n\n) and A(x) ′ = αA(x)F (x) ′2 \" x + 2B sin 2 F (x) x « ( 16\n\n)\n\nwhere, for convenience, we have defined V (x) as\n\nV (x) = x 2 + 2B sin 2 F (x). ( 17\n\n)\n\nB is the baryon number and\n\nI = 1 4π Z \" 1 + |z| 2 1 + |R| 2 ˛dR dz ˛«4 2idzdz (1 + |z| 2 ) 2 ( 18\n\n)\n\nIts value depends on the chosen rational map R. To compute low energy configurations for a given baryon charge B one must find the rational map R or degree B that minimize I.\n\nThis has been done in [10] and [11] for several values of B. Moreover when b is large, one can use the approximation[11] I ≈ 1.28B 2 . The value of I so obtained is then used as a parameter and one can solve equations (14) -(16) for the radial profiles F (x), A(x) and µ(x).\n\nWe should point out here that for the pure Skyrme model the rational map ansatz produce very good approximation to the multi skyrmion solutions [10]: the energies are only 3 or 4 percent higher and the energy densities exhibit the same symmetries and differ by very little. All the solutions computed by Battye and Sutcliffe [11] , when B is not too small, have somehow the shape of a hollow shell. The baryon density is very small everywhere outside the shell, while on the shell itself, it forms a lattice of hexagons and pentagons.\n\nUsing the rational map ansatz, we will now solves the field equations (14) -(16) to compute some low action configurations. These solutions will correspond, initially, to a hollow shell of Skyrmions similar to the configuration obtained with the rational map anstaz for the pure Skyrme model. In the following sections we will show how our ansatz can be generalised to allow for more realistic configuration made out of embedded shells.\n\nThe first thing to note about our solutions is that we again obtain two branches of solutions at each baryon number (Fig. 1 ). Obtaining this same qualitative behaviour is not surprising when one considers that the B = 1 rational map Skyrmion reproduces the usual B = 1 hedgehog. However, the behaviour of the critical coupling itself is drastically altered for the rational map generated configurations. Namely, we observe that it decreases as approximately 0.040378/B 1 2 (Fig. 2 ). In particular this means that for a given value of the coupling, the rational map generated skyrmions can possess a much higher topological charge than their hedgehog counterparts, before there ceases to be any solutions. Quantitatively if B hedgehog is the maximum baryon number for which hedgehog solutions can be found at a given value of the coupling, then the highest baryon number rational map solution found at the same value of α will be approximately B 4 hedgehog . Again we observe that the metric function S(x) is non-zero at the critical coupling for all the solutions we have found and as such a horizon has not formed.\n\nIn Table 1 we present the radius, ADM mass per baryon and minimum value of the metric function, S(x), for configurations up to the maximum baryon number allowed at α = 1 × 10 -6 . These values were obtained by direct numerical solution of equations (14) -(16), where we have used the boundary data as specified in (11) .\n\nWe didn't didn't use the physical value of α (7.3×10 -40 ) because for this value, the ratio between the width of the shell and its radius is so small when we reach the maximum value of B that it becomes very difficult to solve the equation reliably. The value α = 1 × 10 -6 is small enough to allow for a shell with a large baryon number to exist but large enough to make it possible to compute these solution nears the critical value of B for a single shell configuration. 7 50 100 150 200 250 300 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 M ADM α Figure 1: Plot of the two branches of solutions found for B=2 configurations generated with the rational map ansatz.\n\nThe major difference between these configuration and the solutions of Bizon and Chmadj is that the rational map ansatz configurations become more bound when the baryon number increases This suggests the possibility that giant gravitating Skyrmions can be bound and consequently, that the Skyrme model can be used to study baryon stars.\n\nAnother interesting feature of the data is the observed change in the radius of the solutions with increasing baryon number. We note that the radius grows as approximately B 1 2 .\n\nHowever there are two main deviations from this. Firstly, the constant of proportionality relating the radius to the square root of the baryon number decreases slightly but persistently as we increase the baryon number, indicating the gravitational interaction becoming more important as the number of baryons increases.\n\nAs we approach the maximum baryon charge that can exist at α = 1 × 10 -6 , we also notice that the radius of the skyrmion actually decreases as we add more baryons. This shows that the gravitation pull plays a crucial role near the critical value of the skyrmion.\n\nThis is a tantalising property when one considers that generally a neutron star's radius must decrease for an increase in mass in order to achieve sufficient degenerate neutron pressure 8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 2 4 6 8 10 12 14 16 18 20 α crit B Figure 2: Plot of the decrease in α crit with increasing baryon number, for configurations generated with the rational map ansatz. +: α c r for the minimum value of I; curve: α c r = 0.0404B -1/2 .\n\nto support the star.\n\nTo motivate the further approximation that we will introduce in the next section, we now look at the profiles of the configuration that we have computed. First of all, we observe that the profile function F (x) stays approximately at its boundary value, π, for a finite radial distance before decreasing monotonically over some small region and finally attaining its second boundary value, 0. A similar behaviour is seen for both the mass field µ(x) and the metric field A(x) (see Fig. 3 ). Furthermore, as we increase the baryon number the structure becomes more pronounced, with the distance before the fields change (shell radius) increasing significantly, whilst the distance over which the fields change (shell width) settles to a constant size. We conclude that at large baryon numbers, those configurations correspond to hollow shells where the baryons are distributed on a tight lattice over the shell. As such the, structures are nearly spherical, validating our choice of radial metric.\n\nSuch structures immediately pose an interesting question. Can the gravitating Skyrmions exist as shells with more than one layer? To investigate this we note that it is possible to 9 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 1 0.8763 1.2315 1.0000 4 1.7728 1.1365 1.0000 8 2.5065 1.1180 1.0000 100 8.6829 1.0845 0.9999 500 19.3994 1.0827 0.9998 1 × 10 3 27.4314 1.0825 0.9997 1 × 10 4 86.7192 1.0821 0.9989 1 × 10 5 274.0397 1.0814 0.9963 1 × 10 6 864.6968 1.0792 0.9883 1 × 10 7 2715.0729 1.0722 0.9628 1 × 10 8 8377.4601 1.0500 0.88192 1 × 10 9 23585.5315 0.9743 0.6107 1.5 × 10 9 26860.2040 0.9463 0.5020 1.8 × 10 9 27470.2449 0.9302 0.4256 1.81 × 10 9 27456.5804 0.9296 0.4225 1.85 × 10 9 27357.9201 0.9274 0.4090 1.9 × 10 9 27078.6014 0.9246 0.3886 1.95 × 10 9 26126.5508 0.9217 0.3517 1.951 × 10 9 26050.7695 0.9217 0.3495 1.952 × 10 9 25937.4210 0.9216 0.3463 Table 1: Properties of the one shell low energy configuration for α = 1 × 10-6 10 0 0.5 1 1.5 2 2.5 3 3.5 4 1210 1215 1220 1225 1230 1235 F(x) x 0 2 4 6 8 10 1210 1215 1220 1225 1230 1235 µ(x) x 0.99 0.995 1 1.005 1.01 1210 1215 1220 1225 1230 1235 A(x) x Figure 3: Numerical solutions for the profiles F (x), µ(x) and A(x) when B = 2 × 10 6 and α = 1 × 10 -6 .\n\nmodify the boundary condition (11) to read\n\nF (r = 0) = N π ( 19\n\n) F (r = ∞) = 0 ( 20\n\n)\n\nwhilst still ensuring that the Skyrme field is well defined at the origin. This idea was first used in [12] to construct two shell configurations for the pure Skyrme model.\n\nThe baryon charge is now N times the degree of the rational map. Fig. 4 shows the structure of the solutions we find in this case when N = 2. They suggest that the Skyrmion now exists as a N -layered structure. This is exhibited in the form of the profile, mass and metric functions which interpolate between the boundary values in N distinct steps of equal 11 0 1 2 3 4 5 6 7 1200 1205 1210 1215 1220 1225 1230 F(x) x 0 5 10 15 20 1200 1205 1210 1215 1220 1225 1230 µ(x) x 0.995 1 1.005 1.01 1.015 1.02 1200 1205 1210 1215 1220 1225 1230 A(x) x Figure 4: Numerical solutions for for the profiles F (x), µ(x) and A(x) for 2 layers configurations (F (0) = 2π) when B = 2 × 10 6 and α = 1 × 10 -6 .\n\nsize stacked next to each other.\n\nWe can therefore think of this as a naive way of constructing a gravitating Skyrmion.\n\nInstead of using the boundary conditions as in (11) and a rational map of degree B we consider constructing the B-Skyrmion using a rational map of degree B/N (with the associated value of I) and the boundary condition (20). This is a crude construction as we are effectively considering N adjacent shells of baryons, all with the same baryon number. We might realistically expect that the baryon number per shell and distribution of shells may vary significantly for the minimum energy configuration. Nevertheless we shall study the properties of such structures. In fact, in the case where the baryon number is large and the number of shells is small, we expect this crude construction to be quite valid. That is, we do not expect the baryon number to change significantly over the few shells at large radius.\n\n12 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 4 1.2898 1.6179 1.0000 8 1.7858 1.4072 1.0000 100 6.1754 1.1363 0.9999 1 × 10 3 19.4157 1.0913 0.9996 1 × 10 4 61.3207 1.0833 0.9985 1 × 10 5 193.7006 1.0812 0.9949 1 × 10 6 610.6271 1.0779 0.9835 1 × 10 7 1911.3704 1.0680 0.9475 1 × 10 8 5825.2626 1.0362 0.8325 9.0 × 10 8 13736.9982 0.9302 0.4258 9.7 × 10 8 13263.0853 0.9224 0.3644 9.76 × 10 8 12998.2817 0.9217 0.3480 9.764 × 10 8 12931.5189 0.9216 0.3444 9.7647 × 10 8 12895.6984 0.9216 0.3425 9.76472 × 10 8 12891.4247 0.9216 0.3423 9.764724 × 10 8 12889.8645 0.9216 0.3422 Table 2: Table of properties of double layer solutions obtained numerically at α = 1 × 10-6\n\nFor the remainder of this section we will restrict ourselves to the case where N = 2.\n\nTable 2 summarises the properties of double layered gravitating skyrmions up to the maximum baryon charge allowed at α = 1 × 10 -6 . Briefly, we note the main features.\n\nFirstly, for all baryon numbers, the radius of the double layered solutions is significantly less than their single layered counterparts. This is not surprising as the baryon charge exists over a thicker region and so the mean radius can decrease with the baryon density remaining the same. Secondly, when B is large enough, i.e. when the double layer starts to make sense, the double layer solutions are energetically favourable when one compares the ADM mass with the single layer solutions. Finally we note that the maximum baryon number allowed (at the given coupling) is almost twice as much in the case of the single layer skyrmions.\n\nOf course the results of this section are not really the main regime of interest. We clearly 13 need to study configurations of extremely high baryon number (of order 10 58 ) relevant for baryon stars.\n\nWe will now discuss this high baryon number regime.\n\nUnfortunately, at very high baryon numbers, eqns. ((14) -(16)) become difficult to handle numerically. This is largely because the radius of the solutions becomes much larger than the distance over which the fields change. That is, we need to integrate over a region which is much less than 10 -16 radius, and so even double precision data types have insufficient precision.\n\nMoreover, single shell configurations are not physically relevant and multiple shells will only yield configuration that looks like a star if the number of layers is very large, typically well over 10 17 . With such a large number of layers we won't be able to solve the equation numerically as we will need at least 10 times as many sampling points for the profile functions.\n\nWe must thus resort to another level of approximation: approximate the profile functions by profiles that are piecewise linear. This is inspired by the work of Kopeliovich [13] [14] except that our ansatz has to be piecewise linear to be able to generate configurations with a huge number of layers. After defining the ansatz for an arbitrary number of layers, we will show that for a single layer configuration the ansatz produces configurations that are in good agreements with the rational map ansatz configuration. Then we will use the new ansatz to construct configurations that are made out of a very large no of layers.\n\nWe have shown, in the previous section, that one can construct shell like structures with very large Baryon numbers. At large baryon numbers, the Skyrmions resemble shell like structures. That is, the fields are constant nearly everywhere except in a small region corresponding to the shell. In that region, the profile look like linear functions smoothly linked to the constant parts at the edges (cfr. Fig 3 ). Motivated by this we approximate the fields by the ramp-functions\n\nF (x) = N π 2 -(x -x0) π W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 21\n\n) µ(x) = M 2 + (x -x0) M N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 22\n\n) A(x) = (1 + A0) 2 + (x -x0) (1 -A0) N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 23\n\n)\n\nIn the above there are four free parameters, namely the central radius x0 of the shell\n\nover which the fields change, the width of the shell W , the mass field at spatial infinity M and the value of the metric field at the origin A0 such that limx→∞ = 0. N is the number of layers we wish to study and, as such, is treated as an input parameter.\n\nThe picture is of a gravitating skyrmion with very high baryon number existing as N thin layers or shell of small thickness.\n\nThe above ansatz, allow us to find an approximation to the integrated energy. To do this we use the fact that the shell width is much smaller than the radius at large baryon numbers. In particular to evaluate the action integral we can approximate expressions of the type R G(x) sin p F (x) for any function G(x) that varies very little over the width of the shell by R G(x0) sin p F (x). We then use the fact that\n\nZ x 0 +NW/2 x 0 -NW/2 sin p F (x) = N π W Z π 0 sin p y dy. ( 24\n\n)\n\nThis leads to the following expression for the energy:\n\nE = - 16πFπ e » M α \" 1 + A0 2 « - π 2 W » \" 1 + A0 2 « \" x 2 0 -M x0 + W 2 12 - M W 6 « + \" 1 -A0 W « \" W 2 x0 6 - M W 2 12 - M W x0 6 « - + B \" 1 + A0 2 « 1 2 \" M π 2 W x0 -W - π 2 W « - 3IW 16x 2 0 \" 1 + A0 2 « - M α - ( 25\n\n)\n\nTo find the configurations which minimize this energy we first minimised it with respect to A0 and M algebraically in order to find an expression for the energy as a function of the width and radius only. Then we minimised this numerically using Mathematica. We will now discuss the features of these configurations.\n\nFirst of all, we must compare the results obtained with the ramp-profile when N = 1 and compare them to the result obtained with the full profile. Tables. 3 and 4 show the properties of solutions we obtained using the ramp-profile approximation, again at α = 1 × 10 -6 . All the general features of the full numerical solutions are reproduced. In particular, the approximate B 1 2 scaling and then decrease of the radius, the decreasing ADM mass and the differences between the double and single layer solutions are all exhibited by the data obtained using the ramp-profile approximation.\n\nQuantitatively though, there are some differences. The approximation allows a significant increase in the maximum allowed baryon charge. Also, the radius of configurations obtained using the approximation, tend to be smaller than those obtained numerically. If we concentrate on the baryon numbers greater than 10 5 so as to ensure our approximation, 15 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 8.3063 3.1286 1.1023 0.9999 500 18.6031 3.1386 1.1160 0.9997 1 × 10 3 26.313 3.1397 1.1195 0.9996 1 × 10 4 83.206 3.1396 1.1254 0.9987 1 × 10 5 262.94 3.1357 1.1266 0.9960 1 × 10 6 829.60 3.1230 1.1251 0.9872 1 × 10 7 2604.2 3.0825 1.1186 0.9595 1 × 10 8 8032.8 2.9512 1.0972 0.8713 1 × 10 9 22899 2.4837 1.0272 0.5772 2 × 10 9 29121 2.1092 0.9818 0.3645 2.8 × 10 9 29098 1.6623 0.9505 0.1380 2.83 × 10 9 28514 1.6066 0.9495 0.1119 2.839 × 10 9 28024 1.5671 0.94922 0.09373 2.8397 × 10 9 27869.3 1.5556 0.94924 0.08845 2.83975 × 10 9 27869.8 1.5524 0.94925 0.08699 2.839752 × 10 9 27822 1.5521 0.94925 0.08687 Table 3: Table of properties of the single layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6 16 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 5.7924 3.0428 1.0692 0.9999 1 × 10 3 18.5788 3.1305 1.1047 0.9995 1 × 10 4 58.8202 3.1380 1.1201 0.9983 1 × 10 5 185.8420 3.1332 1.1246 0.9944 1 × 10 6 585.7950 3.1153 1.1233 0.9820 1 × 10 7 1833.0500 3.0578 1.1143 0.9428 1 × 10 8 5587.3600 2.8688 1.0840 0.8172 9 × 10 8 14147.1782 2.1859 0.9900 0.4065 9.764724 × 10 8 14472.3851 2.1276 0.9837 0.3746 1 × 10 9 14560.5000 2.1092 0.9818 0.3646 1.4 × 10 9 14549.0000 1.6623 0.9505 0.1381 1.41963 × 10 9 13994.0523 1.5644 0.9492 0.0926 1.419635134 × 10 9 13993.2000 1.5643 0.9492 0.0925 Table 4: Table of properties of the double layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6\n\nthat the width is much smaller than the radius, is valid, then at worst we find a discrepancy in the ADM mass of 11% and in the radius of 7%.\n\nIn general then, the data seems to confirm the reliability the ramp-profile approximation. In fact the approach will be even more reliable at the extremely high values of the baryon number that we are interested in. This is because the radius of solutions is of orders of magnitudes greater than the width in such a regime, consistent with the approximations we have made.\n\nMoreover, whilst searching for minima of the energy does not allow us to probe both branches of solutions, it does allow us to locate the value of αcrit. We again obtain the approximate trend αcrit ∝ B -1 2 , for large B. Now in order to say anything about the possibility of baryon stars in the Skyrme model we need to be able to verify that the decrease in the ADM mass per baryon we observed at low and moderate baryon numbers, extends to baryon numbers of order 10 58 for α = 7.3 × 10 -40 .\n\nTable. 5 summarizes our solutions in such a regime. Firstly we consider constructing a single layer self-gravitating Skyrmion with these values. We do indeed see that the configuration is bound. This is verified by checking that the ADM mass is lower (even at this significantly lower value of α) than for the B = 1 hedgehog. So the possibility of baryon stars in the Einstein-Skyrme model cannot be ruled out on the grounds of energy.\n\nThe Skyrmion exists as a giant thin shell, and the large baryon charge is distributed as a tight lattice over this. However a hollow shell is clearly not a realistic construction for a neutron star. This fact manifests itself in the extremely high radius of the configuration.\n\nTransferring to standard units, the single layer B = 10 58 gravitating Skyrmion has a radius of 2.42 × 10 10 km ! To address this issue, we can use a large number of layered Skyrmions as discussed earlier. This has several benefits. Firstly, as we are distributing the baryon number through a larger volume, then at a given baryon density the necessary radius can decrease. Similar to what we see in the double layer results. On top of this, we expect the radius to decrease further due to extra gravitational compression, as the outer layers of the Skyrmions feel the attraction of inner layers. Finally, the many layer approach is also a more realistic construction of a solid baryon star.\n\nThe results for using more and more layers in the construction (for fixed B and α), are also presented in Fig. 5 . We note that not only does the radius decrease significantly, but\n\nthe added gravitational binding further improves the energies of the configurations, reflected in the low ADM masses obtained. There appears to be a critical number of layers that can be used before there ceases to be any solutions and although the value of Smin is close zero at this point, the star still has not collapsed to form a black hole. Finally, we note that the radius of the Skyrmion at the critical number of layers is approximately 20.91km. This is comparable to a real neutron star, with a typical radius of 10km.\n\nWe reemphasise here that our approach to embedding shells of baryons is quite crude.\n\nFor few shells and large baryon number, we might reasonably believe that baryon number does not chance significantly from one shell to the next. However, when we embed many shells we should really consider that the baryon number of the inner most shells would likely be significantly less than the that of the outer shells. Nevertheless, our naive embedding has produced some interesting properties. In a future work we hope to improve our multi-layer construction to obtain a more realistic description of a baryon star. 19 N Shell R( 2 Fπe ) W M ADM /(6π 2 B) S min 1 × 10 2 8.3236 × 10 27 3.1416 1.1285 1.0000 1 × 10 3 2.6321 × 10 27 3.1416 1.1285 1.0000 1 × 10 4 × 10 26 3.1416 1.1285 1.0000 1 × 10 5 2.6321 × 10 26 3.1416 1.1285 1.0000 1 × 10 6 8.3236 × 10 25 3.1416 1.1285 1.0000 1 × 10 7 2.6321 × 10 25 3.1416 1.1285 1.0000 1 × 10 8 8.3236 × 10 24 3.1416 1.1285 1.0000 1 × 10 9 2.6321 × 10 24 3.1415 1.1285 1.0000 1 × 10 10 8.3234 × 10 23 3.1415 1.1285 0.9999 1 × 10 11 2.6319 × 10 23 3.1412 1.1285 0.9997 1 × 10 12 8.3216 × 10 22 3.1402 1.1283 0.9991 1 × 10 13 2.6301 × 10 22 3.1373 1.1278 0.9971 1 × 10 14 8.3034 × 10 21 3.1280 1.1263 0.9907 1 × 10 15 2.6118 × 10 21 3.0986 1.1213 0.9705 1 × 10 16 8.1147 × 10 20 3.0037 1.1057 0.9063 1 × 10 17 2.4001 × 10 20 2.6810 1.0552 0.6977 5 × 10 17 8.2066 × 10 19 1.7888 0.9552 0.2036 5.3 × 10 17 7.4172 × 10 19 1.6227 0.9491 0.1247 5.33 × 10 17 7.1871 × 10 19 1.5625 0.94866 0.0971 5.3306 × 10 17 7.1597 × 10 19 1.5549 0.94868 0.0936 5.33065 × 10 17 7.1525 × 10 19 1.5528 0.948694 0.0927 5.330657 × 10 17 7.1506 × 10 19 1.5523 0.948692 0.0924 Table 5: Table of properties of the step ansatz configurations for varying the number of embedded shells at fixed B = 10 58 and α = 7.3 × 10 -40 .\n\nPrevious work on the Einstein-Skyrme model highlighted a considerable problem with using the Skyrmions as a model for baryon stars. Namely, multibaryon hedgehog Skyrmions were simply not energetically favourable states. We have shown that this is simply a consequence of a poor ansatz for the true Skyrmion and, having used the more appropriate rational map ansatz, we have generated energetically favourable configurations of multibaryons.\n\nWe also observe the interesting property that near the critical coupling, the Skyrmions can decrease in radius as we add more baryons. This hints towards the similar behaviour exhibited by real neutron stars.\n\nAlthough the rational map ansatz does not have an exact radial symmetry, at large scale it does. The anisotropy only appears at the nucleon scale.\n\nFinally, since we started with the motivation of studying baryon stars within the Skyrme model, it is interesting to compare the features of our configurations with those of neutron stars. For realistic values, B = 10 58 and α = 7.3 × 10 -40 we find a minimal energy single layer configuration with radius=2.42 × 10 10 km. This is clearly too large for a neutron star (which is of order 10km. in radius). This is to be expected however due to the shell model we have taken. Firstly, as we are distributing the baryons over the surface area rather than throughout the volume of the star we naturally must require a much larger star for a given baryon number. This effect is two-fold in that if we were distributing the baryons throughout the volume, outer layers would feel the attraction of inner layers and enhanced radial compression would occur. The loss of such an effect is pronounced when we are considering realistically small values of the coupling.\n\nIt seems therefore that the way to construct baryon stars in the Skyrme model is to consider embedding shells of baryons within shells. This gives rise to more appropriate specifications for the star and is also more realistic. We do indeed observe such improvements for a many layered configuration. In fact the radius of B = 10 58 gravitating Skyrmion (at realistic α), can be decreased in this manner to approximately 20.91km.\n\nWe note however that this approach to shell embedding has only be done naively thus far.\n\nWe have only considered the case where the baryon number is equal for each shell. We really should allow the baryon number(and hence the rational map quantities) to vary over the shells. One approach towards this would be to assume that the baryon density is a constant over the shells. An even better approach would be to allow this to be a smoothly varying function that must be determined by minimising the energy. This will give a more realistic 21 description of baryon stars within the Einstein-Skyrme model, as traditional descriptions of neutron stars also involve many strata, of differing neutron density. We are currently investigating such configurations.\n\n[13] V. B. Kopeliovich \"The Bubbles of Matter from MultiSkyrmions\" JETP Lett. 73 (2001), 587-591; Pisma Zh.Eksp.Teor.Fiz. 73 (2001), 667-671 [14] V. B. Kopeliovich \"MultiSkyrmions and Baryonic Bags\" J.Phys. G28 (2002), 103-120 23\n\nGIP is supported by a PPARC studentship."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"Introduction","text":"The Skyrme model, in its initial form, was proposed and developed by T.H.R. Skyrme in a series of papers as a non-linear field theory of pions [2] , [3] . Skyrme's initial idea was to think of baryons (in particular the nucleons) as secondary structures arising from a more fundamental mesonic fluid. The key property of the model was that the baryons arose as solitons in a topological manner and thus possessed a conserved topological charge identified with the baryon number.\n\nThe lowest energy stable solutions of the model are termed Skyrmions and can be thought of as baryonic solitons. The Skyrme model has been very successful in modelling the structures of various nuclei and has been shown by Witten et al. [4] to possess the general features of a low energy effective field theory for QCD. Some studies of the Skyrme model coupled to gravity have previously been undertaken [1], [5] , [6] , mainly with the motivation of a comparison of its features with those of other nonlinear field theories coupled to gravity. Of particular note is the Einstein-Yang-Mills theory, in which gravitationally bound configurations of non-abelian gauge fields are produced.\n\nOther reasons for studying the Einstein-Skyrme model are cosmological and astrophysical ones. Various authors have studied black hole formation in the model, with the conclusion that the so-called no-hair conjecture may not hold [7] , [8] .\n\nThe purpose of this paper is to study large baryon number Skyrmions or configurations of Skyrmions in the Einstein-Skyrme model. In particular, we wish to investigate if stable solitonic stars could exist within the model and to compare their properties to those of neutron stars. Preliminary studies of Skyrmion stars have predicted instability to single particle decay [1] . However this was done using the hedgehog ansatz for baryon number larger than 1 which is known to lead to unstable solutions even for the usual Skyrme model. Since then, it has been shown that the Skyrme model has stable shell-like solutions [9] which can be well approximated by the so called rational map ansatz [10] .\n\nIn this paper we use the rational map ansatz and its extension to multiple shells to construct configurations in the Gravitating Skyrme model that have a very large number of baryon. We show that those configurations, contrary to the hedgehog ansatz are bound even for very large baryon numbers.\n\nTo construct configurations that have a baryon number comparable to that of neutron star, we have to introduce a further approximation, which we call the ramp ansatz. We show 2 that this anstaz introduces further errors of only a few percent and we use it to compute very large Skyrmion configurations.\n\nThe paper is organised as follow: first we outline the Einstein-Skyrme model and discuss the main features of the results on static gravitating SU (2) hedgehogs obtained by Bizon and Chmaj [1] . We then use the rational map ansatz to construct shell like gravitating multibaryon configurations and show that for a fixed value of the coupling constant, the configurations exist only when the baryon number is below a certain critical value. Finally we introduce a ramp profile approximation to construct solutions with extremely high baryon numbers. We show how accurate it is and use it to construct Skyrmion stars configuration."},{"section_type":"OTHER","section_title":"The Einstein-Skyrme Model","text":"The action for gravitating Skyrmions is formed from the standard Skyrme action for the matter field and the Einstein-Hilbert action for the gravitational field.\n\nS = Z M √ -g \" L Sk - R 16πG « d 4 x. ( 1\n\n)\n\nHere L Sk is the Lagrangian density for the Skyrme model defined on the manifold M :\n\nL Sk = F 2 π 16\n\nT r(∇µU ∇ µ U -1 ) + 1 32e 2 T r[(∇µU )U -1 , (∇νU )U -1 ] 2 , (\n\n) 2\n\nwhere U belongs to SU (2). As we eventually wish to study baryon stars, we take a spherically symmetric metric, such as associated with the line element ds 2 = -A 2 (r) \" 1 -2m(r) r « dt 2 + \" 1 -2m(r) r\n\n« -1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), ( 3\n\n)\n\nwhere A(r) and m(r) are two profile functions that must be determined by solving the Einstein equations for the model. Our choice of ansatz is motivated by the fact that although in some cases we will be studying non-spherical Skyrmion configurations, the regime we are primarily interested in (i.e. Skyrmions of extremely high baryon number) will be shown to admit quasi spherical solutions. Also, for realistic values of the couplings, the gravitational interaction is small compared to the Skyrme interaction and thus the use of a spherical metric even with non-spherical configurations, is not a great problem.\n\nFrom (3), it can be shown that the Ricci scalar is\n\nR = -2 Ar 2 `-A ′′ r 2 -2A ′ r + 2A ′′ rm + A ′ m + 3A ′ rm ′ + Arm ′′ + 2Am ′ ´( 4\n\n)\n\nwhich, after integrating various terms by parts and noting that asymptotic flatness requires both A(r) and m(r) to take a constant value at spatial infinity, reduces the gravitational part of the action to Sgr = Z A(r) \" -m ′ (r) G « dr + m(∞) G . (5) For what follows, it will be convenient to scale to dimensionless variables by defining x = eFπr and µ(x) = eFπm(r)/2, resulting in one dimensionless coupling parameter for the model, α = πF 2 π G. We note that taking Fπ = 186M ev and G = 6.72 × 10 -45 M ev -2 , then the physical value of the coupling is α = 7.3 × 10 -40 .\n\nAs the Skyrme field is an SU (2) valued scalar field, at any given time one can think of it as a map from R 3 to the SU (2) manifold. Finite energy considerations impose that the field at spatial infinity should map to the same point on SU (2), say the identity. Thus, one can simply think of the Skyrme field as a map between three-spheres. All such maps fall into disjoint homotopy classes characterised by their winding number. This winding number is a conserved topological charge because no continuous deformation of the field and thus no time evolution, can allow transitions between homotopy classes. It is this topological charge that is interpreted as the baryon number."},{"section_type":"OTHER","section_title":"Gravitating Hedgehog Skyrmions","text":"Gravitating Skyrmions were first studied by Bizon and Chmaj[1] who analysed the properties of static spherically symmetric gravitating SU (2) skyrmions. Taking the Hedgehog Ansatz for the Skyrme field\n\nU = exp(i -→ σ .rF (r)) ( 6\n\n)\n\nsubject to the boundary conditions\n\nF (r = 0) = Bπ ( 7\n\n) F (r = ∞) = 0 ( 8\n\n)\n\nwhere B is the Baryon number associated with the Skyrmion configuration, they derived the Euler-Lagrange equation for the profiles F (r), (A(r) and m(r) and found that the model admit two branches of global solitonic solutions at each given baryon number, which annihilate at a critical value of the coupling parameter. Above αcrit no further solutions were found. In particular the value of the critical coupling decreased quite considerably with increasing baryon number as αcrit ≈ 0.040378/B 2 . It appears that the existence of a 4 critical coupling does not signal the collapse of a Skyrmion to form a black hole. In fact the metric factor S(x) = (1 -2µ(x) x ) is non-zero at αcrit; there simply ceases to be any stationary points of the action above the critical coupling.\n\nThe major problem with the ansatz (7) is that it leads to unstable solutions, i.e. for any given value of α, MADM (B = N ) > N MADM (B = 1). This is actually the case for the pure Skyrme model as well where the hedgehog anstaz (7) with B > 1 does not correspond to the lowest energy solution for the model. The solutions of the pure Skyrme model when B > 1 are known not to be spherically symmetric [11] but are stable i.e. E(B = N ) < N * E(B = 1).\n\nIt was actually shown by Houghton et al [10] , [12] that the multi-baryon solutions of the pure Skyrme model can be well approximated by the so called rational maps ansatz which is a generalisation of the hedgehog ansatz. While not radially symmetric, the ansatz separates its radial and angular dependence through a profile function and a rational map respectively.\n\nIn the following sections we will generalise the construction of Houghton et al to approximate the solution of the Einstein-Skyrme model."},{"section_type":"OTHER","section_title":"The Rational Map Ansatz","text":"The rational map ansatz introduced by Houghton et al.[10] works by decomposing the field into angular and radial parts. Using the polar coordinates in R 3 and defining the stereographic coordinates z = tan(θ/2) exp iφ the ansatz reads [10]\n\nU = exp (i σ • nRF (r, t)) ( 9\n\n) where nR = 1 1 + |R| 2 `2ℜ(R), 2ℑ(R), 1 -|R| 2 ´( 10\n\n)\n\nis a unit vector where R is a rational function of z.\n\nIt can be shown that the baryon number for Skyrmions constructed in this way, is equal to the degree of the rational map providing we take the boundary conditions\n\nF (r = 0) = π F (r = ∞) = 0. ( 11\n\n)\n\nSubstituting the ansatz (9) into the action for the model and scaling to dimensionless\n\n5\n\nvariables as earlier, we obtain the following reduced Hamiltonian\n\nH = 16πFπ e »Z ∞ 0 A(x) \" 1 2 S(x)F (x) ′2 x 2 + Bsin 2 F (x)(1 + S(x)F (x) ′2 ) + Isin 4 F (x) 2x 2 - µ ′ (x) α « dx + µ(∞) α - ( 12\n\n)\n\nwhere\n\nS(x) = 1 - 2µ(x) x ( 13\n\n)\n\nFrom which one obtains the following field equations\n\nµ(x) ′ = α \" 1 2 S(x)x 2 F (x) ′2 + B sin 2 F (x) + S(x)BF (x) ′2 sin 2 F (x) + I sin 4 F (x) 2x 2 « ( 14\n\n) F (x) ′′ = 1 S(x)V (x) » sin 2F (x) \" B + S(x)BF (x) ′2 + I sin 2 F (x) x 2 « - αS(x)F (x) ′3 V (x) 2 x -S(x) ′ F (x) ′ V (x) -S(x)F (x) ′ V (x) ′ ˜( 15\n\n) and A(x) ′ = αA(x)F (x) ′2 \" x + 2B sin 2 F (x) x « ( 16\n\n)\n\nwhere, for convenience, we have defined V (x) as\n\nV (x) = x 2 + 2B sin 2 F (x). ( 17\n\n)\n\nB is the baryon number and\n\nI = 1 4π Z \" 1 + |z| 2 1 + |R| 2 ˛dR dz ˛«4 2idzdz (1 + |z| 2 ) 2 ( 18\n\n)\n\nIts value depends on the chosen rational map R. To compute low energy configurations for a given baryon charge B one must find the rational map R or degree B that minimize I.\n\nThis has been done in [10] and [11] for several values of B. Moreover when b is large, one can use the approximation[11] I ≈ 1.28B 2 . The value of I so obtained is then used as a parameter and one can solve equations (14) -(16) for the radial profiles F (x), A(x) and µ(x).\n\nWe should point out here that for the pure Skyrme model the rational map ansatz produce very good approximation to the multi skyrmion solutions [10]: the energies are only 3 or 4 percent higher and the energy densities exhibit the same symmetries and differ by very little. All the solutions computed by Battye and Sutcliffe [11] , when B is not too small, have somehow the shape of a hollow shell. The baryon density is very small everywhere outside the shell, while on the shell itself, it forms a lattice of hexagons and pentagons."},{"section_type":"OTHER","section_title":"5 Hollow Skyrmion Shells","text":"Using the rational map ansatz, we will now solves the field equations (14) -(16) to compute some low action configurations. These solutions will correspond, initially, to a hollow shell of Skyrmions similar to the configuration obtained with the rational map anstaz for the pure Skyrme model. In the following sections we will show how our ansatz can be generalised to allow for more realistic configuration made out of embedded shells.\n\nThe first thing to note about our solutions is that we again obtain two branches of solutions at each baryon number (Fig. 1 ). Obtaining this same qualitative behaviour is not surprising when one considers that the B = 1 rational map Skyrmion reproduces the usual B = 1 hedgehog. However, the behaviour of the critical coupling itself is drastically altered for the rational map generated configurations. Namely, we observe that it decreases as approximately 0.040378/B 1 2 (Fig. 2 ). In particular this means that for a given value of the coupling, the rational map generated skyrmions can possess a much higher topological charge than their hedgehog counterparts, before there ceases to be any solutions. Quantitatively if B hedgehog is the maximum baryon number for which hedgehog solutions can be found at a given value of the coupling, then the highest baryon number rational map solution found at the same value of α will be approximately B 4 hedgehog . Again we observe that the metric function S(x) is non-zero at the critical coupling for all the solutions we have found and as such a horizon has not formed.\n\nIn Table 1 we present the radius, ADM mass per baryon and minimum value of the metric function, S(x), for configurations up to the maximum baryon number allowed at α = 1 × 10 -6 . These values were obtained by direct numerical solution of equations (14) -(16), where we have used the boundary data as specified in (11) .\n\nWe didn't didn't use the physical value of α (7.3×10 -40 ) because for this value, the ratio between the width of the shell and its radius is so small when we reach the maximum value of B that it becomes very difficult to solve the equation reliably. The value α = 1 × 10 -6 is small enough to allow for a shell with a large baryon number to exist but large enough to make it possible to compute these solution nears the critical value of B for a single shell configuration. 7 50 100 150 200 250 300 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 M ADM α Figure 1: Plot of the two branches of solutions found for B=2 configurations generated with the rational map ansatz.\n\nThe major difference between these configuration and the solutions of Bizon and Chmadj is that the rational map ansatz configurations become more bound when the baryon number increases This suggests the possibility that giant gravitating Skyrmions can be bound and consequently, that the Skyrme model can be used to study baryon stars.\n\nAnother interesting feature of the data is the observed change in the radius of the solutions with increasing baryon number. We note that the radius grows as approximately B 1 2 .\n\nHowever there are two main deviations from this. Firstly, the constant of proportionality relating the radius to the square root of the baryon number decreases slightly but persistently as we increase the baryon number, indicating the gravitational interaction becoming more important as the number of baryons increases.\n\nAs we approach the maximum baryon charge that can exist at α = 1 × 10 -6 , we also notice that the radius of the skyrmion actually decreases as we add more baryons. This shows that the gravitation pull plays a crucial role near the critical value of the skyrmion.\n\nThis is a tantalising property when one considers that generally a neutron star's radius must decrease for an increase in mass in order to achieve sufficient degenerate neutron pressure 8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 2 4 6 8 10 12 14 16 18 20 α crit B Figure 2: Plot of the decrease in α crit with increasing baryon number, for configurations generated with the rational map ansatz. +: α c r for the minimum value of I; curve: α c r = 0.0404B -1/2 .\n\nto support the star.\n\nTo motivate the further approximation that we will introduce in the next section, we now look at the profiles of the configuration that we have computed. First of all, we observe that the profile function F (x) stays approximately at its boundary value, π, for a finite radial distance before decreasing monotonically over some small region and finally attaining its second boundary value, 0. A similar behaviour is seen for both the mass field µ(x) and the metric field A(x) (see Fig. 3 ). Furthermore, as we increase the baryon number the structure becomes more pronounced, with the distance before the fields change (shell radius) increasing significantly, whilst the distance over which the fields change (shell width) settles to a constant size. We conclude that at large baryon numbers, those configurations correspond to hollow shells where the baryons are distributed on a tight lattice over the shell. As such the, structures are nearly spherical, validating our choice of radial metric.\n\nSuch structures immediately pose an interesting question. Can the gravitating Skyrmions exist as shells with more than one layer? To investigate this we note that it is possible to 9 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 1 0.8763 1.2315 1.0000 4 1.7728 1.1365 1.0000 8 2.5065 1.1180 1.0000 100 8.6829 1.0845 0.9999 500 19.3994 1.0827 0.9998 1 × 10 3 27.4314 1.0825 0.9997 1 × 10 4 86.7192 1.0821 0.9989 1 × 10 5 274.0397 1.0814 0.9963 1 × 10 6 864.6968 1.0792 0.9883 1 × 10 7 2715.0729 1.0722 0.9628 1 × 10 8 8377.4601 1.0500 0.88192 1 × 10 9 23585.5315 0.9743 0.6107 1.5 × 10 9 26860.2040 0.9463 0.5020 1.8 × 10 9 27470.2449 0.9302 0.4256 1.81 × 10 9 27456.5804 0.9296 0.4225 1.85 × 10 9 27357.9201 0.9274 0.4090 1.9 × 10 9 27078.6014 0.9246 0.3886 1.95 × 10 9 26126.5508 0.9217 0.3517 1.951 × 10 9 26050.7695 0.9217 0.3495 1.952 × 10 9 25937.4210 0.9216 0.3463 Table 1: Properties of the one shell low energy configuration for α = 1 × 10-6 10 0 0.5 1 1.5 2 2.5 3 3.5 4 1210 1215 1220 1225 1230 1235 F(x) x 0 2 4 6 8 10 1210 1215 1220 1225 1230 1235 µ(x) x 0.99 0.995 1 1.005 1.01 1210 1215 1220 1225 1230 1235 A(x) x Figure 3: Numerical solutions for the profiles F (x), µ(x) and A(x) when B = 2 × 10 6 and α = 1 × 10 -6 .\n\nmodify the boundary condition (11) to read\n\nF (r = 0) = N π ( 19\n\n) F (r = ∞) = 0 ( 20\n\n)\n\nwhilst still ensuring that the Skyrme field is well defined at the origin. This idea was first used in [12] to construct two shell configurations for the pure Skyrme model.\n\nThe baryon charge is now N times the degree of the rational map. Fig. 4 shows the structure of the solutions we find in this case when N = 2. They suggest that the Skyrmion now exists as a N -layered structure. This is exhibited in the form of the profile, mass and metric functions which interpolate between the boundary values in N distinct steps of equal 11 0 1 2 3 4 5 6 7 1200 1205 1210 1215 1220 1225 1230 F(x) x 0 5 10 15 20 1200 1205 1210 1215 1220 1225 1230 µ(x) x 0.995 1 1.005 1.01 1.015 1.02 1200 1205 1210 1215 1220 1225 1230 A(x) x Figure 4: Numerical solutions for for the profiles F (x), µ(x) and A(x) for 2 layers configurations (F (0) = 2π) when B = 2 × 10 6 and α = 1 × 10 -6 .\n\nsize stacked next to each other.\n\nWe can therefore think of this as a naive way of constructing a gravitating Skyrmion.\n\nInstead of using the boundary conditions as in (11) and a rational map of degree B we consider constructing the B-Skyrmion using a rational map of degree B/N (with the associated value of I) and the boundary condition (20). This is a crude construction as we are effectively considering N adjacent shells of baryons, all with the same baryon number. We might realistically expect that the baryon number per shell and distribution of shells may vary significantly for the minimum energy configuration. Nevertheless we shall study the properties of such structures. In fact, in the case where the baryon number is large and the number of shells is small, we expect this crude construction to be quite valid. That is, we do not expect the baryon number to change significantly over the few shells at large radius.\n\n12 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 4 1.2898 1.6179 1.0000 8 1.7858 1.4072 1.0000 100 6.1754 1.1363 0.9999 1 × 10 3 19.4157 1.0913 0.9996 1 × 10 4 61.3207 1.0833 0.9985 1 × 10 5 193.7006 1.0812 0.9949 1 × 10 6 610.6271 1.0779 0.9835 1 × 10 7 1911.3704 1.0680 0.9475 1 × 10 8 5825.2626 1.0362 0.8325 9.0 × 10 8 13736.9982 0.9302 0.4258 9.7 × 10 8 13263.0853 0.9224 0.3644 9.76 × 10 8 12998.2817 0.9217 0.3480 9.764 × 10 8 12931.5189 0.9216 0.3444 9.7647 × 10 8 12895.6984 0.9216 0.3425 9.76472 × 10 8 12891.4247 0.9216 0.3423 9.764724 × 10 8 12889.8645 0.9216 0.3422 Table 2: Table of properties of double layer solutions obtained numerically at α = 1 × 10-6\n\nFor the remainder of this section we will restrict ourselves to the case where N = 2.\n\nTable 2 summarises the properties of double layered gravitating skyrmions up to the maximum baryon charge allowed at α = 1 × 10 -6 . Briefly, we note the main features.\n\nFirstly, for all baryon numbers, the radius of the double layered solutions is significantly less than their single layered counterparts. This is not surprising as the baryon charge exists over a thicker region and so the mean radius can decrease with the baryon density remaining the same. Secondly, when B is large enough, i.e. when the double layer starts to make sense, the double layer solutions are energetically favourable when one compares the ADM mass with the single layer solutions. Finally we note that the maximum baryon number allowed (at the given coupling) is almost twice as much in the case of the single layer skyrmions.\n\nOf course the results of this section are not really the main regime of interest. We clearly 13 need to study configurations of extremely high baryon number (of order 10 58 ) relevant for baryon stars.\n\nWe will now discuss this high baryon number regime."},{"section_type":"OTHER","section_title":"The Ramp-profile Approximation","text":"Unfortunately, at very high baryon numbers, eqns. ((14) -(16)) become difficult to handle numerically. This is largely because the radius of the solutions becomes much larger than the distance over which the fields change. That is, we need to integrate over a region which is much less than 10 -16 radius, and so even double precision data types have insufficient precision.\n\nMoreover, single shell configurations are not physically relevant and multiple shells will only yield configuration that looks like a star if the number of layers is very large, typically well over 10 17 . With such a large number of layers we won't be able to solve the equation numerically as we will need at least 10 times as many sampling points for the profile functions.\n\nWe must thus resort to another level of approximation: approximate the profile functions by profiles that are piecewise linear. This is inspired by the work of Kopeliovich [13] [14] except that our ansatz has to be piecewise linear to be able to generate configurations with a huge number of layers. After defining the ansatz for an arbitrary number of layers, we will show that for a single layer configuration the ansatz produces configurations that are in good agreements with the rational map ansatz configuration. Then we will use the new ansatz to construct configurations that are made out of a very large no of layers.\n\nWe have shown, in the previous section, that one can construct shell like structures with very large Baryon numbers. At large baryon numbers, the Skyrmions resemble shell like structures. That is, the fields are constant nearly everywhere except in a small region corresponding to the shell. In that region, the profile look like linear functions smoothly linked to the constant parts at the edges (cfr. Fig 3 ). Motivated by this we approximate the fields by the ramp-functions\n\nF (x) = N π 2 -(x -x0) π W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 21\n\n) µ(x) = M 2 + (x -x0) M N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 22\n\n) A(x) = (1 + A0) 2 + (x -x0) (1 -A0) N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 23\n\n)\n\nIn the above there are four free parameters, namely the central radius x0 of the shell\n\nover which the fields change, the width of the shell W , the mass field at spatial infinity M and the value of the metric field at the origin A0 such that limx→∞ = 0. N is the number of layers we wish to study and, as such, is treated as an input parameter.\n\nThe picture is of a gravitating skyrmion with very high baryon number existing as N thin layers or shell of small thickness.\n\nThe above ansatz, allow us to find an approximation to the integrated energy. To do this we use the fact that the shell width is much smaller than the radius at large baryon numbers. In particular to evaluate the action integral we can approximate expressions of the type R G(x) sin p F (x) for any function G(x) that varies very little over the width of the shell by R G(x0) sin p F (x). We then use the fact that\n\nZ x 0 +NW/2 x 0 -NW/2 sin p F (x) = N π W Z π 0 sin p y dy. ( 24\n\n)\n\nThis leads to the following expression for the energy:\n\nE = - 16πFπ e » M α \" 1 + A0 2 « - π 2 W » \" 1 + A0 2 « \" x 2 0 -M x0 + W 2 12 - M W 6 « + \" 1 -A0 W « \" W 2 x0 6 - M W 2 12 - M W x0 6 « - + B \" 1 + A0 2 « 1 2 \" M π 2 W x0 -W - π 2 W « - 3IW 16x 2 0 \" 1 + A0 2 « - M α - ( 25\n\n)\n\nTo find the configurations which minimize this energy we first minimised it with respect to A0 and M algebraically in order to find an expression for the energy as a function of the width and radius only. Then we minimised this numerically using Mathematica. We will now discuss the features of these configurations.\n\nFirst of all, we must compare the results obtained with the ramp-profile when N = 1 and compare them to the result obtained with the full profile. Tables. 3 and 4 show the properties of solutions we obtained using the ramp-profile approximation, again at α = 1 × 10 -6 . All the general features of the full numerical solutions are reproduced. In particular, the approximate B 1 2 scaling and then decrease of the radius, the decreasing ADM mass and the differences between the double and single layer solutions are all exhibited by the data obtained using the ramp-profile approximation.\n\nQuantitatively though, there are some differences. The approximation allows a significant increase in the maximum allowed baryon charge. Also, the radius of configurations obtained using the approximation, tend to be smaller than those obtained numerically. If we concentrate on the baryon numbers greater than 10 5 so as to ensure our approximation, 15 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 8.3063 3.1286 1.1023 0.9999 500 18.6031 3.1386 1.1160 0.9997 1 × 10 3 26.313 3.1397 1.1195 0.9996 1 × 10 4 83.206 3.1396 1.1254 0.9987 1 × 10 5 262.94 3.1357 1.1266 0.9960 1 × 10 6 829.60 3.1230 1.1251 0.9872 1 × 10 7 2604.2 3.0825 1.1186 0.9595 1 × 10 8 8032.8 2.9512 1.0972 0.8713 1 × 10 9 22899 2.4837 1.0272 0.5772 2 × 10 9 29121 2.1092 0.9818 0.3645 2.8 × 10 9 29098 1.6623 0.9505 0.1380 2.83 × 10 9 28514 1.6066 0.9495 0.1119 2.839 × 10 9 28024 1.5671 0.94922 0.09373 2.8397 × 10 9 27869.3 1.5556 0.94924 0.08845 2.83975 × 10 9 27869.8 1.5524 0.94925 0.08699 2.839752 × 10 9 27822 1.5521 0.94925 0.08687 Table 3: Table of properties of the single layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6 16 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 5.7924 3.0428 1.0692 0.9999 1 × 10 3 18.5788 3.1305 1.1047 0.9995 1 × 10 4 58.8202 3.1380 1.1201 0.9983 1 × 10 5 185.8420 3.1332 1.1246 0.9944 1 × 10 6 585.7950 3.1153 1.1233 0.9820 1 × 10 7 1833.0500 3.0578 1.1143 0.9428 1 × 10 8 5587.3600 2.8688 1.0840 0.8172 9 × 10 8 14147.1782 2.1859 0.9900 0.4065 9.764724 × 10 8 14472.3851 2.1276 0.9837 0.3746 1 × 10 9 14560.5000 2.1092 0.9818 0.3646 1.4 × 10 9 14549.0000 1.6623 0.9505 0.1381 1.41963 × 10 9 13994.0523 1.5644 0.9492 0.0926 1.419635134 × 10 9 13993.2000 1.5643 0.9492 0.0925 Table 4: Table of properties of the double layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6\n\nthat the width is much smaller than the radius, is valid, then at worst we find a discrepancy in the ADM mass of 11% and in the radius of 7%.\n\nIn general then, the data seems to confirm the reliability the ramp-profile approximation. In fact the approach will be even more reliable at the extremely high values of the baryon number that we are interested in. This is because the radius of solutions is of orders of magnitudes greater than the width in such a regime, consistent with the approximations we have made.\n\nMoreover, whilst searching for minima of the energy does not allow us to probe both branches of solutions, it does allow us to locate the value of αcrit. We again obtain the approximate trend αcrit ∝ B -1 2 , for large B. Now in order to say anything about the possibility of baryon stars in the Skyrme model we need to be able to verify that the decrease in the ADM mass per baryon we observed at low and moderate baryon numbers, extends to baryon numbers of order 10 58 for α = 7.3 × 10 -40 .\n\nTable. 5 summarizes our solutions in such a regime. Firstly we consider constructing a single layer self-gravitating Skyrmion with these values. We do indeed see that the configuration is bound. This is verified by checking that the ADM mass is lower (even at this significantly lower value of α) than for the B = 1 hedgehog. So the possibility of baryon stars in the Einstein-Skyrme model cannot be ruled out on the grounds of energy.\n\nThe Skyrmion exists as a giant thin shell, and the large baryon charge is distributed as a tight lattice over this. However a hollow shell is clearly not a realistic construction for a neutron star. This fact manifests itself in the extremely high radius of the configuration.\n\nTransferring to standard units, the single layer B = 10 58 gravitating Skyrmion has a radius of 2.42 × 10 10 km ! To address this issue, we can use a large number of layered Skyrmions as discussed earlier. This has several benefits. Firstly, as we are distributing the baryon number through a larger volume, then at a given baryon density the necessary radius can decrease. Similar to what we see in the double layer results. On top of this, we expect the radius to decrease further due to extra gravitational compression, as the outer layers of the Skyrmions feel the attraction of inner layers. Finally, the many layer approach is also a more realistic construction of a solid baryon star.\n\nThe results for using more and more layers in the construction (for fixed B and α), are also presented in Fig. 5 . We note that not only does the radius decrease significantly, but\n\nthe added gravitational binding further improves the energies of the configurations, reflected in the low ADM masses obtained. There appears to be a critical number of layers that can be used before there ceases to be any solutions and although the value of Smin is close zero at this point, the star still has not collapsed to form a black hole. Finally, we note that the radius of the Skyrmion at the critical number of layers is approximately 20.91km. This is comparable to a real neutron star, with a typical radius of 10km.\n\nWe reemphasise here that our approach to embedding shells of baryons is quite crude.\n\nFor few shells and large baryon number, we might reasonably believe that baryon number does not chance significantly from one shell to the next. However, when we embed many shells we should really consider that the baryon number of the inner most shells would likely be significantly less than the that of the outer shells. Nevertheless, our naive embedding has produced some interesting properties. In a future work we hope to improve our multi-layer construction to obtain a more realistic description of a baryon star. 19 N Shell R( 2 Fπe ) W M ADM /(6π 2 B) S min 1 × 10 2 8.3236 × 10 27 3.1416 1.1285 1.0000 1 × 10 3 2.6321 × 10 27 3.1416 1.1285 1.0000 1 × 10 4 × 10 26 3.1416 1.1285 1.0000 1 × 10 5 2.6321 × 10 26 3.1416 1.1285 1.0000 1 × 10 6 8.3236 × 10 25 3.1416 1.1285 1.0000 1 × 10 7 2.6321 × 10 25 3.1416 1.1285 1.0000 1 × 10 8 8.3236 × 10 24 3.1416 1.1285 1.0000 1 × 10 9 2.6321 × 10 24 3.1415 1.1285 1.0000 1 × 10 10 8.3234 × 10 23 3.1415 1.1285 0.9999 1 × 10 11 2.6319 × 10 23 3.1412 1.1285 0.9997 1 × 10 12 8.3216 × 10 22 3.1402 1.1283 0.9991 1 × 10 13 2.6301 × 10 22 3.1373 1.1278 0.9971 1 × 10 14 8.3034 × 10 21 3.1280 1.1263 0.9907 1 × 10 15 2.6118 × 10 21 3.0986 1.1213 0.9705 1 × 10 16 8.1147 × 10 20 3.0037 1.1057 0.9063 1 × 10 17 2.4001 × 10 20 2.6810 1.0552 0.6977 5 × 10 17 8.2066 × 10 19 1.7888 0.9552 0.2036 5.3 × 10 17 7.4172 × 10 19 1.6227 0.9491 0.1247 5.33 × 10 17 7.1871 × 10 19 1.5625 0.94866 0.0971 5.3306 × 10 17 7.1597 × 10 19 1.5549 0.94868 0.0936 5.33065 × 10 17 7.1525 × 10 19 1.5528 0.948694 0.0927 5.330657 × 10 17 7.1506 × 10 19 1.5523 0.948692 0.0924 Table 5: Table of properties of the step ansatz configurations for varying the number of embedded shells at fixed B = 10 58 and α = 7.3 × 10 -40 ."},{"section_type":"CONCLUSION","section_title":"7 Conclusions - Acknowledgements","text":"Previous work on the Einstein-Skyrme model highlighted a considerable problem with using the Skyrmions as a model for baryon stars. Namely, multibaryon hedgehog Skyrmions were simply not energetically favourable states. We have shown that this is simply a consequence of a poor ansatz for the true Skyrmion and, having used the more appropriate rational map ansatz, we have generated energetically favourable configurations of multibaryons.\n\nWe also observe the interesting property that near the critical coupling, the Skyrmions can decrease in radius as we add more baryons. This hints towards the similar behaviour exhibited by real neutron stars.\n\nAlthough the rational map ansatz does not have an exact radial symmetry, at large scale it does. The anisotropy only appears at the nucleon scale.\n\nFinally, since we started with the motivation of studying baryon stars within the Skyrme model, it is interesting to compare the features of our configurations with those of neutron stars. For realistic values, B = 10 58 and α = 7.3 × 10 -40 we find a minimal energy single layer configuration with radius=2.42 × 10 10 km. This is clearly too large for a neutron star (which is of order 10km. in radius). This is to be expected however due to the shell model we have taken. Firstly, as we are distributing the baryons over the surface area rather than throughout the volume of the star we naturally must require a much larger star for a given baryon number. This effect is two-fold in that if we were distributing the baryons throughout the volume, outer layers would feel the attraction of inner layers and enhanced radial compression would occur. The loss of such an effect is pronounced when we are considering realistically small values of the coupling.\n\nIt seems therefore that the way to construct baryon stars in the Skyrme model is to consider embedding shells of baryons within shells. This gives rise to more appropriate specifications for the star and is also more realistic. We do indeed observe such improvements for a many layered configuration. In fact the radius of B = 10 58 gravitating Skyrmion (at realistic α), can be decreased in this manner to approximately 20.91km.\n\nWe note however that this approach to shell embedding has only be done naively thus far.\n\nWe have only considered the case where the baryon number is equal for each shell. We really should allow the baryon number(and hence the rational map quantities) to vary over the shells. One approach towards this would be to assume that the baryon density is a constant over the shells. An even better approach would be to allow this to be a smoothly varying function that must be determined by minimising the energy. This will give a more realistic 21 description of baryon stars within the Einstein-Skyrme model, as traditional descriptions of neutron stars also involve many strata, of differing neutron density. We are currently investigating such configurations.\n\n[13] V. B. Kopeliovich \"The Bubbles of Matter from MultiSkyrmions\" JETP Lett. 73 (2001), 587-591; Pisma Zh.Eksp.Teor.Fiz. 73 (2001), 667-671 [14] V. B. Kopeliovich \"MultiSkyrmions and Baryonic Bags\" J.Phys. G28 (2002), 103-120 23\n\nGIP is supported by a PPARC studentship."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction\",\n \"text\": \"The Skyrme model, in its initial form, was proposed and developed by T.H.R. Skyrme in a series of papers as a non-linear field theory of pions [2] , [3] . Skyrme's initial idea was to think of baryons (in particular the nucleons) as secondary structures arising from a more fundamental mesonic fluid. The key property of the model was that the baryons arose as solitons in a topological manner and thus possessed a conserved topological charge identified with the baryon number.\\n\\nThe lowest energy stable solutions of the model are termed Skyrmions and can be thought of as baryonic solitons. The Skyrme model has been very successful in modelling the structures of various nuclei and has been shown by Witten et al. [4] to possess the general features of a low energy effective field theory for QCD. Some studies of the Skyrme model coupled to gravity have previously been undertaken [1], [5] , [6] , mainly with the motivation of a comparison of its features with those of other nonlinear field theories coupled to gravity. Of particular note is the Einstein-Yang-Mills theory, in which gravitationally bound configurations of non-abelian gauge fields are produced.\\n\\nOther reasons for studying the Einstein-Skyrme model are cosmological and astrophysical ones. Various authors have studied black hole formation in the model, with the conclusion that the so-called no-hair conjecture may not hold [7] , [8] .\\n\\nThe purpose of this paper is to study large baryon number Skyrmions or configurations of Skyrmions in the Einstein-Skyrme model. In particular, we wish to investigate if stable solitonic stars could exist within the model and to compare their properties to those of neutron stars. Preliminary studies of Skyrmion stars have predicted instability to single particle decay [1] . However this was done using the hedgehog ansatz for baryon number larger than 1 which is known to lead to unstable solutions even for the usual Skyrme model. Since then, it has been shown that the Skyrme model has stable shell-like solutions [9] which can be well approximated by the so called rational map ansatz [10] .\\n\\nIn this paper we use the rational map ansatz and its extension to multiple shells to construct configurations in the Gravitating Skyrme model that have a very large number of baryon. We show that those configurations, contrary to the hedgehog ansatz are bound even for very large baryon numbers.\\n\\nTo construct configurations that have a baryon number comparable to that of neutron star, we have to introduce a further approximation, which we call the ramp ansatz. We show 2 that this anstaz introduces further errors of only a few percent and we use it to compute very large Skyrmion configurations.\\n\\nThe paper is organised as follow: first we outline the Einstein-Skyrme model and discuss the main features of the results on static gravitating SU (2) hedgehogs obtained by Bizon and Chmaj [1] . We then use the rational map ansatz to construct shell like gravitating multibaryon configurations and show that for a fixed value of the coupling constant, the configurations exist only when the baryon number is below a certain critical value. Finally we introduce a ramp profile approximation to construct solutions with extremely high baryon numbers. We show how accurate it is and use it to construct Skyrmion stars configuration.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The Einstein-Skyrme Model\",\n \"text\": \"The action for gravitating Skyrmions is formed from the standard Skyrme action for the matter field and the Einstein-Hilbert action for the gravitational field.\\n\\nS = Z M √ -g \\\" L Sk - R 16πG « d 4 x. ( 1\\n\\n)\\n\\nHere L Sk is the Lagrangian density for the Skyrme model defined on the manifold M :\\n\\nL Sk = F 2 π 16\\n\\nT r(∇µU ∇ µ U -1 ) + 1 32e 2 T r[(∇µU )U -1 , (∇νU )U -1 ] 2 , (\\n\\n) 2\\n\\nwhere U belongs to SU (2). As we eventually wish to study baryon stars, we take a spherically symmetric metric, such as associated with the line element ds 2 = -A 2 (r) \\\" 1 -2m(r) r « dt 2 + \\\" 1 -2m(r) r\\n\\n« -1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), ( 3\\n\\n)\\n\\nwhere A(r) and m(r) are two profile functions that must be determined by solving the Einstein equations for the model. Our choice of ansatz is motivated by the fact that although in some cases we will be studying non-spherical Skyrmion configurations, the regime we are primarily interested in (i.e. Skyrmions of extremely high baryon number) will be shown to admit quasi spherical solutions. Also, for realistic values of the couplings, the gravitational interaction is small compared to the Skyrme interaction and thus the use of a spherical metric even with non-spherical configurations, is not a great problem.\\n\\nFrom (3), it can be shown that the Ricci scalar is\\n\\nR = -2 Ar 2 `-A ′′ r 2 -2A ′ r + 2A ′′ rm + A ′ m + 3A ′ rm ′ + Arm ′′ + 2Am ′ ´( 4\\n\\n)\\n\\nwhich, after integrating various terms by parts and noting that asymptotic flatness requires both A(r) and m(r) to take a constant value at spatial infinity, reduces the gravitational part of the action to Sgr = Z A(r) \\\" -m ′ (r) G « dr + m(∞) G . (5) For what follows, it will be convenient to scale to dimensionless variables by defining x = eFπr and µ(x) = eFπm(r)/2, resulting in one dimensionless coupling parameter for the model, α = πF 2 π G. We note that taking Fπ = 186M ev and G = 6.72 × 10 -45 M ev -2 , then the physical value of the coupling is α = 7.3 × 10 -40 .\\n\\nAs the Skyrme field is an SU (2) valued scalar field, at any given time one can think of it as a map from R 3 to the SU (2) manifold. Finite energy considerations impose that the field at spatial infinity should map to the same point on SU (2), say the identity. Thus, one can simply think of the Skyrme field as a map between three-spheres. All such maps fall into disjoint homotopy classes characterised by their winding number. This winding number is a conserved topological charge because no continuous deformation of the field and thus no time evolution, can allow transitions between homotopy classes. It is this topological charge that is interpreted as the baryon number.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Gravitating Hedgehog Skyrmions\",\n \"text\": \"Gravitating Skyrmions were first studied by Bizon and Chmaj[1] who analysed the properties of static spherically symmetric gravitating SU (2) skyrmions. Taking the Hedgehog Ansatz for the Skyrme field\\n\\nU = exp(i -→ σ .rF (r)) ( 6\\n\\n)\\n\\nsubject to the boundary conditions\\n\\nF (r = 0) = Bπ ( 7\\n\\n) F (r = ∞) = 0 ( 8\\n\\n)\\n\\nwhere B is the Baryon number associated with the Skyrmion configuration, they derived the Euler-Lagrange equation for the profiles F (r), (A(r) and m(r) and found that the model admit two branches of global solitonic solutions at each given baryon number, which annihilate at a critical value of the coupling parameter. Above αcrit no further solutions were found. In particular the value of the critical coupling decreased quite considerably with increasing baryon number as αcrit ≈ 0.040378/B 2 . It appears that the existence of a 4 critical coupling does not signal the collapse of a Skyrmion to form a black hole. In fact the metric factor S(x) = (1 -2µ(x) x ) is non-zero at αcrit; there simply ceases to be any stationary points of the action above the critical coupling.\\n\\nThe major problem with the ansatz (7) is that it leads to unstable solutions, i.e. for any given value of α, MADM (B = N ) > N MADM (B = 1). This is actually the case for the pure Skyrme model as well where the hedgehog anstaz (7) with B > 1 does not correspond to the lowest energy solution for the model. The solutions of the pure Skyrme model when B > 1 are known not to be spherically symmetric [11] but are stable i.e. E(B = N ) < N * E(B = 1).\\n\\nIt was actually shown by Houghton et al [10] , [12] that the multi-baryon solutions of the pure Skyrme model can be well approximated by the so called rational maps ansatz which is a generalisation of the hedgehog ansatz. While not radially symmetric, the ansatz separates its radial and angular dependence through a profile function and a rational map respectively.\\n\\nIn the following sections we will generalise the construction of Houghton et al to approximate the solution of the Einstein-Skyrme model.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The Rational Map Ansatz\",\n \"text\": \"The rational map ansatz introduced by Houghton et al.[10] works by decomposing the field into angular and radial parts. Using the polar coordinates in R 3 and defining the stereographic coordinates z = tan(θ/2) exp iφ the ansatz reads [10]\\n\\nU = exp (i σ • nRF (r, t)) ( 9\\n\\n) where nR = 1 1 + |R| 2 `2ℜ(R), 2ℑ(R), 1 -|R| 2 ´( 10\\n\\n)\\n\\nis a unit vector where R is a rational function of z.\\n\\nIt can be shown that the baryon number for Skyrmions constructed in this way, is equal to the degree of the rational map providing we take the boundary conditions\\n\\nF (r = 0) = π F (r = ∞) = 0. ( 11\\n\\n)\\n\\nSubstituting the ansatz (9) into the action for the model and scaling to dimensionless\\n\\n5\\n\\nvariables as earlier, we obtain the following reduced Hamiltonian\\n\\nH = 16πFπ e »Z ∞ 0 A(x) \\\" 1 2 S(x)F (x) ′2 x 2 + Bsin 2 F (x)(1 + S(x)F (x) ′2 ) + Isin 4 F (x) 2x 2 - µ ′ (x) α « dx + µ(∞) α - ( 12\\n\\n)\\n\\nwhere\\n\\nS(x) = 1 - 2µ(x) x ( 13\\n\\n)\\n\\nFrom which one obtains the following field equations\\n\\nµ(x) ′ = α \\\" 1 2 S(x)x 2 F (x) ′2 + B sin 2 F (x) + S(x)BF (x) ′2 sin 2 F (x) + I sin 4 F (x) 2x 2 « ( 14\\n\\n) F (x) ′′ = 1 S(x)V (x) » sin 2F (x) \\\" B + S(x)BF (x) ′2 + I sin 2 F (x) x 2 « - αS(x)F (x) ′3 V (x) 2 x -S(x) ′ F (x) ′ V (x) -S(x)F (x) ′ V (x) ′ ˜( 15\\n\\n) and A(x) ′ = αA(x)F (x) ′2 \\\" x + 2B sin 2 F (x) x « ( 16\\n\\n)\\n\\nwhere, for convenience, we have defined V (x) as\\n\\nV (x) = x 2 + 2B sin 2 F (x). ( 17\\n\\n)\\n\\nB is the baryon number and\\n\\nI = 1 4π Z \\\" 1 + |z| 2 1 + |R| 2 ˛dR dz ˛«4 2idzdz (1 + |z| 2 ) 2 ( 18\\n\\n)\\n\\nIts value depends on the chosen rational map R. To compute low energy configurations for a given baryon charge B one must find the rational map R or degree B that minimize I.\\n\\nThis has been done in [10] and [11] for several values of B. Moreover when b is large, one can use the approximation[11] I ≈ 1.28B 2 . The value of I so obtained is then used as a parameter and one can solve equations (14) -(16) for the radial profiles F (x), A(x) and µ(x).\\n\\nWe should point out here that for the pure Skyrme model the rational map ansatz produce very good approximation to the multi skyrmion solutions [10]: the energies are only 3 or 4 percent higher and the energy densities exhibit the same symmetries and differ by very little. All the solutions computed by Battye and Sutcliffe [11] , when B is not too small, have somehow the shape of a hollow shell. The baryon density is very small everywhere outside the shell, while on the shell itself, it forms a lattice of hexagons and pentagons.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"5 Hollow Skyrmion Shells\",\n \"text\": \"Using the rational map ansatz, we will now solves the field equations (14) -(16) to compute some low action configurations. These solutions will correspond, initially, to a hollow shell of Skyrmions similar to the configuration obtained with the rational map anstaz for the pure Skyrme model. In the following sections we will show how our ansatz can be generalised to allow for more realistic configuration made out of embedded shells.\\n\\nThe first thing to note about our solutions is that we again obtain two branches of solutions at each baryon number (Fig. 1 ). Obtaining this same qualitative behaviour is not surprising when one considers that the B = 1 rational map Skyrmion reproduces the usual B = 1 hedgehog. However, the behaviour of the critical coupling itself is drastically altered for the rational map generated configurations. Namely, we observe that it decreases as approximately 0.040378/B 1 2 (Fig. 2 ). In particular this means that for a given value of the coupling, the rational map generated skyrmions can possess a much higher topological charge than their hedgehog counterparts, before there ceases to be any solutions. Quantitatively if B hedgehog is the maximum baryon number for which hedgehog solutions can be found at a given value of the coupling, then the highest baryon number rational map solution found at the same value of α will be approximately B 4 hedgehog . Again we observe that the metric function S(x) is non-zero at the critical coupling for all the solutions we have found and as such a horizon has not formed.\\n\\nIn Table 1 we present the radius, ADM mass per baryon and minimum value of the metric function, S(x), for configurations up to the maximum baryon number allowed at α = 1 × 10 -6 . These values were obtained by direct numerical solution of equations (14) -(16), where we have used the boundary data as specified in (11) .\\n\\nWe didn't didn't use the physical value of α (7.3×10 -40 ) because for this value, the ratio between the width of the shell and its radius is so small when we reach the maximum value of B that it becomes very difficult to solve the equation reliably. The value α = 1 × 10 -6 is small enough to allow for a shell with a large baryon number to exist but large enough to make it possible to compute these solution nears the critical value of B for a single shell configuration. 7 50 100 150 200 250 300 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 M ADM α Figure 1: Plot of the two branches of solutions found for B=2 configurations generated with the rational map ansatz.\\n\\nThe major difference between these configuration and the solutions of Bizon and Chmadj is that the rational map ansatz configurations become more bound when the baryon number increases This suggests the possibility that giant gravitating Skyrmions can be bound and consequently, that the Skyrme model can be used to study baryon stars.\\n\\nAnother interesting feature of the data is the observed change in the radius of the solutions with increasing baryon number. We note that the radius grows as approximately B 1 2 .\\n\\nHowever there are two main deviations from this. Firstly, the constant of proportionality relating the radius to the square root of the baryon number decreases slightly but persistently as we increase the baryon number, indicating the gravitational interaction becoming more important as the number of baryons increases.\\n\\nAs we approach the maximum baryon charge that can exist at α = 1 × 10 -6 , we also notice that the radius of the skyrmion actually decreases as we add more baryons. This shows that the gravitation pull plays a crucial role near the critical value of the skyrmion.\\n\\nThis is a tantalising property when one considers that generally a neutron star's radius must decrease for an increase in mass in order to achieve sufficient degenerate neutron pressure 8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 2 4 6 8 10 12 14 16 18 20 α crit B Figure 2: Plot of the decrease in α crit with increasing baryon number, for configurations generated with the rational map ansatz. +: α c r for the minimum value of I; curve: α c r = 0.0404B -1/2 .\\n\\nto support the star.\\n\\nTo motivate the further approximation that we will introduce in the next section, we now look at the profiles of the configuration that we have computed. First of all, we observe that the profile function F (x) stays approximately at its boundary value, π, for a finite radial distance before decreasing monotonically over some small region and finally attaining its second boundary value, 0. A similar behaviour is seen for both the mass field µ(x) and the metric field A(x) (see Fig. 3 ). Furthermore, as we increase the baryon number the structure becomes more pronounced, with the distance before the fields change (shell radius) increasing significantly, whilst the distance over which the fields change (shell width) settles to a constant size. We conclude that at large baryon numbers, those configurations correspond to hollow shells where the baryons are distributed on a tight lattice over the shell. As such the, structures are nearly spherical, validating our choice of radial metric.\\n\\nSuch structures immediately pose an interesting question. Can the gravitating Skyrmions exist as shells with more than one layer? To investigate this we note that it is possible to 9 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 1 0.8763 1.2315 1.0000 4 1.7728 1.1365 1.0000 8 2.5065 1.1180 1.0000 100 8.6829 1.0845 0.9999 500 19.3994 1.0827 0.9998 1 × 10 3 27.4314 1.0825 0.9997 1 × 10 4 86.7192 1.0821 0.9989 1 × 10 5 274.0397 1.0814 0.9963 1 × 10 6 864.6968 1.0792 0.9883 1 × 10 7 2715.0729 1.0722 0.9628 1 × 10 8 8377.4601 1.0500 0.88192 1 × 10 9 23585.5315 0.9743 0.6107 1.5 × 10 9 26860.2040 0.9463 0.5020 1.8 × 10 9 27470.2449 0.9302 0.4256 1.81 × 10 9 27456.5804 0.9296 0.4225 1.85 × 10 9 27357.9201 0.9274 0.4090 1.9 × 10 9 27078.6014 0.9246 0.3886 1.95 × 10 9 26126.5508 0.9217 0.3517 1.951 × 10 9 26050.7695 0.9217 0.3495 1.952 × 10 9 25937.4210 0.9216 0.3463 Table 1: Properties of the one shell low energy configuration for α = 1 × 10-6 10 0 0.5 1 1.5 2 2.5 3 3.5 4 1210 1215 1220 1225 1230 1235 F(x) x 0 2 4 6 8 10 1210 1215 1220 1225 1230 1235 µ(x) x 0.99 0.995 1 1.005 1.01 1210 1215 1220 1225 1230 1235 A(x) x Figure 3: Numerical solutions for the profiles F (x), µ(x) and A(x) when B = 2 × 10 6 and α = 1 × 10 -6 .\\n\\nmodify the boundary condition (11) to read\\n\\nF (r = 0) = N π ( 19\\n\\n) F (r = ∞) = 0 ( 20\\n\\n)\\n\\nwhilst still ensuring that the Skyrme field is well defined at the origin. This idea was first used in [12] to construct two shell configurations for the pure Skyrme model.\\n\\nThe baryon charge is now N times the degree of the rational map. Fig. 4 shows the structure of the solutions we find in this case when N = 2. They suggest that the Skyrmion now exists as a N -layered structure. This is exhibited in the form of the profile, mass and metric functions which interpolate between the boundary values in N distinct steps of equal 11 0 1 2 3 4 5 6 7 1200 1205 1210 1215 1220 1225 1230 F(x) x 0 5 10 15 20 1200 1205 1210 1215 1220 1225 1230 µ(x) x 0.995 1 1.005 1.01 1.015 1.02 1200 1205 1210 1215 1220 1225 1230 A(x) x Figure 4: Numerical solutions for for the profiles F (x), µ(x) and A(x) for 2 layers configurations (F (0) = 2π) when B = 2 × 10 6 and α = 1 × 10 -6 .\\n\\nsize stacked next to each other.\\n\\nWe can therefore think of this as a naive way of constructing a gravitating Skyrmion.\\n\\nInstead of using the boundary conditions as in (11) and a rational map of degree B we consider constructing the B-Skyrmion using a rational map of degree B/N (with the associated value of I) and the boundary condition (20). This is a crude construction as we are effectively considering N adjacent shells of baryons, all with the same baryon number. We might realistically expect that the baryon number per shell and distribution of shells may vary significantly for the minimum energy configuration. Nevertheless we shall study the properties of such structures. In fact, in the case where the baryon number is large and the number of shells is small, we expect this crude construction to be quite valid. That is, we do not expect the baryon number to change significantly over the few shells at large radius.\\n\\n12 B R( 2 Fπ e ) M ADM ( Fπ 2etopconv ) S min 4 1.2898 1.6179 1.0000 8 1.7858 1.4072 1.0000 100 6.1754 1.1363 0.9999 1 × 10 3 19.4157 1.0913 0.9996 1 × 10 4 61.3207 1.0833 0.9985 1 × 10 5 193.7006 1.0812 0.9949 1 × 10 6 610.6271 1.0779 0.9835 1 × 10 7 1911.3704 1.0680 0.9475 1 × 10 8 5825.2626 1.0362 0.8325 9.0 × 10 8 13736.9982 0.9302 0.4258 9.7 × 10 8 13263.0853 0.9224 0.3644 9.76 × 10 8 12998.2817 0.9217 0.3480 9.764 × 10 8 12931.5189 0.9216 0.3444 9.7647 × 10 8 12895.6984 0.9216 0.3425 9.76472 × 10 8 12891.4247 0.9216 0.3423 9.764724 × 10 8 12889.8645 0.9216 0.3422 Table 2: Table of properties of double layer solutions obtained numerically at α = 1 × 10-6\\n\\nFor the remainder of this section we will restrict ourselves to the case where N = 2.\\n\\nTable 2 summarises the properties of double layered gravitating skyrmions up to the maximum baryon charge allowed at α = 1 × 10 -6 . Briefly, we note the main features.\\n\\nFirstly, for all baryon numbers, the radius of the double layered solutions is significantly less than their single layered counterparts. This is not surprising as the baryon charge exists over a thicker region and so the mean radius can decrease with the baryon density remaining the same. Secondly, when B is large enough, i.e. when the double layer starts to make sense, the double layer solutions are energetically favourable when one compares the ADM mass with the single layer solutions. Finally we note that the maximum baryon number allowed (at the given coupling) is almost twice as much in the case of the single layer skyrmions.\\n\\nOf course the results of this section are not really the main regime of interest. We clearly 13 need to study configurations of extremely high baryon number (of order 10 58 ) relevant for baryon stars.\\n\\nWe will now discuss this high baryon number regime.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The Ramp-profile Approximation\",\n \"text\": \"Unfortunately, at very high baryon numbers, eqns. ((14) -(16)) become difficult to handle numerically. This is largely because the radius of the solutions becomes much larger than the distance over which the fields change. That is, we need to integrate over a region which is much less than 10 -16 radius, and so even double precision data types have insufficient precision.\\n\\nMoreover, single shell configurations are not physically relevant and multiple shells will only yield configuration that looks like a star if the number of layers is very large, typically well over 10 17 . With such a large number of layers we won't be able to solve the equation numerically as we will need at least 10 times as many sampling points for the profile functions.\\n\\nWe must thus resort to another level of approximation: approximate the profile functions by profiles that are piecewise linear. This is inspired by the work of Kopeliovich [13] [14] except that our ansatz has to be piecewise linear to be able to generate configurations with a huge number of layers. After defining the ansatz for an arbitrary number of layers, we will show that for a single layer configuration the ansatz produces configurations that are in good agreements with the rational map ansatz configuration. Then we will use the new ansatz to construct configurations that are made out of a very large no of layers.\\n\\nWe have shown, in the previous section, that one can construct shell like structures with very large Baryon numbers. At large baryon numbers, the Skyrmions resemble shell like structures. That is, the fields are constant nearly everywhere except in a small region corresponding to the shell. In that region, the profile look like linear functions smoothly linked to the constant parts at the edges (cfr. Fig 3 ). Motivated by this we approximate the fields by the ramp-functions\\n\\nF (x) = N π 2 -(x -x0) π W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 21\\n\\n) µ(x) = M 2 + (x -x0) M N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 22\\n\\n) A(x) = (1 + A0) 2 + (x -x0) (1 -A0) N W , (x0 -N W/2) ≤ x ≤ (x0 + N W/2) ( 23\\n\\n)\\n\\nIn the above there are four free parameters, namely the central radius x0 of the shell\\n\\nover which the fields change, the width of the shell W , the mass field at spatial infinity M and the value of the metric field at the origin A0 such that limx→∞ = 0. N is the number of layers we wish to study and, as such, is treated as an input parameter.\\n\\nThe picture is of a gravitating skyrmion with very high baryon number existing as N thin layers or shell of small thickness.\\n\\nThe above ansatz, allow us to find an approximation to the integrated energy. To do this we use the fact that the shell width is much smaller than the radius at large baryon numbers. In particular to evaluate the action integral we can approximate expressions of the type R G(x) sin p F (x) for any function G(x) that varies very little over the width of the shell by R G(x0) sin p F (x). We then use the fact that\\n\\nZ x 0 +NW/2 x 0 -NW/2 sin p F (x) = N π W Z π 0 sin p y dy. ( 24\\n\\n)\\n\\nThis leads to the following expression for the energy:\\n\\nE = - 16πFπ e » M α \\\" 1 + A0 2 « - π 2 W » \\\" 1 + A0 2 « \\\" x 2 0 -M x0 + W 2 12 - M W 6 « + \\\" 1 -A0 W « \\\" W 2 x0 6 - M W 2 12 - M W x0 6 « - + B \\\" 1 + A0 2 « 1 2 \\\" M π 2 W x0 -W - π 2 W « - 3IW 16x 2 0 \\\" 1 + A0 2 « - M α - ( 25\\n\\n)\\n\\nTo find the configurations which minimize this energy we first minimised it with respect to A0 and M algebraically in order to find an expression for the energy as a function of the width and radius only. Then we minimised this numerically using Mathematica. We will now discuss the features of these configurations.\\n\\nFirst of all, we must compare the results obtained with the ramp-profile when N = 1 and compare them to the result obtained with the full profile. Tables. 3 and 4 show the properties of solutions we obtained using the ramp-profile approximation, again at α = 1 × 10 -6 . All the general features of the full numerical solutions are reproduced. In particular, the approximate B 1 2 scaling and then decrease of the radius, the decreasing ADM mass and the differences between the double and single layer solutions are all exhibited by the data obtained using the ramp-profile approximation.\\n\\nQuantitatively though, there are some differences. The approximation allows a significant increase in the maximum allowed baryon charge. Also, the radius of configurations obtained using the approximation, tend to be smaller than those obtained numerically. If we concentrate on the baryon numbers greater than 10 5 so as to ensure our approximation, 15 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 8.3063 3.1286 1.1023 0.9999 500 18.6031 3.1386 1.1160 0.9997 1 × 10 3 26.313 3.1397 1.1195 0.9996 1 × 10 4 83.206 3.1396 1.1254 0.9987 1 × 10 5 262.94 3.1357 1.1266 0.9960 1 × 10 6 829.60 3.1230 1.1251 0.9872 1 × 10 7 2604.2 3.0825 1.1186 0.9595 1 × 10 8 8032.8 2.9512 1.0972 0.8713 1 × 10 9 22899 2.4837 1.0272 0.5772 2 × 10 9 29121 2.1092 0.9818 0.3645 2.8 × 10 9 29098 1.6623 0.9505 0.1380 2.83 × 10 9 28514 1.6066 0.9495 0.1119 2.839 × 10 9 28024 1.5671 0.94922 0.09373 2.8397 × 10 9 27869.3 1.5556 0.94924 0.08845 2.83975 × 10 9 27869.8 1.5524 0.94925 0.08699 2.839752 × 10 9 27822 1.5521 0.94925 0.08687 Table 3: Table of properties of the single layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6 16 B R( 2 Fπe ) W M ADM ( Fπ 2e ) S min 100 5.7924 3.0428 1.0692 0.9999 1 × 10 3 18.5788 3.1305 1.1047 0.9995 1 × 10 4 58.8202 3.1380 1.1201 0.9983 1 × 10 5 185.8420 3.1332 1.1246 0.9944 1 × 10 6 585.7950 3.1153 1.1233 0.9820 1 × 10 7 1833.0500 3.0578 1.1143 0.9428 1 × 10 8 5587.3600 2.8688 1.0840 0.8172 9 × 10 8 14147.1782 2.1859 0.9900 0.4065 9.764724 × 10 8 14472.3851 2.1276 0.9837 0.3746 1 × 10 9 14560.5000 2.1092 0.9818 0.3646 1.4 × 10 9 14549.0000 1.6623 0.9505 0.1381 1.41963 × 10 9 13994.0523 1.5644 0.9492 0.0926 1.419635134 × 10 9 13993.2000 1.5643 0.9492 0.0925 Table 4: Table of properties of the double layer step ansatz configurations for varying the baryon number at fixed α = 1 × 10 -6\\n\\nthat the width is much smaller than the radius, is valid, then at worst we find a discrepancy in the ADM mass of 11% and in the radius of 7%.\\n\\nIn general then, the data seems to confirm the reliability the ramp-profile approximation. In fact the approach will be even more reliable at the extremely high values of the baryon number that we are interested in. This is because the radius of solutions is of orders of magnitudes greater than the width in such a regime, consistent with the approximations we have made.\\n\\nMoreover, whilst searching for minima of the energy does not allow us to probe both branches of solutions, it does allow us to locate the value of αcrit. We again obtain the approximate trend αcrit ∝ B -1 2 , for large B. Now in order to say anything about the possibility of baryon stars in the Skyrme model we need to be able to verify that the decrease in the ADM mass per baryon we observed at low and moderate baryon numbers, extends to baryon numbers of order 10 58 for α = 7.3 × 10 -40 .\\n\\nTable. 5 summarizes our solutions in such a regime. Firstly we consider constructing a single layer self-gravitating Skyrmion with these values. We do indeed see that the configuration is bound. This is verified by checking that the ADM mass is lower (even at this significantly lower value of α) than for the B = 1 hedgehog. So the possibility of baryon stars in the Einstein-Skyrme model cannot be ruled out on the grounds of energy.\\n\\nThe Skyrmion exists as a giant thin shell, and the large baryon charge is distributed as a tight lattice over this. However a hollow shell is clearly not a realistic construction for a neutron star. This fact manifests itself in the extremely high radius of the configuration.\\n\\nTransferring to standard units, the single layer B = 10 58 gravitating Skyrmion has a radius of 2.42 × 10 10 km ! To address this issue, we can use a large number of layered Skyrmions as discussed earlier. This has several benefits. Firstly, as we are distributing the baryon number through a larger volume, then at a given baryon density the necessary radius can decrease. Similar to what we see in the double layer results. On top of this, we expect the radius to decrease further due to extra gravitational compression, as the outer layers of the Skyrmions feel the attraction of inner layers. Finally, the many layer approach is also a more realistic construction of a solid baryon star.\\n\\nThe results for using more and more layers in the construction (for fixed B and α), are also presented in Fig. 5 . We note that not only does the radius decrease significantly, but\\n\\nthe added gravitational binding further improves the energies of the configurations, reflected in the low ADM masses obtained. There appears to be a critical number of layers that can be used before there ceases to be any solutions and although the value of Smin is close zero at this point, the star still has not collapsed to form a black hole. Finally, we note that the radius of the Skyrmion at the critical number of layers is approximately 20.91km. This is comparable to a real neutron star, with a typical radius of 10km.\\n\\nWe reemphasise here that our approach to embedding shells of baryons is quite crude.\\n\\nFor few shells and large baryon number, we might reasonably believe that baryon number does not chance significantly from one shell to the next. However, when we embed many shells we should really consider that the baryon number of the inner most shells would likely be significantly less than the that of the outer shells. Nevertheless, our naive embedding has produced some interesting properties. In a future work we hope to improve our multi-layer construction to obtain a more realistic description of a baryon star. 19 N Shell R( 2 Fπe ) W M ADM /(6π 2 B) S min 1 × 10 2 8.3236 × 10 27 3.1416 1.1285 1.0000 1 × 10 3 2.6321 × 10 27 3.1416 1.1285 1.0000 1 × 10 4 × 10 26 3.1416 1.1285 1.0000 1 × 10 5 2.6321 × 10 26 3.1416 1.1285 1.0000 1 × 10 6 8.3236 × 10 25 3.1416 1.1285 1.0000 1 × 10 7 2.6321 × 10 25 3.1416 1.1285 1.0000 1 × 10 8 8.3236 × 10 24 3.1416 1.1285 1.0000 1 × 10 9 2.6321 × 10 24 3.1415 1.1285 1.0000 1 × 10 10 8.3234 × 10 23 3.1415 1.1285 0.9999 1 × 10 11 2.6319 × 10 23 3.1412 1.1285 0.9997 1 × 10 12 8.3216 × 10 22 3.1402 1.1283 0.9991 1 × 10 13 2.6301 × 10 22 3.1373 1.1278 0.9971 1 × 10 14 8.3034 × 10 21 3.1280 1.1263 0.9907 1 × 10 15 2.6118 × 10 21 3.0986 1.1213 0.9705 1 × 10 16 8.1147 × 10 20 3.0037 1.1057 0.9063 1 × 10 17 2.4001 × 10 20 2.6810 1.0552 0.6977 5 × 10 17 8.2066 × 10 19 1.7888 0.9552 0.2036 5.3 × 10 17 7.4172 × 10 19 1.6227 0.9491 0.1247 5.33 × 10 17 7.1871 × 10 19 1.5625 0.94866 0.0971 5.3306 × 10 17 7.1597 × 10 19 1.5549 0.94868 0.0936 5.33065 × 10 17 7.1525 × 10 19 1.5528 0.948694 0.0927 5.330657 × 10 17 7.1506 × 10 19 1.5523 0.948692 0.0924 Table 5: Table of properties of the step ansatz configurations for varying the number of embedded shells at fixed B = 10 58 and α = 7.3 × 10 -40 .\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"7 Conclusions - Acknowledgements\",\n \"text\": \"Previous work on the Einstein-Skyrme model highlighted a considerable problem with using the Skyrmions as a model for baryon stars. Namely, multibaryon hedgehog Skyrmions were simply not energetically favourable states. We have shown that this is simply a consequence of a poor ansatz for the true Skyrmion and, having used the more appropriate rational map ansatz, we have generated energetically favourable configurations of multibaryons.\\n\\nWe also observe the interesting property that near the critical coupling, the Skyrmions can decrease in radius as we add more baryons. This hints towards the similar behaviour exhibited by real neutron stars.\\n\\nAlthough the rational map ansatz does not have an exact radial symmetry, at large scale it does. The anisotropy only appears at the nucleon scale.\\n\\nFinally, since we started with the motivation of studying baryon stars within the Skyrme model, it is interesting to compare the features of our configurations with those of neutron stars. For realistic values, B = 10 58 and α = 7.3 × 10 -40 we find a minimal energy single layer configuration with radius=2.42 × 10 10 km. This is clearly too large for a neutron star (which is of order 10km. in radius). This is to be expected however due to the shell model we have taken. Firstly, as we are distributing the baryons over the surface area rather than throughout the volume of the star we naturally must require a much larger star for a given baryon number. This effect is two-fold in that if we were distributing the baryons throughout the volume, outer layers would feel the attraction of inner layers and enhanced radial compression would occur. The loss of such an effect is pronounced when we are considering realistically small values of the coupling.\\n\\nIt seems therefore that the way to construct baryon stars in the Skyrme model is to consider embedding shells of baryons within shells. This gives rise to more appropriate specifications for the star and is also more realistic. We do indeed observe such improvements for a many layered configuration. In fact the radius of B = 10 58 gravitating Skyrmion (at realistic α), can be decreased in this manner to approximately 20.91km.\\n\\nWe note however that this approach to shell embedding has only be done naively thus far.\\n\\nWe have only considered the case where the baryon number is equal for each shell. We really should allow the baryon number(and hence the rational map quantities) to vary over the shells. One approach towards this would be to assume that the baryon density is a constant over the shells. An even better approach would be to allow this to be a smoothly varying function that must be determined by minimising the energy. This will give a more realistic 21 description of baryon stars within the Einstein-Skyrme model, as traditional descriptions of neutron stars also involve many strata, of differing neutron density. We are currently investigating such configurations.\\n\\n[13] V. B. Kopeliovich \\\"The Bubbles of Matter from MultiSkyrmions\\\" JETP Lett. 73 (2001), 587-591; Pisma Zh.Eksp.Teor.Fiz. 73 (2001), 667-671 [14] V. B. Kopeliovich \\\"MultiSkyrmions and Baryonic Bags\\\" J.Phys. G28 (2002), 103-120 23\\n\\nGIP is supported by a PPARC studentship.\"\n }\n]","isConversational":false}}},{"rowIdx":98,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0530"},"arxiv_id":{"kind":"string","value":"0704.0530"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1088/1126-6708/2007/06/065"},"content_hash":{"kind":"string","value":"e09094677ad52079a512d808ae2511bbf4a65c00c4c0f9afba29ba8ee99b8a1e"},"title":{"kind":"string","value":"Noncommutative Solitons in a Supersymmetric Chiral Model in 2+1\n Dimensions"},"abstract":{"kind":"string","value":"We consider a supersymmetric Bogomolny-type model in 2+1 dimensions\noriginating from twistor string theory. By a gauge fixing this model is reduced\nto a modified U(n) chiral model with N<=8 supersymmetries in 2+1 dimensions.\nAfter a Moyal-type deformation of the model, we employ the dressing method to\nexplicitly construct multi-soliton configurations on noncommutative R^{2,1} and\nanalyze some of their properties."},"authors":{"kind":"list like","value":["Olaf Lechtenfeld","Alexander D. Popov"],"string":"[\n \"Olaf Lechtenfeld\",\n \"Alexander D. Popov\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["hep-th"],"string":"[\n \"hep-th\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"We consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from twistor string theory. By a gauge fixing this model is reduced to a modified U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions. After a Moyal-type deformation of the model, we employ the dressing method to explicitly construct multi-soliton configurations on noncommutative R 2,1 and analyze some of their properties.\n\nIn the low-energy limit string theory with D-branes gives rise to noncommutative field theory on the branes when the string propagates in a nontrivial NS-NS two-form (B-field) background [1, 2, 3, 4] . In particular, if the open string has N =2 worldsheet supersymmetry, the tree-level target space dynamics is described by a noncommutative self-dual Yang-Mills (SDYM) theory in 2+2 dimensions [5] . Furthermore, open N =2 strings in a B-field background induce on the worldvolume of n coincident D2-branes a noncommutative Yang-Mills-Higgs Bogomolny-type system in 2+1 dimensions which is equivalent to a noncommutative generalization [6] of the modified U(n) chiral model known as the Ward model [7] . The topological nature of N =2 strings and the integrability of their tree-level dynamics [8] render this noncommutative sigma model integrable. 1 Being integrable, the commutative U(n≥2) Ward model features a plethora of exact scattering and no-scattering multi-soliton and wave solutions, i.e. time-dependent stable configurations on R foot_1 . These are not only a rich testing ground for physical properties such as adiabatic dynamics or quantization, but also descend to more standard multi-solitons of various integrable systems in 2+0 and 1+1 dimensions, such as sine-Gordon, upon dimensional and algebraic reduction. There is a price to pay however: Nonlinear sigma models in 2+1 dimensions may be Lorentz-invariant or integrable but not both [7, 11] . In fact, Derrick's theorem prohibits the existence of stable solitons in Lorentz-invariant scalar field theories above 1+1 dimensions. A Moyal deformation, however, overcomes this hurdle, but of course replaces Lorentz invariance by a Drinfeld-twisted version. There is another gain: The deformed Ward model possesses not only deformed versions of the just-mentioned multi-solitons, but in addition allows for a whole new class of genuinely noncommutative (multi-)solitons, in particular for the U(1) group [12, 13] ! Moreover, this class is related to the generic but perturbatively constructed noncommutative scalar-field solitons [14, 15] by an infinite-stiffness limit of the potential [16] .\n\nIn [12, 13] and [17] - [20] families of multi-solitons as well as their reduction to solitons of the noncommutative sine-Gordon equations were described and studied. In the nonabelian case both scattering and nonscattering configurations were obtained. For static configurations the issue of their stability was analyzed [21] . The full moduli space metric for the abelian model was computed and its adiabatic two-soliton dynamics was discussed [16] .\n\nRecall that the critical N =2 string theory has a four-dimensional target space, and its open string effective field theory is self-dual Yang-Mills [8] , which gets deformed noncommutatively in the presence of a B-field [5] . Conversely, the noncommutative SDYM equations are contained [19] in the equations of motion of N =2 string field theory (SFT) [22] in a B-field background. This SFT formulation is based on the N =4 topological string description [23] . It is well known that the SDYM model can be described in terms of holomorphic bundles over (an open subset of) the twistor space 2 [26] CP 3 and the topological N =4 string theory contains twistors from the outset. The Lax pair, integrability and the solutions to the equations of motion by twistor and dressing methods were incorporated into the N =2 open SFT in [27, 28] . However, this theory reproduces only bosonic SDYM theory, its symmetries (see e.g. [29, 30, 31] ) and integrability properties. It is natural to ask: What string theory can describe supersymmetric SDYM theory [32, 33] in four dimensions?\n\nThere are some proposals [33, 34, 35, 36] for extending N =2 open string theory (and its SFT) to be space-time supersymmetric. Moreover, it was shown by Witten [37] that N =4 supersymmetric SDYM theory appears in twistor string theory, which is a B-type open topological string with the supertwistor space CP 3|4 as a target space. 3 Note that N <4 SDYM theory forms a BPS subsector of N -extended super Yang-Mills theory, and N =4 SDYM can be considered as a truncation of the full N =4 super Yang-Mills theory [37] . It is believed [43, 39] that twistor string theory is related with the previous proposals [33, 34, 35, 36] for a Lorentz-invariant supersymmetric extension of N =2 (and topological N =4) string theory which also leads to the N =4 SDYM model.\n\nA dimensional reduction of the above relations between twistor strings and N =4 super Yang-Mills and SDYM models was considered in [44, 45, 46, 47] . The corresponding twistor string theory after this reduction is the topological B-model on the mini-supertwistor space P 2|4 . In [47] it was shown that the 2N =8 supersymmetric extension of the Bogomolny-type model in 2+1 dimensions is equivalent to an 2N =8 supersymmetric modified U(n) chiral model on R 2,1 . The subject of the current paper is an 2N ≤8 version of the above supersymmetric Bogomolny-type Yang-Mills-Higgs model in signature (-+ +), its relation with an N -extended supersymmetric modified integrable U(n) chiral model (to be defined) in 2+1 dimensions and the Moyal-type noncommutative deformation of this chiral model. We go on to explicitly construct multi-soliton configurations on noncommutative R 2,1 for the corresponding supersymmetric sigma model field equations. By studying the scattering properties of the constructed configurations, we prove their asymptotic factorization without scattering for large times. We also briefly discuss a D-brane interpretation of these soliton configurations from the viewpoint of twistor string theory.\n\nSpace R 2,2 . Let us consider the four-dimensional space R 2,2 = (R 4 , g) with the metric\n\nds 2 = g µν dx µ dx ν = det(dx α α) = dx 1 1dx 2 2 -dx 2 1dx 1 2 (2.1)\n\nwith (g µν ) = diag(-1, +1, +1, -1), where µ, ν, . . . = 1, . . . , 4 are space-time indices and α = 1, 2, α = 1, 2 are spinor indices. We choose the coordinates 4\n\n(x µ ) = (x a , t) = (t, x, y, t) with a, b, . . . = 1, 2, 3 , (2.2)\n\nand the signature (-+ + -) allows us to introduce real isotropic coordinates (cf. [19, 6] )\n\nx 1 1 = 1 2 (t -y) , x 1 2 = 1 2 (x + t) , x 2 1 = 1 2 (x -t) , x 2 2 = 1 2 (t + y) . (2.3)\n\nSDYM. Recall that the SDYM equations for a field strength tensor\n\nF µν on R 2,2 read 1 2 ε µνρσ F ρσ = F µν , (2.4)\n\nwhere ε µνρσ is a completely antisymmetric tensor on R 2,2 and ε 1234 = 1. In the coordinates (2.3) we have the decomposition\n\nF α α,β β = ∂ α αA β β -∂ β β A α α + [A α α, A β β ] = ε αβ F α β + ε α β F αβ (2.5) with F α β := -1 2 ε αβ F α α,β β and F αβ := -1 2 ε α β F α α,β β , (2.6)\n\nwhere ε αβ is antisymmetric, ε αβ ε βγ = δ γ α , and similar for ε α β , with ε 12 = ε 1 2 = 1. The gauge potential (A α α) will appear in the covariant derivative\n\nD α β = ∂ α β + [A α β , • ] . (2.7)\n\nIn spinor notation, (2.4) is equivalently written as\n\nF α β = 0 ⇔ F α α,β β = ε α β F αβ . (2.8)\n\nSolutions {A α α} to these equations form a subset (a BPS sector) of the solution space of Yang-Mills theory on R 2,2 .\n\nN -extended SDYM in component fields. The field content of N -extended super SDYM is foot_4\n\nN = 0 A α α (2.9a) N = 1 A α α, χ i α with i = 1 (2.9b) N = 2 A α α, χ i α , φ [ij] with i, j = 1, 2 (2.9c)\n\nN = 3 A α α, χ i α , φ [ij] , χ[ijk] α with i, j, k = 1, 2, 3 (2.9d) N = 4 A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl] α β\n\nwith i, j, k, l = 1, 2, 3, 4 .\n\n(2.9e)\n\nHere (A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl]\n\nα β ) are fields of helicities (+1, + 1 2 , 0, -1 2 , -1). These fields obey the field equations of the N = 4 SDYM model, namely [33, 37]\n\nF α β = 0 , (2.10a)\n\nD α αχ iα = 0 , (2.10b)\n\nD α αD α αφ ij + 2{χ iα , χ j α } = 0 , (2.10c)\n\nD α α χ α[ijk] -6[χ [i α , φ jk] ] = 0 , (2.10d)\n\nD γ α G [ijkl] γ β + 12{χ [i α , χjkl] β } -18[φ [ij , D α β φ kl] ] = 0 . (2.10e)\n\nNote that the N < 4 SDYM field equations are governed by the first N +1 equations of (2.10), where F α β = 0 is counted as one equation and so on.\n\nN -extended SDYM Superspace R 4|4N .\n\nRecall that in the space R 2,2 = (R 4 , g) with the metric g given in (2.1) one may introduce purely real Majorana-Weyl spinors foot_5 θ α and η α of helicities + 1 2 and -1 2 as anticommuting (Grassmann-algebra) objects. Using 2N such spinors with components θ iα and η α i for i = 1, . . . , N , one can define the N -extended superspace R 4|4N and the N -extended supersymmetry algebra generated by the supertranslation operators\n\nP α α = ∂ α α , Q iα = ∂ iα -η α i ∂ α α and Q i α = ∂ i α -θ iα ∂ α α , (2.11)\n\nwhere\n\n∂ α α := ∂ ∂x α α , ∂ iα := ∂ ∂θ iα and ∂ i α := ∂ ∂η α i .\n\n(2.12)\n\nThe commutation relations for the generators (2.11) read\n\n{Q iα , Q j α} = -2δ j i P α α , [P α α, Q iβ ] = 0 and [P α α, Q i β ] = 0 . (2.13)\n\nTo rewrite equations of motion in terms of R 4|4N superfields one uses the additional operators\n\nD iα = ∂ iα + η α i ∂ α α and D i α = ∂ i α + θ iα ∂ α α , (2.14)\n\nwhich (anti)commute with the operators (2.11) and satisfy\n\n{D iα , D j β } = 2δ j i P α β , [P α α, D iβ ] = 0 and [P α α, D j β ] = 0 . (2.15)\n\nAntichiral superspace R 4|2N . On the superspace R 4|4N one may introduce tensor fields depending on bosonic and fermionic coordinates (superfields), differential forms, Lie derivatives L X etc.. Furthermore, on any such superfield A one can impose the constraint equations L D iα A = 0, which for a scalar superfield f reduce to the so-called antichirality conditions\n\nD iα f = 0 . (2.16)\n\nThese are easily solved by using a coordinate transformation on R 4|4N ,\n\n(x α α, η α i , θ iα ) → (x α α = x α α-θ iα η α i , η α i , θ iα ) , (2.17)\n\nunder which ∂ α α, D iα and D i α transform to the operators\n\n∂α α = ∂ α α , Diα = ∂ iα and Di α = ∂ i α + 2θ iα ∂ α α . ( 2\n\n.18) Then (2.16) simply means that f is defined on a sub-superspace R 4|2N ⊂ R 4|4N with coordinates xα α and η α i . (2.19)\n\nThis space is called antichiral superspace. In the following we will usually omit the tildes when working on the antichiral superspace.\n\nN -extended SDYM in superfields. The N -extended SDYM equations can be rewritten in terms of superfields on the antichiral superspace R 4|2N [33, 48] . Namely, for any given 0 ≤ N ≤ 4, fields of a proper multiplet from (2.9) can be combined into superfields A α α and A i α depending on x α α, η α i ∈ R 4|2N and giving rise to covariant derivatives\n\n∇ α α := ∂ α α + A α α and ∇ i α := ∂ i α + A i α .\n\n(2.20)\n\nIn such terms the N -extended SDYM equations (2.10) read\n\n[∇ α α, ∇ β β ] + [∇ α β , ∇ β α] = 0 , [∇ i α, ∇ β β ] + [∇ i β , ∇ β α] = 0 , {∇ i α, ∇ j β } + {∇ i β , ∇ j α} = 0 , (2.21)\n\nwhich is equivalent to\n\n[∇ α α, ∇ β β ] = ε α β F αβ , [∇ i α, ∇ β β ] = ε α β F i β and {∇ i α, ∇ j β } = ε α β F ij , (2.22)\n\nwhere F ij is antisymmetric and F αβ is symmetric in their indices.\n\nThe above gauge potential superfields (A α α, A i α) as well as the gauge strength superfields (F αβ , F i α , F ij ) contain all physical component fields of the N -extended SDYM model. For instance, the lowest component of the triple (F αβ , F i α , F ij ) in an η-expansion is (F αβ , χ i α , φ ij ), with zeros in case N is too small. By employing Bianchi identities for the gauge strength superfields, one successively obtains [48] the superfield expansions and the field equations (2.10) for all component fields.\n\nIt is instructive to extend the antichiral combination in (2.18) to potentials and covariant derivatives, Di\n\nα = ∂ i α + 2 θ iα ∂ α α + + + Ãi α := A i α + 2 θ iα A α α ∇i α := ∇ i α + 2 θ iα ∇ α α ( 2\n\n.23) where ∇ α α, ∇ i α and Di α are given by (2.20) and (2.18), while A i α and A α α depend on x α α and η α i only. With the antichiral covariant derivatives, one may condense (2.21) or (2.22) into the single set { ∇i α, ∇j β\n\n} + { ∇i β , ∇j α} = 0 ⇔ { ∇i α, ∇j β } = ε α β Fij , (2.24)\n\nwith Fij =\n\nF ij + 4 θ [iα F j] α + 4 θ iα θ jβ F αβ .\n\nThe concise form (2.24) of the N -extended SDYM equations is quite convenient, and we will use it interchangeable with (2.21).\n\nLinear system for N -extended SDYM. It is well known that the superfield SDYM equations (2.21) can be seen as the compatibility conditions for the linear system of differential equations\n\nζ α(∂ α α + A α α) ψ = 0 and ζ α(∂ i α + A i α) ψ = 0 , (2.25)\n\nwhere\n\n(ζ β ) = 1 ζ and ζ α = ε α β ζ β . The extra (spectral) parameter 7 ζ lies in the extended complex plane C ∪ ∞ = CP 1 .\n\nHere ψ is a matrix-valued function depending not only on x α α and η α i but also (meromorphically) on ζ ∈ CP 1 . We subject the n×n matrix ψ to the following reality condition:\n\nψ(x α α, η α i , ζ) ψ(x α α, η α i , ζ) † = 1l , (2.26)\n\nwhere \" †\" denotes hermitian conjugation and ζ is complex conjugate to ζ. This condition guarantees that all physical fields of the N -extended SDYM model will take values in the adjoint representation of the algebra u(n). In the concise form the linear system (2.25) is written as\n\nζ α(∇ i α + 2θ iα ∇ α α) ψ = 0 ⇔ ζ α( Di α + Ãi α) ψ = 0 ⇔ ζ α ∇i α ψ = 0 . (2.27) 2.3 Reduction of N -extended SDYM to 2+1 dimensions\n\nThe supersymmetric Bogomolny-type Yang-Mills-Higgs equations in 2+1 dimensions are obtained from the described N -extended super SDYM equations by a dimensional reduction R 2,2 → R 2,1 .\n\nIn particular, for the N =0 sector we demand the components A µ of a gauge potential to be independent of x 4 and put A 4 =: ϕ. Here, ϕ is a Lie-algebra valued scalar field in three dimensions (the Higgs field) which enters into the Bogomolny-type equations. Similarly, for N ≥ 1 one can reduce the N -extended SDYM equations on R 2,2 by imposing the ∂ 4 -invariance condition on all the fields (A α α,\n\nχ i α , φ [ij] , χ[ijk] α , G [ijkl]\n\nα β ) from the N =4 supermultiplet or its truncation to N <4 and obtain supersymmetric Bogomolny-type equations on R 2,1 .\n\nSpinors in R 2,1 .\n\nRecall that on R 2,2 both N =4 SDYM theory and full N =4 super Yang-Mills theory have an SL(4, R) ∼ = Spin(3,3) R-symmetry group [33] . A dimensional reduction to R 2,1 enlarges the supersymmetry and R-symmetry to 2N =8 and Spin(4,4), respectively, for both theories (cf. [49] for Minkowski signature). More generally, any number N of supersymmetries gets doubled to 2N in the reduction. Since dimensional reduction collapses the rotation group Spin(2,2)\n\n∼ = Spin(2,1) L ×Spin(2,1) R of R 2,2\n\nto its diagonal subgroup Spin(2,1) D as the local rotation group of R 2,1 , the distinction between undotted and dotted indices disappears. We shall use undotted indices henceforth.\n\nCoordinates and derivatives in R 2,1 . The above discussion implies that one can relabel the bosonic coordinates x α β from (2.3) by x αβ and split them as\n\nx αβ = 1 2 (x αβ + x βα ) + 1 2 (x αβ -x βα ) = x (αβ) + x [αβ]\n\n(2.28) into antisymmetric and symmetric parts,\n\nx [αβ] = 1 2 ε αβ x 4 = 1 2 ε αβ t and x (αβ) =: y αβ , (2.29)\n\nrespectively, with\n\ny 11 = x 11 = 1 2 (t -y) , y 12 = 1 2 (x 12 + x 21 ) = 1 2 x , y 22 = x 22 = 1 2 (t + y) . (2.30)\n\nWe also have θ iα → θ iα and η α i → η α i for the fermionic coordinates on R 4|4N reduced to R 3|4N . Bosonic coordinate derivatives reduce in 2+1 dimensions to the operators\n\n∂ (αβ) = 1 2 (∂ αβ + ∂ βα ) (2.31)\n\nwhich read explicitly as\n\n∂ (11) = ∂ ∂y 11 = ∂ t -∂ y , ∂ (12) = ∂ (21) = 1 2 ∂ ∂y 12 = ∂ x , ∂ (22) = ∂ ∂y 22 = ∂ t + ∂ y . (2.32)\n\nWe thus have\n\n∂ ∂x αβ = ∂ (αβ) -ε αβ ∂ 4 = ∂ (αβ) -ε αβ ∂ t , (2.33)\n\nwhere\n\nε 12 = -ε 21 = -1, ∂ 4 = ∂/∂x 4 and ∂ t = ∂/∂ t.\n\nThe operators D iα and D i α acting on t-independent superfields reduce to\n\nD iα = ∂ iα + η β i ∂ (αβ) and D i α = ∂ i α + θ iβ ∂ (αβ) , (2.34)\n\nwhere\n\n∂ iα = ∂/∂θ iα and ∂ i α = ∂/∂η α i .\n\nSimilarly, the antichiral operators Diα and Di α in (2.18) become Diα = ∂ iα and Di α = ∂ i α + 2θ iβ ∂ (αβ) . (2.35) Supersymmetric Bogomolny-type equations in component fields. According to (2.33), the components A α β of a gauge potential in four dimensions split into the components A (αβ) of a gauge potential in three dimensions and a Higgs field\n\nA [αβ] = -ε αβ ϕ, i.e. A αβ = A (αβ) + A [αβ] = A (αβ) -ε αβ ϕ . (2.36)\n\nThen the covariant derivatives D α β reduced to three dimensions become the differential operators\n\nD αβ -ε αβ ϕ = ∂ (αβ) + [A (αβ) , • ] -ε αβ [ϕ, • ] , (2.37)\n\nand the Yang-Mills field strength on R 2,1 decomposes as\n\nF αβ, γδ = [D αβ , D γδ ] = ε αγ f βδ + ε βδ f αγ with f αβ = f βα . (2.38)\n\nSubstituting (2.36) and (2.37) into (2.10), i.e. demanding that all fields in (2.10) are independent of x 4 = t, we obtain the following supersymmetric Bogomolny-type equations on R 2,1 :\n\nf αβ + D αβ ϕ = 0 , (2.39a)\n\nD αβ χ iβ + ε αβ [ϕ, χ iβ ] = 0 , (2.39b)\n\nD αβ D αβ φ ij + 2[ϕ, [ϕ, φ ij ]] + 2{χ iα , χ j α } = 0 , (2.39c)\n\nD αβ χβ[ijk] -ε αβ [ϕ, χβ[ijk] ] -6[χ [i α , φ jk] ] = 0 , (2.39d)\n\nD γ α G [ijkl] γβ + [ϕ, G [ijkl] αβ ] + 12{χ [i α , χjkl] β } -18[φ [ij , D αβ φ kl] ] -18ε αβ [φ [ij , [φ kl] , ϕ]] = 0 .(2.\n\nSupersymmetric Bogomolny-type equations in terms of superfields. Translations generated by the vector field ∂ 4 = ∂ t are isometries of superspaces R 4|4N and R 4|2N . By taking the quotient with respect to the action of the abelian group G generated by ∂ 4 , we obtain the reduced full superspace R 3|4N ∼ = R 4|4N /G and the reduced antichiral superspace R 3|2N ∼ = R 4|2N /G. In the following, we shall work on R 3|2N and R 3|2N × CP 1 , since the reduced ψ-function from (2.25) and (2.27) is defined on the latter space.\n\nThe linear system stays in the center of the superfield approach to the N -extended SDYM equations. After imposing t-independence on all fields in the linear system (2.27), we arrive at the linear equations\n\nζ α ∇i α ψ ≡ ζ α ( Di α + Âi α ) ψ = 0 (2.40)\n\nof the same form but with\n\nDi α = ∂ i α + 2θ iβ ∂ (αβ) and Âi α = A i α + 2θ iβ (A (αβ) -ε αβ Ξ) , (2.41)\n\nwhere A i α , A (αβ) and Ξ are superfields depending on y αβ and η α i only. These linear equations expand again to the pair (cf. (2.25))\n\nζ β (∂ (αβ) + A (αβ) -ε αβ Ξ) ψ = 0 and ζ α (∂ i α + A i α ) ψ = 0 . (2.42)\n\nThe compatibility conditions for the linear system (2.40) read\n\n{ ∇i α , ∇j β } + { ∇i β , ∇j α } = 0 ⇔ { ∇i α , ∇j β } = ε αβ Fij (2.43)\n\nand present a condensed form of (2.39) rewritten in terms of R 3|2N superfields. Similarly, these equations can also be written in more expanded forms analogously to (2.21) or using the superfield analog of (2.37). However, we will not do this since all these sets of equations are equivalent.\n\nAs has been known for some time, nonlinear sigma models in 2 + 1 dimensions may be Lorentzinvariant or integrable but not both [7, 11] . We will show that the super Bogomolny-type model discussed in Section 2 after a gauge fixing is equivalent to a super extension of the modified U(n) chiral model (so as to be integrable) first formulated by Ward [7] . Since integrability is compatible with noncommutative deformation (if introduced properly, see e.g. [9] - [20] ) we choose from the beginning to formulate our super extension of this chiral model on Moyal-deformed R 2,1 with noncommutativity parameter θ ≥ 0. Ordinary space-time R 2,1 can always be restored by taking the commutative limit θ → 0.\n\nStar-product formulation. Classical field theory on noncommutative spaces may be realized in a star-product formulation or in an operator formalism 8 . The first approach is closer to the commutative field theory: it is obtained by simply deforming the ordinary product of classical fields (or their components) to the noncommutative star product\n\n(f ⋆ g)(x) = f (x) exp{ i 2 ← - ∂ a θ ab -→ ∂ b } g(x) ⇒ x a ⋆ x b -x b ⋆ x a = iθ ab (3.1)\n\nwith a constant antisymmetric tensor θ ab . Specializing to R 2,1 , we use real coordinates (x a ) = (t, x, y) in which the Minkowski metric g on R 3 reads (g ab ) = diag(-1, +1, +1) with a, b, . . . = 1, 2, 3 (cf. Section 2). It is straightforward to generalize the Moyal deformation (3.1) to the superspaces introduced in the previous section, allowing in particular for non-anticommuting Grassmann-odd coordinates. Deferring general superspace deformations and their consequences to future work, we here content ourselves with the simple embedding of the \"bosonic\" Moyal deformation into superspace, meaning that (3.1) is also valid for superfields f and g depending on Grassmann variables θ iα and η α i . For later use we consider not only isotropic coordinates and vector fields\n\nu := 1 2 (t+y) = y 22 , v := 1 2 (t-y) = y 11 , ∂ u = ∂ t + ∂ y = ∂ (22) , ∂ v = ∂ t -∂ y = ∂ (11) (3.2)\n\nintroduced in Section 2, but also the complex combinations\n\nz := x + iy , z := x -iy , ∂ z = 1 2 (∂ x -i∂ y ) , ∂ z = 1 2 (∂ x + i∂ y ) . (3.3)\n\nSince the time coordinate t remains commutative, the only nonvanishing component of the noncommutativity tensor θ ab is Operator formalism. The nonlocality of the star products renders explicit computation cumbersome. We therefore pass to the operator formalism, which trades the star product for operatorvalued spatial coordinates (x, ŷ) or their complex combinations (ẑ, ẑ), subject to\n\nθ xy = -θ yx =: θ > 0 ⇒ θ z z = -θ zz = -2i θ . ( 3\n\n[t, x] = [t, ŷ] = 0 but [x, ŷ] = iθ ⇒ [ẑ, ẑ] = 2 θ . (3.6)\n\nThe latter equation suggests the introduction of annihilation and creation operators,\n\na = 1 √ 2θ ẑ and a † = 1 √ 2θ ẑ with [a , a † ] = 1 , (3.7)\n\nwhich act on a harmonic-oscillator Fock space H with an orthonormal basis { |ℓ , ℓ = 0, 1, 2, . . .} such that\n\na |ℓ = √ ℓ |ℓ-1 and a † |ℓ = √ ℓ+1 |ℓ+1 . (3.8)\n\nAny superfield f (t, z, z, η α i ) on R 3|2N can be related to an operator-valued superfield f (t, η α i ) ≡ F (t, a, a † , η α i ) on R 1|2N acting in H, with the help of the Moyal-Weyl map\n\nf (t, z, z, η α i ) → f (t, η α i ) = Weyl-ordered f t, √ 2θa, √ 2θa † , η α i . (3.9)\n\nThe inverse transformation recovers the ordinary superfield,\n\nf (t, η α i ) ≡ F (t, a, a † , η α i ) → f (t, z, z, η α i ) = F ⋆ t, z √ 2θ , z √ 2θ , η α i , (3.10)\n\nwhere F ⋆ is obtained from F by replacing ordinary with star products. Under the Moyal-Weyl map, we have\n\nf ⋆ g → f ĝ and dx dy f = 2π θ Tr f = 2π θ ℓ≥0 ℓ| f |ℓ , (3.11)\n\nand the spatial derivatives are mapped into commutators,\n\n∂ z f → ∂z f = -1 √ 2θ [a † , f ] and ∂ z f → ∂z f = 1 √ 2θ [a , f ] . (3.12)\n\nFor notational simplicity we will from now on omit the hats over the operators except when confusion may arise.\n\nGauge fixing for ψ. Note that the linear system (2.40) and the compatibility conditions (2.43) are invariant under a gauge transformation\n\nψ → ψ ′ = g -1 ψ , (3.13a)\n\nA → A ′ = g -1 A g + g -1 ∂ g (with appropriate indices) , (3.13b)\n\nΞ → Ξ ′ = g -1 Ξ g , (3.13c)\n\nwhere g = g(x a , η α i ) is a U(n)-valued superfield globally defined on the deformed superspace R 3|2N θ × CP 1 . Using a gauge transformation of the form (3.13), we can choose ψ such that it will satisfy the standard asymptotic conditions (see e.g. [51] )\n\nψ = Φ -1 + O(ζ) for ζ → 0 , (3.14a\n\n)\n\nψ = 1l + ζ -1 Υ + O(ζ -2 ) for ζ → ∞ , (3.14b)\n\nwhere the U(n)-valued function Φ and u(n)-valued function Υ depend on x a and η α i . This \"unitary\" gauge is compatible with the reality condition for ψ,\n\nψ(x a , η α i , ζ) ψ(x a , η α i , ζ) † = 1l , (3.15)\n\nobtained by reduction from (2.26).\n\nA i 1 = 0 and A i 2 = Φ -1 ∂ i 2 Φ = ∂ i 1 Υ , (3.17)\n\nA (11) = 0 and\n\nA (12) + Ξ = Φ -1 ∂ (12) Φ = ∂ (11) Υ , (3.18)\n\nA (21) -Ξ = 0 and\n\nA (22) = Φ -1 ∂ (22) Φ = ∂ (12) Υ . (3.19)\n\nUsing (2.32), we can rewrite the nonzero components as\n\nA := Φ -1 ∂ u Φ = ∂ x Υ , B := Φ -1 ∂ x Φ = ∂ v Υ , C i := Φ -1 ∂ i 2 Φ = ∂ i 1 Υ . (3.20)\n\nRecall that the superfields Φ and Υ depend on x a and η α i .\n\nLinear system. In the above-introduced unitary gauge the linear system (2.42) reads\n\n(ζ∂ x -∂ u -A) ψ = 0 , (ζ∂ v -∂ x -B) ψ = 0 , (ζ∂ i 1 -∂ i 2 -C i ) ψ = 0 , (3.21)\n\nwhich adds the last equation to the linear system of the Ward model [7] and generalizes it to superfields A(x a , η α j ), B(x a , η α j ) and C i (x a , η α j ). The concise form of (3.21) reads\n\nζ Di 1 -Di 2 -Âi 2 ψ = 0 (3.22)\n\nor, in more explicit form,\n\nζ ∂ i 1 + 2θ i1 ∂ v + 2θ i2 ∂ x -∂ i 2 + C i + 2θ i1 (∂ x + B) + 2θ i2 (∂ u + A) ψ = 0 . (3.23)\n\nN -extended sigma model.\n\nThe compatibility conditions of this linear system are the Nextended noncommutative sigma model equations Di\n\n1 (Φ -1 Dj 2 Φ) + Dj 1 (Φ -1 Di 2 Φ) = 0 (3.24)\n\nwhich in expanded form reads\n\n(g ab + v c ε cab ) ∂ a (Φ -1 ∂ b Φ) = 0 ⇔ ∂ x (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ u Φ) = 0 , (3.25a) ∂ i 1 (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ i 2 Φ) = 0 , ∂ i 1 (Φ -1 ∂ u Φ) -∂ x (Φ -1 ∂ i 2 Φ) = 0 , (3.25b)\n\n∂ i 1 (Φ -1 ∂ j 2 Φ) + ∂ j 1 (Φ -1 ∂ i 2 Φ) = 0 . (3.25c)\n\nHere, the first line contains the Wess-Zumino-Witten term with a constant vector (v c ) = (0, 1, 0) which spoils the standard Lorentz invariance but yields an integrable chiral model in 2+1 dimensions. Recall that Φ is a U(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . As discussed in Section 2, the compatibility conditions of the linear equations (3.22) (or (3.21)) are equivalent to the N -extended Bogomolnytype equations (2.39) for the component (physical) fields. Thus, chiral model field equations (3.25) are equivalent to a gauge fixed form of equations (2.39). Υ-formulation. Instead of Φ-parametrization of (A, B, C i ) given in (3.17)-(3.20) we may use the equivalent Υ-parametrization also given there. In this case, the compatibility conditions for the linear system (3.21) reduce to\n\n(∂ 2 x -∂ u ∂ v )Υ + [∂ v Υ , ∂ x Υ] = 0 , (3.26a)\n\n(∂ i 2 ∂ v -∂ i 1 ∂ x )Υ + [∂ i 1 Υ , ∂ v Υ] = 0 , (∂ i 2 ∂ x -∂ i 1 ∂ u )Υ + [∂ i 1 Υ , ∂ x Υ] = 0 , (3.26b) (∂ i 2 ∂ j 1 + ∂ j 2 ∂ i 1 )Υ + {∂ i 1 Υ , ∂ j 1 Υ} = 0 , (3.26c)\n\nwhich in concise form read\n\n( Di 2 Dj 1 + Dj 2 Di 1 ) Υ + { Di 1 Υ , Dj 1 Υ} = 0 . (3.27)\n\nRecall that Υ is a u(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . For N =4, the commutative limit of (3.27) can be considered as Siegel's equation [33] reduced to 2+1 dimensions. According to Siegel, one can extract the multiplet of physical fields appearing in (2.39) from the prepotential Υ via\n\n∂ i 1 Υ = A i 2 , ∂ i 1 ∂ j 1 Υ = φ ij , ∂ i 1 ∂ j 1 ∂ k 1 Υ = χ[ijk] 2 , ∂ i 1 ∂ j 1 ∂ k 1 ∂ l 1 Υ = G [ijkl] 22 , (3.28a)\n\n∂ (α1) Υ = A (α2) -ε α2 ϕ , ∂ (α1) ∂ i 1 Υ = χ i α , ∂ (α1) ∂ (β1) Υ = f αβ , (3.28b)\n\nwhere one takes Υ and its derivatives at η 2 i = 0. The other components of the physical fields, i.e. χ[ijk]\n\n1 , G [ijkl] 11 , G [ijkl]\n\n21 , A (11) and A (21) -ϕ, vanish in this light-cone gauge.\n\nSupersymmetry transformations. The 4N supercharges given in (2.11) reduce in 2+1 dimensions to the form\n\nQ iα = ∂ iα -η β i ∂ (αβ) and Q i α = ∂ i α -θ iβ ∂ (αβ) . (3.29)\n\nTheir antichiral version, matching to Diα and Dj β of (2.35), reads\n\nQiα = ∂ iα -2η β i ∂ (αβ) and Qj β = ∂ j β , (3.30)\n\nso that { Qiα , Qj β } = -2 δ j i ∂ (αβ) . (3.31)\n\nOn a (scalar) R 3|2N superfield Σ these supersymmetry transformations act as\n\nδ Σ := ε iα Qiα Σ + ε α i Qi α Σ (3.32)\n\nand are induced by the coordinate shifts\n\nδ y αβ = -2ε i(α η β) i and δ η α i = ε α i , (3.33)\n\nwhere ε iα and ε α i are 4N real Grassmann parameters. It is easy to see that our equations (3.24) and (3.27) are invariant under the supersymmetry transformations (3.32) (applied to Φ or Υ). This is simply because the operators Diα and Dj β anticommute with the supersymmetry generators Qiα and Qj β . Therefore, the equations of motion (3.25) of the modified N -extended chiral model in 2+1 dimensions as well as their reductions to 2+0 and 1+1 dimensions carry 2N supersymmetries and are genuine supersymmetric extensions of the corresponding bosonic equations. Note that this type of extension is not the standard one since the R-symmetry groups are Spin(N , N ) in 2+1 and Spin(N , N )× Spin(N , N ) in 1+1 dimensions, which differ from the compact unitary R-symmetry groups of standard sigma models. Contrary to the standard case of two-dimensional sigma models the above \"noncompact\" 2N supersymmetries do not impose any constraints on the geometry of the target space, e.g. they do not demand it to be Kähler [52] or hyper-Kähler [53] . This may be of interest and deserves further study.\n\nAction functionals. In either formulation of the N -extended supersymmetric SDYM model on R 2,2 there are difficulties with finding a proper action functional generalizing the one [54, 55] for the purely bosonic case. These difficulties persist after the reduction to 2+1 dimensions, i.e. for the equations (3.25) and (3.26) describing our supersymmetric modified U(n) chiral model. It is the price to be paid for overcoming the no-go barrier N ≤ 4 and the absence of geometric target-space constraints. On a more formal level, the problem is related to the chiral character of (3.24) as well as (3.27) , where only the operators Di α but not Diα appear. Note however, that for N = 4 one can write an action functional in component fields producing the equations (2.39), which are equivalent to the superspace equations (3.24) when i, j = 1, . . . , 4 (see e.g. [47] ).\n\nOne proposal for an action functional stems from Siegel's idea [33] for the Υ-formulation of the N -extended SDYM equations. Namely, one sees that ∂ i 2 Υ enters only linearly into the last two lines in (3.26) . Therefore, if we introduce\n\nΥ (1) := Υ| η 2 i =0 (3.34)\n\nthen it must satisfy the first equation from (3.26) , and the remaining equations iteratively define the dependence of Υ on η 2 i starting from Υ (1) . Hence, all information is contained in Υ (1) , as can also be seen from (3.28) . In other words, the dependence of Υ on η 2 i is not 'dynamical'. For an action one can then take (cf. [33] )\n\nS = d 3 x d N η 1 Υ (1) ∂ (αβ) ∂ (αβ) Υ (1) + 2 3 Υ (1) ε αβ ∂ (α1) Υ (1) ∂ (β1) Υ (1) . (3.35)\n\nExtremizing this functional yields the first line of (3.26) at η 2 i = 0. Except for the Grassmann integration, this action has the same form as the purely bosonic one [55] . One may apply the same logic to the Φ-formulation where the action for the purely bosonic case is also known [54, 56] .\n\nThe existence of the linear system (3.22) (equivalent to (3.21)) encoding solutions of the N -extended U(n) chiral model in an auxiliary matrix ψ allows for powerful methods to systematically construct explicit solutions for ψ and hence for Φ † = ψ| ζ=0 and Υ = lim ζ→∞ ζ (ψ-1l). For our purposes the so-called dressing method [57, 51] proves to be the most practical [12] - [20] , and so we shall use it here for our linear system, i.e. already in the N -extended noncommutative case.\n\nMulti-pole ansatz for ψ. The dressing method is a recursive procedure for generating a new solution from an old one. More concretely, we rewrite the linear system (3.21) in the form\n\nψ(∂ u -ζ∂ x )ψ † = A , ψ(∂ x -ζ∂ v )ψ † = B , ψ(∂ i 2 -ζ∂ i 1 )ψ † = C i . (4.1)\n\nRecall that ψ † := (ψ(x a , η α i , ζ)) † and (A, B, C i ) depend only on x a and η α i . The central idea is to demand analyticity in the spectral parameter ζ, which strongly restricts the possible form of ψ. One way to exploit this constraint starts from the observation that the left hand sides of (4.1) as well as of the reality condition (3.15) do not depend on ζ while ψ is expected to be a nontrivial function of ζ globally defined on CP 1 . Therefore, it must be a meromorphic function on CP 1 possessing some poles which we choose to lie at finite points with constant coordinates µ k ∈ CP 1 .\n\nHere we will build a (multi-soliton) solution ψ m featuring m simple poles at positions µ 1 , . . . , µ m with foot_8 Im µ k < 0 by left-multiplying an (m-1)-pole solution ψ m-1 with a single-pole factor of the form 1l\n\n+ µ m -μm ζ -µ m P m (x a , η α i ) , (4.2)\n\nwhere the n×n matrix function P m is yet to be determined. Starting from the trivial (vacuum) solution ψ 0 = 1l, the iteration ψ 0 → ψ 1 → . . . → ψ m yields a multiplicative ansatz for ψ m ,\n\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ , (4.3)\n\nwhich, via partial fraction decomposition, may be rewritten in the additive form\n\nψ m = 1l + m k=1 Λ mk S † k ζ -µ k , (4.4)\n\nwhere Λ mk and S k are some n×r k matrices depending on x a and η α i , with r k ≤ n.\n\nEquations for S k . Let us first consider the additive parametrization (4.4) of ψ m . This ansatz must satisfy the reality condition (3.15) as well as our linear equations in the form (4.1). In particular, the poles at ζ = μk on the left hand sides of these equations have to be removable since the right hand sides are independent of ζ. Inserting the ansatz (4.4) and putting to zero the corresponding residues, we learn from (3.15) that\n\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ S k = 0 , (4.5)\n\nwhile from (4.1) we obtain the differential equations\n\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ LA,B,i k S k = 0 , (4.6)\n\nwhere LA,B,i\n\nk stands for either LA k = ∂ u -μk ∂ x , LB k = µ k (∂ x -μk ∂ v ) or Li k = ∂ i 2 -μk ∂ i 1 . (4.7)\n\nNote that we consider a recursive procedure starting from m=1, and operators (4.7) will appear with k = 1, . . . , m if we consider poles at ζ = μk .\n\nBecause the LA,B,i k for k = 1, . . . , m are linear differential operators, it is easy to write down the general solution for (4.6) at any given k, by passing from the coordinates (u, v, x; η 1 i , η 2 i ) to \"co-moving coordinates\" (w k , wk , s k ; η i k , ηi k ). The precise relation for k = 1, . . . , m is [12, 58]\n\nw k := x + μk u + μ-1 k v = x + 1 2 (μ k -μ -1 k )y + 1 2 (μ k +μ -1 k )t and η i k := η 1 i + μk η 2 i , (4.8)\n\nwith wk and ηi k obtained by complex conjugation and the co-moving time s k being inessential because by definition nothing will depend on it. The kth moving frame travels with a constant velocity\n\n(v x , v y ) k = - µ k + μk µ k μk + 1 , µ k μk -1 µ k μk + 1 , (4.9)\n\nso that the static case w k =z is recovered for µ k = -i. On functions of (w k , η i k , wk , ηi k ) alone the operators (4.7) act as\n\nLA k = LB k = (µ k -μ k ) ∂ ∂ wk =: Lk and Li k = (µ k -μ k ) ∂ ∂ ηi k . (4.10)\n\nBy induction in k = 1, . . . , m we learn that, due to (4.5), a necessary and sufficient condition for a solution of (4.6) is Lk\n\nS k = S k Zk and Li k S k = S k Zi k (4.11)\n\nwith some r k ×r k matrices Zk and Zi k depending on (w k , wk , η j k , ηj k ). Passing to the noncommutative bosonic coordinates we obtain ŵk , ŵk = 2θ ν k νk with\n\nν k νk = 4i µ k -μ k -µ -1 k +μ -1 k . (4.12)\n\nThus, we can introduce annihilation and creation operators\n\nc k = 1 √ 2θ ŵk ν k and c † k = 1 √ 2θ ŵk νk so that [c k , c † k ] = 1 (4.13)\n\nfor k = 1, . . . , m. Naturally, this Heisenberg algebra is realized on a \"co-moving\" Fock space H k , with basis states |ℓ k and a \"co-moving\" vacuum |0 k subject to c k |0 k = 0. Each co-moving vacuum |0 k (annihilated by c k ) is related to the static vacuum |0 (annihilated by a) through an ISU(1,1) squeezing transformation (cf. [12] ) which is time-dependent. The fermionic coordinates η i k and ηi k remain spectators in the deformation. Coordinate derivatives are represented in the standard fashion as\n\nν k √ 2θ ∂ ∂w k → -[c † k , • ] and νk √ 2θ ∂ ∂ wk → [c k , • ] . (4.14)\n\nAfter the Moyal deformation, the n×r k matrices S k have become operator-valued, but are still functions of the Grassmann coordinates η i k and ηi k . The noncommutative version of the BPS conditions (4.11) naturally reads\n\nc k S k = S k Z k and ∂ ∂ ηi k S k = S k Z i k (4.15)\n\nwhere Z k and Z i k are some operator-valued r k ×r k matrix functions of η j k and ηj k .\n\nNonabelian solutions for S k . For general data Z k and Z i k it is difficult to solve (4.15), but it is also unnecessary because the final expression ψ m turns out not to depend on them. Therefore, we conveniently choose\n\nZ k = c k ⊗ 1l r k ×r k and Z i k = 0 ⇒ S k = R k (c k , η i k ) , (4.16)\n\nwhere R k is an arbitrary n×r k matrix function independent of c † k and ηi k . 10 It is known that nonabelian (multi-) solitons arise for algebraic functions R k (cf. e.g. [7] for the commutative and [12] for the noncommutative N =0 case). Their common feature is a smooth commutative limit. The only novelty of the supersymmetric extension is the η i k dependence, i.e.\n\nR k = R k,0 + η i k R k,i + η i k η j k R k,ij + η i k η j k η p k R k,ijp + η i k η j k η p k η q k R k,ijpq . (4.17)\n\nAbelian solutions for S k . It is useful to view S k as a map from C r k ⊗H k to C n ⊗H k (momentarily suppressing the η dependence). The noncommutative setup now allows us to generalize the domain of this map to any subspace of C n ⊗ H k . In particular, we may choose it to be finite-dimensional, say C q k , and represent the map by an n×q k array |S k of kets in H. In this situation, Z k and Z i k in (4.15) are just number -valued q k ×q k matrix functions of η j k and ηj k . In case they do not depend on ηj k , we can write down the most general solution as\n\n|S k = R k (c k , η j k ) |Z k exp i Z i k (η j k ) ηi k with |Z k := exp Z k (η j k ) c † k |0 k . (4.18)\n\nAs before, we may put Z i k = 0 without loss of generality, but now the choice of Z k does matter. For any given k generically there exists a q k -dimensional basis change which diagonalizes the ket-valued matrix\n\n|Z k → diag e α 1 k c † , e α 2 k c † , . . . , e α q k k c † |0 k = diag |α 1 k , |α 2 k , . . . , |α q k k , (4.19)\n\nwhere we defined coherent states\n\n|α l k := e α l k c † |0 k so that c k |α l k = α l k |α l k for l = 1, . . . , q k and α l k ∈ C . (4.20)\n\nNote that not only the entries of R k but also the α l k are holomorphic functions of the co-moving Grassmann parameters η j k and thus can be expanded like in (4.17). In the U(1) model, we must use ket-valued 1×q k matrices |S k for all k, yielding rows\n\n|S k = R 1 k |α 1 k , R 2 k |α 2 k , . . . , R q k k |α q k k for k = 1, . . . , m , (4.21)\n\nwith functions α l k (η j k ). Here, the R l k only affect the states' normalization and can be collected in a diagonal matrix to the right, hence will drop out later and thus may all be put to one. Formally, we have recovered the known abelian (multi-) soliton solutions, but the supersymmetric extension has generalized |S k → |S k (η j k ) . Explicit form of P k . Let us now consider the multiplicative parametrization (4.3) of ψ m which also allows us to solve (4.5) . First of all, note that the reality condition (3.15) is satisfied if\n\nP k = P † k = P 2 k ⇔ P k = T k (T † k T k ) -1 T † k for k = 1, . . . , m , (4.22)\n\nmeaning that P k is an operator-valued hermitian projector (of group-space rank r k ≤ n) built from an n×r k matrix function T k (the abelian case of n=1 is included). The reality condition follows just because The r k columns of T k span the image of P k and obey\n\n1l + µ k -μk ζ -µ k P k 1l + μk -µ k ζ -μk P k =\n\nP k T k = T k ⇔ (1l-P k ) T k = 0 . (4.24)\n\nFurthermore, the equation (4.5) with m = k (induction) rewritten in the form\n\n(1l-P k ) k-1 ℓ=1 1l + µ k-ℓ -μk-ℓ μk -µ k-ℓ P k-ℓ S k = 0 (4.25)\n\nreveals that (cf. (4.24))\n\nT 1 = S 1 and T k = k-1 ℓ=1 1l - µ k-ℓ -μk-ℓ µ k-ℓ -μk P k-ℓ S k for k ≥ 2 , (4.26)\n\nwhere the explicit form of S k for k = 1, . . . , m is given in (4.16) or (4.18) . The final result reads\n\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ = 1l + m k=1 Λ mk S † k ζ -µ k (4.27)\n\nwith hermitian projectors P k given by (4.22), T k given by (4.26) and S k given by (4.16) or (4.18) . The explicit form of Λ mk (which we do not need) can be found in [12] . The corresponding superfields Φ and Υ are\n\nΦ m = ψ † m | ζ=0 = m k=1 (1l -ρ k P k ) with ρ k = 1 - µ k μk , (4.28a)\n\nΥ m = lim ζ→∞ ζ (ψ m -1l) = m k=1 (µ k -μ k ) P k . (4.28b)\n\nFrom (4.22) it is obvious that P k is invariant under a similarity transformation\n\nT k → T k Λ k ⇔ S k → S k Λ k (4.29)\n\nfor an invertible operator-valued r k ×r k matrix Λ k . This justifies putting Z i k = 0 from the beginning and also the restriction to Z k = c k ⊗ 1l r k ×r k in the nonabelian case, both without loss of generality. Hence, the nonabelian solution space constructed here is parametrized by the set {R k } m 1 of matrixvalued functions of c k and η i k and the pole positions µ k . The abelian moduli space, however, is larger by the set {Z k } m 1 of matrix-values functions of η i k which generically contain the coherentstate parameter functions {α l k (η i k )}. Restricting to η i k =0 reproduces the soliton configurations of the bosonic model [12] .\n\nStatic solutions. Let us consider the reduction to 2+0 dimensions, i.e. the static case. Recall that static solutions correspond to the choice m = 1 and µ 1 ≡ µ = -i implying w 1 = z, so we drop the index k. Specializing (4.27), we have\n\nψ = 1l - 2 i ζ + i P so that Φ = Φ † = 1l -2P , (4.30)\n\nwhere a hermitian projector P of group-space rank r satisfies the BPS equations (1l-P ) a P = 0 ⇒ (1l-P ) a T = 0 ,\n\n(1l-P ) ∂ ∂ ηi P = 0 ⇒ (1l-P ) ∂ ∂ ηi T = 0 , (4.31a)\n\nwith P = T (T † T ) -1 T † and η i = η 1 i + iη 2 i . In this case T = S, and for a nonabelian r=1 projector P we get T = T (a, η i ) as an n×1 column. For the simplest case of N =1 we just have (cf. [59] )\n\nT = T e (a) + η T o (a) with η = η 1 + iη 2 , (4.32)\n\nwhere T e (a) and T o (a) are rational functions of a (e.g. polynomials) taking values in the even and odd parts of the Grassmann algebra. Similarly, an abelian N =1 projector (for n=1) is built from\n\n|T = |α 1 , |α 2 , . . . , |α q . (4.33)\n\nAt θ=0, the static solution (4.32) of our supersymmetric U(n) sigma model is also a solution of the standard N =1 supersymmetric CP n-1 sigma model in two dimensions (see e.g. [59] ). 11 For this reason, one can overcome the previously mentioned difficulty with constructing an action (or energy from the viewpoint of 2+1 dimensions) for static configurations. Moreover, on solutions obeying the BPS conditions (4.31) the topological charge\n\nQ = 2πθ dη 1 dη 2 Tr tr Φ D + Φ , D -Φ (4.34)\n\nis proportional to the action (BPS bound)\n\nS = 2πθ dη 1 dη 2 Tr tr D + Φ , D -Φ (4.35)\n\nand is finite for algebraic functions T e and T o . Here, the standard superderivatives D ± are defined as\n\nD + = ∂ ∂η + iη ∂ z and D -= ∂ ∂ η + iη ∂ z . ( 4\n\nOne-soliton configuration. For one moving soliton, from (4.27) and (4.28) we obtain\n\nψ 1 = 1l + µ - μ ζ -µ P with P = T (T † T ) -1 T † (4.37) and Φ = 1l -ρ P with ρ = 1 - µ μ . (4.38)\n\nNow our n×r matrix T must satisfy (putting\n\nZ i = 0 and Z = c ⊗ 1l r×r ) [c , T ] = 0 and ∂ ∂ ηi T = 0 with η i = η 1 i + μ η 2 i , (4.39)\n\nwhere c is the moving-frame annihilation operator given by (4.13) for k=1.\n\nRecall that the operators c and c † and therefore the matrix T and the projector P can be expressed in terms of the corresponding static objects by a unitary squeezing transformation (see e.g. (4.8) and (4.13)). For simplicity we again consider the case N =1 and a nonabelian projector with r=1. Then (4.39) tells us that T is a holomorphic function of c and η, i.e.\n\nT = T e (c) + η T o (c) = T 1 e (c) + η T 1 o (c)\n\n. . .\n\nT n e (c) + η T n o (c) (4.40)\n\nwith polynomials T a e and T a o of order q, say, analogously to the static case (4.32). Note that, for T a o to be Grassmann-odd and nonzero, some extraneous Grassmann parameter must appear. Similarly, abelian projectors for a moving one-soliton obtain by subjecting (4.33) to a squeezing transformation.\n\nFor N =1 the moving frame was defined in (4.8) (dropping the index k) via\n\nw = x + 1 2 (μ-μ -1 )y + 1 2 (μ+μ -1 )t and η = η 1 + μη 2 hence ∂ t η = 0 . (4.41)\n\nConsider the moving frame with the coordinates (w, w, s; η, η) with the choice s = t and the related change of the derivatives (see [12, 58] )\n\n∂ x = ∂ w + ∂ w , (4.42a)\n\n∂ y = 1 2 (μ-μ -1 ) ∂ w + 1 2 (µ-µ -1 ) ∂ w , (4.42b)\n\n∂ t = 1 2 (μ+μ -1 ) ∂ w + 1 2 (µ+µ -1 ) ∂ w + ∂ s , (4.42c)\n\n∂ η 1 = ∂ η + ∂ η , (4.42d)\n\n∂ η 2 = μ ∂ η + µ ∂ η . (4.42e)\n\nIn the moving frame our solution (4.38) is static, i.e. ∂ s Φ = 0, and the projector P has the same form as in the static case. The only difference is the coefficient ρ instead of 2 in (4.38). Therefore, by computing the action (4.35) in (w, w; η 1 , η 2 ) coordinates, we obtain for algebraic functions T in (4.40) a finite answer, which differs from the static one by a kinematical prefactor depending on µ (cf. [12] for the bosonic case).\n\nLarge-time asymptotics. Note that in the distinguished (z, z, t) coordinate frame (4.41) implies that at large times w → κ t with κ = 1 2 (μ+μ -1 ). As a consequence, the t q term in each polynomial in (4.40) will dominate, i.e.\n\nT → t q a 1 + η b 1 . . .\n\nan + η bn =: t q Γ , (4.43)\n\nwhere Γ is a fixed vector in C n . It is easy to see that in the distinguished frame the large-time limit of Φ given by (4. being the projector on the constant vector Γ.\n\nConsider now the m-soliton configuration (4.28). By induction of the above argument one easily arrives at the m-soliton generalization of (4.44). Namely, in the frame moving with the ℓth lump we have lim t→±∞ Φ m = (1lρ 1 Π 1 ) . . . (1lρ ℓ-1 Π ℓ-1 )(1lρ ℓ P ℓ )(1lρ ℓ+1 Π ℓ+1 ) . . . (1lρ m Π m ) , (4.45) where the Π m are constant projectors. This large-time factorization of multi-soliton solutions provides a proof of the no-scattering property because the asymptotic configurations are identical for large negative and large positive times.\n\nIn this paper we introduced a generalization of the modified integrable U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions and considered a Moyal deformation of this model. It was shown that this N -extended chiral model is equivalent to a gauge-fixed BPS subsector of an N -extended super Yang-Mills model in 2+1 dimensions originating from twistor string theory. The dressing method was applied to generate a wide class of multi-soliton configurations, which are time-dependent finite-energy solutions to the equations of motion. Compared to the N =0 model, the supersymmetric extension was seen to promote the configurations' building blocks to holomorphic functions of suitable Grassmann coordinates. By considering the large-time asymptotic factorization into a product of single soliton solutions we have shown that no scattering occurs within the dressing ansatz chosen here.\n\nThe considered model does not stand alone but is motivated by twistor string theory [37] with a target space reduced to the mini-supertwistor space [44, 45, 47] . In this context, the obtained multi-soliton solutions are to be regarded as D(0|2N )-branes moving inside D(2|2N )-branes [60] . Here 2N appears due to fermionic worldvolume directions of our branes in the superspace description [60] . Switching on a constant B-field simply deforms the sigma model and D-brane worldvolumes noncommutatively, thereby admitting also regular supersymmetric noncommutative abelian solutions.\n\nRestricting to static configurations, the models can be specialized to Grassmannian supersymmetric sigma models, where the superfield Φ takes values in Gr(r, n), and the field equations are invariant under 2N supersymmetry transformations with 0 ≤ N ≤ 4. This differs from the results for standard 2D sigma models [52, 53] where the target spaces have to be Kähler or hyper-Kähler for admitting two or four supersymmetries, respectively. This difference will be discussed in more details elsewhere.\n\nWe derived the supersymmetric chiral model in 2+1 dimensions through dimensional reduction and gauge fixing of the N -extended supersymmetric SDYM equations in 2+2 dimensions. Recall that for the purely bosonic case most (if not all) integrable equations in three and fewer dimensions can be obtained from the SDYM equations (or their hierarchy [25] ) by suitable dimensional reductions (see e.g. [61] - [65] and references therein). Moreover, this Ward conjecture [61] was extended to the noncommutative case (see e.g. [66, 67] ). It will be interesting to consider similar reductions of the N -extended supersymmetric SDYM equations (and their hierarchy [68] ) to supersymmetric integrable equations in three and two dimensions generalizing earlier results [69] ."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"We consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from twistor string theory. By a gauge fixing this model is reduced to a modified U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions. After a Moyal-type deformation of the model, we employ the dressing method to explicitly construct multi-soliton configurations on noncommutative R 2,1 and analyze some of their properties."},{"section_type":"BACKGROUND","section_title":"Introduction","text":"In the low-energy limit string theory with D-branes gives rise to noncommutative field theory on the branes when the string propagates in a nontrivial NS-NS two-form (B-field) background [1, 2, 3, 4] . In particular, if the open string has N =2 worldsheet supersymmetry, the tree-level target space dynamics is described by a noncommutative self-dual Yang-Mills (SDYM) theory in 2+2 dimensions [5] . Furthermore, open N =2 strings in a B-field background induce on the worldvolume of n coincident D2-branes a noncommutative Yang-Mills-Higgs Bogomolny-type system in 2+1 dimensions which is equivalent to a noncommutative generalization [6] of the modified U(n) chiral model known as the Ward model [7] . The topological nature of N =2 strings and the integrability of their tree-level dynamics [8] render this noncommutative sigma model integrable. 1 Being integrable, the commutative U(n≥2) Ward model features a plethora of exact scattering and no-scattering multi-soliton and wave solutions, i.e. time-dependent stable configurations on R foot_1 . These are not only a rich testing ground for physical properties such as adiabatic dynamics or quantization, but also descend to more standard multi-solitons of various integrable systems in 2+0 and 1+1 dimensions, such as sine-Gordon, upon dimensional and algebraic reduction. There is a price to pay however: Nonlinear sigma models in 2+1 dimensions may be Lorentz-invariant or integrable but not both [7, 11] . In fact, Derrick's theorem prohibits the existence of stable solitons in Lorentz-invariant scalar field theories above 1+1 dimensions. A Moyal deformation, however, overcomes this hurdle, but of course replaces Lorentz invariance by a Drinfeld-twisted version. There is another gain: The deformed Ward model possesses not only deformed versions of the just-mentioned multi-solitons, but in addition allows for a whole new class of genuinely noncommutative (multi-)solitons, in particular for the U(1) group [12, 13] ! Moreover, this class is related to the generic but perturbatively constructed noncommutative scalar-field solitons [14, 15] by an infinite-stiffness limit of the potential [16] .\n\nIn [12, 13] and [17] - [20] families of multi-solitons as well as their reduction to solitons of the noncommutative sine-Gordon equations were described and studied. In the nonabelian case both scattering and nonscattering configurations were obtained. For static configurations the issue of their stability was analyzed [21] . The full moduli space metric for the abelian model was computed and its adiabatic two-soliton dynamics was discussed [16] .\n\nRecall that the critical N =2 string theory has a four-dimensional target space, and its open string effective field theory is self-dual Yang-Mills [8] , which gets deformed noncommutatively in the presence of a B-field [5] . Conversely, the noncommutative SDYM equations are contained [19] in the equations of motion of N =2 string field theory (SFT) [22] in a B-field background. This SFT formulation is based on the N =4 topological string description [23] . It is well known that the SDYM model can be described in terms of holomorphic bundles over (an open subset of) the twistor space 2 [26] CP 3 and the topological N =4 string theory contains twistors from the outset. The Lax pair, integrability and the solutions to the equations of motion by twistor and dressing methods were incorporated into the N =2 open SFT in [27, 28] . However, this theory reproduces only bosonic SDYM theory, its symmetries (see e.g. [29, 30, 31] ) and integrability properties. It is natural to ask: What string theory can describe supersymmetric SDYM theory [32, 33] in four dimensions?\n\nThere are some proposals [33, 34, 35, 36] for extending N =2 open string theory (and its SFT) to be space-time supersymmetric. Moreover, it was shown by Witten [37] that N =4 supersymmetric SDYM theory appears in twistor string theory, which is a B-type open topological string with the supertwistor space CP 3|4 as a target space. 3 Note that N <4 SDYM theory forms a BPS subsector of N -extended super Yang-Mills theory, and N =4 SDYM can be considered as a truncation of the full N =4 super Yang-Mills theory [37] . It is believed [43, 39] that twistor string theory is related with the previous proposals [33, 34, 35, 36] for a Lorentz-invariant supersymmetric extension of N =2 (and topological N =4) string theory which also leads to the N =4 SDYM model.\n\nA dimensional reduction of the above relations between twistor strings and N =4 super Yang-Mills and SDYM models was considered in [44, 45, 46, 47] . The corresponding twistor string theory after this reduction is the topological B-model on the mini-supertwistor space P 2|4 . In [47] it was shown that the 2N =8 supersymmetric extension of the Bogomolny-type model in 2+1 dimensions is equivalent to an 2N =8 supersymmetric modified U(n) chiral model on R 2,1 . The subject of the current paper is an 2N ≤8 version of the above supersymmetric Bogomolny-type Yang-Mills-Higgs model in signature (-+ +), its relation with an N -extended supersymmetric modified integrable U(n) chiral model (to be defined) in 2+1 dimensions and the Moyal-type noncommutative deformation of this chiral model. We go on to explicitly construct multi-soliton configurations on noncommutative R 2,1 for the corresponding supersymmetric sigma model field equations. By studying the scattering properties of the constructed configurations, we prove their asymptotic factorization without scattering for large times. We also briefly discuss a D-brane interpretation of these soliton configurations from the viewpoint of twistor string theory."},{"section_type":"OTHER","section_title":"N -extended SDYM equations in 2+2 dimensions","text":"Space R 2,2 . Let us consider the four-dimensional space R 2,2 = (R 4 , g) with the metric\n\nds 2 = g µν dx µ dx ν = det(dx α α) = dx 1 1dx 2 2 -dx 2 1dx 1 2 (2.1)\n\nwith (g µν ) = diag(-1, +1, +1, -1), where µ, ν, . . . = 1, . . . , 4 are space-time indices and α = 1, 2, α = 1, 2 are spinor indices. We choose the coordinates 4\n\n(x µ ) = (x a , t) = (t, x, y, t) with a, b, . . . = 1, 2, 3 , (2.2)\n\nand the signature (-+ + -) allows us to introduce real isotropic coordinates (cf. [19, 6] )\n\nx 1 1 = 1 2 (t -y) , x 1 2 = 1 2 (x + t) , x 2 1 = 1 2 (x -t) , x 2 2 = 1 2 (t + y) . (2.3)\n\nSDYM. Recall that the SDYM equations for a field strength tensor\n\nF µν on R 2,2 read 1 2 ε µνρσ F ρσ = F µν , (2.4)\n\nwhere ε µνρσ is a completely antisymmetric tensor on R 2,2 and ε 1234 = 1. In the coordinates (2.3) we have the decomposition\n\nF α α,β β = ∂ α αA β β -∂ β β A α α + [A α α, A β β ] = ε αβ F α β + ε α β F αβ (2.5) with F α β := -1 2 ε αβ F α α,β β and F αβ := -1 2 ε α β F α α,β β , (2.6)\n\nwhere ε αβ is antisymmetric, ε αβ ε βγ = δ γ α , and similar for ε α β , with ε 12 = ε 1 2 = 1. The gauge potential (A α α) will appear in the covariant derivative\n\nD α β = ∂ α β + [A α β , • ] . (2.7)\n\nIn spinor notation, (2.4) is equivalently written as\n\nF α β = 0 ⇔ F α α,β β = ε α β F αβ . (2.8)\n\nSolutions {A α α} to these equations form a subset (a BPS sector) of the solution space of Yang-Mills theory on R 2,2 .\n\nN -extended SDYM in component fields. The field content of N -extended super SDYM is foot_4\n\nN = 0 A α α (2.9a) N = 1 A α α, χ i α with i = 1 (2.9b) N = 2 A α α, χ i α , φ [ij] with i, j = 1, 2 (2.9c)\n\nN = 3 A α α, χ i α , φ [ij] , χ[ijk] α with i, j, k = 1, 2, 3 (2.9d) N = 4 A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl] α β\n\nwith i, j, k, l = 1, 2, 3, 4 .\n\n(2.9e)\n\nHere (A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl]\n\nα β ) are fields of helicities (+1, + 1 2 , 0, -1 2 , -1). These fields obey the field equations of the N = 4 SDYM model, namely [33, 37]\n\nF α β = 0 , (2.10a)\n\nD α αχ iα = 0 , (2.10b)\n\nD α αD α αφ ij + 2{χ iα , χ j α } = 0 , (2.10c)\n\nD α α χ α[ijk] -6[χ [i α , φ jk] ] = 0 , (2.10d)\n\nD γ α G [ijkl] γ β + 12{χ [i α , χjkl] β } -18[φ [ij , D α β φ kl] ] = 0 . (2.10e)\n\nNote that the N < 4 SDYM field equations are governed by the first N +1 equations of (2.10), where F α β = 0 is counted as one equation and so on."},{"section_type":"OTHER","section_title":"Superfield formulation of","text":"N -extended SDYM Superspace R 4|4N .\n\nRecall that in the space R 2,2 = (R 4 , g) with the metric g given in (2.1) one may introduce purely real Majorana-Weyl spinors foot_5 θ α and η α of helicities + 1 2 and -1 2 as anticommuting (Grassmann-algebra) objects. Using 2N such spinors with components θ iα and η α i for i = 1, . . . , N , one can define the N -extended superspace R 4|4N and the N -extended supersymmetry algebra generated by the supertranslation operators\n\nP α α = ∂ α α , Q iα = ∂ iα -η α i ∂ α α and Q i α = ∂ i α -θ iα ∂ α α , (2.11)\n\nwhere\n\n∂ α α := ∂ ∂x α α , ∂ iα := ∂ ∂θ iα and ∂ i α := ∂ ∂η α i .\n\n(2.12)\n\nThe commutation relations for the generators (2.11) read\n\n{Q iα , Q j α} = -2δ j i P α α , [P α α, Q iβ ] = 0 and [P α α, Q i β ] = 0 . (2.13)\n\nTo rewrite equations of motion in terms of R 4|4N superfields one uses the additional operators\n\nD iα = ∂ iα + η α i ∂ α α and D i α = ∂ i α + θ iα ∂ α α , (2.14)\n\nwhich (anti)commute with the operators (2.11) and satisfy\n\n{D iα , D j β } = 2δ j i P α β , [P α α, D iβ ] = 0 and [P α α, D j β ] = 0 . (2.15)\n\nAntichiral superspace R 4|2N . On the superspace R 4|4N one may introduce tensor fields depending on bosonic and fermionic coordinates (superfields), differential forms, Lie derivatives L X etc.. Furthermore, on any such superfield A one can impose the constraint equations L D iα A = 0, which for a scalar superfield f reduce to the so-called antichirality conditions\n\nD iα f = 0 . (2.16)\n\nThese are easily solved by using a coordinate transformation on R 4|4N ,\n\n(x α α, η α i , θ iα ) → (x α α = x α α-θ iα η α i , η α i , θ iα ) , (2.17)\n\nunder which ∂ α α, D iα and D i α transform to the operators\n\n∂α α = ∂ α α , Diα = ∂ iα and Di α = ∂ i α + 2θ iα ∂ α α . ( 2\n\n.18) Then (2.16) simply means that f is defined on a sub-superspace R 4|2N ⊂ R 4|4N with coordinates xα α and η α i . (2.19)\n\nThis space is called antichiral superspace. In the following we will usually omit the tildes when working on the antichiral superspace.\n\nN -extended SDYM in superfields. The N -extended SDYM equations can be rewritten in terms of superfields on the antichiral superspace R 4|2N [33, 48] . Namely, for any given 0 ≤ N ≤ 4, fields of a proper multiplet from (2.9) can be combined into superfields A α α and A i α depending on x α α, η α i ∈ R 4|2N and giving rise to covariant derivatives\n\n∇ α α := ∂ α α + A α α and ∇ i α := ∂ i α + A i α .\n\n(2.20)\n\nIn such terms the N -extended SDYM equations (2.10) read\n\n[∇ α α, ∇ β β ] + [∇ α β , ∇ β α] = 0 , [∇ i α, ∇ β β ] + [∇ i β , ∇ β α] = 0 , {∇ i α, ∇ j β } + {∇ i β , ∇ j α} = 0 , (2.21)\n\nwhich is equivalent to\n\n[∇ α α, ∇ β β ] = ε α β F αβ , [∇ i α, ∇ β β ] = ε α β F i β and {∇ i α, ∇ j β } = ε α β F ij , (2.22)\n\nwhere F ij is antisymmetric and F αβ is symmetric in their indices.\n\nThe above gauge potential superfields (A α α, A i α) as well as the gauge strength superfields (F αβ , F i α , F ij ) contain all physical component fields of the N -extended SDYM model. For instance, the lowest component of the triple (F αβ , F i α , F ij ) in an η-expansion is (F αβ , χ i α , φ ij ), with zeros in case N is too small. By employing Bianchi identities for the gauge strength superfields, one successively obtains [48] the superfield expansions and the field equations (2.10) for all component fields.\n\nIt is instructive to extend the antichiral combination in (2.18) to potentials and covariant derivatives, Di\n\nα = ∂ i α + 2 θ iα ∂ α α + + + Ãi α := A i α + 2 θ iα A α α ∇i α := ∇ i α + 2 θ iα ∇ α α ( 2\n\n.23) where ∇ α α, ∇ i α and Di α are given by (2.20) and (2.18), while A i α and A α α depend on x α α and η α i only. With the antichiral covariant derivatives, one may condense (2.21) or (2.22) into the single set { ∇i α, ∇j β\n\n} + { ∇i β , ∇j α} = 0 ⇔ { ∇i α, ∇j β } = ε α β Fij , (2.24)\n\nwith Fij =\n\nF ij + 4 θ [iα F j] α + 4 θ iα θ jβ F αβ .\n\nThe concise form (2.24) of the N -extended SDYM equations is quite convenient, and we will use it interchangeable with (2.21).\n\nLinear system for N -extended SDYM. It is well known that the superfield SDYM equations (2.21) can be seen as the compatibility conditions for the linear system of differential equations\n\nζ α(∂ α α + A α α) ψ = 0 and ζ α(∂ i α + A i α) ψ = 0 , (2.25)\n\nwhere\n\n(ζ β ) = 1 ζ and ζ α = ε α β ζ β . The extra (spectral) parameter 7 ζ lies in the extended complex plane C ∪ ∞ = CP 1 .\n\nHere ψ is a matrix-valued function depending not only on x α α and η α i but also (meromorphically) on ζ ∈ CP 1 . We subject the n×n matrix ψ to the following reality condition:\n\nψ(x α α, η α i , ζ) ψ(x α α, η α i , ζ) † = 1l , (2.26)\n\nwhere \" †\" denotes hermitian conjugation and ζ is complex conjugate to ζ. This condition guarantees that all physical fields of the N -extended SDYM model will take values in the adjoint representation of the algebra u(n). In the concise form the linear system (2.25) is written as\n\nζ α(∇ i α + 2θ iα ∇ α α) ψ = 0 ⇔ ζ α( Di α + Ãi α) ψ = 0 ⇔ ζ α ∇i α ψ = 0 . (2.27) 2.3 Reduction of N -extended SDYM to 2+1 dimensions\n\nThe supersymmetric Bogomolny-type Yang-Mills-Higgs equations in 2+1 dimensions are obtained from the described N -extended super SDYM equations by a dimensional reduction R 2,2 → R 2,1 .\n\nIn particular, for the N =0 sector we demand the components A µ of a gauge potential to be independent of x 4 and put A 4 =: ϕ. Here, ϕ is a Lie-algebra valued scalar field in three dimensions (the Higgs field) which enters into the Bogomolny-type equations. Similarly, for N ≥ 1 one can reduce the N -extended SDYM equations on R 2,2 by imposing the ∂ 4 -invariance condition on all the fields (A α α,\n\nχ i α , φ [ij] , χ[ijk] α , G [ijkl]\n\nα β ) from the N =4 supermultiplet or its truncation to N <4 and obtain supersymmetric Bogomolny-type equations on R 2,1 .\n\nSpinors in R 2,1 .\n\nRecall that on R 2,2 both N =4 SDYM theory and full N =4 super Yang-Mills theory have an SL(4, R) ∼ = Spin(3,3) R-symmetry group [33] . A dimensional reduction to R 2,1 enlarges the supersymmetry and R-symmetry to 2N =8 and Spin(4,4), respectively, for both theories (cf. [49] for Minkowski signature). More generally, any number N of supersymmetries gets doubled to 2N in the reduction. Since dimensional reduction collapses the rotation group Spin(2,2)\n\n∼ = Spin(2,1) L ×Spin(2,1) R of R 2,2\n\nto its diagonal subgroup Spin(2,1) D as the local rotation group of R 2,1 , the distinction between undotted and dotted indices disappears. We shall use undotted indices henceforth.\n\nCoordinates and derivatives in R 2,1 . The above discussion implies that one can relabel the bosonic coordinates x α β from (2.3) by x αβ and split them as\n\nx αβ = 1 2 (x αβ + x βα ) + 1 2 (x αβ -x βα ) = x (αβ) + x [αβ]\n\n(2.28) into antisymmetric and symmetric parts,\n\nx [αβ] = 1 2 ε αβ x 4 = 1 2 ε αβ t and x (αβ) =: y αβ , (2.29)\n\nrespectively, with\n\ny 11 = x 11 = 1 2 (t -y) , y 12 = 1 2 (x 12 + x 21 ) = 1 2 x , y 22 = x 22 = 1 2 (t + y) . (2.30)\n\nWe also have θ iα → θ iα and η α i → η α i for the fermionic coordinates on R 4|4N reduced to R 3|4N . Bosonic coordinate derivatives reduce in 2+1 dimensions to the operators\n\n∂ (αβ) = 1 2 (∂ αβ + ∂ βα ) (2.31)\n\nwhich read explicitly as\n\n∂ (11) = ∂ ∂y 11 = ∂ t -∂ y , ∂ (12) = ∂ (21) = 1 2 ∂ ∂y 12 = ∂ x , ∂ (22) = ∂ ∂y 22 = ∂ t + ∂ y . (2.32)\n\nWe thus have\n\n∂ ∂x αβ = ∂ (αβ) -ε αβ ∂ 4 = ∂ (αβ) -ε αβ ∂ t , (2.33)\n\nwhere\n\nε 12 = -ε 21 = -1, ∂ 4 = ∂/∂x 4 and ∂ t = ∂/∂ t.\n\nThe operators D iα and D i α acting on t-independent superfields reduce to\n\nD iα = ∂ iα + η β i ∂ (αβ) and D i α = ∂ i α + θ iβ ∂ (αβ) , (2.34)\n\nwhere\n\n∂ iα = ∂/∂θ iα and ∂ i α = ∂/∂η α i .\n\nSimilarly, the antichiral operators Diα and Di α in (2.18) become Diα = ∂ iα and Di α = ∂ i α + 2θ iβ ∂ (αβ) . (2.35) Supersymmetric Bogomolny-type equations in component fields. According to (2.33), the components A α β of a gauge potential in four dimensions split into the components A (αβ) of a gauge potential in three dimensions and a Higgs field\n\nA [αβ] = -ε αβ ϕ, i.e. A αβ = A (αβ) + A [αβ] = A (αβ) -ε αβ ϕ . (2.36)\n\nThen the covariant derivatives D α β reduced to three dimensions become the differential operators\n\nD αβ -ε αβ ϕ = ∂ (αβ) + [A (αβ) , • ] -ε αβ [ϕ, • ] , (2.37)\n\nand the Yang-Mills field strength on R 2,1 decomposes as\n\nF αβ, γδ = [D αβ , D γδ ] = ε αγ f βδ + ε βδ f αγ with f αβ = f βα . (2.38)\n\nSubstituting (2.36) and (2.37) into (2.10), i.e. demanding that all fields in (2.10) are independent of x 4 = t, we obtain the following supersymmetric Bogomolny-type equations on R 2,1 :\n\nf αβ + D αβ ϕ = 0 , (2.39a)\n\nD αβ χ iβ + ε αβ [ϕ, χ iβ ] = 0 , (2.39b)\n\nD αβ D αβ φ ij + 2[ϕ, [ϕ, φ ij ]] + 2{χ iα , χ j α } = 0 , (2.39c)\n\nD αβ χβ[ijk] -ε αβ [ϕ, χβ[ijk] ] -6[χ [i α , φ jk] ] = 0 , (2.39d)\n\nD γ α G [ijkl] γβ + [ϕ, G [ijkl] αβ ] + 12{χ [i α , χjkl] β } -18[φ [ij , D αβ φ kl] ] -18ε αβ [φ [ij , [φ kl] , ϕ]] = 0 .(2.\n\nSupersymmetric Bogomolny-type equations in terms of superfields. Translations generated by the vector field ∂ 4 = ∂ t are isometries of superspaces R 4|4N and R 4|2N . By taking the quotient with respect to the action of the abelian group G generated by ∂ 4 , we obtain the reduced full superspace R 3|4N ∼ = R 4|4N /G and the reduced antichiral superspace R 3|2N ∼ = R 4|2N /G. In the following, we shall work on R 3|2N and R 3|2N × CP 1 , since the reduced ψ-function from (2.25) and (2.27) is defined on the latter space.\n\nThe linear system stays in the center of the superfield approach to the N -extended SDYM equations. After imposing t-independence on all fields in the linear system (2.27), we arrive at the linear equations\n\nζ α ∇i α ψ ≡ ζ α ( Di α + Âi α ) ψ = 0 (2.40)\n\nof the same form but with\n\nDi α = ∂ i α + 2θ iβ ∂ (αβ) and Âi α = A i α + 2θ iβ (A (αβ) -ε αβ Ξ) , (2.41)\n\nwhere A i α , A (αβ) and Ξ are superfields depending on y αβ and η α i only. These linear equations expand again to the pair (cf. (2.25))\n\nζ β (∂ (αβ) + A (αβ) -ε αβ Ξ) ψ = 0 and ζ α (∂ i α + A i α ) ψ = 0 . (2.42)\n\nThe compatibility conditions for the linear system (2.40) read\n\n{ ∇i α , ∇j β } + { ∇i β , ∇j α } = 0 ⇔ { ∇i α , ∇j β } = ε αβ Fij (2.43)\n\nand present a condensed form of (2.39) rewritten in terms of R 3|2N superfields. Similarly, these equations can also be written in more expanded forms analogously to (2.21) or using the superfield analog of (2.37). However, we will not do this since all these sets of equations are equivalent."},{"section_type":"OTHER","section_title":"Noncommutative N -extended U(n) chiral model in 2+1 dimensions","text":"As has been known for some time, nonlinear sigma models in 2 + 1 dimensions may be Lorentzinvariant or integrable but not both [7, 11] . We will show that the super Bogomolny-type model discussed in Section 2 after a gauge fixing is equivalent to a super extension of the modified U(n) chiral model (so as to be integrable) first formulated by Ward [7] . Since integrability is compatible with noncommutative deformation (if introduced properly, see e.g. [9] - [20] ) we choose from the beginning to formulate our super extension of this chiral model on Moyal-deformed R 2,1 with noncommutativity parameter θ ≥ 0. Ordinary space-time R 2,1 can always be restored by taking the commutative limit θ → 0.\n\nStar-product formulation. Classical field theory on noncommutative spaces may be realized in a star-product formulation or in an operator formalism 8 . The first approach is closer to the commutative field theory: it is obtained by simply deforming the ordinary product of classical fields (or their components) to the noncommutative star product\n\n(f ⋆ g)(x) = f (x) exp{ i 2 ← - ∂ a θ ab -→ ∂ b } g(x) ⇒ x a ⋆ x b -x b ⋆ x a = iθ ab (3.1)\n\nwith a constant antisymmetric tensor θ ab . Specializing to R 2,1 , we use real coordinates (x a ) = (t, x, y) in which the Minkowski metric g on R 3 reads (g ab ) = diag(-1, +1, +1) with a, b, . . . = 1, 2, 3 (cf. Section 2). It is straightforward to generalize the Moyal deformation (3.1) to the superspaces introduced in the previous section, allowing in particular for non-anticommuting Grassmann-odd coordinates. Deferring general superspace deformations and their consequences to future work, we here content ourselves with the simple embedding of the \"bosonic\" Moyal deformation into superspace, meaning that (3.1) is also valid for superfields f and g depending on Grassmann variables θ iα and η α i . For later use we consider not only isotropic coordinates and vector fields\n\nu := 1 2 (t+y) = y 22 , v := 1 2 (t-y) = y 11 , ∂ u = ∂ t + ∂ y = ∂ (22) , ∂ v = ∂ t -∂ y = ∂ (11) (3.2)\n\nintroduced in Section 2, but also the complex combinations\n\nz := x + iy , z := x -iy , ∂ z = 1 2 (∂ x -i∂ y ) , ∂ z = 1 2 (∂ x + i∂ y ) . (3.3)\n\nSince the time coordinate t remains commutative, the only nonvanishing component of the noncommutativity tensor θ ab is Operator formalism. The nonlocality of the star products renders explicit computation cumbersome. We therefore pass to the operator formalism, which trades the star product for operatorvalued spatial coordinates (x, ŷ) or their complex combinations (ẑ, ẑ), subject to\n\nθ xy = -θ yx =: θ > 0 ⇒ θ z z = -θ zz = -2i θ . ( 3\n\n[t, x] = [t, ŷ] = 0 but [x, ŷ] = iθ ⇒ [ẑ, ẑ] = 2 θ . (3.6)\n\nThe latter equation suggests the introduction of annihilation and creation operators,\n\na = 1 √ 2θ ẑ and a † = 1 √ 2θ ẑ with [a , a † ] = 1 , (3.7)\n\nwhich act on a harmonic-oscillator Fock space H with an orthonormal basis { |ℓ , ℓ = 0, 1, 2, . . .} such that\n\na |ℓ = √ ℓ |ℓ-1 and a † |ℓ = √ ℓ+1 |ℓ+1 . (3.8)\n\nAny superfield f (t, z, z, η α i ) on R 3|2N can be related to an operator-valued superfield f (t, η α i ) ≡ F (t, a, a † , η α i ) on R 1|2N acting in H, with the help of the Moyal-Weyl map\n\nf (t, z, z, η α i ) → f (t, η α i ) = Weyl-ordered f t, √ 2θa, √ 2θa † , η α i . (3.9)\n\nThe inverse transformation recovers the ordinary superfield,\n\nf (t, η α i ) ≡ F (t, a, a † , η α i ) → f (t, z, z, η α i ) = F ⋆ t, z √ 2θ , z √ 2θ , η α i , (3.10)\n\nwhere F ⋆ is obtained from F by replacing ordinary with star products. Under the Moyal-Weyl map, we have\n\nf ⋆ g → f ĝ and dx dy f = 2π θ Tr f = 2π θ ℓ≥0 ℓ| f |ℓ , (3.11)\n\nand the spatial derivatives are mapped into commutators,\n\n∂ z f → ∂z f = -1 √ 2θ [a † , f ] and ∂ z f → ∂z f = 1 √ 2θ [a , f ] . (3.12)\n\nFor notational simplicity we will from now on omit the hats over the operators except when confusion may arise.\n\nGauge fixing for ψ. Note that the linear system (2.40) and the compatibility conditions (2.43) are invariant under a gauge transformation\n\nψ → ψ ′ = g -1 ψ , (3.13a)\n\nA → A ′ = g -1 A g + g -1 ∂ g (with appropriate indices) , (3.13b)\n\nΞ → Ξ ′ = g -1 Ξ g , (3.13c)\n\nwhere g = g(x a , η α i ) is a U(n)-valued superfield globally defined on the deformed superspace R 3|2N θ × CP 1 . Using a gauge transformation of the form (3.13), we can choose ψ such that it will satisfy the standard asymptotic conditions (see e.g. [51] )\n\nψ = Φ -1 + O(ζ) for ζ → 0 , (3.14a\n\n)\n\nψ = 1l + ζ -1 Υ + O(ζ -2 ) for ζ → ∞ , (3.14b)\n\nwhere the U(n)-valued function Φ and u(n)-valued function Υ depend on x a and η α i . This \"unitary\" gauge is compatible with the reality condition for ψ,\n\nψ(x a , η α i , ζ) ψ(x a , η α i , ζ) † = 1l , (3.15)\n\nobtained by reduction from (2.26)."},{"section_type":"OTHER","section_title":"Gauge fixing for","text":"A i 1 = 0 and A i 2 = Φ -1 ∂ i 2 Φ = ∂ i 1 Υ , (3.17)\n\nA (11) = 0 and\n\nA (12) + Ξ = Φ -1 ∂ (12) Φ = ∂ (11) Υ , (3.18)\n\nA (21) -Ξ = 0 and\n\nA (22) = Φ -1 ∂ (22) Φ = ∂ (12) Υ . (3.19)\n\nUsing (2.32), we can rewrite the nonzero components as\n\nA := Φ -1 ∂ u Φ = ∂ x Υ , B := Φ -1 ∂ x Φ = ∂ v Υ , C i := Φ -1 ∂ i 2 Φ = ∂ i 1 Υ . (3.20)\n\nRecall that the superfields Φ and Υ depend on x a and η α i .\n\nLinear system. In the above-introduced unitary gauge the linear system (2.42) reads\n\n(ζ∂ x -∂ u -A) ψ = 0 , (ζ∂ v -∂ x -B) ψ = 0 , (ζ∂ i 1 -∂ i 2 -C i ) ψ = 0 , (3.21)\n\nwhich adds the last equation to the linear system of the Ward model [7] and generalizes it to superfields A(x a , η α j ), B(x a , η α j ) and C i (x a , η α j ). The concise form of (3.21) reads\n\nζ Di 1 -Di 2 -Âi 2 ψ = 0 (3.22)\n\nor, in more explicit form,\n\nζ ∂ i 1 + 2θ i1 ∂ v + 2θ i2 ∂ x -∂ i 2 + C i + 2θ i1 (∂ x + B) + 2θ i2 (∂ u + A) ψ = 0 . (3.23)\n\nN -extended sigma model.\n\nThe compatibility conditions of this linear system are the Nextended noncommutative sigma model equations Di\n\n1 (Φ -1 Dj 2 Φ) + Dj 1 (Φ -1 Di 2 Φ) = 0 (3.24)\n\nwhich in expanded form reads\n\n(g ab + v c ε cab ) ∂ a (Φ -1 ∂ b Φ) = 0 ⇔ ∂ x (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ u Φ) = 0 , (3.25a) ∂ i 1 (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ i 2 Φ) = 0 , ∂ i 1 (Φ -1 ∂ u Φ) -∂ x (Φ -1 ∂ i 2 Φ) = 0 , (3.25b)\n\n∂ i 1 (Φ -1 ∂ j 2 Φ) + ∂ j 1 (Φ -1 ∂ i 2 Φ) = 0 . (3.25c)\n\nHere, the first line contains the Wess-Zumino-Witten term with a constant vector (v c ) = (0, 1, 0) which spoils the standard Lorentz invariance but yields an integrable chiral model in 2+1 dimensions. Recall that Φ is a U(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . As discussed in Section 2, the compatibility conditions of the linear equations (3.22) (or (3.21)) are equivalent to the N -extended Bogomolnytype equations (2.39) for the component (physical) fields. Thus, chiral model field equations (3.25) are equivalent to a gauge fixed form of equations (2.39). Υ-formulation. Instead of Φ-parametrization of (A, B, C i ) given in (3.17)-(3.20) we may use the equivalent Υ-parametrization also given there. In this case, the compatibility conditions for the linear system (3.21) reduce to\n\n(∂ 2 x -∂ u ∂ v )Υ + [∂ v Υ , ∂ x Υ] = 0 , (3.26a)\n\n(∂ i 2 ∂ v -∂ i 1 ∂ x )Υ + [∂ i 1 Υ , ∂ v Υ] = 0 , (∂ i 2 ∂ x -∂ i 1 ∂ u )Υ + [∂ i 1 Υ , ∂ x Υ] = 0 , (3.26b) (∂ i 2 ∂ j 1 + ∂ j 2 ∂ i 1 )Υ + {∂ i 1 Υ , ∂ j 1 Υ} = 0 , (3.26c)\n\nwhich in concise form read\n\n( Di 2 Dj 1 + Dj 2 Di 1 ) Υ + { Di 1 Υ , Dj 1 Υ} = 0 . (3.27)\n\nRecall that Υ is a u(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . For N =4, the commutative limit of (3.27) can be considered as Siegel's equation [33] reduced to 2+1 dimensions. According to Siegel, one can extract the multiplet of physical fields appearing in (2.39) from the prepotential Υ via\n\n∂ i 1 Υ = A i 2 , ∂ i 1 ∂ j 1 Υ = φ ij , ∂ i 1 ∂ j 1 ∂ k 1 Υ = χ[ijk] 2 , ∂ i 1 ∂ j 1 ∂ k 1 ∂ l 1 Υ = G [ijkl] 22 , (3.28a)\n\n∂ (α1) Υ = A (α2) -ε α2 ϕ , ∂ (α1) ∂ i 1 Υ = χ i α , ∂ (α1) ∂ (β1) Υ = f αβ , (3.28b)\n\nwhere one takes Υ and its derivatives at η 2 i = 0. The other components of the physical fields, i.e. χ[ijk]\n\n1 , G [ijkl] 11 , G [ijkl]\n\n21 , A (11) and A (21) -ϕ, vanish in this light-cone gauge.\n\nSupersymmetry transformations. The 4N supercharges given in (2.11) reduce in 2+1 dimensions to the form\n\nQ iα = ∂ iα -η β i ∂ (αβ) and Q i α = ∂ i α -θ iβ ∂ (αβ) . (3.29)\n\nTheir antichiral version, matching to Diα and Dj β of (2.35), reads\n\nQiα = ∂ iα -2η β i ∂ (αβ) and Qj β = ∂ j β , (3.30)\n\nso that { Qiα , Qj β } = -2 δ j i ∂ (αβ) . (3.31)\n\nOn a (scalar) R 3|2N superfield Σ these supersymmetry transformations act as\n\nδ Σ := ε iα Qiα Σ + ε α i Qi α Σ (3.32)\n\nand are induced by the coordinate shifts\n\nδ y αβ = -2ε i(α η β) i and δ η α i = ε α i , (3.33)\n\nwhere ε iα and ε α i are 4N real Grassmann parameters. It is easy to see that our equations (3.24) and (3.27) are invariant under the supersymmetry transformations (3.32) (applied to Φ or Υ). This is simply because the operators Diα and Dj β anticommute with the supersymmetry generators Qiα and Qj β . Therefore, the equations of motion (3.25) of the modified N -extended chiral model in 2+1 dimensions as well as their reductions to 2+0 and 1+1 dimensions carry 2N supersymmetries and are genuine supersymmetric extensions of the corresponding bosonic equations. Note that this type of extension is not the standard one since the R-symmetry groups are Spin(N , N ) in 2+1 and Spin(N , N )× Spin(N , N ) in 1+1 dimensions, which differ from the compact unitary R-symmetry groups of standard sigma models. Contrary to the standard case of two-dimensional sigma models the above \"noncompact\" 2N supersymmetries do not impose any constraints on the geometry of the target space, e.g. they do not demand it to be Kähler [52] or hyper-Kähler [53] . This may be of interest and deserves further study.\n\nAction functionals. In either formulation of the N -extended supersymmetric SDYM model on R 2,2 there are difficulties with finding a proper action functional generalizing the one [54, 55] for the purely bosonic case. These difficulties persist after the reduction to 2+1 dimensions, i.e. for the equations (3.25) and (3.26) describing our supersymmetric modified U(n) chiral model. It is the price to be paid for overcoming the no-go barrier N ≤ 4 and the absence of geometric target-space constraints. On a more formal level, the problem is related to the chiral character of (3.24) as well as (3.27) , where only the operators Di α but not Diα appear. Note however, that for N = 4 one can write an action functional in component fields producing the equations (2.39), which are equivalent to the superspace equations (3.24) when i, j = 1, . . . , 4 (see e.g. [47] ).\n\nOne proposal for an action functional stems from Siegel's idea [33] for the Υ-formulation of the N -extended SDYM equations. Namely, one sees that ∂ i 2 Υ enters only linearly into the last two lines in (3.26) . Therefore, if we introduce\n\nΥ (1) := Υ| η 2 i =0 (3.34)\n\nthen it must satisfy the first equation from (3.26) , and the remaining equations iteratively define the dependence of Υ on η 2 i starting from Υ (1) . Hence, all information is contained in Υ (1) , as can also be seen from (3.28) . In other words, the dependence of Υ on η 2 i is not 'dynamical'. For an action one can then take (cf. [33] )\n\nS = d 3 x d N η 1 Υ (1) ∂ (αβ) ∂ (αβ) Υ (1) + 2 3 Υ (1) ε αβ ∂ (α1) Υ (1) ∂ (β1) Υ (1) . (3.35)\n\nExtremizing this functional yields the first line of (3.26) at η 2 i = 0. Except for the Grassmann integration, this action has the same form as the purely bosonic one [55] . One may apply the same logic to the Φ-formulation where the action for the purely bosonic case is also known [54, 56] ."},{"section_type":"OTHER","section_title":"N -extended multi-soliton configurations via dressing","text":"The existence of the linear system (3.22) (equivalent to (3.21)) encoding solutions of the N -extended U(n) chiral model in an auxiliary matrix ψ allows for powerful methods to systematically construct explicit solutions for ψ and hence for Φ † = ψ| ζ=0 and Υ = lim ζ→∞ ζ (ψ-1l). For our purposes the so-called dressing method [57, 51] proves to be the most practical [12] - [20] , and so we shall use it here for our linear system, i.e. already in the N -extended noncommutative case.\n\nMulti-pole ansatz for ψ. The dressing method is a recursive procedure for generating a new solution from an old one. More concretely, we rewrite the linear system (3.21) in the form\n\nψ(∂ u -ζ∂ x )ψ † = A , ψ(∂ x -ζ∂ v )ψ † = B , ψ(∂ i 2 -ζ∂ i 1 )ψ † = C i . (4.1)\n\nRecall that ψ † := (ψ(x a , η α i , ζ)) † and (A, B, C i ) depend only on x a and η α i . The central idea is to demand analyticity in the spectral parameter ζ, which strongly restricts the possible form of ψ. One way to exploit this constraint starts from the observation that the left hand sides of (4.1) as well as of the reality condition (3.15) do not depend on ζ while ψ is expected to be a nontrivial function of ζ globally defined on CP 1 . Therefore, it must be a meromorphic function on CP 1 possessing some poles which we choose to lie at finite points with constant coordinates µ k ∈ CP 1 .\n\nHere we will build a (multi-soliton) solution ψ m featuring m simple poles at positions µ 1 , . . . , µ m with foot_8 Im µ k < 0 by left-multiplying an (m-1)-pole solution ψ m-1 with a single-pole factor of the form 1l\n\n+ µ m -μm ζ -µ m P m (x a , η α i ) , (4.2)\n\nwhere the n×n matrix function P m is yet to be determined. Starting from the trivial (vacuum) solution ψ 0 = 1l, the iteration ψ 0 → ψ 1 → . . . → ψ m yields a multiplicative ansatz for ψ m ,\n\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ , (4.3)\n\nwhich, via partial fraction decomposition, may be rewritten in the additive form\n\nψ m = 1l + m k=1 Λ mk S † k ζ -µ k , (4.4)\n\nwhere Λ mk and S k are some n×r k matrices depending on x a and η α i , with r k ≤ n.\n\nEquations for S k . Let us first consider the additive parametrization (4.4) of ψ m . This ansatz must satisfy the reality condition (3.15) as well as our linear equations in the form (4.1). In particular, the poles at ζ = μk on the left hand sides of these equations have to be removable since the right hand sides are independent of ζ. Inserting the ansatz (4.4) and putting to zero the corresponding residues, we learn from (3.15) that\n\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ S k = 0 , (4.5)\n\nwhile from (4.1) we obtain the differential equations\n\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ LA,B,i k S k = 0 , (4.6)\n\nwhere LA,B,i\n\nk stands for either LA k = ∂ u -μk ∂ x , LB k = µ k (∂ x -μk ∂ v ) or Li k = ∂ i 2 -μk ∂ i 1 . (4.7)\n\nNote that we consider a recursive procedure starting from m=1, and operators (4.7) will appear with k = 1, . . . , m if we consider poles at ζ = μk .\n\nBecause the LA,B,i k for k = 1, . . . , m are linear differential operators, it is easy to write down the general solution for (4.6) at any given k, by passing from the coordinates (u, v, x; η 1 i , η 2 i ) to \"co-moving coordinates\" (w k , wk , s k ; η i k , ηi k ). The precise relation for k = 1, . . . , m is [12, 58]\n\nw k := x + μk u + μ-1 k v = x + 1 2 (μ k -μ -1 k )y + 1 2 (μ k +μ -1 k )t and η i k := η 1 i + μk η 2 i , (4.8)\n\nwith wk and ηi k obtained by complex conjugation and the co-moving time s k being inessential because by definition nothing will depend on it. The kth moving frame travels with a constant velocity\n\n(v x , v y ) k = - µ k + μk µ k μk + 1 , µ k μk -1 µ k μk + 1 , (4.9)\n\nso that the static case w k =z is recovered for µ k = -i. On functions of (w k , η i k , wk , ηi k ) alone the operators (4.7) act as\n\nLA k = LB k = (µ k -μ k ) ∂ ∂ wk =: Lk and Li k = (µ k -μ k ) ∂ ∂ ηi k . (4.10)\n\nBy induction in k = 1, . . . , m we learn that, due to (4.5), a necessary and sufficient condition for a solution of (4.6) is Lk\n\nS k = S k Zk and Li k S k = S k Zi k (4.11)\n\nwith some r k ×r k matrices Zk and Zi k depending on (w k , wk , η j k , ηj k ). Passing to the noncommutative bosonic coordinates we obtain ŵk , ŵk = 2θ ν k νk with\n\nν k νk = 4i µ k -μ k -µ -1 k +μ -1 k . (4.12)\n\nThus, we can introduce annihilation and creation operators\n\nc k = 1 √ 2θ ŵk ν k and c † k = 1 √ 2θ ŵk νk so that [c k , c † k ] = 1 (4.13)\n\nfor k = 1, . . . , m. Naturally, this Heisenberg algebra is realized on a \"co-moving\" Fock space H k , with basis states |ℓ k and a \"co-moving\" vacuum |0 k subject to c k |0 k = 0. Each co-moving vacuum |0 k (annihilated by c k ) is related to the static vacuum |0 (annihilated by a) through an ISU(1,1) squeezing transformation (cf. [12] ) which is time-dependent. The fermionic coordinates η i k and ηi k remain spectators in the deformation. Coordinate derivatives are represented in the standard fashion as\n\nν k √ 2θ ∂ ∂w k → -[c † k , • ] and νk √ 2θ ∂ ∂ wk → [c k , • ] . (4.14)\n\nAfter the Moyal deformation, the n×r k matrices S k have become operator-valued, but are still functions of the Grassmann coordinates η i k and ηi k . The noncommutative version of the BPS conditions (4.11) naturally reads\n\nc k S k = S k Z k and ∂ ∂ ηi k S k = S k Z i k (4.15)\n\nwhere Z k and Z i k are some operator-valued r k ×r k matrix functions of η j k and ηj k .\n\nNonabelian solutions for S k . For general data Z k and Z i k it is difficult to solve (4.15), but it is also unnecessary because the final expression ψ m turns out not to depend on them. Therefore, we conveniently choose\n\nZ k = c k ⊗ 1l r k ×r k and Z i k = 0 ⇒ S k = R k (c k , η i k ) , (4.16)\n\nwhere R k is an arbitrary n×r k matrix function independent of c † k and ηi k . 10 It is known that nonabelian (multi-) solitons arise for algebraic functions R k (cf. e.g. [7] for the commutative and [12] for the noncommutative N =0 case). Their common feature is a smooth commutative limit. The only novelty of the supersymmetric extension is the η i k dependence, i.e.\n\nR k = R k,0 + η i k R k,i + η i k η j k R k,ij + η i k η j k η p k R k,ijp + η i k η j k η p k η q k R k,ijpq . (4.17)\n\nAbelian solutions for S k . It is useful to view S k as a map from C r k ⊗H k to C n ⊗H k (momentarily suppressing the η dependence). The noncommutative setup now allows us to generalize the domain of this map to any subspace of C n ⊗ H k . In particular, we may choose it to be finite-dimensional, say C q k , and represent the map by an n×q k array |S k of kets in H. In this situation, Z k and Z i k in (4.15) are just number -valued q k ×q k matrix functions of η j k and ηj k . In case they do not depend on ηj k , we can write down the most general solution as\n\n|S k = R k (c k , η j k ) |Z k exp i Z i k (η j k ) ηi k with |Z k := exp Z k (η j k ) c † k |0 k . (4.18)\n\nAs before, we may put Z i k = 0 without loss of generality, but now the choice of Z k does matter. For any given k generically there exists a q k -dimensional basis change which diagonalizes the ket-valued matrix\n\n|Z k → diag e α 1 k c † , e α 2 k c † , . . . , e α q k k c † |0 k = diag |α 1 k , |α 2 k , . . . , |α q k k , (4.19)\n\nwhere we defined coherent states\n\n|α l k := e α l k c † |0 k so that c k |α l k = α l k |α l k for l = 1, . . . , q k and α l k ∈ C . (4.20)\n\nNote that not only the entries of R k but also the α l k are holomorphic functions of the co-moving Grassmann parameters η j k and thus can be expanded like in (4.17). In the U(1) model, we must use ket-valued 1×q k matrices |S k for all k, yielding rows\n\n|S k = R 1 k |α 1 k , R 2 k |α 2 k , . . . , R q k k |α q k k for k = 1, . . . , m , (4.21)\n\nwith functions α l k (η j k ). Here, the R l k only affect the states' normalization and can be collected in a diagonal matrix to the right, hence will drop out later and thus may all be put to one. Formally, we have recovered the known abelian (multi-) soliton solutions, but the supersymmetric extension has generalized |S k → |S k (η j k ) . Explicit form of P k . Let us now consider the multiplicative parametrization (4.3) of ψ m which also allows us to solve (4.5) . First of all, note that the reality condition (3.15) is satisfied if\n\nP k = P † k = P 2 k ⇔ P k = T k (T † k T k ) -1 T † k for k = 1, . . . , m , (4.22)\n\nmeaning that P k is an operator-valued hermitian projector (of group-space rank r k ≤ n) built from an n×r k matrix function T k (the abelian case of n=1 is included). The reality condition follows just because The r k columns of T k span the image of P k and obey\n\n1l + µ k -μk ζ -µ k P k 1l + μk -µ k ζ -μk P k =\n\nP k T k = T k ⇔ (1l-P k ) T k = 0 . (4.24)\n\nFurthermore, the equation (4.5) with m = k (induction) rewritten in the form\n\n(1l-P k ) k-1 ℓ=1 1l + µ k-ℓ -μk-ℓ μk -µ k-ℓ P k-ℓ S k = 0 (4.25)\n\nreveals that (cf. (4.24))\n\nT 1 = S 1 and T k = k-1 ℓ=1 1l - µ k-ℓ -μk-ℓ µ k-ℓ -μk P k-ℓ S k for k ≥ 2 , (4.26)\n\nwhere the explicit form of S k for k = 1, . . . , m is given in (4.16) or (4.18) . The final result reads\n\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ = 1l + m k=1 Λ mk S † k ζ -µ k (4.27)\n\nwith hermitian projectors P k given by (4.22), T k given by (4.26) and S k given by (4.16) or (4.18) . The explicit form of Λ mk (which we do not need) can be found in [12] . The corresponding superfields Φ and Υ are\n\nΦ m = ψ † m | ζ=0 = m k=1 (1l -ρ k P k ) with ρ k = 1 - µ k μk , (4.28a)\n\nΥ m = lim ζ→∞ ζ (ψ m -1l) = m k=1 (µ k -μ k ) P k . (4.28b)\n\nFrom (4.22) it is obvious that P k is invariant under a similarity transformation\n\nT k → T k Λ k ⇔ S k → S k Λ k (4.29)\n\nfor an invertible operator-valued r k ×r k matrix Λ k . This justifies putting Z i k = 0 from the beginning and also the restriction to Z k = c k ⊗ 1l r k ×r k in the nonabelian case, both without loss of generality. Hence, the nonabelian solution space constructed here is parametrized by the set {R k } m 1 of matrixvalued functions of c k and η i k and the pole positions µ k . The abelian moduli space, however, is larger by the set {Z k } m 1 of matrix-values functions of η i k which generically contain the coherentstate parameter functions {α l k (η i k )}. Restricting to η i k =0 reproduces the soliton configurations of the bosonic model [12] .\n\nStatic solutions. Let us consider the reduction to 2+0 dimensions, i.e. the static case. Recall that static solutions correspond to the choice m = 1 and µ 1 ≡ µ = -i implying w 1 = z, so we drop the index k. Specializing (4.27), we have\n\nψ = 1l - 2 i ζ + i P so that Φ = Φ † = 1l -2P , (4.30)\n\nwhere a hermitian projector P of group-space rank r satisfies the BPS equations (1l-P ) a P = 0 ⇒ (1l-P ) a T = 0 ,\n\n(1l-P ) ∂ ∂ ηi P = 0 ⇒ (1l-P ) ∂ ∂ ηi T = 0 , (4.31a)\n\nwith P = T (T † T ) -1 T † and η i = η 1 i + iη 2 i . In this case T = S, and for a nonabelian r=1 projector P we get T = T (a, η i ) as an n×1 column. For the simplest case of N =1 we just have (cf. [59] )\n\nT = T e (a) + η T o (a) with η = η 1 + iη 2 , (4.32)\n\nwhere T e (a) and T o (a) are rational functions of a (e.g. polynomials) taking values in the even and odd parts of the Grassmann algebra. Similarly, an abelian N =1 projector (for n=1) is built from\n\n|T = |α 1 , |α 2 , . . . , |α q . (4.33)\n\nAt θ=0, the static solution (4.32) of our supersymmetric U(n) sigma model is also a solution of the standard N =1 supersymmetric CP n-1 sigma model in two dimensions (see e.g. [59] ). 11 For this reason, one can overcome the previously mentioned difficulty with constructing an action (or energy from the viewpoint of 2+1 dimensions) for static configurations. Moreover, on solutions obeying the BPS conditions (4.31) the topological charge\n\nQ = 2πθ dη 1 dη 2 Tr tr Φ D + Φ , D -Φ (4.34)\n\nis proportional to the action (BPS bound)\n\nS = 2πθ dη 1 dη 2 Tr tr D + Φ , D -Φ (4.35)\n\nand is finite for algebraic functions T e and T o . Here, the standard superderivatives D ± are defined as\n\nD + = ∂ ∂η + iη ∂ z and D -= ∂ ∂ η + iη ∂ z . ( 4\n\nOne-soliton configuration. For one moving soliton, from (4.27) and (4.28) we obtain\n\nψ 1 = 1l + µ - μ ζ -µ P with P = T (T † T ) -1 T † (4.37) and Φ = 1l -ρ P with ρ = 1 - µ μ . (4.38)\n\nNow our n×r matrix T must satisfy (putting\n\nZ i = 0 and Z = c ⊗ 1l r×r ) [c , T ] = 0 and ∂ ∂ ηi T = 0 with η i = η 1 i + μ η 2 i , (4.39)\n\nwhere c is the moving-frame annihilation operator given by (4.13) for k=1.\n\nRecall that the operators c and c † and therefore the matrix T and the projector P can be expressed in terms of the corresponding static objects by a unitary squeezing transformation (see e.g. (4.8) and (4.13)). For simplicity we again consider the case N =1 and a nonabelian projector with r=1. Then (4.39) tells us that T is a holomorphic function of c and η, i.e.\n\nT = T e (c) + η T o (c) = T 1 e (c) + η T 1 o (c)\n\n. . .\n\nT n e (c) + η T n o (c) (4.40)\n\nwith polynomials T a e and T a o of order q, say, analogously to the static case (4.32). Note that, for T a o to be Grassmann-odd and nonzero, some extraneous Grassmann parameter must appear. Similarly, abelian projectors for a moving one-soliton obtain by subjecting (4.33) to a squeezing transformation.\n\nFor N =1 the moving frame was defined in (4.8) (dropping the index k) via\n\nw = x + 1 2 (μ-μ -1 )y + 1 2 (μ+μ -1 )t and η = η 1 + μη 2 hence ∂ t η = 0 . (4.41)\n\nConsider the moving frame with the coordinates (w, w, s; η, η) with the choice s = t and the related change of the derivatives (see [12, 58] )\n\n∂ x = ∂ w + ∂ w , (4.42a)\n\n∂ y = 1 2 (μ-μ -1 ) ∂ w + 1 2 (µ-µ -1 ) ∂ w , (4.42b)\n\n∂ t = 1 2 (μ+μ -1 ) ∂ w + 1 2 (µ+µ -1 ) ∂ w + ∂ s , (4.42c)\n\n∂ η 1 = ∂ η + ∂ η , (4.42d)\n\n∂ η 2 = μ ∂ η + µ ∂ η . (4.42e)\n\nIn the moving frame our solution (4.38) is static, i.e. ∂ s Φ = 0, and the projector P has the same form as in the static case. The only difference is the coefficient ρ instead of 2 in (4.38). Therefore, by computing the action (4.35) in (w, w; η 1 , η 2 ) coordinates, we obtain for algebraic functions T in (4.40) a finite answer, which differs from the static one by a kinematical prefactor depending on µ (cf. [12] for the bosonic case).\n\nLarge-time asymptotics. Note that in the distinguished (z, z, t) coordinate frame (4.41) implies that at large times w → κ t with κ = 1 2 (μ+μ -1 ). As a consequence, the t q term in each polynomial in (4.40) will dominate, i.e.\n\nT → t q a 1 + η b 1 . . .\n\nan + η bn =: t q Γ , (4.43)\n\nwhere Γ is a fixed vector in C n . It is easy to see that in the distinguished frame the large-time limit of Φ given by (4. being the projector on the constant vector Γ.\n\nConsider now the m-soliton configuration (4.28). By induction of the above argument one easily arrives at the m-soliton generalization of (4.44). Namely, in the frame moving with the ℓth lump we have lim t→±∞ Φ m = (1lρ 1 Π 1 ) . . . (1lρ ℓ-1 Π ℓ-1 )(1lρ ℓ P ℓ )(1lρ ℓ+1 Π ℓ+1 ) . . . (1lρ m Π m ) , (4.45) where the Π m are constant projectors. This large-time factorization of multi-soliton solutions provides a proof of the no-scattering property because the asymptotic configurations are identical for large negative and large positive times."},{"section_type":"CONCLUSION","section_title":"Conclusions","text":"In this paper we introduced a generalization of the modified integrable U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions and considered a Moyal deformation of this model. It was shown that this N -extended chiral model is equivalent to a gauge-fixed BPS subsector of an N -extended super Yang-Mills model in 2+1 dimensions originating from twistor string theory. The dressing method was applied to generate a wide class of multi-soliton configurations, which are time-dependent finite-energy solutions to the equations of motion. Compared to the N =0 model, the supersymmetric extension was seen to promote the configurations' building blocks to holomorphic functions of suitable Grassmann coordinates. By considering the large-time asymptotic factorization into a product of single soliton solutions we have shown that no scattering occurs within the dressing ansatz chosen here.\n\nThe considered model does not stand alone but is motivated by twistor string theory [37] with a target space reduced to the mini-supertwistor space [44, 45, 47] . In this context, the obtained multi-soliton solutions are to be regarded as D(0|2N )-branes moving inside D(2|2N )-branes [60] . Here 2N appears due to fermionic worldvolume directions of our branes in the superspace description [60] . Switching on a constant B-field simply deforms the sigma model and D-brane worldvolumes noncommutatively, thereby admitting also regular supersymmetric noncommutative abelian solutions.\n\nRestricting to static configurations, the models can be specialized to Grassmannian supersymmetric sigma models, where the superfield Φ takes values in Gr(r, n), and the field equations are invariant under 2N supersymmetry transformations with 0 ≤ N ≤ 4. This differs from the results for standard 2D sigma models [52, 53] where the target spaces have to be Kähler or hyper-Kähler for admitting two or four supersymmetries, respectively. This difference will be discussed in more details elsewhere.\n\nWe derived the supersymmetric chiral model in 2+1 dimensions through dimensional reduction and gauge fixing of the N -extended supersymmetric SDYM equations in 2+2 dimensions. Recall that for the purely bosonic case most (if not all) integrable equations in three and fewer dimensions can be obtained from the SDYM equations (or their hierarchy [25] ) by suitable dimensional reductions (see e.g. [61] - [65] and references therein). Moreover, this Ward conjecture [61] was extended to the noncommutative case (see e.g. [66, 67] ). It will be interesting to consider similar reductions of the N -extended supersymmetric SDYM equations (and their hierarchy [68] ) to supersymmetric integrable equations in three and two dimensions generalizing earlier results [69] ."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"We consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from twistor string theory. By a gauge fixing this model is reduced to a modified U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions. After a Moyal-type deformation of the model, we employ the dressing method to explicitly construct multi-soliton configurations on noncommutative R 2,1 and analyze some of their properties.\"\n },\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction\",\n \"text\": \"In the low-energy limit string theory with D-branes gives rise to noncommutative field theory on the branes when the string propagates in a nontrivial NS-NS two-form (B-field) background [1, 2, 3, 4] . In particular, if the open string has N =2 worldsheet supersymmetry, the tree-level target space dynamics is described by a noncommutative self-dual Yang-Mills (SDYM) theory in 2+2 dimensions [5] . Furthermore, open N =2 strings in a B-field background induce on the worldvolume of n coincident D2-branes a noncommutative Yang-Mills-Higgs Bogomolny-type system in 2+1 dimensions which is equivalent to a noncommutative generalization [6] of the modified U(n) chiral model known as the Ward model [7] . The topological nature of N =2 strings and the integrability of their tree-level dynamics [8] render this noncommutative sigma model integrable. 1 Being integrable, the commutative U(n≥2) Ward model features a plethora of exact scattering and no-scattering multi-soliton and wave solutions, i.e. time-dependent stable configurations on R foot_1 . These are not only a rich testing ground for physical properties such as adiabatic dynamics or quantization, but also descend to more standard multi-solitons of various integrable systems in 2+0 and 1+1 dimensions, such as sine-Gordon, upon dimensional and algebraic reduction. There is a price to pay however: Nonlinear sigma models in 2+1 dimensions may be Lorentz-invariant or integrable but not both [7, 11] . In fact, Derrick's theorem prohibits the existence of stable solitons in Lorentz-invariant scalar field theories above 1+1 dimensions. A Moyal deformation, however, overcomes this hurdle, but of course replaces Lorentz invariance by a Drinfeld-twisted version. There is another gain: The deformed Ward model possesses not only deformed versions of the just-mentioned multi-solitons, but in addition allows for a whole new class of genuinely noncommutative (multi-)solitons, in particular for the U(1) group [12, 13] ! Moreover, this class is related to the generic but perturbatively constructed noncommutative scalar-field solitons [14, 15] by an infinite-stiffness limit of the potential [16] .\\n\\nIn [12, 13] and [17] - [20] families of multi-solitons as well as their reduction to solitons of the noncommutative sine-Gordon equations were described and studied. In the nonabelian case both scattering and nonscattering configurations were obtained. For static configurations the issue of their stability was analyzed [21] . The full moduli space metric for the abelian model was computed and its adiabatic two-soliton dynamics was discussed [16] .\\n\\nRecall that the critical N =2 string theory has a four-dimensional target space, and its open string effective field theory is self-dual Yang-Mills [8] , which gets deformed noncommutatively in the presence of a B-field [5] . Conversely, the noncommutative SDYM equations are contained [19] in the equations of motion of N =2 string field theory (SFT) [22] in a B-field background. This SFT formulation is based on the N =4 topological string description [23] . It is well known that the SDYM model can be described in terms of holomorphic bundles over (an open subset of) the twistor space 2 [26] CP 3 and the topological N =4 string theory contains twistors from the outset. The Lax pair, integrability and the solutions to the equations of motion by twistor and dressing methods were incorporated into the N =2 open SFT in [27, 28] . However, this theory reproduces only bosonic SDYM theory, its symmetries (see e.g. [29, 30, 31] ) and integrability properties. It is natural to ask: What string theory can describe supersymmetric SDYM theory [32, 33] in four dimensions?\\n\\nThere are some proposals [33, 34, 35, 36] for extending N =2 open string theory (and its SFT) to be space-time supersymmetric. Moreover, it was shown by Witten [37] that N =4 supersymmetric SDYM theory appears in twistor string theory, which is a B-type open topological string with the supertwistor space CP 3|4 as a target space. 3 Note that N <4 SDYM theory forms a BPS subsector of N -extended super Yang-Mills theory, and N =4 SDYM can be considered as a truncation of the full N =4 super Yang-Mills theory [37] . It is believed [43, 39] that twistor string theory is related with the previous proposals [33, 34, 35, 36] for a Lorentz-invariant supersymmetric extension of N =2 (and topological N =4) string theory which also leads to the N =4 SDYM model.\\n\\nA dimensional reduction of the above relations between twistor strings and N =4 super Yang-Mills and SDYM models was considered in [44, 45, 46, 47] . The corresponding twistor string theory after this reduction is the topological B-model on the mini-supertwistor space P 2|4 . In [47] it was shown that the 2N =8 supersymmetric extension of the Bogomolny-type model in 2+1 dimensions is equivalent to an 2N =8 supersymmetric modified U(n) chiral model on R 2,1 . The subject of the current paper is an 2N ≤8 version of the above supersymmetric Bogomolny-type Yang-Mills-Higgs model in signature (-+ +), its relation with an N -extended supersymmetric modified integrable U(n) chiral model (to be defined) in 2+1 dimensions and the Moyal-type noncommutative deformation of this chiral model. We go on to explicitly construct multi-soliton configurations on noncommutative R 2,1 for the corresponding supersymmetric sigma model field equations. By studying the scattering properties of the constructed configurations, we prove their asymptotic factorization without scattering for large times. We also briefly discuss a D-brane interpretation of these soliton configurations from the viewpoint of twistor string theory.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"N -extended SDYM equations in 2+2 dimensions\",\n \"text\": \"Space R 2,2 . Let us consider the four-dimensional space R 2,2 = (R 4 , g) with the metric\\n\\nds 2 = g µν dx µ dx ν = det(dx α α) = dx 1 1dx 2 2 -dx 2 1dx 1 2 (2.1)\\n\\nwith (g µν ) = diag(-1, +1, +1, -1), where µ, ν, . . . = 1, . . . , 4 are space-time indices and α = 1, 2, α = 1, 2 are spinor indices. We choose the coordinates 4\\n\\n(x µ ) = (x a , t) = (t, x, y, t) with a, b, . . . = 1, 2, 3 , (2.2)\\n\\nand the signature (-+ + -) allows us to introduce real isotropic coordinates (cf. [19, 6] )\\n\\nx 1 1 = 1 2 (t -y) , x 1 2 = 1 2 (x + t) , x 2 1 = 1 2 (x -t) , x 2 2 = 1 2 (t + y) . (2.3)\\n\\nSDYM. Recall that the SDYM equations for a field strength tensor\\n\\nF µν on R 2,2 read 1 2 ε µνρσ F ρσ = F µν , (2.4)\\n\\nwhere ε µνρσ is a completely antisymmetric tensor on R 2,2 and ε 1234 = 1. In the coordinates (2.3) we have the decomposition\\n\\nF α α,β β = ∂ α αA β β -∂ β β A α α + [A α α, A β β ] = ε αβ F α β + ε α β F αβ (2.5) with F α β := -1 2 ε αβ F α α,β β and F αβ := -1 2 ε α β F α α,β β , (2.6)\\n\\nwhere ε αβ is antisymmetric, ε αβ ε βγ = δ γ α , and similar for ε α β , with ε 12 = ε 1 2 = 1. The gauge potential (A α α) will appear in the covariant derivative\\n\\nD α β = ∂ α β + [A α β , • ] . (2.7)\\n\\nIn spinor notation, (2.4) is equivalently written as\\n\\nF α β = 0 ⇔ F α α,β β = ε α β F αβ . (2.8)\\n\\nSolutions {A α α} to these equations form a subset (a BPS sector) of the solution space of Yang-Mills theory on R 2,2 .\\n\\nN -extended SDYM in component fields. The field content of N -extended super SDYM is foot_4\\n\\nN = 0 A α α (2.9a) N = 1 A α α, χ i α with i = 1 (2.9b) N = 2 A α α, χ i α , φ [ij] with i, j = 1, 2 (2.9c)\\n\\nN = 3 A α α, χ i α , φ [ij] , χ[ijk] α with i, j, k = 1, 2, 3 (2.9d) N = 4 A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl] α β\\n\\nwith i, j, k, l = 1, 2, 3, 4 .\\n\\n(2.9e)\\n\\nHere (A α α, χ i α , φ [ij] , χ[ijk] α , G [ijkl]\\n\\nα β ) are fields of helicities (+1, + 1 2 , 0, -1 2 , -1). These fields obey the field equations of the N = 4 SDYM model, namely [33, 37]\\n\\nF α β = 0 , (2.10a)\\n\\nD α αχ iα = 0 , (2.10b)\\n\\nD α αD α αφ ij + 2{χ iα , χ j α } = 0 , (2.10c)\\n\\nD α α χ α[ijk] -6[χ [i α , φ jk] ] = 0 , (2.10d)\\n\\nD γ α G [ijkl] γ β + 12{χ [i α , χjkl] β } -18[φ [ij , D α β φ kl] ] = 0 . (2.10e)\\n\\nNote that the N < 4 SDYM field equations are governed by the first N +1 equations of (2.10), where F α β = 0 is counted as one equation and so on.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Superfield formulation of\",\n \"text\": \"N -extended SDYM Superspace R 4|4N .\\n\\nRecall that in the space R 2,2 = (R 4 , g) with the metric g given in (2.1) one may introduce purely real Majorana-Weyl spinors foot_5 θ α and η α of helicities + 1 2 and -1 2 as anticommuting (Grassmann-algebra) objects. Using 2N such spinors with components θ iα and η α i for i = 1, . . . , N , one can define the N -extended superspace R 4|4N and the N -extended supersymmetry algebra generated by the supertranslation operators\\n\\nP α α = ∂ α α , Q iα = ∂ iα -η α i ∂ α α and Q i α = ∂ i α -θ iα ∂ α α , (2.11)\\n\\nwhere\\n\\n∂ α α := ∂ ∂x α α , ∂ iα := ∂ ∂θ iα and ∂ i α := ∂ ∂η α i .\\n\\n(2.12)\\n\\nThe commutation relations for the generators (2.11) read\\n\\n{Q iα , Q j α} = -2δ j i P α α , [P α α, Q iβ ] = 0 and [P α α, Q i β ] = 0 . (2.13)\\n\\nTo rewrite equations of motion in terms of R 4|4N superfields one uses the additional operators\\n\\nD iα = ∂ iα + η α i ∂ α α and D i α = ∂ i α + θ iα ∂ α α , (2.14)\\n\\nwhich (anti)commute with the operators (2.11) and satisfy\\n\\n{D iα , D j β } = 2δ j i P α β , [P α α, D iβ ] = 0 and [P α α, D j β ] = 0 . (2.15)\\n\\nAntichiral superspace R 4|2N . On the superspace R 4|4N one may introduce tensor fields depending on bosonic and fermionic coordinates (superfields), differential forms, Lie derivatives L X etc.. Furthermore, on any such superfield A one can impose the constraint equations L D iα A = 0, which for a scalar superfield f reduce to the so-called antichirality conditions\\n\\nD iα f = 0 . (2.16)\\n\\nThese are easily solved by using a coordinate transformation on R 4|4N ,\\n\\n(x α α, η α i , θ iα ) → (x α α = x α α-θ iα η α i , η α i , θ iα ) , (2.17)\\n\\nunder which ∂ α α, D iα and D i α transform to the operators\\n\\n∂α α = ∂ α α , Diα = ∂ iα and Di α = ∂ i α + 2θ iα ∂ α α . ( 2\\n\\n.18) Then (2.16) simply means that f is defined on a sub-superspace R 4|2N ⊂ R 4|4N with coordinates xα α and η α i . (2.19)\\n\\nThis space is called antichiral superspace. In the following we will usually omit the tildes when working on the antichiral superspace.\\n\\nN -extended SDYM in superfields. The N -extended SDYM equations can be rewritten in terms of superfields on the antichiral superspace R 4|2N [33, 48] . Namely, for any given 0 ≤ N ≤ 4, fields of a proper multiplet from (2.9) can be combined into superfields A α α and A i α depending on x α α, η α i ∈ R 4|2N and giving rise to covariant derivatives\\n\\n∇ α α := ∂ α α + A α α and ∇ i α := ∂ i α + A i α .\\n\\n(2.20)\\n\\nIn such terms the N -extended SDYM equations (2.10) read\\n\\n[∇ α α, ∇ β β ] + [∇ α β , ∇ β α] = 0 , [∇ i α, ∇ β β ] + [∇ i β , ∇ β α] = 0 , {∇ i α, ∇ j β } + {∇ i β , ∇ j α} = 0 , (2.21)\\n\\nwhich is equivalent to\\n\\n[∇ α α, ∇ β β ] = ε α β F αβ , [∇ i α, ∇ β β ] = ε α β F i β and {∇ i α, ∇ j β } = ε α β F ij , (2.22)\\n\\nwhere F ij is antisymmetric and F αβ is symmetric in their indices.\\n\\nThe above gauge potential superfields (A α α, A i α) as well as the gauge strength superfields (F αβ , F i α , F ij ) contain all physical component fields of the N -extended SDYM model. For instance, the lowest component of the triple (F αβ , F i α , F ij ) in an η-expansion is (F αβ , χ i α , φ ij ), with zeros in case N is too small. By employing Bianchi identities for the gauge strength superfields, one successively obtains [48] the superfield expansions and the field equations (2.10) for all component fields.\\n\\nIt is instructive to extend the antichiral combination in (2.18) to potentials and covariant derivatives, Di\\n\\nα = ∂ i α + 2 θ iα ∂ α α + + + Ãi α := A i α + 2 θ iα A α α ∇i α := ∇ i α + 2 θ iα ∇ α α ( 2\\n\\n.23) where ∇ α α, ∇ i α and Di α are given by (2.20) and (2.18), while A i α and A α α depend on x α α and η α i only. With the antichiral covariant derivatives, one may condense (2.21) or (2.22) into the single set { ∇i α, ∇j β\\n\\n} + { ∇i β , ∇j α} = 0 ⇔ { ∇i α, ∇j β } = ε α β Fij , (2.24)\\n\\nwith Fij =\\n\\nF ij + 4 θ [iα F j] α + 4 θ iα θ jβ F αβ .\\n\\nThe concise form (2.24) of the N -extended SDYM equations is quite convenient, and we will use it interchangeable with (2.21).\\n\\nLinear system for N -extended SDYM. It is well known that the superfield SDYM equations (2.21) can be seen as the compatibility conditions for the linear system of differential equations\\n\\nζ α(∂ α α + A α α) ψ = 0 and ζ α(∂ i α + A i α) ψ = 0 , (2.25)\\n\\nwhere\\n\\n(ζ β ) = 1 ζ and ζ α = ε α β ζ β . The extra (spectral) parameter 7 ζ lies in the extended complex plane C ∪ ∞ = CP 1 .\\n\\nHere ψ is a matrix-valued function depending not only on x α α and η α i but also (meromorphically) on ζ ∈ CP 1 . We subject the n×n matrix ψ to the following reality condition:\\n\\nψ(x α α, η α i , ζ) ψ(x α α, η α i , ζ) † = 1l , (2.26)\\n\\nwhere \\\" †\\\" denotes hermitian conjugation and ζ is complex conjugate to ζ. This condition guarantees that all physical fields of the N -extended SDYM model will take values in the adjoint representation of the algebra u(n). In the concise form the linear system (2.25) is written as\\n\\nζ α(∇ i α + 2θ iα ∇ α α) ψ = 0 ⇔ ζ α( Di α + Ãi α) ψ = 0 ⇔ ζ α ∇i α ψ = 0 . (2.27) 2.3 Reduction of N -extended SDYM to 2+1 dimensions\\n\\nThe supersymmetric Bogomolny-type Yang-Mills-Higgs equations in 2+1 dimensions are obtained from the described N -extended super SDYM equations by a dimensional reduction R 2,2 → R 2,1 .\\n\\nIn particular, for the N =0 sector we demand the components A µ of a gauge potential to be independent of x 4 and put A 4 =: ϕ. Here, ϕ is a Lie-algebra valued scalar field in three dimensions (the Higgs field) which enters into the Bogomolny-type equations. Similarly, for N ≥ 1 one can reduce the N -extended SDYM equations on R 2,2 by imposing the ∂ 4 -invariance condition on all the fields (A α α,\\n\\nχ i α , φ [ij] , χ[ijk] α , G [ijkl]\\n\\nα β ) from the N =4 supermultiplet or its truncation to N <4 and obtain supersymmetric Bogomolny-type equations on R 2,1 .\\n\\nSpinors in R 2,1 .\\n\\nRecall that on R 2,2 both N =4 SDYM theory and full N =4 super Yang-Mills theory have an SL(4, R) ∼ = Spin(3,3) R-symmetry group [33] . A dimensional reduction to R 2,1 enlarges the supersymmetry and R-symmetry to 2N =8 and Spin(4,4), respectively, for both theories (cf. [49] for Minkowski signature). More generally, any number N of supersymmetries gets doubled to 2N in the reduction. Since dimensional reduction collapses the rotation group Spin(2,2)\\n\\n∼ = Spin(2,1) L ×Spin(2,1) R of R 2,2\\n\\nto its diagonal subgroup Spin(2,1) D as the local rotation group of R 2,1 , the distinction between undotted and dotted indices disappears. We shall use undotted indices henceforth.\\n\\nCoordinates and derivatives in R 2,1 . The above discussion implies that one can relabel the bosonic coordinates x α β from (2.3) by x αβ and split them as\\n\\nx αβ = 1 2 (x αβ + x βα ) + 1 2 (x αβ -x βα ) = x (αβ) + x [αβ]\\n\\n(2.28) into antisymmetric and symmetric parts,\\n\\nx [αβ] = 1 2 ε αβ x 4 = 1 2 ε αβ t and x (αβ) =: y αβ , (2.29)\\n\\nrespectively, with\\n\\ny 11 = x 11 = 1 2 (t -y) , y 12 = 1 2 (x 12 + x 21 ) = 1 2 x , y 22 = x 22 = 1 2 (t + y) . (2.30)\\n\\nWe also have θ iα → θ iα and η α i → η α i for the fermionic coordinates on R 4|4N reduced to R 3|4N . Bosonic coordinate derivatives reduce in 2+1 dimensions to the operators\\n\\n∂ (αβ) = 1 2 (∂ αβ + ∂ βα ) (2.31)\\n\\nwhich read explicitly as\\n\\n∂ (11) = ∂ ∂y 11 = ∂ t -∂ y , ∂ (12) = ∂ (21) = 1 2 ∂ ∂y 12 = ∂ x , ∂ (22) = ∂ ∂y 22 = ∂ t + ∂ y . (2.32)\\n\\nWe thus have\\n\\n∂ ∂x αβ = ∂ (αβ) -ε αβ ∂ 4 = ∂ (αβ) -ε αβ ∂ t , (2.33)\\n\\nwhere\\n\\nε 12 = -ε 21 = -1, ∂ 4 = ∂/∂x 4 and ∂ t = ∂/∂ t.\\n\\nThe operators D iα and D i α acting on t-independent superfields reduce to\\n\\nD iα = ∂ iα + η β i ∂ (αβ) and D i α = ∂ i α + θ iβ ∂ (αβ) , (2.34)\\n\\nwhere\\n\\n∂ iα = ∂/∂θ iα and ∂ i α = ∂/∂η α i .\\n\\nSimilarly, the antichiral operators Diα and Di α in (2.18) become Diα = ∂ iα and Di α = ∂ i α + 2θ iβ ∂ (αβ) . (2.35) Supersymmetric Bogomolny-type equations in component fields. According to (2.33), the components A α β of a gauge potential in four dimensions split into the components A (αβ) of a gauge potential in three dimensions and a Higgs field\\n\\nA [αβ] = -ε αβ ϕ, i.e. A αβ = A (αβ) + A [αβ] = A (αβ) -ε αβ ϕ . (2.36)\\n\\nThen the covariant derivatives D α β reduced to three dimensions become the differential operators\\n\\nD αβ -ε αβ ϕ = ∂ (αβ) + [A (αβ) , • ] -ε αβ [ϕ, • ] , (2.37)\\n\\nand the Yang-Mills field strength on R 2,1 decomposes as\\n\\nF αβ, γδ = [D αβ , D γδ ] = ε αγ f βδ + ε βδ f αγ with f αβ = f βα . (2.38)\\n\\nSubstituting (2.36) and (2.37) into (2.10), i.e. demanding that all fields in (2.10) are independent of x 4 = t, we obtain the following supersymmetric Bogomolny-type equations on R 2,1 :\\n\\nf αβ + D αβ ϕ = 0 , (2.39a)\\n\\nD αβ χ iβ + ε αβ [ϕ, χ iβ ] = 0 , (2.39b)\\n\\nD αβ D αβ φ ij + 2[ϕ, [ϕ, φ ij ]] + 2{χ iα , χ j α } = 0 , (2.39c)\\n\\nD αβ χβ[ijk] -ε αβ [ϕ, χβ[ijk] ] -6[χ [i α , φ jk] ] = 0 , (2.39d)\\n\\nD γ α G [ijkl] γβ + [ϕ, G [ijkl] αβ ] + 12{χ [i α , χjkl] β } -18[φ [ij , D αβ φ kl] ] -18ε αβ [φ [ij , [φ kl] , ϕ]] = 0 .(2.\\n\\nSupersymmetric Bogomolny-type equations in terms of superfields. Translations generated by the vector field ∂ 4 = ∂ t are isometries of superspaces R 4|4N and R 4|2N . By taking the quotient with respect to the action of the abelian group G generated by ∂ 4 , we obtain the reduced full superspace R 3|4N ∼ = R 4|4N /G and the reduced antichiral superspace R 3|2N ∼ = R 4|2N /G. In the following, we shall work on R 3|2N and R 3|2N × CP 1 , since the reduced ψ-function from (2.25) and (2.27) is defined on the latter space.\\n\\nThe linear system stays in the center of the superfield approach to the N -extended SDYM equations. After imposing t-independence on all fields in the linear system (2.27), we arrive at the linear equations\\n\\nζ α ∇i α ψ ≡ ζ α ( Di α + Âi α ) ψ = 0 (2.40)\\n\\nof the same form but with\\n\\nDi α = ∂ i α + 2θ iβ ∂ (αβ) and Âi α = A i α + 2θ iβ (A (αβ) -ε αβ Ξ) , (2.41)\\n\\nwhere A i α , A (αβ) and Ξ are superfields depending on y αβ and η α i only. These linear equations expand again to the pair (cf. (2.25))\\n\\nζ β (∂ (αβ) + A (αβ) -ε αβ Ξ) ψ = 0 and ζ α (∂ i α + A i α ) ψ = 0 . (2.42)\\n\\nThe compatibility conditions for the linear system (2.40) read\\n\\n{ ∇i α , ∇j β } + { ∇i β , ∇j α } = 0 ⇔ { ∇i α , ∇j β } = ε αβ Fij (2.43)\\n\\nand present a condensed form of (2.39) rewritten in terms of R 3|2N superfields. Similarly, these equations can also be written in more expanded forms analogously to (2.21) or using the superfield analog of (2.37). However, we will not do this since all these sets of equations are equivalent.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Noncommutative N -extended U(n) chiral model in 2+1 dimensions\",\n \"text\": \"As has been known for some time, nonlinear sigma models in 2 + 1 dimensions may be Lorentzinvariant or integrable but not both [7, 11] . We will show that the super Bogomolny-type model discussed in Section 2 after a gauge fixing is equivalent to a super extension of the modified U(n) chiral model (so as to be integrable) first formulated by Ward [7] . Since integrability is compatible with noncommutative deformation (if introduced properly, see e.g. [9] - [20] ) we choose from the beginning to formulate our super extension of this chiral model on Moyal-deformed R 2,1 with noncommutativity parameter θ ≥ 0. Ordinary space-time R 2,1 can always be restored by taking the commutative limit θ → 0.\\n\\nStar-product formulation. Classical field theory on noncommutative spaces may be realized in a star-product formulation or in an operator formalism 8 . The first approach is closer to the commutative field theory: it is obtained by simply deforming the ordinary product of classical fields (or their components) to the noncommutative star product\\n\\n(f ⋆ g)(x) = f (x) exp{ i 2 ← - ∂ a θ ab -→ ∂ b } g(x) ⇒ x a ⋆ x b -x b ⋆ x a = iθ ab (3.1)\\n\\nwith a constant antisymmetric tensor θ ab . Specializing to R 2,1 , we use real coordinates (x a ) = (t, x, y) in which the Minkowski metric g on R 3 reads (g ab ) = diag(-1, +1, +1) with a, b, . . . = 1, 2, 3 (cf. Section 2). It is straightforward to generalize the Moyal deformation (3.1) to the superspaces introduced in the previous section, allowing in particular for non-anticommuting Grassmann-odd coordinates. Deferring general superspace deformations and their consequences to future work, we here content ourselves with the simple embedding of the \\\"bosonic\\\" Moyal deformation into superspace, meaning that (3.1) is also valid for superfields f and g depending on Grassmann variables θ iα and η α i . For later use we consider not only isotropic coordinates and vector fields\\n\\nu := 1 2 (t+y) = y 22 , v := 1 2 (t-y) = y 11 , ∂ u = ∂ t + ∂ y = ∂ (22) , ∂ v = ∂ t -∂ y = ∂ (11) (3.2)\\n\\nintroduced in Section 2, but also the complex combinations\\n\\nz := x + iy , z := x -iy , ∂ z = 1 2 (∂ x -i∂ y ) , ∂ z = 1 2 (∂ x + i∂ y ) . (3.3)\\n\\nSince the time coordinate t remains commutative, the only nonvanishing component of the noncommutativity tensor θ ab is Operator formalism. The nonlocality of the star products renders explicit computation cumbersome. We therefore pass to the operator formalism, which trades the star product for operatorvalued spatial coordinates (x, ŷ) or their complex combinations (ẑ, ẑ), subject to\\n\\nθ xy = -θ yx =: θ > 0 ⇒ θ z z = -θ zz = -2i θ . ( 3\\n\\n[t, x] = [t, ŷ] = 0 but [x, ŷ] = iθ ⇒ [ẑ, ẑ] = 2 θ . (3.6)\\n\\nThe latter equation suggests the introduction of annihilation and creation operators,\\n\\na = 1 √ 2θ ẑ and a † = 1 √ 2θ ẑ with [a , a † ] = 1 , (3.7)\\n\\nwhich act on a harmonic-oscillator Fock space H with an orthonormal basis { |ℓ , ℓ = 0, 1, 2, . . .} such that\\n\\na |ℓ = √ ℓ |ℓ-1 and a † |ℓ = √ ℓ+1 |ℓ+1 . (3.8)\\n\\nAny superfield f (t, z, z, η α i ) on R 3|2N can be related to an operator-valued superfield f (t, η α i ) ≡ F (t, a, a † , η α i ) on R 1|2N acting in H, with the help of the Moyal-Weyl map\\n\\nf (t, z, z, η α i ) → f (t, η α i ) = Weyl-ordered f t, √ 2θa, √ 2θa † , η α i . (3.9)\\n\\nThe inverse transformation recovers the ordinary superfield,\\n\\nf (t, η α i ) ≡ F (t, a, a † , η α i ) → f (t, z, z, η α i ) = F ⋆ t, z √ 2θ , z √ 2θ , η α i , (3.10)\\n\\nwhere F ⋆ is obtained from F by replacing ordinary with star products. Under the Moyal-Weyl map, we have\\n\\nf ⋆ g → f ĝ and dx dy f = 2π θ Tr f = 2π θ ℓ≥0 ℓ| f |ℓ , (3.11)\\n\\nand the spatial derivatives are mapped into commutators,\\n\\n∂ z f → ∂z f = -1 √ 2θ [a † , f ] and ∂ z f → ∂z f = 1 √ 2θ [a , f ] . (3.12)\\n\\nFor notational simplicity we will from now on omit the hats over the operators except when confusion may arise.\\n\\nGauge fixing for ψ. Note that the linear system (2.40) and the compatibility conditions (2.43) are invariant under a gauge transformation\\n\\nψ → ψ ′ = g -1 ψ , (3.13a)\\n\\nA → A ′ = g -1 A g + g -1 ∂ g (with appropriate indices) , (3.13b)\\n\\nΞ → Ξ ′ = g -1 Ξ g , (3.13c)\\n\\nwhere g = g(x a , η α i ) is a U(n)-valued superfield globally defined on the deformed superspace R 3|2N θ × CP 1 . Using a gauge transformation of the form (3.13), we can choose ψ such that it will satisfy the standard asymptotic conditions (see e.g. [51] )\\n\\nψ = Φ -1 + O(ζ) for ζ → 0 , (3.14a\\n\\n)\\n\\nψ = 1l + ζ -1 Υ + O(ζ -2 ) for ζ → ∞ , (3.14b)\\n\\nwhere the U(n)-valued function Φ and u(n)-valued function Υ depend on x a and η α i . This \\\"unitary\\\" gauge is compatible with the reality condition for ψ,\\n\\nψ(x a , η α i , ζ) ψ(x a , η α i , ζ) † = 1l , (3.15)\\n\\nobtained by reduction from (2.26).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Gauge fixing for\",\n \"text\": \"A i 1 = 0 and A i 2 = Φ -1 ∂ i 2 Φ = ∂ i 1 Υ , (3.17)\\n\\nA (11) = 0 and\\n\\nA (12) + Ξ = Φ -1 ∂ (12) Φ = ∂ (11) Υ , (3.18)\\n\\nA (21) -Ξ = 0 and\\n\\nA (22) = Φ -1 ∂ (22) Φ = ∂ (12) Υ . (3.19)\\n\\nUsing (2.32), we can rewrite the nonzero components as\\n\\nA := Φ -1 ∂ u Φ = ∂ x Υ , B := Φ -1 ∂ x Φ = ∂ v Υ , C i := Φ -1 ∂ i 2 Φ = ∂ i 1 Υ . (3.20)\\n\\nRecall that the superfields Φ and Υ depend on x a and η α i .\\n\\nLinear system. In the above-introduced unitary gauge the linear system (2.42) reads\\n\\n(ζ∂ x -∂ u -A) ψ = 0 , (ζ∂ v -∂ x -B) ψ = 0 , (ζ∂ i 1 -∂ i 2 -C i ) ψ = 0 , (3.21)\\n\\nwhich adds the last equation to the linear system of the Ward model [7] and generalizes it to superfields A(x a , η α j ), B(x a , η α j ) and C i (x a , η α j ). The concise form of (3.21) reads\\n\\nζ Di 1 -Di 2 -Âi 2 ψ = 0 (3.22)\\n\\nor, in more explicit form,\\n\\nζ ∂ i 1 + 2θ i1 ∂ v + 2θ i2 ∂ x -∂ i 2 + C i + 2θ i1 (∂ x + B) + 2θ i2 (∂ u + A) ψ = 0 . (3.23)\\n\\nN -extended sigma model.\\n\\nThe compatibility conditions of this linear system are the Nextended noncommutative sigma model equations Di\\n\\n1 (Φ -1 Dj 2 Φ) + Dj 1 (Φ -1 Di 2 Φ) = 0 (3.24)\\n\\nwhich in expanded form reads\\n\\n(g ab + v c ε cab ) ∂ a (Φ -1 ∂ b Φ) = 0 ⇔ ∂ x (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ u Φ) = 0 , (3.25a) ∂ i 1 (Φ -1 ∂ x Φ) -∂ v (Φ -1 ∂ i 2 Φ) = 0 , ∂ i 1 (Φ -1 ∂ u Φ) -∂ x (Φ -1 ∂ i 2 Φ) = 0 , (3.25b)\\n\\n∂ i 1 (Φ -1 ∂ j 2 Φ) + ∂ j 1 (Φ -1 ∂ i 2 Φ) = 0 . (3.25c)\\n\\nHere, the first line contains the Wess-Zumino-Witten term with a constant vector (v c ) = (0, 1, 0) which spoils the standard Lorentz invariance but yields an integrable chiral model in 2+1 dimensions. Recall that Φ is a U(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . As discussed in Section 2, the compatibility conditions of the linear equations (3.22) (or (3.21)) are equivalent to the N -extended Bogomolnytype equations (2.39) for the component (physical) fields. Thus, chiral model field equations (3.25) are equivalent to a gauge fixed form of equations (2.39). Υ-formulation. Instead of Φ-parametrization of (A, B, C i ) given in (3.17)-(3.20) we may use the equivalent Υ-parametrization also given there. In this case, the compatibility conditions for the linear system (3.21) reduce to\\n\\n(∂ 2 x -∂ u ∂ v )Υ + [∂ v Υ , ∂ x Υ] = 0 , (3.26a)\\n\\n(∂ i 2 ∂ v -∂ i 1 ∂ x )Υ + [∂ i 1 Υ , ∂ v Υ] = 0 , (∂ i 2 ∂ x -∂ i 1 ∂ u )Υ + [∂ i 1 Υ , ∂ x Υ] = 0 , (3.26b) (∂ i 2 ∂ j 1 + ∂ j 2 ∂ i 1 )Υ + {∂ i 1 Υ , ∂ j 1 Υ} = 0 , (3.26c)\\n\\nwhich in concise form read\\n\\n( Di 2 Dj 1 + Dj 2 Di 1 ) Υ + { Di 1 Υ , Dj 1 Υ} = 0 . (3.27)\\n\\nRecall that Υ is a u(n)-valued matrix whose elements act as operators in the Fock space H and depend on x a and 2N Grassmann variables η α i . For N =4, the commutative limit of (3.27) can be considered as Siegel's equation [33] reduced to 2+1 dimensions. According to Siegel, one can extract the multiplet of physical fields appearing in (2.39) from the prepotential Υ via\\n\\n∂ i 1 Υ = A i 2 , ∂ i 1 ∂ j 1 Υ = φ ij , ∂ i 1 ∂ j 1 ∂ k 1 Υ = χ[ijk] 2 , ∂ i 1 ∂ j 1 ∂ k 1 ∂ l 1 Υ = G [ijkl] 22 , (3.28a)\\n\\n∂ (α1) Υ = A (α2) -ε α2 ϕ , ∂ (α1) ∂ i 1 Υ = χ i α , ∂ (α1) ∂ (β1) Υ = f αβ , (3.28b)\\n\\nwhere one takes Υ and its derivatives at η 2 i = 0. The other components of the physical fields, i.e. χ[ijk]\\n\\n1 , G [ijkl] 11 , G [ijkl]\\n\\n21 , A (11) and A (21) -ϕ, vanish in this light-cone gauge.\\n\\nSupersymmetry transformations. The 4N supercharges given in (2.11) reduce in 2+1 dimensions to the form\\n\\nQ iα = ∂ iα -η β i ∂ (αβ) and Q i α = ∂ i α -θ iβ ∂ (αβ) . (3.29)\\n\\nTheir antichiral version, matching to Diα and Dj β of (2.35), reads\\n\\nQiα = ∂ iα -2η β i ∂ (αβ) and Qj β = ∂ j β , (3.30)\\n\\nso that { Qiα , Qj β } = -2 δ j i ∂ (αβ) . (3.31)\\n\\nOn a (scalar) R 3|2N superfield Σ these supersymmetry transformations act as\\n\\nδ Σ := ε iα Qiα Σ + ε α i Qi α Σ (3.32)\\n\\nand are induced by the coordinate shifts\\n\\nδ y αβ = -2ε i(α η β) i and δ η α i = ε α i , (3.33)\\n\\nwhere ε iα and ε α i are 4N real Grassmann parameters. It is easy to see that our equations (3.24) and (3.27) are invariant under the supersymmetry transformations (3.32) (applied to Φ or Υ). This is simply because the operators Diα and Dj β anticommute with the supersymmetry generators Qiα and Qj β . Therefore, the equations of motion (3.25) of the modified N -extended chiral model in 2+1 dimensions as well as their reductions to 2+0 and 1+1 dimensions carry 2N supersymmetries and are genuine supersymmetric extensions of the corresponding bosonic equations. Note that this type of extension is not the standard one since the R-symmetry groups are Spin(N , N ) in 2+1 and Spin(N , N )× Spin(N , N ) in 1+1 dimensions, which differ from the compact unitary R-symmetry groups of standard sigma models. Contrary to the standard case of two-dimensional sigma models the above \\\"noncompact\\\" 2N supersymmetries do not impose any constraints on the geometry of the target space, e.g. they do not demand it to be Kähler [52] or hyper-Kähler [53] . This may be of interest and deserves further study.\\n\\nAction functionals. In either formulation of the N -extended supersymmetric SDYM model on R 2,2 there are difficulties with finding a proper action functional generalizing the one [54, 55] for the purely bosonic case. These difficulties persist after the reduction to 2+1 dimensions, i.e. for the equations (3.25) and (3.26) describing our supersymmetric modified U(n) chiral model. It is the price to be paid for overcoming the no-go barrier N ≤ 4 and the absence of geometric target-space constraints. On a more formal level, the problem is related to the chiral character of (3.24) as well as (3.27) , where only the operators Di α but not Diα appear. Note however, that for N = 4 one can write an action functional in component fields producing the equations (2.39), which are equivalent to the superspace equations (3.24) when i, j = 1, . . . , 4 (see e.g. [47] ).\\n\\nOne proposal for an action functional stems from Siegel's idea [33] for the Υ-formulation of the N -extended SDYM equations. Namely, one sees that ∂ i 2 Υ enters only linearly into the last two lines in (3.26) . Therefore, if we introduce\\n\\nΥ (1) := Υ| η 2 i =0 (3.34)\\n\\nthen it must satisfy the first equation from (3.26) , and the remaining equations iteratively define the dependence of Υ on η 2 i starting from Υ (1) . Hence, all information is contained in Υ (1) , as can also be seen from (3.28) . In other words, the dependence of Υ on η 2 i is not 'dynamical'. For an action one can then take (cf. [33] )\\n\\nS = d 3 x d N η 1 Υ (1) ∂ (αβ) ∂ (αβ) Υ (1) + 2 3 Υ (1) ε αβ ∂ (α1) Υ (1) ∂ (β1) Υ (1) . (3.35)\\n\\nExtremizing this functional yields the first line of (3.26) at η 2 i = 0. Except for the Grassmann integration, this action has the same form as the purely bosonic one [55] . One may apply the same logic to the Φ-formulation where the action for the purely bosonic case is also known [54, 56] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"N -extended multi-soliton configurations via dressing\",\n \"text\": \"The existence of the linear system (3.22) (equivalent to (3.21)) encoding solutions of the N -extended U(n) chiral model in an auxiliary matrix ψ allows for powerful methods to systematically construct explicit solutions for ψ and hence for Φ † = ψ| ζ=0 and Υ = lim ζ→∞ ζ (ψ-1l). For our purposes the so-called dressing method [57, 51] proves to be the most practical [12] - [20] , and so we shall use it here for our linear system, i.e. already in the N -extended noncommutative case.\\n\\nMulti-pole ansatz for ψ. The dressing method is a recursive procedure for generating a new solution from an old one. More concretely, we rewrite the linear system (3.21) in the form\\n\\nψ(∂ u -ζ∂ x )ψ † = A , ψ(∂ x -ζ∂ v )ψ † = B , ψ(∂ i 2 -ζ∂ i 1 )ψ † = C i . (4.1)\\n\\nRecall that ψ † := (ψ(x a , η α i , ζ)) † and (A, B, C i ) depend only on x a and η α i . The central idea is to demand analyticity in the spectral parameter ζ, which strongly restricts the possible form of ψ. One way to exploit this constraint starts from the observation that the left hand sides of (4.1) as well as of the reality condition (3.15) do not depend on ζ while ψ is expected to be a nontrivial function of ζ globally defined on CP 1 . Therefore, it must be a meromorphic function on CP 1 possessing some poles which we choose to lie at finite points with constant coordinates µ k ∈ CP 1 .\\n\\nHere we will build a (multi-soliton) solution ψ m featuring m simple poles at positions µ 1 , . . . , µ m with foot_8 Im µ k < 0 by left-multiplying an (m-1)-pole solution ψ m-1 with a single-pole factor of the form 1l\\n\\n+ µ m -μm ζ -µ m P m (x a , η α i ) , (4.2)\\n\\nwhere the n×n matrix function P m is yet to be determined. Starting from the trivial (vacuum) solution ψ 0 = 1l, the iteration ψ 0 → ψ 1 → . . . → ψ m yields a multiplicative ansatz for ψ m ,\\n\\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ , (4.3)\\n\\nwhich, via partial fraction decomposition, may be rewritten in the additive form\\n\\nψ m = 1l + m k=1 Λ mk S † k ζ -µ k , (4.4)\\n\\nwhere Λ mk and S k are some n×r k matrices depending on x a and η α i , with r k ≤ n.\\n\\nEquations for S k . Let us first consider the additive parametrization (4.4) of ψ m . This ansatz must satisfy the reality condition (3.15) as well as our linear equations in the form (4.1). In particular, the poles at ζ = μk on the left hand sides of these equations have to be removable since the right hand sides are independent of ζ. Inserting the ansatz (4.4) and putting to zero the corresponding residues, we learn from (3.15) that\\n\\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ S k = 0 , (4.5)\\n\\nwhile from (4.1) we obtain the differential equations\\n\\n1l + m ℓ=1 Λ mℓ S † ℓ μk -µ ℓ LA,B,i k S k = 0 , (4.6)\\n\\nwhere LA,B,i\\n\\nk stands for either LA k = ∂ u -μk ∂ x , LB k = µ k (∂ x -μk ∂ v ) or Li k = ∂ i 2 -μk ∂ i 1 . (4.7)\\n\\nNote that we consider a recursive procedure starting from m=1, and operators (4.7) will appear with k = 1, . . . , m if we consider poles at ζ = μk .\\n\\nBecause the LA,B,i k for k = 1, . . . , m are linear differential operators, it is easy to write down the general solution for (4.6) at any given k, by passing from the coordinates (u, v, x; η 1 i , η 2 i ) to \\\"co-moving coordinates\\\" (w k , wk , s k ; η i k , ηi k ). The precise relation for k = 1, . . . , m is [12, 58]\\n\\nw k := x + μk u + μ-1 k v = x + 1 2 (μ k -μ -1 k )y + 1 2 (μ k +μ -1 k )t and η i k := η 1 i + μk η 2 i , (4.8)\\n\\nwith wk and ηi k obtained by complex conjugation and the co-moving time s k being inessential because by definition nothing will depend on it. The kth moving frame travels with a constant velocity\\n\\n(v x , v y ) k = - µ k + μk µ k μk + 1 , µ k μk -1 µ k μk + 1 , (4.9)\\n\\nso that the static case w k =z is recovered for µ k = -i. On functions of (w k , η i k , wk , ηi k ) alone the operators (4.7) act as\\n\\nLA k = LB k = (µ k -μ k ) ∂ ∂ wk =: Lk and Li k = (µ k -μ k ) ∂ ∂ ηi k . (4.10)\\n\\nBy induction in k = 1, . . . , m we learn that, due to (4.5), a necessary and sufficient condition for a solution of (4.6) is Lk\\n\\nS k = S k Zk and Li k S k = S k Zi k (4.11)\\n\\nwith some r k ×r k matrices Zk and Zi k depending on (w k , wk , η j k , ηj k ). Passing to the noncommutative bosonic coordinates we obtain ŵk , ŵk = 2θ ν k νk with\\n\\nν k νk = 4i µ k -μ k -µ -1 k +μ -1 k . (4.12)\\n\\nThus, we can introduce annihilation and creation operators\\n\\nc k = 1 √ 2θ ŵk ν k and c † k = 1 √ 2θ ŵk νk so that [c k , c † k ] = 1 (4.13)\\n\\nfor k = 1, . . . , m. Naturally, this Heisenberg algebra is realized on a \\\"co-moving\\\" Fock space H k , with basis states |ℓ k and a \\\"co-moving\\\" vacuum |0 k subject to c k |0 k = 0. Each co-moving vacuum |0 k (annihilated by c k ) is related to the static vacuum |0 (annihilated by a) through an ISU(1,1) squeezing transformation (cf. [12] ) which is time-dependent. The fermionic coordinates η i k and ηi k remain spectators in the deformation. Coordinate derivatives are represented in the standard fashion as\\n\\nν k √ 2θ ∂ ∂w k → -[c † k , • ] and νk √ 2θ ∂ ∂ wk → [c k , • ] . (4.14)\\n\\nAfter the Moyal deformation, the n×r k matrices S k have become operator-valued, but are still functions of the Grassmann coordinates η i k and ηi k . The noncommutative version of the BPS conditions (4.11) naturally reads\\n\\nc k S k = S k Z k and ∂ ∂ ηi k S k = S k Z i k (4.15)\\n\\nwhere Z k and Z i k are some operator-valued r k ×r k matrix functions of η j k and ηj k .\\n\\nNonabelian solutions for S k . For general data Z k and Z i k it is difficult to solve (4.15), but it is also unnecessary because the final expression ψ m turns out not to depend on them. Therefore, we conveniently choose\\n\\nZ k = c k ⊗ 1l r k ×r k and Z i k = 0 ⇒ S k = R k (c k , η i k ) , (4.16)\\n\\nwhere R k is an arbitrary n×r k matrix function independent of c † k and ηi k . 10 It is known that nonabelian (multi-) solitons arise for algebraic functions R k (cf. e.g. [7] for the commutative and [12] for the noncommutative N =0 case). Their common feature is a smooth commutative limit. The only novelty of the supersymmetric extension is the η i k dependence, i.e.\\n\\nR k = R k,0 + η i k R k,i + η i k η j k R k,ij + η i k η j k η p k R k,ijp + η i k η j k η p k η q k R k,ijpq . (4.17)\\n\\nAbelian solutions for S k . It is useful to view S k as a map from C r k ⊗H k to C n ⊗H k (momentarily suppressing the η dependence). The noncommutative setup now allows us to generalize the domain of this map to any subspace of C n ⊗ H k . In particular, we may choose it to be finite-dimensional, say C q k , and represent the map by an n×q k array |S k of kets in H. In this situation, Z k and Z i k in (4.15) are just number -valued q k ×q k matrix functions of η j k and ηj k . In case they do not depend on ηj k , we can write down the most general solution as\\n\\n|S k = R k (c k , η j k ) |Z k exp i Z i k (η j k ) ηi k with |Z k := exp Z k (η j k ) c † k |0 k . (4.18)\\n\\nAs before, we may put Z i k = 0 without loss of generality, but now the choice of Z k does matter. For any given k generically there exists a q k -dimensional basis change which diagonalizes the ket-valued matrix\\n\\n|Z k → diag e α 1 k c † , e α 2 k c † , . . . , e α q k k c † |0 k = diag |α 1 k , |α 2 k , . . . , |α q k k , (4.19)\\n\\nwhere we defined coherent states\\n\\n|α l k := e α l k c † |0 k so that c k |α l k = α l k |α l k for l = 1, . . . , q k and α l k ∈ C . (4.20)\\n\\nNote that not only the entries of R k but also the α l k are holomorphic functions of the co-moving Grassmann parameters η j k and thus can be expanded like in (4.17). In the U(1) model, we must use ket-valued 1×q k matrices |S k for all k, yielding rows\\n\\n|S k = R 1 k |α 1 k , R 2 k |α 2 k , . . . , R q k k |α q k k for k = 1, . . . , m , (4.21)\\n\\nwith functions α l k (η j k ). Here, the R l k only affect the states' normalization and can be collected in a diagonal matrix to the right, hence will drop out later and thus may all be put to one. Formally, we have recovered the known abelian (multi-) soliton solutions, but the supersymmetric extension has generalized |S k → |S k (η j k ) . Explicit form of P k . Let us now consider the multiplicative parametrization (4.3) of ψ m which also allows us to solve (4.5) . First of all, note that the reality condition (3.15) is satisfied if\\n\\nP k = P † k = P 2 k ⇔ P k = T k (T † k T k ) -1 T † k for k = 1, . . . , m , (4.22)\\n\\nmeaning that P k is an operator-valued hermitian projector (of group-space rank r k ≤ n) built from an n×r k matrix function T k (the abelian case of n=1 is included). The reality condition follows just because The r k columns of T k span the image of P k and obey\\n\\n1l + µ k -μk ζ -µ k P k 1l + μk -µ k ζ -μk P k =\\n\\nP k T k = T k ⇔ (1l-P k ) T k = 0 . (4.24)\\n\\nFurthermore, the equation (4.5) with m = k (induction) rewritten in the form\\n\\n(1l-P k ) k-1 ℓ=1 1l + µ k-ℓ -μk-ℓ μk -µ k-ℓ P k-ℓ S k = 0 (4.25)\\n\\nreveals that (cf. (4.24))\\n\\nT 1 = S 1 and T k = k-1 ℓ=1 1l - µ k-ℓ -μk-ℓ µ k-ℓ -μk P k-ℓ S k for k ≥ 2 , (4.26)\\n\\nwhere the explicit form of S k for k = 1, . . . , m is given in (4.16) or (4.18) . The final result reads\\n\\nψ m = m-1 ℓ=0 1l + µ m-ℓ -μm-ℓ ζ -µ m-ℓ P m-ℓ = 1l + m k=1 Λ mk S † k ζ -µ k (4.27)\\n\\nwith hermitian projectors P k given by (4.22), T k given by (4.26) and S k given by (4.16) or (4.18) . The explicit form of Λ mk (which we do not need) can be found in [12] . The corresponding superfields Φ and Υ are\\n\\nΦ m = ψ † m | ζ=0 = m k=1 (1l -ρ k P k ) with ρ k = 1 - µ k μk , (4.28a)\\n\\nΥ m = lim ζ→∞ ζ (ψ m -1l) = m k=1 (µ k -μ k ) P k . (4.28b)\\n\\nFrom (4.22) it is obvious that P k is invariant under a similarity transformation\\n\\nT k → T k Λ k ⇔ S k → S k Λ k (4.29)\\n\\nfor an invertible operator-valued r k ×r k matrix Λ k . This justifies putting Z i k = 0 from the beginning and also the restriction to Z k = c k ⊗ 1l r k ×r k in the nonabelian case, both without loss of generality. Hence, the nonabelian solution space constructed here is parametrized by the set {R k } m 1 of matrixvalued functions of c k and η i k and the pole positions µ k . The abelian moduli space, however, is larger by the set {Z k } m 1 of matrix-values functions of η i k which generically contain the coherentstate parameter functions {α l k (η i k )}. Restricting to η i k =0 reproduces the soliton configurations of the bosonic model [12] .\\n\\nStatic solutions. Let us consider the reduction to 2+0 dimensions, i.e. the static case. Recall that static solutions correspond to the choice m = 1 and µ 1 ≡ µ = -i implying w 1 = z, so we drop the index k. Specializing (4.27), we have\\n\\nψ = 1l - 2 i ζ + i P so that Φ = Φ † = 1l -2P , (4.30)\\n\\nwhere a hermitian projector P of group-space rank r satisfies the BPS equations (1l-P ) a P = 0 ⇒ (1l-P ) a T = 0 ,\\n\\n(1l-P ) ∂ ∂ ηi P = 0 ⇒ (1l-P ) ∂ ∂ ηi T = 0 , (4.31a)\\n\\nwith P = T (T † T ) -1 T † and η i = η 1 i + iη 2 i . In this case T = S, and for a nonabelian r=1 projector P we get T = T (a, η i ) as an n×1 column. For the simplest case of N =1 we just have (cf. [59] )\\n\\nT = T e (a) + η T o (a) with η = η 1 + iη 2 , (4.32)\\n\\nwhere T e (a) and T o (a) are rational functions of a (e.g. polynomials) taking values in the even and odd parts of the Grassmann algebra. Similarly, an abelian N =1 projector (for n=1) is built from\\n\\n|T = |α 1 , |α 2 , . . . , |α q . (4.33)\\n\\nAt θ=0, the static solution (4.32) of our supersymmetric U(n) sigma model is also a solution of the standard N =1 supersymmetric CP n-1 sigma model in two dimensions (see e.g. [59] ). 11 For this reason, one can overcome the previously mentioned difficulty with constructing an action (or energy from the viewpoint of 2+1 dimensions) for static configurations. Moreover, on solutions obeying the BPS conditions (4.31) the topological charge\\n\\nQ = 2πθ dη 1 dη 2 Tr tr Φ D + Φ , D -Φ (4.34)\\n\\nis proportional to the action (BPS bound)\\n\\nS = 2πθ dη 1 dη 2 Tr tr D + Φ , D -Φ (4.35)\\n\\nand is finite for algebraic functions T e and T o . Here, the standard superderivatives D ± are defined as\\n\\nD + = ∂ ∂η + iη ∂ z and D -= ∂ ∂ η + iη ∂ z . ( 4\\n\\nOne-soliton configuration. For one moving soliton, from (4.27) and (4.28) we obtain\\n\\nψ 1 = 1l + µ - μ ζ -µ P with P = T (T † T ) -1 T † (4.37) and Φ = 1l -ρ P with ρ = 1 - µ μ . (4.38)\\n\\nNow our n×r matrix T must satisfy (putting\\n\\nZ i = 0 and Z = c ⊗ 1l r×r ) [c , T ] = 0 and ∂ ∂ ηi T = 0 with η i = η 1 i + μ η 2 i , (4.39)\\n\\nwhere c is the moving-frame annihilation operator given by (4.13) for k=1.\\n\\nRecall that the operators c and c † and therefore the matrix T and the projector P can be expressed in terms of the corresponding static objects by a unitary squeezing transformation (see e.g. (4.8) and (4.13)). For simplicity we again consider the case N =1 and a nonabelian projector with r=1. Then (4.39) tells us that T is a holomorphic function of c and η, i.e.\\n\\nT = T e (c) + η T o (c) = T 1 e (c) + η T 1 o (c)\\n\\n. . .\\n\\nT n e (c) + η T n o (c) (4.40)\\n\\nwith polynomials T a e and T a o of order q, say, analogously to the static case (4.32). Note that, for T a o to be Grassmann-odd and nonzero, some extraneous Grassmann parameter must appear. Similarly, abelian projectors for a moving one-soliton obtain by subjecting (4.33) to a squeezing transformation.\\n\\nFor N =1 the moving frame was defined in (4.8) (dropping the index k) via\\n\\nw = x + 1 2 (μ-μ -1 )y + 1 2 (μ+μ -1 )t and η = η 1 + μη 2 hence ∂ t η = 0 . (4.41)\\n\\nConsider the moving frame with the coordinates (w, w, s; η, η) with the choice s = t and the related change of the derivatives (see [12, 58] )\\n\\n∂ x = ∂ w + ∂ w , (4.42a)\\n\\n∂ y = 1 2 (μ-μ -1 ) ∂ w + 1 2 (µ-µ -1 ) ∂ w , (4.42b)\\n\\n∂ t = 1 2 (μ+μ -1 ) ∂ w + 1 2 (µ+µ -1 ) ∂ w + ∂ s , (4.42c)\\n\\n∂ η 1 = ∂ η + ∂ η , (4.42d)\\n\\n∂ η 2 = μ ∂ η + µ ∂ η . (4.42e)\\n\\nIn the moving frame our solution (4.38) is static, i.e. ∂ s Φ = 0, and the projector P has the same form as in the static case. The only difference is the coefficient ρ instead of 2 in (4.38). Therefore, by computing the action (4.35) in (w, w; η 1 , η 2 ) coordinates, we obtain for algebraic functions T in (4.40) a finite answer, which differs from the static one by a kinematical prefactor depending on µ (cf. [12] for the bosonic case).\\n\\nLarge-time asymptotics. Note that in the distinguished (z, z, t) coordinate frame (4.41) implies that at large times w → κ t with κ = 1 2 (μ+μ -1 ). As a consequence, the t q term in each polynomial in (4.40) will dominate, i.e.\\n\\nT → t q a 1 + η b 1 . . .\\n\\nan + η bn =: t q Γ , (4.43)\\n\\nwhere Γ is a fixed vector in C n . It is easy to see that in the distinguished frame the large-time limit of Φ given by (4. being the projector on the constant vector Γ.\\n\\nConsider now the m-soliton configuration (4.28). By induction of the above argument one easily arrives at the m-soliton generalization of (4.44). Namely, in the frame moving with the ℓth lump we have lim t→±∞ Φ m = (1lρ 1 Π 1 ) . . . (1lρ ℓ-1 Π ℓ-1 )(1lρ ℓ P ℓ )(1lρ ℓ+1 Π ℓ+1 ) . . . (1lρ m Π m ) , (4.45) where the Π m are constant projectors. This large-time factorization of multi-soliton solutions provides a proof of the no-scattering property because the asymptotic configurations are identical for large negative and large positive times.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"Conclusions\",\n \"text\": \"In this paper we introduced a generalization of the modified integrable U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions and considered a Moyal deformation of this model. It was shown that this N -extended chiral model is equivalent to a gauge-fixed BPS subsector of an N -extended super Yang-Mills model in 2+1 dimensions originating from twistor string theory. The dressing method was applied to generate a wide class of multi-soliton configurations, which are time-dependent finite-energy solutions to the equations of motion. Compared to the N =0 model, the supersymmetric extension was seen to promote the configurations' building blocks to holomorphic functions of suitable Grassmann coordinates. By considering the large-time asymptotic factorization into a product of single soliton solutions we have shown that no scattering occurs within the dressing ansatz chosen here.\\n\\nThe considered model does not stand alone but is motivated by twistor string theory [37] with a target space reduced to the mini-supertwistor space [44, 45, 47] . In this context, the obtained multi-soliton solutions are to be regarded as D(0|2N )-branes moving inside D(2|2N )-branes [60] . Here 2N appears due to fermionic worldvolume directions of our branes in the superspace description [60] . Switching on a constant B-field simply deforms the sigma model and D-brane worldvolumes noncommutatively, thereby admitting also regular supersymmetric noncommutative abelian solutions.\\n\\nRestricting to static configurations, the models can be specialized to Grassmannian supersymmetric sigma models, where the superfield Φ takes values in Gr(r, n), and the field equations are invariant under 2N supersymmetry transformations with 0 ≤ N ≤ 4. This differs from the results for standard 2D sigma models [52, 53] where the target spaces have to be Kähler or hyper-Kähler for admitting two or four supersymmetries, respectively. This difference will be discussed in more details elsewhere.\\n\\nWe derived the supersymmetric chiral model in 2+1 dimensions through dimensional reduction and gauge fixing of the N -extended supersymmetric SDYM equations in 2+2 dimensions. Recall that for the purely bosonic case most (if not all) integrable equations in three and fewer dimensions can be obtained from the SDYM equations (or their hierarchy [25] ) by suitable dimensional reductions (see e.g. [61] - [65] and references therein). Moreover, this Ward conjecture [61] was extended to the noncommutative case (see e.g. [66, 67] ). It will be interesting to consider similar reductions of the N -extended supersymmetric SDYM equations (and their hierarchy [68] ) to supersymmetric integrable equations in three and two dimensions generalizing earlier results [69] .\"\n }\n]","isConversational":false}}},{"rowIdx":99,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.0531"},"arxiv_id":{"kind":"string","value":"0704.0531"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1088/1126-6708/2007/11/079"},"content_hash":{"kind":"string","value":"b0e695f3b0dd96c805ccd0cc547d6cced9bf6b91d8fab221fb029b978f81b498"},"title":{"kind":"string","value":"Gravitational Duality Transformations on (A)dS4"},"abstract":{"kind":"string","value":"We discuss the implementation of electric-magnetic duality transformations in\nfour-dimensional gravity linearized around Minkowski or (A)dS4 backgrounds. In\nthe presence of a cosmological constant duality generically modifies the\nHamiltonian, nevertheless the bulk dynamics is unchanged. We pay particular\nattention to the boundary terms generated by the duality transformations and\ndiscuss their implications for holography."},"authors":{"kind":"list like","value":["Robert G. Leigh","Anastasios C. Petkou"],"string":"[\n \"Robert G. Leigh\",\n \"Anastasios C. Petkou\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["hep-th"],"string":"[\n \"hep-th\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-04"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"We discuss the implementation of electric-magnetic duality transformations in fourdimensional gravity linearized around Minkowski or (A)dS 4 backgrounds. In the presence of a cosmological constant duality generically modifies the Hamiltonian, nevertheless the bulk dynamics is unchanged. We pay particular attention to the boundary terms generated by the duality transformations and discuss their implications for holography. 1 For reviews of higher-spin theories see e.g. [1]. 2 See [4] and [5] for more recent works.\n\nContents 1. Introduction and Summary 2. Action and Hamiltonian 2.1 The 3 + 1 split 2.2 Shifted Variables 2.3 Linearization 3. Linearized Gravitational Duality and Holography 3.1 Duality and Holography 3.2 Linearized gravitational duality 3.3 Linearized Constraints and Bianchi Identities 3.4 Connection with other known dualities 4. The Effect on the Boundary Theory 5. Conclusions and Outlook 6. Appendix: other duality mappings\n\nDuality has played an important role in our understanding of Yang-Mill theories and it is believed that it will play an important role also in gravity and in higher-spin gauge theories. Indeed, although it is less clear what could be the implications of duality for theories whose quantum versions are still unknown, gravity and higher-spin gauge theories 1 are intimately connected to a quantum string theory where certainly duality plays a crucial role. The recent advent of holography raises some intriguing questions for duality. For example one may wonder what is the holographic image of a duality invariant spectrum, a duality transformation or a possible quantization condition that usually duality implies for charges. Some of these issues were raised by Witten in [2] where it was argued that the standard electric-magnetic duality of a U(1) gauge theory on AdS 4 is responsible for a \"natural\" SL(2, Z) action on current two-point functions in three-dimensional CFTs. 2 Shortly afterwards it was shown in [3] that such an SL(2, Z) action is intimately related to certain \"double-trace\" deformations in the boundary, assuming suitable large-N limits and existence of non-trivial fixed points. The latter assumptions are strengthened by the fact that there exist models (e.g., see [6] and references therein) which exhibit the required behavior. In particular, it was shown in [3] that certain \"double-trace\" deformations induce an SL(2, Z) action on two-point functions of higher-spin (i.e. spin s ≥ 2) currents. This has led to the Duality Conjecture of [3] : linearized higher-spin theories on AdS 4 spaces possess a generalization of electric-magnetic duality whose holographic image is the natural SL(2, Z) action on boundary two-point functions.\n\nSurprisingly, even the duality for linearized spin-2 gauge fields (linearized gravity) was not widely known by the time of this conjecture. 3 Second order linearized gravitational duality was discussed among other in [8, 9, 10, 11, 12] . More recently, the duality properties of linearized gravity around flat space were studied in [13] and were further discussed in [14] . The duality of linearized gravity around dS 4 was later studied in [15] .\n\nIn this note we present our calculations regarding the duality properties of gravity in the presence of a cosmological constant. Having in mind applications to higher-spin gauge theories we use forms and work in the first order formalism where duality is also manifested at the level of the action [16] . Moreover, the first order formalism is relevant for applications of duality to holography, since the correlation functions of the boundary theory are essentially determined by the bulk canonical momenta (see e.g. [17] ).\n\nOur aim in this work is to formulate linearized first order gravity using suitable \"electric\" and \"magnetic\" variables, in close analogy with electromagnetism. We find that this is possible only when the background geometry is Minkowski or (A)dS 4 . Then we implement the standard electric-magnetic duality rotations. We find that, up to \"boundary\" terms, the linearized Hamiltonian changes by terms that do not alter the bulk dynamics i.e. do not alter the second order bulk equations of motion. Moreover, the duality rotation interchanges the (linearized) constraints with the (linearized) Bianchi identitites. The \"boundary\" terms have important holographic consequences since they correspond to marginal \"double-trace\" deformations [3] that induce the boundary SL(2, Z) action. In the Appendix we exhibit a modified duality rotation that leaves the bulk Hamiltonian invariant and induces \"boundary\" terms that correspond to relevant deformations as in [3] .\n\nHaving in mind the extension of our results to higher-spin gauge theorieswe start from the MacDowell-Mansouri form [18] of the gravitational action foot_1\n\nI M M = 1 2Λ M ǫ abcd R ab ∧ R cd + 2Λe a ∧ e b ∧ R cd + Λ 2 e a ∧ e b ∧ e c ∧ e d , (2.1)\n\nwhere a, b, ... are Lorentz indices. In this formalism, the vierbein e a and the spin connection ω a b are initially thought of as independent variables. The curvature 2-form is\n\nR a b = dω a b + ω a c ∧ ω c b = 1 2 R a bcd e c ∧ e d .\n\nVarying the action with respect to e a and ω a b , we find\n\nR ab + Λe a ∧ e b = 0 , (2.2)\n\nT a = de a + ω a b ∧ e b = 0 . (2.\n\n3)\n\nThe relation to gravity is established via the vanishing torsion equation (2.3), which relates e and ω in the familiar way. The above equations are equivalent to the Einstein equation in metric variables R µν -\n\n1 2 Rg µν = +3Λg µν . (2.4)\n\nand the scalar curvature is R = -d(d -1)Λ = -12Λ. Note that our Λ is related to the cosmological constant in its usual definition via Λ cosm = -6Λ. Λ > 0 corresponds to AdS. Note that this is actually SO(3, 2) covariant, as we can combine ω, e into a super-connection. Note that Λ has units (Length) -2 . In the SO(3, 2)-invariant formalism, I M M arises from\n\nI M M = 1 2Λ ǫ ABCDE V E R AB ∧ R CD , (2.5)\n\nwhere V E is a non-dynamical 0-form field (that we take to have value V -1 = 1 to gauge back to the SO(3, 1) formalism) and R A B is the curvature of Ω A B ≡ {e a , ω a b }. There are also quasitopological terms of the form\n\nI top = θ 2Λ R A B ∧ R B A + α Λ R A B ∧ R AC V B V C (2.6)\n\nthat we could add to the action. In the stated gauge, this reduces to\n\nI top = θ 2Λ P 2 + (θ + α)C N Y + α R ab ∧ e a ∧ e b (2.7)\n\nwhere\n\nP 2 = R a b ∧ R b a is the Pontryagin class, C N Y = (T a ∧ T a -R ab ∧ e a ∧ e b\n\n) is the Nieh-Yan class and we also note the Euler class E 2 = ǫ abcd R ab ∧ R cd . Note that in the presence of torsion, the action (2.7) contains the non-topological term R ab ∧ e a ∧ e b with \"Immirzi parameter\" γ = -2/α. In the absence of torsion, this term is a total derivative.\n\nThe Hilbert-Palatini action is\n\nI HP = I M M - 1 2Λ E 2 .\n\n(2.8)\n\nIt differs from I M M by a boundary term, is smooth as Λ → 0 but is not manifestly SO(3, 2)invariant.\n\nNext, we carefully consider the 3 + 1 split. Although much of the discussion here is familiar from the ADM formalism, we feel it is important to set notation carefully, as we will introduce some new ingredients. To accommodate both AdS and dS signatures simultaneously, we will introduce a 'time' function t and a foliation of space-time Σ t ֒→ M. In dS, t is time-like, and this corresponds to the usual Hamiltonian foliation; in AdS on the other hand, we will take t to be the (space-like) radial coordinate. We will keep track of the resulting signs by a parameter σ ⊥ , equal to ±1 in dS(AdS).\n\nProceeding as usual then, we get a vector field t that satisfies ∇ t t = 1 ≡ t(t) (so t = ∂ ∂t ) and a 1-form dt. Given a 4-metric, we can introduce the normal 1-form n as\n\nn = σ ⊥ Ndt , (2.9)\n\nwhich is normalized as (n, n) = σ ⊥ . The dual vector field n can be expanded as\n\nn = 1 N t - 1 N N , (2.10)\n\nwhere the shift N satisfies (N, n) = 0, and thus (t, n) = σ ⊥ N.\n\nNext, we will locally choose a basis of 1-forms\n\ne 0 = σ ⊥ n = Ndt , (2.11)\n\ne α = ẽα + N α dt .\n\n(2.12)\n\nThe ẽα span T * Σ t , and correspond to a 3-metric h ij = ẽα i ẽβ j η αβ . The quantities N α are the components of N: N α = e α i N i . These basis 1-forms are dual to {e 0 = n, e α = ẽα }, with e a (e b ) = δ b a . We expand the spin connection in the same basis 5\n\nω a b = q a b dt + ωa b , (2.13)\n\nwhich leads to\n\nR a b = Ra b + dt ∧ r a b , (2.14)\n\nwhere R is formed from ω and d only, and\n\nr a b = ωa b -dq a b -ωa c q c b + q a c ωc b . (2.15)\n\nNote that these quantities are merely decompositions along T * Σ t in the 4-geometry; we will introduce the intrinsically defined objects shortly.\n\nWe then find\n\nI HP = 2ǫ αβγ dt ∧ N( Rαβ + Λẽ α ∧ ẽβ ) ∧ ẽγ -2N α ( R0β ) ∧ ẽγ + r 0α ∧ ẽβ ∧ ẽγ .\n\n(2.16) 5 We have\n\nq a b = N ω 0 a b + N α ω α a b and ωa b ≡ ω α a b ẽα .\n\nAs is familiar, the lapse and shift appear as Lagrange multipliers. The constraints that they multiply are of course zero in any background (i.e. vacuum solution), such as (A)dS 4 . The final term in the action contains the real dynamics -r 0α depends on the components R 0α 0β of the Riemann tensor.\n\nNote though that the tensors used here are 4-dimensional. Let us define the \"electric field\"\n\nK α = σ ⊥ ω0 α = K βα ẽβ .\n\n(2.17)\n\nIn the case that ω is the torsion-free Levi-Civita connection, this agrees with the standard definition for extrinsic curvature, regarded as a vector-valued one-form. We then find\n\nRα β = (3) R α β -σ ⊥ K α ∧ K β , (2.18)\n\nand\n\nR0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ ( DK) α . (2.19)\n\nThese equations amount to the Gauss-Codazzi relations. Furthermore, r 0α contains time derivatives of ω0α as well as terms linear in components of q. We find\n\n2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = 2ǫ αβγ σ ⊥ Kα -( Dq) 0α ∧ ẽβ ∧ ẽγ , (2.20)\n\n= 2σ ⊥ ǫ αβγ Kα + q αδ K δ ∧ ẽβ ∧ ẽγ + 4q 0α ǫ αβγ T β ∧ ẽγ up to a total 3-derivative. We have defined the intrinsic 3-torsion T α = dẽ α + ωα β ∧ ẽβ . Since we wish to regard the ẽ as coordinate variables, 6 we integrate the first term by parts to obtain (up to the total time-derivative\n\n∂ ∂t 2σ ⊥ K α ∧ ẽβ ∧ ẽγ ǫ αβγ ) 2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = Π α ∧ ėα + 4q 0α ǫ αβγ T β ∧ ẽγ + 2σ ⊥ q αδ ǫ αβγ K δ ∧ ẽβ ∧ ẽγ . (2.21)\n\nwhere we have defined the momentum 2-form\n\nΠ α = -4σ ⊥ ǫ αβγ K β ∧ ẽγ . (2.22)\n\nThe q ab appear as Lagrange multipliers. In particular, the q αβ constraint precisely sets the antisymmetric (torsional) part of the extrinsic curvature tensor K [αβ] to zero. Next, we define the \"magnetic field\"\n\nB α = 1 2 σ ⊥ ǫ αβγ ωβγ , ω αβ = -ǫ αβγ B γ . (2.23)\n\nand we find that the q 0α constraint\n\nǫ αβγ T β ∧ ẽγ = ǫ αβγ dẽ β ∧ ẽγ -σ ⊥ B β ∧ ẽβ ∧ ẽα = 0 , (2.24)\n\ninvolves only the antisymmetric part B [α,β] of the magnetic field B α = B αβ ẽβ . The antisymmetric part of B α spoils the gauge covariance of the constraint (2.24) under an SO(3) rotation of the dreibein ẽα , hence it represents degrees of freedom that can be gauged fixed to zero by an SO(3) rotation. On the other hand, an algebraic equation of motion connects the symmetric part of B αβ to derivatives of ẽα as dẽ α + ǫ αβγ B β ∧ ẽγ = 0 (2.25)\n\nAt the end, one is left with the canonically conjugate variables ẽα and Π α . These results are familiar from the metric formalism.\n\nDropping the torsional terms, we then arrive at the action\n\nI HP = dt ∧ ėα ∧ Π α + 2Nǫ αβγ ( (3) R αβ -σ ⊥ K α ∧ K β + Λẽ α ∧ ẽβ ) ∧ ẽγ -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ . (2.26) Furthermore, using * 3 ẽα ∧ ẽβ = 1 2 η γδ ǫ αβγ ẽδ , we have Πα = * 3 Π α = -2(K αβ -η αβ trK)ẽ β , (2.27)\n\nwhere trK = η αβ K αβ . We can solve the above equation to get\n\nK α = - 1 2 ( Παβ - 1 2 η αβ tr Π)ẽ β .\n\n(2.28)\n\nAs stated above, K αβ (and Παβ ) is symmetric when the torsion vanishes. Finally, with the definition (2.23) we find foot_3\n\nǫ αβγ (3) R αβ ∧ ẽγ = ǫ αβγ dω αβ + ωα δ ∧ ωδβ ∧ ẽγ = σ ⊥ 2 dB γ + ǫ αβγ B α ∧ B β ∧ ẽγ . (2.29)\n\nIntroducing B α is an unusual thing to do but it will play a role in duality: in this form, the Hamiltonian contains terms which are reminiscent of those of the Maxwell theory. The full HP action is of the form\n\nI HP = dt ∧ ėα ∧ Π α -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ +2σ ⊥ N(2 dB γ + ǫ αβγ B α ∧ B β -ǫ αβγ K α ∧ K β + σ ⊥ Λǫ αβγ ẽα ∧ ẽβ ) ∧ ẽγ . (2.30)\n\nNote that the entire contribution of the cosmological constant appears in the last term of the Hamiltonian constraint.\n\nIt is possible to make a transformation of the canonical variables in order to absorb the cosmological constant term in (2.30 ). This can be achieved by introducing the new variables\n\nKα = K α -ρẽ α , (2.31)\n\nand requiring that\n\nρ 2 = σ ⊥ Λ . (2.32)\n\nThis is positive only when σ ⊥ and Λ are simultaneously positive or negative, as it is the case for both AdS 4 (Λ > 0) and dS 4 (Λ < 0). We will often write Λ = σ ⊥ /L 2 where L is a length scale. Under (2.31) the momentum 2-form becomes\n\nΠ α → P α -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ . (2.33)\n\nThe last term in (2.33) contributes a total time derivative to the action (of the form of a boundary cosmological term). We have introduced a new momentum variable\n\nP α = -4σ ⊥ ǫ αβγ Kβ ∧ ẽγ .\n\nThen, we get the action\n\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K + ρ T ) β ∧ ẽγ - 4 3 σ ⊥ ρǫ αβγ ∂ ∂t (ẽ α ∧ ẽβ ∧ ẽγ ) +2σ ⊥ N 2 d(B α ∧ ẽα ) + 2B γ ∧ T γ -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ .(2.34)\n\nNote that the shift constraint is still written in terms of the ordinary covariant derivative, and thus involves a non-linear term coupling B to K. Consistent with our previous discussion, we drop the terms involving the torsion T , and disregard the boundary term to obtain\n\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K) β ∧ ẽγ +2σ ⊥ N 2 d(B α ∧ ẽα ) -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ . (2.35)\n\nWe note that the parameter ρ can be of either sign (although, this sign does not appear in the second order equations of motion).\n\nNext, we linearize the above action around an appropriate fixed background. We expand as\n\nẽα = ẽα + E α , N = 1 + n, N α = n α , B α = B α + b α , Kα = Kα + k α . (2.36)\n\nThe background values should satisfy the constraints. The simplest choice is the background where Kα = 0 = B α .\n\n(2.37)\n\nIn fact, reaching this simple form was a motivation for the shift (2.31). Then, to quadratic order in the fluctuating fields the Hamiltonian gives\n\nI HP = dt ∧ Ėα ∧ p α -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + 4σ ⊥ n d(b α ∧ ẽα ) -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β + 2ρk α ∧ E β ∧ ẽγ , (2.38)\n\nwhere\n\np α = -4σ ⊥ ǫ αβγ k β ∧ ẽγ (2.39)\n\nare the linearized momentum variables conjugate to E α . In order to reach the form (2.38) the linear terms in the fluctuations must vanish. For this to happen we find the relationships ėα + ρẽ α = 0 .\n\n(2.40)\n\nNotice that we can also write the linearized action in the form\n\nI HP = dt ∧ ( Ėα + ρE α ) ∧ p α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + n 4σ ⊥ db γ + ρp γ ∧ ẽγ . (2.41)\n\nThe form of the first term, involving the momentum, makes clear that longitudinal fluctuations are non-dynamical. The natural time dependence of E α is of the form e -ρt (correspondingly, the natural time dependence of p α is e +ρt ). Other than that, we see that in comparing to the flat space action, in these variables, the only change is that the Hamiltonian constraint is modified. The solutions of (2.40) and (2.37) are components of (A)dS 4 spacetimes. We can solve (2.40) to obtain e 0 = dt, e α = e -ρt dx α .\n\n(2.42)\n\nWith these we construct the usual Poincaré metric on (A)dS which, however, covers only half of the space even though the parameter t runs from -∞ to +∞. The conformal boundary in these coordinates is at t = +∞. Then we derive\n\nω α 0 = -ρe -ρt dx α = -ρe α , (2.43)\n\nand so\n\nR α β = -σ ⊥ L 2 e α ∧ e β R α 0 = -1 L 2 e α ∧ e 0 R a b = - σ ⊥ L 2 e a ∧ e b . ( 2\n\nHence Ric ab = -3σ ⊥ L 2 η ab and R = -12σ ⊥ /L 2 = -12Λ. We also evaluate Π α = -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ , Πα = 4ρẽ α , tr Π = 12ρ (2.45)\n\nB α = 0, K α = ρẽ α ⇒ Kα = 0 (2.46)\n\nNote that in this gauge, ( DK) α = 1 L T α = 0, which solves the shift constraint, while the Hamiltonian constraint is satisfied through a cancellation between the K 2 term and the cosmological term.\n\nLet us summarize what we have obtained so far. In the presence of a cosmological constant we have defined variables such that the action resembles most closely the action without the cosmological constant. This was done in order to look for a suitable background around which linear fluctuations are as simple as possible. Requiring that K (the \"electric field\") and B (the \"magnetic field\") vanish in such a background -as they do around flat space -we found that the background should be (A)dS 4 . Quite satisfactorily, both sign choices for ρ in the change of variables (2.31) lead to (A)dS 4 spacetimes.\n\nThis is the appropriate point to recall some salient features of duality rotations. In simple Hamiltonian systems the effect of the canonical transformation p → q and q → -p to the action is (see e.g. [19] )\n\nI = t 2 t 1 dt[p q -H(p, q)] → I D = t 2 t 1 dt[-q ṗ -H(q, -p)] . (3.1)\n\nNotice that I D involves the dual variables, for which we have however kept the same notation for simplicity. The transformed Hamiltonian H(q, -p) is in general not related to H(p, q). However, if H(q, -p) = H(p, q) we call the above transformation a duality. It then holds\n\nI D = I -qp t 2 t 1 . (3.2)\n\nThe dual action describes exactly the same dynamics as the initial one, up to a modification of the boundary conditions. For example, if I is stationary on the e.o.m for fixed q in the boundary, I D is stationary on the same e.o.m. for fixed p in the boundary. This simple example illustrates the role of duality in holography; a bulk duality transformation corresponds to a particular modification of the boundary conditions. This property of duality transformations is behind the remarkable holographic properties of electormagnetism in (A)dS 4 [2, 3] .\n\nClearly, the crucial properties of a duality transformation are to be canonical and to leave the Hamiltonian unchanged. However, consider a slight generalization\n\nS = t 2 t 1 dt[p q - 1 2 (p 2 + q 2 + 2λpq)] (3.3)\n\nwhere λ is an arbitrary parameter. The Hamiltonian now is not invariant under the canonical transformation p → q and q → -p -the pq term changes sign. Consequently, the first order form of the equations of motion are also not duality invariant. Nevertheless, the second order equation of motion is invariant. We will find that gravity in the presence of a cosmological constant follows precisely this model. Of course, gravity is a much more complicated constrained system, but as we will show, the constraints and Bianchi identities transform appropriately.\n\nWe also note that the canonical transformation (implemented by a generating functional of the first kind) p → q + 2λp , q → -p . (3.4) is of interest here. The above does not change the Hamiltonian and the transformed action differs from the initial one by total time derivative terms foot_4 S → S D = S -pq\n\nt 2 t 1 -λp 2 t 2 t 1 . (3.5)\n\nAs a preamble to gravity we recall the duality properties of Maxwell theory\n\nI M ax = 1 2g 2 dt ∧ Ȧ ∧ * 3 E - 1 2 (E ∧ * 3 E + B ∧ * 3 B) -A 0 d * 3 E , (3.6)\n\nUnder the duality E → - * 3 B, B → * 3 E, Ã → ÃD , we find\n\nI M ax → I M ax,D = 1 2g 2 dt ∧ -ȦD ∧ B - 1 2 (E ∧ * 3 E + B ∧ * 3 B) + A 0 dB . (3.7)\n\nE and B in (3.7) should be expressed through ÃD . We observe that the kinetic term has changed sign, while the Hamiltonian remains invariant. In addition, the (Gauss) constraint is dualized to the trivial 'Bianchi' identity dB = 0 for the dual magnetic field.\n\nNext we try to apply a Maxwell-type duality map in gravity. We consider the following transformation around the fixed background (2.40)\n\nk α → -b α , b α → k α . (3.8)\n\nTo implement the map (3.8) we need to specify the mapping of E α to a 'dual 3-bein' E α . We do that using the linearized form of (2.25) as\n\nǫ αβγ b β ∧ ẽγ + dE α = 0 → ǫ αβγ k β ∧ ẽγ + dE α = 0 = dE α - 1 4σ ⊥ p α (3.9) Since p α = 4σ ⊥ dE α , it is natural to define p D,α = 4σ ⊥ dE α = -4σ ⊥ ǫ αβγ b β ∧ ẽγ , (3.10)\n\nand thus the mapping (3.8) is supplemented by\n\nE → E , E → -E , p → -p D , p D → p (3.11)\n\nNow, let us see the effects of the above duality mapping. The action transforms to\n\nI HP → I HP,D = dt ∧ -Ėα ∧ p D,α -ρE α ∧ p D,α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ (3.12) +4σ ⊥ n α ǫ αβγ db β ∧ ẽγ + n 4σ ⊥ dk γ + ρp D,α ∧ ẽγ\n\nwhere now k α and b α should be expressed in terms of the dual variables E α and p D,α via (3.9) and (3.10). We notice that the 'kinetic' part Ė ∧ p of the action changes sign under the duality map, in direct analogy with the Maxwell case. However, the Hamiltonian is not invariant due to the change of sign of the second term in the first line of (3.12). We will discuss this further in a later section. For now, we note that this sign change would not show up in the equations of motion, written in second order form. It is important to also note that the constraints are transformed into quantities which in the next subsection we will recognize as the linearized Bianchi identities. This is to be expected since the duality transformations are canonical. We also note that it may be possible to choose an alternative canonical transformation, designed to leave the Hamiltonian invariant. The latter is presumably related to the work of Julia et. al. [15] and is considered in the Appendix.\n\nBy virtue of the discussion above we may now demonstrate that under the duality mapping (3.8) the linearized constraints transform to the linearized Bianchi identities as\n\nC α ≡ ǫ αβγ dk β ∧ ẽγ → -ǫ αβγ db β ∧ ẽγ (3.13) C 0 ≡ -σ ⊥ db γ -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ → -σ ⊥ dk γ + ρǫ αβγ b α ∧ ẽβ ∧ ẽγ (3.14)\n\nTo identify the right hand sides, we first note that the Bianchi identities are\n\nB R a b = dR a b -R a c ∧ ω c b + ω a c ∧ R c b = 0 (3.15) B a T = dT a -R a b ∧ e b + ω a b ∧ T b = 0 (3.16)\n\nwhich are obtained from the definitions of R a b and T a by exterior differentiation. The first equation is satisfied identically. Since the torsion vanishes, the second equation tells us only that R a b ∧ e b = 0. If we do the 3+1 split, we find two equations. The first is\n\nB T α = -( (3) R α β -σ ⊥ K α ∧ K β ) ∧ ẽβ = 0 (3.17)\n\nwhich upon using the symmetry of K α linearizes to\n\nB T α = -ǫ αβγ db β ∧ ẽγ + . . . (3.18)\n\nNote that this is the image under duality of the shift constraint as in (3.13) .\n\nThe second identity is\n\nB T 0 = -R0 α ∧ ẽα = -σ ⊥ ( DK) α = -σ ⊥ dk α + ρǫ αβγ b β ∧ ẽγ ∧ ẽα = 0 (3.19)\n\nwhere to arrive in the second line we used (2.46) . This is the image of the Hamiltonian constraint as in (3.14) . Summarizing, the duality transformations between linearized constraints and Bianchi identities are\n\nC α → B T,α C 0 → B 0 T (3.20) B T,α → -C α B 0 T → -C 0 (3.21)\n\nThe Maxwell-type duality operation (3. We begin with the spatial 2-forms when we have\n\nRαβ = -ǫ αβγ dB γ + σ ⊥ (B α ∧ B β -K α ∧ K β ) (3.23) R0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ (\n\nDK) α (3.24) and S0 γ = 1 2 σ ⊥ ǫ αβγ Rαβ (3.25) Sαβ = ǫ αβγ R0γ (3.26) If we linearize these expressions, we find under the duality transformation (3.8) Rab → -σ ⊥ Sab (3.27) Because the expressions (3.24) involve derivatives of B and K, the duality (3.8) is an 'integrated form' of the usual Riemann tensor duality, but implies it. Similarly, if we investigate the spatial 1-forms, we find r ab → -σ ⊥ s ab (3.28)\n\nTo arrive at this result we have set to zero the Lagrange multiplier field q.\n\nIt is well known that AdS is holographic. We may well ask, in the context of AdS/CFT, how the duality transformation that we have defined here acts in the boundary. We are instructed to consider the on-shell bulk action as a function of bulk fields. So, we evaluate the action on a solution to the equation of motion, resulting in a pure boundary term which is of the form\n\nS bdy = ∂M p α ∧ E α (4.1)\n\nApplying the duality transformation to the bulk theory, although the bulk action is not invariant as we have discussed above, nevertheless it may be easily shown that it induces a simple transformation on the (linearized) boundary term: it simply changes its sign.\n\nS dual bdy = - ∂M p D,α ∧ E α (4.2)\n\nThis transformation is exactly analogous to what happens in the Maxwell case: it amounts to the result [21] .\n\n10 G 2 G dual 2 = -1 . (4.3)\n\nMotivated by possible application in holography and in higher-spin gauge theory we have studied the duality properties of gravity in the Hamiltonian formulation. We have presented the gravity action in terms of suitable variables that closely resemble the electric and magnetic fields in Maxwell theory. We have found suitable \"electric\" and \"magnetic\" field variables, such that at the linearized level first order gravity most closely resembles electromagnetism. This can be done only around Minkowksi and (A)dS 4 backgrounds. We have implemented duality transformations in the linearized gravity fluctuations around these backgrounds. In the presence of a cosmological constant, the Hamiltonian changes, nevertheless the bulk dynamics remains unaltered, while the linearized lapse and shift constraints are mapped into the linearized Bianchi identities. Moreover, the duality transformations induce boundary terms whose relevance in holography we have briefly discussed. Finally, we have exhibited a modified duality rotation that leaves the bulk Hamiltonian invariant, while it induces boundary terms corresponding to relevant deformations.\n\nThe main implication of our results is that certain properties of correlations functions in three-dimensional CFTs mimic the duality of gravity. It would be interesting to extend our results to black-hole backgrounds and also when topological terms are present in the bulk. We also expect that one can analyze the duality of higher-spin gauge theories based on our first-order approach.\n\nThe work of A. C. P. was partially supported by the research program \"PYTHAGORAS II\" of the Greek Ministry of Education. RGL was supported in part by the U.S. Department of Energy under contract DE-FG02-91ER40709.\n\nIt is possible to find a transformation that leaves the Hamiltonian unchanged. Consider the following transformation in the fixed background (2.40)\n\nk α → -b α -2ρE α , b α → k α . (6.1)\n\nThe mapping to the dual dreibein is still specified by (3.9). A straightforward calculation reveals that the action transforms as The transformations (6.1) leaves unchanged the Hamiltonian and changes the action by the total \"time\" derivative terms shown in the first line of (6.2). Moreover, the linearized constraints transform into the linearized Bianchi identities. Let us see that in some detail. The second term in the shift constraint is zero since k α is a symmetric one form k α = k αβ ẽβ with k αβ = k βα ; see (2.21) . The term proportional to Λ in the lapse constraint is also zero. This is slightly more involved to see and it is based on the possibility of solving (3.9) for E α after gauge fixing. 11 One way to see this is in components. Write E α = E α β ẽβ and (3.9) becomes\n\nI\n\n∂ α E β γ -∂ γ E β α = ǫ β δα k δ γ -ǫ β δγ k δ α (6.3)\n\nIn the \"Lorentz gauge\" where ∂ α E β α = 0 = ∂ α k β α the above can be inverted as\n\nE α β = 1 ∂ 2 ǫ α δγ ∂ γ k δ β (6.4)\n\nUsing (6.4) one verifies that the last term in the lapse constraint vanishes. This modified duality transformation is probably related to the one considered by Julia et. al. in [15] ."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"We discuss the implementation of electric-magnetic duality transformations in fourdimensional gravity linearized around Minkowski or (A)dS 4 backgrounds. In the presence of a cosmological constant duality generically modifies the Hamiltonian, nevertheless the bulk dynamics is unchanged. We pay particular attention to the boundary terms generated by the duality transformations and discuss their implications for holography. 1 For reviews of higher-spin theories see e.g. [1]. 2 See [4] and [5] for more recent works."},{"section_type":"OTHER","section_title":"Untitled Section","text":"Contents 1. Introduction and Summary 2. Action and Hamiltonian 2.1 The 3 + 1 split 2.2 Shifted Variables 2.3 Linearization 3. Linearized Gravitational Duality and Holography 3.1 Duality and Holography 3.2 Linearized gravitational duality 3.3 Linearized Constraints and Bianchi Identities 3.4 Connection with other known dualities 4. The Effect on the Boundary Theory 5. Conclusions and Outlook 6. Appendix: other duality mappings"},{"section_type":"BACKGROUND","section_title":"Introduction and Summary","text":"Duality has played an important role in our understanding of Yang-Mill theories and it is believed that it will play an important role also in gravity and in higher-spin gauge theories. Indeed, although it is less clear what could be the implications of duality for theories whose quantum versions are still unknown, gravity and higher-spin gauge theories 1 are intimately connected to a quantum string theory where certainly duality plays a crucial role. The recent advent of holography raises some intriguing questions for duality. For example one may wonder what is the holographic image of a duality invariant spectrum, a duality transformation or a possible quantization condition that usually duality implies for charges. Some of these issues were raised by Witten in [2] where it was argued that the standard electric-magnetic duality of a U(1) gauge theory on AdS 4 is responsible for a \"natural\" SL(2, Z) action on current two-point functions in three-dimensional CFTs. 2 Shortly afterwards it was shown in [3] that such an SL(2, Z) action is intimately related to certain \"double-trace\" deformations in the boundary, assuming suitable large-N limits and existence of non-trivial fixed points. The latter assumptions are strengthened by the fact that there exist models (e.g., see [6] and references therein) which exhibit the required behavior. In particular, it was shown in [3] that certain \"double-trace\" deformations induce an SL(2, Z) action on two-point functions of higher-spin (i.e. spin s ≥ 2) currents. This has led to the Duality Conjecture of [3] : linearized higher-spin theories on AdS 4 spaces possess a generalization of electric-magnetic duality whose holographic image is the natural SL(2, Z) action on boundary two-point functions.\n\nSurprisingly, even the duality for linearized spin-2 gauge fields (linearized gravity) was not widely known by the time of this conjecture. 3 Second order linearized gravitational duality was discussed among other in [8, 9, 10, 11, 12] . More recently, the duality properties of linearized gravity around flat space were studied in [13] and were further discussed in [14] . The duality of linearized gravity around dS 4 was later studied in [15] .\n\nIn this note we present our calculations regarding the duality properties of gravity in the presence of a cosmological constant. Having in mind applications to higher-spin gauge theories we use forms and work in the first order formalism where duality is also manifested at the level of the action [16] . Moreover, the first order formalism is relevant for applications of duality to holography, since the correlation functions of the boundary theory are essentially determined by the bulk canonical momenta (see e.g. [17] ).\n\nOur aim in this work is to formulate linearized first order gravity using suitable \"electric\" and \"magnetic\" variables, in close analogy with electromagnetism. We find that this is possible only when the background geometry is Minkowski or (A)dS 4 . Then we implement the standard electric-magnetic duality rotations. We find that, up to \"boundary\" terms, the linearized Hamiltonian changes by terms that do not alter the bulk dynamics i.e. do not alter the second order bulk equations of motion. Moreover, the duality rotation interchanges the (linearized) constraints with the (linearized) Bianchi identitites. The \"boundary\" terms have important holographic consequences since they correspond to marginal \"double-trace\" deformations [3] that induce the boundary SL(2, Z) action. In the Appendix we exhibit a modified duality rotation that leaves the bulk Hamiltonian invariant and induces \"boundary\" terms that correspond to relevant deformations as in [3] ."},{"section_type":"OTHER","section_title":"Action and Hamiltonian","text":"Having in mind the extension of our results to higher-spin gauge theorieswe start from the MacDowell-Mansouri form [18] of the gravitational action foot_1\n\nI M M = 1 2Λ M ǫ abcd R ab ∧ R cd + 2Λe a ∧ e b ∧ R cd + Λ 2 e a ∧ e b ∧ e c ∧ e d , (2.1)\n\nwhere a, b, ... are Lorentz indices. In this formalism, the vierbein e a and the spin connection ω a b are initially thought of as independent variables. The curvature 2-form is\n\nR a b = dω a b + ω a c ∧ ω c b = 1 2 R a bcd e c ∧ e d .\n\nVarying the action with respect to e a and ω a b , we find\n\nR ab + Λe a ∧ e b = 0 , (2.2)\n\nT a = de a + ω a b ∧ e b = 0 . (2.\n\n3)\n\nThe relation to gravity is established via the vanishing torsion equation (2.3), which relates e and ω in the familiar way. The above equations are equivalent to the Einstein equation in metric variables R µν -\n\n1 2 Rg µν = +3Λg µν . (2.4)\n\nand the scalar curvature is R = -d(d -1)Λ = -12Λ. Note that our Λ is related to the cosmological constant in its usual definition via Λ cosm = -6Λ. Λ > 0 corresponds to AdS. Note that this is actually SO(3, 2) covariant, as we can combine ω, e into a super-connection. Note that Λ has units (Length) -2 . In the SO(3, 2)-invariant formalism, I M M arises from\n\nI M M = 1 2Λ ǫ ABCDE V E R AB ∧ R CD , (2.5)\n\nwhere V E is a non-dynamical 0-form field (that we take to have value V -1 = 1 to gauge back to the SO(3, 1) formalism) and R A B is the curvature of Ω A B ≡ {e a , ω a b }. There are also quasitopological terms of the form\n\nI top = θ 2Λ R A B ∧ R B A + α Λ R A B ∧ R AC V B V C (2.6)\n\nthat we could add to the action. In the stated gauge, this reduces to\n\nI top = θ 2Λ P 2 + (θ + α)C N Y + α R ab ∧ e a ∧ e b (2.7)\n\nwhere\n\nP 2 = R a b ∧ R b a is the Pontryagin class, C N Y = (T a ∧ T a -R ab ∧ e a ∧ e b\n\n) is the Nieh-Yan class and we also note the Euler class E 2 = ǫ abcd R ab ∧ R cd . Note that in the presence of torsion, the action (2.7) contains the non-topological term R ab ∧ e a ∧ e b with \"Immirzi parameter\" γ = -2/α. In the absence of torsion, this term is a total derivative.\n\nThe Hilbert-Palatini action is\n\nI HP = I M M - 1 2Λ E 2 .\n\n(2.8)\n\nIt differs from I M M by a boundary term, is smooth as Λ → 0 but is not manifestly SO(3, 2)invariant."},{"section_type":"OTHER","section_title":"The 3 + 1 split","text":"Next, we carefully consider the 3 + 1 split. Although much of the discussion here is familiar from the ADM formalism, we feel it is important to set notation carefully, as we will introduce some new ingredients. To accommodate both AdS and dS signatures simultaneously, we will introduce a 'time' function t and a foliation of space-time Σ t ֒→ M. In dS, t is time-like, and this corresponds to the usual Hamiltonian foliation; in AdS on the other hand, we will take t to be the (space-like) radial coordinate. We will keep track of the resulting signs by a parameter σ ⊥ , equal to ±1 in dS(AdS).\n\nProceeding as usual then, we get a vector field t that satisfies ∇ t t = 1 ≡ t(t) (so t = ∂ ∂t ) and a 1-form dt. Given a 4-metric, we can introduce the normal 1-form n as\n\nn = σ ⊥ Ndt , (2.9)\n\nwhich is normalized as (n, n) = σ ⊥ . The dual vector field n can be expanded as\n\nn = 1 N t - 1 N N , (2.10)\n\nwhere the shift N satisfies (N, n) = 0, and thus (t, n) = σ ⊥ N.\n\nNext, we will locally choose a basis of 1-forms\n\ne 0 = σ ⊥ n = Ndt , (2.11)\n\ne α = ẽα + N α dt .\n\n(2.12)\n\nThe ẽα span T * Σ t , and correspond to a 3-metric h ij = ẽα i ẽβ j η αβ . The quantities N α are the components of N: N α = e α i N i . These basis 1-forms are dual to {e 0 = n, e α = ẽα }, with e a (e b ) = δ b a . We expand the spin connection in the same basis 5\n\nω a b = q a b dt + ωa b , (2.13)\n\nwhich leads to\n\nR a b = Ra b + dt ∧ r a b , (2.14)\n\nwhere R is formed from ω and d only, and\n\nr a b = ωa b -dq a b -ωa c q c b + q a c ωc b . (2.15)\n\nNote that these quantities are merely decompositions along T * Σ t in the 4-geometry; we will introduce the intrinsically defined objects shortly.\n\nWe then find\n\nI HP = 2ǫ αβγ dt ∧ N( Rαβ + Λẽ α ∧ ẽβ ) ∧ ẽγ -2N α ( R0β ) ∧ ẽγ + r 0α ∧ ẽβ ∧ ẽγ .\n\n(2.16) 5 We have\n\nq a b = N ω 0 a b + N α ω α a b and ωa b ≡ ω α a b ẽα .\n\nAs is familiar, the lapse and shift appear as Lagrange multipliers. The constraints that they multiply are of course zero in any background (i.e. vacuum solution), such as (A)dS 4 . The final term in the action contains the real dynamics -r 0α depends on the components R 0α 0β of the Riemann tensor.\n\nNote though that the tensors used here are 4-dimensional. Let us define the \"electric field\"\n\nK α = σ ⊥ ω0 α = K βα ẽβ .\n\n(2.17)\n\nIn the case that ω is the torsion-free Levi-Civita connection, this agrees with the standard definition for extrinsic curvature, regarded as a vector-valued one-form. We then find\n\nRα β = (3) R α β -σ ⊥ K α ∧ K β , (2.18)\n\nand\n\nR0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ ( DK) α . (2.19)\n\nThese equations amount to the Gauss-Codazzi relations. Furthermore, r 0α contains time derivatives of ω0α as well as terms linear in components of q. We find\n\n2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = 2ǫ αβγ σ ⊥ Kα -( Dq) 0α ∧ ẽβ ∧ ẽγ , (2.20)\n\n= 2σ ⊥ ǫ αβγ Kα + q αδ K δ ∧ ẽβ ∧ ẽγ + 4q 0α ǫ αβγ T β ∧ ẽγ up to a total 3-derivative. We have defined the intrinsic 3-torsion T α = dẽ α + ωα β ∧ ẽβ . Since we wish to regard the ẽ as coordinate variables, 6 we integrate the first term by parts to obtain (up to the total time-derivative\n\n∂ ∂t 2σ ⊥ K α ∧ ẽβ ∧ ẽγ ǫ αβγ ) 2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = Π α ∧ ėα + 4q 0α ǫ αβγ T β ∧ ẽγ + 2σ ⊥ q αδ ǫ αβγ K δ ∧ ẽβ ∧ ẽγ . (2.21)\n\nwhere we have defined the momentum 2-form\n\nΠ α = -4σ ⊥ ǫ αβγ K β ∧ ẽγ . (2.22)\n\nThe q ab appear as Lagrange multipliers. In particular, the q αβ constraint precisely sets the antisymmetric (torsional) part of the extrinsic curvature tensor K [αβ] to zero. Next, we define the \"magnetic field\"\n\nB α = 1 2 σ ⊥ ǫ αβγ ωβγ , ω αβ = -ǫ αβγ B γ . (2.23)\n\nand we find that the q 0α constraint\n\nǫ αβγ T β ∧ ẽγ = ǫ αβγ dẽ β ∧ ẽγ -σ ⊥ B β ∧ ẽβ ∧ ẽα = 0 , (2.24)\n\ninvolves only the antisymmetric part B [α,β] of the magnetic field B α = B αβ ẽβ . The antisymmetric part of B α spoils the gauge covariance of the constraint (2.24) under an SO(3) rotation of the dreibein ẽα , hence it represents degrees of freedom that can be gauged fixed to zero by an SO(3) rotation. On the other hand, an algebraic equation of motion connects the symmetric part of B αβ to derivatives of ẽα as dẽ α + ǫ αβγ B β ∧ ẽγ = 0 (2.25)\n\nAt the end, one is left with the canonically conjugate variables ẽα and Π α . These results are familiar from the metric formalism.\n\nDropping the torsional terms, we then arrive at the action\n\nI HP = dt ∧ ėα ∧ Π α + 2Nǫ αβγ ( (3) R αβ -σ ⊥ K α ∧ K β + Λẽ α ∧ ẽβ ) ∧ ẽγ -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ . (2.26) Furthermore, using * 3 ẽα ∧ ẽβ = 1 2 η γδ ǫ αβγ ẽδ , we have Πα = * 3 Π α = -2(K αβ -η αβ trK)ẽ β , (2.27)\n\nwhere trK = η αβ K αβ . We can solve the above equation to get\n\nK α = - 1 2 ( Παβ - 1 2 η αβ tr Π)ẽ β .\n\n(2.28)\n\nAs stated above, K αβ (and Παβ ) is symmetric when the torsion vanishes. Finally, with the definition (2.23) we find foot_3\n\nǫ αβγ (3) R αβ ∧ ẽγ = ǫ αβγ dω αβ + ωα δ ∧ ωδβ ∧ ẽγ = σ ⊥ 2 dB γ + ǫ αβγ B α ∧ B β ∧ ẽγ . (2.29)\n\nIntroducing B α is an unusual thing to do but it will play a role in duality: in this form, the Hamiltonian contains terms which are reminiscent of those of the Maxwell theory. The full HP action is of the form\n\nI HP = dt ∧ ėα ∧ Π α -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ +2σ ⊥ N(2 dB γ + ǫ αβγ B α ∧ B β -ǫ αβγ K α ∧ K β + σ ⊥ Λǫ αβγ ẽα ∧ ẽβ ) ∧ ẽγ . (2.30)\n\nNote that the entire contribution of the cosmological constant appears in the last term of the Hamiltonian constraint."},{"section_type":"OTHER","section_title":"Shifted Variables","text":"It is possible to make a transformation of the canonical variables in order to absorb the cosmological constant term in (2.30 ). This can be achieved by introducing the new variables\n\nKα = K α -ρẽ α , (2.31)\n\nand requiring that\n\nρ 2 = σ ⊥ Λ . (2.32)\n\nThis is positive only when σ ⊥ and Λ are simultaneously positive or negative, as it is the case for both AdS 4 (Λ > 0) and dS 4 (Λ < 0). We will often write Λ = σ ⊥ /L 2 where L is a length scale. Under (2.31) the momentum 2-form becomes\n\nΠ α → P α -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ . (2.33)\n\nThe last term in (2.33) contributes a total time derivative to the action (of the form of a boundary cosmological term). We have introduced a new momentum variable\n\nP α = -4σ ⊥ ǫ αβγ Kβ ∧ ẽγ .\n\nThen, we get the action\n\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K + ρ T ) β ∧ ẽγ - 4 3 σ ⊥ ρǫ αβγ ∂ ∂t (ẽ α ∧ ẽβ ∧ ẽγ ) +2σ ⊥ N 2 d(B α ∧ ẽα ) + 2B γ ∧ T γ -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ .(2.34)\n\nNote that the shift constraint is still written in terms of the ordinary covariant derivative, and thus involves a non-linear term coupling B to K. Consistent with our previous discussion, we drop the terms involving the torsion T , and disregard the boundary term to obtain\n\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K) β ∧ ẽγ +2σ ⊥ N 2 d(B α ∧ ẽα ) -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ . (2.35)\n\nWe note that the parameter ρ can be of either sign (although, this sign does not appear in the second order equations of motion)."},{"section_type":"OTHER","section_title":"Linearization","text":"Next, we linearize the above action around an appropriate fixed background. We expand as\n\nẽα = ẽα + E α , N = 1 + n, N α = n α , B α = B α + b α , Kα = Kα + k α . (2.36)\n\nThe background values should satisfy the constraints. The simplest choice is the background where Kα = 0 = B α .\n\n(2.37)\n\nIn fact, reaching this simple form was a motivation for the shift (2.31). Then, to quadratic order in the fluctuating fields the Hamiltonian gives\n\nI HP = dt ∧ Ėα ∧ p α -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + 4σ ⊥ n d(b α ∧ ẽα ) -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β + 2ρk α ∧ E β ∧ ẽγ , (2.38)\n\nwhere\n\np α = -4σ ⊥ ǫ αβγ k β ∧ ẽγ (2.39)\n\nare the linearized momentum variables conjugate to E α . In order to reach the form (2.38) the linear terms in the fluctuations must vanish. For this to happen we find the relationships ėα + ρẽ α = 0 .\n\n(2.40)\n\nNotice that we can also write the linearized action in the form\n\nI HP = dt ∧ ( Ėα + ρE α ) ∧ p α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + n 4σ ⊥ db γ + ρp γ ∧ ẽγ . (2.41)\n\nThe form of the first term, involving the momentum, makes clear that longitudinal fluctuations are non-dynamical. The natural time dependence of E α is of the form e -ρt (correspondingly, the natural time dependence of p α is e +ρt ). Other than that, we see that in comparing to the flat space action, in these variables, the only change is that the Hamiltonian constraint is modified. The solutions of (2.40) and (2.37) are components of (A)dS 4 spacetimes. We can solve (2.40) to obtain e 0 = dt, e α = e -ρt dx α .\n\n(2.42)\n\nWith these we construct the usual Poincaré metric on (A)dS which, however, covers only half of the space even though the parameter t runs from -∞ to +∞. The conformal boundary in these coordinates is at t = +∞. Then we derive\n\nω α 0 = -ρe -ρt dx α = -ρe α , (2.43)\n\nand so\n\nR α β = -σ ⊥ L 2 e α ∧ e β R α 0 = -1 L 2 e α ∧ e 0 R a b = - σ ⊥ L 2 e a ∧ e b . ( 2\n\nHence Ric ab = -3σ ⊥ L 2 η ab and R = -12σ ⊥ /L 2 = -12Λ. We also evaluate Π α = -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ , Πα = 4ρẽ α , tr Π = 12ρ (2.45)\n\nB α = 0, K α = ρẽ α ⇒ Kα = 0 (2.46)\n\nNote that in this gauge, ( DK) α = 1 L T α = 0, which solves the shift constraint, while the Hamiltonian constraint is satisfied through a cancellation between the K 2 term and the cosmological term."},{"section_type":"OTHER","section_title":"Linearized Gravitational Duality and Holography","text":"Let us summarize what we have obtained so far. In the presence of a cosmological constant we have defined variables such that the action resembles most closely the action without the cosmological constant. This was done in order to look for a suitable background around which linear fluctuations are as simple as possible. Requiring that K (the \"electric field\") and B (the \"magnetic field\") vanish in such a background -as they do around flat space -we found that the background should be (A)dS 4 . Quite satisfactorily, both sign choices for ρ in the change of variables (2.31) lead to (A)dS 4 spacetimes."},{"section_type":"OTHER","section_title":"Duality and Holography","text":"This is the appropriate point to recall some salient features of duality rotations. In simple Hamiltonian systems the effect of the canonical transformation p → q and q → -p to the action is (see e.g. [19] )\n\nI = t 2 t 1 dt[p q -H(p, q)] → I D = t 2 t 1 dt[-q ṗ -H(q, -p)] . (3.1)\n\nNotice that I D involves the dual variables, for which we have however kept the same notation for simplicity. The transformed Hamiltonian H(q, -p) is in general not related to H(p, q). However, if H(q, -p) = H(p, q) we call the above transformation a duality. It then holds\n\nI D = I -qp t 2 t 1 . (3.2)\n\nThe dual action describes exactly the same dynamics as the initial one, up to a modification of the boundary conditions. For example, if I is stationary on the e.o.m for fixed q in the boundary, I D is stationary on the same e.o.m. for fixed p in the boundary. This simple example illustrates the role of duality in holography; a bulk duality transformation corresponds to a particular modification of the boundary conditions. This property of duality transformations is behind the remarkable holographic properties of electormagnetism in (A)dS 4 [2, 3] .\n\nClearly, the crucial properties of a duality transformation are to be canonical and to leave the Hamiltonian unchanged. However, consider a slight generalization\n\nS = t 2 t 1 dt[p q - 1 2 (p 2 + q 2 + 2λpq)] (3.3)\n\nwhere λ is an arbitrary parameter. The Hamiltonian now is not invariant under the canonical transformation p → q and q → -p -the pq term changes sign. Consequently, the first order form of the equations of motion are also not duality invariant. Nevertheless, the second order equation of motion is invariant. We will find that gravity in the presence of a cosmological constant follows precisely this model. Of course, gravity is a much more complicated constrained system, but as we will show, the constraints and Bianchi identities transform appropriately.\n\nWe also note that the canonical transformation (implemented by a generating functional of the first kind) p → q + 2λp , q → -p . (3.4) is of interest here. The above does not change the Hamiltonian and the transformed action differs from the initial one by total time derivative terms foot_4 S → S D = S -pq\n\nt 2 t 1 -λp 2 t 2 t 1 . (3.5)"},{"section_type":"OTHER","section_title":"Linearized gravitational duality","text":"As a preamble to gravity we recall the duality properties of Maxwell theory\n\nI M ax = 1 2g 2 dt ∧ Ȧ ∧ * 3 E - 1 2 (E ∧ * 3 E + B ∧ * 3 B) -A 0 d * 3 E , (3.6)\n\nUnder the duality E → - * 3 B, B → * 3 E, Ã → ÃD , we find\n\nI M ax → I M ax,D = 1 2g 2 dt ∧ -ȦD ∧ B - 1 2 (E ∧ * 3 E + B ∧ * 3 B) + A 0 dB . (3.7)\n\nE and B in (3.7) should be expressed through ÃD . We observe that the kinetic term has changed sign, while the Hamiltonian remains invariant. In addition, the (Gauss) constraint is dualized to the trivial 'Bianchi' identity dB = 0 for the dual magnetic field.\n\nNext we try to apply a Maxwell-type duality map in gravity. We consider the following transformation around the fixed background (2.40)\n\nk α → -b α , b α → k α . (3.8)\n\nTo implement the map (3.8) we need to specify the mapping of E α to a 'dual 3-bein' E α . We do that using the linearized form of (2.25) as\n\nǫ αβγ b β ∧ ẽγ + dE α = 0 → ǫ αβγ k β ∧ ẽγ + dE α = 0 = dE α - 1 4σ ⊥ p α (3.9) Since p α = 4σ ⊥ dE α , it is natural to define p D,α = 4σ ⊥ dE α = -4σ ⊥ ǫ αβγ b β ∧ ẽγ , (3.10)\n\nand thus the mapping (3.8) is supplemented by\n\nE → E , E → -E , p → -p D , p D → p (3.11)\n\nNow, let us see the effects of the above duality mapping. The action transforms to\n\nI HP → I HP,D = dt ∧ -Ėα ∧ p D,α -ρE α ∧ p D,α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ (3.12) +4σ ⊥ n α ǫ αβγ db β ∧ ẽγ + n 4σ ⊥ dk γ + ρp D,α ∧ ẽγ\n\nwhere now k α and b α should be expressed in terms of the dual variables E α and p D,α via (3.9) and (3.10). We notice that the 'kinetic' part Ė ∧ p of the action changes sign under the duality map, in direct analogy with the Maxwell case. However, the Hamiltonian is not invariant due to the change of sign of the second term in the first line of (3.12). We will discuss this further in a later section. For now, we note that this sign change would not show up in the equations of motion, written in second order form. It is important to also note that the constraints are transformed into quantities which in the next subsection we will recognize as the linearized Bianchi identities. This is to be expected since the duality transformations are canonical. We also note that it may be possible to choose an alternative canonical transformation, designed to leave the Hamiltonian invariant. The latter is presumably related to the work of Julia et. al. [15] and is considered in the Appendix."},{"section_type":"OTHER","section_title":"Linearized Constraints and Bianchi Identities","text":"By virtue of the discussion above we may now demonstrate that under the duality mapping (3.8) the linearized constraints transform to the linearized Bianchi identities as\n\nC α ≡ ǫ αβγ dk β ∧ ẽγ → -ǫ αβγ db β ∧ ẽγ (3.13) C 0 ≡ -σ ⊥ db γ -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ → -σ ⊥ dk γ + ρǫ αβγ b α ∧ ẽβ ∧ ẽγ (3.14)\n\nTo identify the right hand sides, we first note that the Bianchi identities are\n\nB R a b = dR a b -R a c ∧ ω c b + ω a c ∧ R c b = 0 (3.15) B a T = dT a -R a b ∧ e b + ω a b ∧ T b = 0 (3.16)\n\nwhich are obtained from the definitions of R a b and T a by exterior differentiation. The first equation is satisfied identically. Since the torsion vanishes, the second equation tells us only that R a b ∧ e b = 0. If we do the 3+1 split, we find two equations. The first is\n\nB T α = -( (3) R α β -σ ⊥ K α ∧ K β ) ∧ ẽβ = 0 (3.17)\n\nwhich upon using the symmetry of K α linearizes to\n\nB T α = -ǫ αβγ db β ∧ ẽγ + . . . (3.18)\n\nNote that this is the image under duality of the shift constraint as in (3.13) .\n\nThe second identity is\n\nB T 0 = -R0 α ∧ ẽα = -σ ⊥ ( DK) α = -σ ⊥ dk α + ρǫ αβγ b β ∧ ẽγ ∧ ẽα = 0 (3.19)\n\nwhere to arrive in the second line we used (2.46) . This is the image of the Hamiltonian constraint as in (3.14) . Summarizing, the duality transformations between linearized constraints and Bianchi identities are\n\nC α → B T,α C 0 → B 0 T (3.20) B T,α → -C α B 0 T → -C 0 (3.21)"},{"section_type":"OTHER","section_title":"Connection with other known dualities","text":"The Maxwell-type duality operation (3. We begin with the spatial 2-forms when we have\n\nRαβ = -ǫ αβγ dB γ + σ ⊥ (B α ∧ B β -K α ∧ K β ) (3.23) R0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ (\n\nDK) α (3.24) and S0 γ = 1 2 σ ⊥ ǫ αβγ Rαβ (3.25) Sαβ = ǫ αβγ R0γ (3.26) If we linearize these expressions, we find under the duality transformation (3.8) Rab → -σ ⊥ Sab (3.27) Because the expressions (3.24) involve derivatives of B and K, the duality (3.8) is an 'integrated form' of the usual Riemann tensor duality, but implies it. Similarly, if we investigate the spatial 1-forms, we find r ab → -σ ⊥ s ab (3.28)\n\nTo arrive at this result we have set to zero the Lagrange multiplier field q."},{"section_type":"OTHER","section_title":"The Effect on the Boundary Theory","text":"It is well known that AdS is holographic. We may well ask, in the context of AdS/CFT, how the duality transformation that we have defined here acts in the boundary. We are instructed to consider the on-shell bulk action as a function of bulk fields. So, we evaluate the action on a solution to the equation of motion, resulting in a pure boundary term which is of the form\n\nS bdy = ∂M p α ∧ E α (4.1)\n\nApplying the duality transformation to the bulk theory, although the bulk action is not invariant as we have discussed above, nevertheless it may be easily shown that it induces a simple transformation on the (linearized) boundary term: it simply changes its sign.\n\nS dual bdy = - ∂M p D,α ∧ E α (4.2)\n\nThis transformation is exactly analogous to what happens in the Maxwell case: it amounts to the result [21] .\n\n10 G 2 G dual 2 = -1 . (4.3)"},{"section_type":"CONCLUSION","section_title":"Conclusions and Outlook","text":"Motivated by possible application in holography and in higher-spin gauge theory we have studied the duality properties of gravity in the Hamiltonian formulation. We have presented the gravity action in terms of suitable variables that closely resemble the electric and magnetic fields in Maxwell theory. We have found suitable \"electric\" and \"magnetic\" field variables, such that at the linearized level first order gravity most closely resembles electromagnetism. This can be done only around Minkowksi and (A)dS 4 backgrounds. We have implemented duality transformations in the linearized gravity fluctuations around these backgrounds. In the presence of a cosmological constant, the Hamiltonian changes, nevertheless the bulk dynamics remains unaltered, while the linearized lapse and shift constraints are mapped into the linearized Bianchi identities. Moreover, the duality transformations induce boundary terms whose relevance in holography we have briefly discussed. Finally, we have exhibited a modified duality rotation that leaves the bulk Hamiltonian invariant, while it induces boundary terms corresponding to relevant deformations.\n\nThe main implication of our results is that certain properties of correlations functions in three-dimensional CFTs mimic the duality of gravity. It would be interesting to extend our results to black-hole backgrounds and also when topological terms are present in the bulk. We also expect that one can analyze the duality of higher-spin gauge theories based on our first-order approach.\n\nThe work of A. C. P. was partially supported by the research program \"PYTHAGORAS II\" of the Greek Ministry of Education. RGL was supported in part by the U.S. Department of Energy under contract DE-FG02-91ER40709."},{"section_type":"OTHER","section_title":"Appendix: other duality mappings","text":"It is possible to find a transformation that leaves the Hamiltonian unchanged. Consider the following transformation in the fixed background (2.40)\n\nk α → -b α -2ρE α , b α → k α . (6.1)\n\nThe mapping to the dual dreibein is still specified by (3.9). A straightforward calculation reveals that the action transforms as The transformations (6.1) leaves unchanged the Hamiltonian and changes the action by the total \"time\" derivative terms shown in the first line of (6.2). Moreover, the linearized constraints transform into the linearized Bianchi identities. Let us see that in some detail. The second term in the shift constraint is zero since k α is a symmetric one form k α = k αβ ẽβ with k αβ = k βα ; see (2.21) . The term proportional to Λ in the lapse constraint is also zero. This is slightly more involved to see and it is based on the possibility of solving (3.9) for E α after gauge fixing. 11 One way to see this is in components. Write E α = E α β ẽβ and (3.9) becomes\n\nI\n\n∂ α E β γ -∂ γ E β α = ǫ β δα k δ γ -ǫ β δγ k δ α (6.3)\n\nIn the \"Lorentz gauge\" where ∂ α E β α = 0 = ∂ α k β α the above can be inverted as\n\nE α β = 1 ∂ 2 ǫ α δγ ∂ γ k δ β (6.4)\n\nUsing (6.4) one verifies that the last term in the lapse constraint vanishes. This modified duality transformation is probably related to the one considered by Julia et. al. in [15] ."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"We discuss the implementation of electric-magnetic duality transformations in fourdimensional gravity linearized around Minkowski or (A)dS 4 backgrounds. In the presence of a cosmological constant duality generically modifies the Hamiltonian, nevertheless the bulk dynamics is unchanged. We pay particular attention to the boundary terms generated by the duality transformations and discuss their implications for holography. 1 For reviews of higher-spin theories see e.g. [1]. 2 See [4] and [5] for more recent works.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Untitled Section\",\n \"text\": \"Contents 1. Introduction and Summary 2. Action and Hamiltonian 2.1 The 3 + 1 split 2.2 Shifted Variables 2.3 Linearization 3. Linearized Gravitational Duality and Holography 3.1 Duality and Holography 3.2 Linearized gravitational duality 3.3 Linearized Constraints and Bianchi Identities 3.4 Connection with other known dualities 4. The Effect on the Boundary Theory 5. Conclusions and Outlook 6. Appendix: other duality mappings\"\n },\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction and Summary\",\n \"text\": \"Duality has played an important role in our understanding of Yang-Mill theories and it is believed that it will play an important role also in gravity and in higher-spin gauge theories. Indeed, although it is less clear what could be the implications of duality for theories whose quantum versions are still unknown, gravity and higher-spin gauge theories 1 are intimately connected to a quantum string theory where certainly duality plays a crucial role. The recent advent of holography raises some intriguing questions for duality. For example one may wonder what is the holographic image of a duality invariant spectrum, a duality transformation or a possible quantization condition that usually duality implies for charges. Some of these issues were raised by Witten in [2] where it was argued that the standard electric-magnetic duality of a U(1) gauge theory on AdS 4 is responsible for a \\\"natural\\\" SL(2, Z) action on current two-point functions in three-dimensional CFTs. 2 Shortly afterwards it was shown in [3] that such an SL(2, Z) action is intimately related to certain \\\"double-trace\\\" deformations in the boundary, assuming suitable large-N limits and existence of non-trivial fixed points. The latter assumptions are strengthened by the fact that there exist models (e.g., see [6] and references therein) which exhibit the required behavior. In particular, it was shown in [3] that certain \\\"double-trace\\\" deformations induce an SL(2, Z) action on two-point functions of higher-spin (i.e. spin s ≥ 2) currents. This has led to the Duality Conjecture of [3] : linearized higher-spin theories on AdS 4 spaces possess a generalization of electric-magnetic duality whose holographic image is the natural SL(2, Z) action on boundary two-point functions.\\n\\nSurprisingly, even the duality for linearized spin-2 gauge fields (linearized gravity) was not widely known by the time of this conjecture. 3 Second order linearized gravitational duality was discussed among other in [8, 9, 10, 11, 12] . More recently, the duality properties of linearized gravity around flat space were studied in [13] and were further discussed in [14] . The duality of linearized gravity around dS 4 was later studied in [15] .\\n\\nIn this note we present our calculations regarding the duality properties of gravity in the presence of a cosmological constant. Having in mind applications to higher-spin gauge theories we use forms and work in the first order formalism where duality is also manifested at the level of the action [16] . Moreover, the first order formalism is relevant for applications of duality to holography, since the correlation functions of the boundary theory are essentially determined by the bulk canonical momenta (see e.g. [17] ).\\n\\nOur aim in this work is to formulate linearized first order gravity using suitable \\\"electric\\\" and \\\"magnetic\\\" variables, in close analogy with electromagnetism. We find that this is possible only when the background geometry is Minkowski or (A)dS 4 . Then we implement the standard electric-magnetic duality rotations. We find that, up to \\\"boundary\\\" terms, the linearized Hamiltonian changes by terms that do not alter the bulk dynamics i.e. do not alter the second order bulk equations of motion. Moreover, the duality rotation interchanges the (linearized) constraints with the (linearized) Bianchi identitites. The \\\"boundary\\\" terms have important holographic consequences since they correspond to marginal \\\"double-trace\\\" deformations [3] that induce the boundary SL(2, Z) action. In the Appendix we exhibit a modified duality rotation that leaves the bulk Hamiltonian invariant and induces \\\"boundary\\\" terms that correspond to relevant deformations as in [3] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Action and Hamiltonian\",\n \"text\": \"Having in mind the extension of our results to higher-spin gauge theorieswe start from the MacDowell-Mansouri form [18] of the gravitational action foot_1\\n\\nI M M = 1 2Λ M ǫ abcd R ab ∧ R cd + 2Λe a ∧ e b ∧ R cd + Λ 2 e a ∧ e b ∧ e c ∧ e d , (2.1)\\n\\nwhere a, b, ... are Lorentz indices. In this formalism, the vierbein e a and the spin connection ω a b are initially thought of as independent variables. The curvature 2-form is\\n\\nR a b = dω a b + ω a c ∧ ω c b = 1 2 R a bcd e c ∧ e d .\\n\\nVarying the action with respect to e a and ω a b , we find\\n\\nR ab + Λe a ∧ e b = 0 , (2.2)\\n\\nT a = de a + ω a b ∧ e b = 0 . (2.\\n\\n3)\\n\\nThe relation to gravity is established via the vanishing torsion equation (2.3), which relates e and ω in the familiar way. The above equations are equivalent to the Einstein equation in metric variables R µν -\\n\\n1 2 Rg µν = +3Λg µν . (2.4)\\n\\nand the scalar curvature is R = -d(d -1)Λ = -12Λ. Note that our Λ is related to the cosmological constant in its usual definition via Λ cosm = -6Λ. Λ > 0 corresponds to AdS. Note that this is actually SO(3, 2) covariant, as we can combine ω, e into a super-connection. Note that Λ has units (Length) -2 . In the SO(3, 2)-invariant formalism, I M M arises from\\n\\nI M M = 1 2Λ ǫ ABCDE V E R AB ∧ R CD , (2.5)\\n\\nwhere V E is a non-dynamical 0-form field (that we take to have value V -1 = 1 to gauge back to the SO(3, 1) formalism) and R A B is the curvature of Ω A B ≡ {e a , ω a b }. There are also quasitopological terms of the form\\n\\nI top = θ 2Λ R A B ∧ R B A + α Λ R A B ∧ R AC V B V C (2.6)\\n\\nthat we could add to the action. In the stated gauge, this reduces to\\n\\nI top = θ 2Λ P 2 + (θ + α)C N Y + α R ab ∧ e a ∧ e b (2.7)\\n\\nwhere\\n\\nP 2 = R a b ∧ R b a is the Pontryagin class, C N Y = (T a ∧ T a -R ab ∧ e a ∧ e b\\n\\n) is the Nieh-Yan class and we also note the Euler class E 2 = ǫ abcd R ab ∧ R cd . Note that in the presence of torsion, the action (2.7) contains the non-topological term R ab ∧ e a ∧ e b with \\\"Immirzi parameter\\\" γ = -2/α. In the absence of torsion, this term is a total derivative.\\n\\nThe Hilbert-Palatini action is\\n\\nI HP = I M M - 1 2Λ E 2 .\\n\\n(2.8)\\n\\nIt differs from I M M by a boundary term, is smooth as Λ → 0 but is not manifestly SO(3, 2)invariant.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The 3 + 1 split\",\n \"text\": \"Next, we carefully consider the 3 + 1 split. Although much of the discussion here is familiar from the ADM formalism, we feel it is important to set notation carefully, as we will introduce some new ingredients. To accommodate both AdS and dS signatures simultaneously, we will introduce a 'time' function t and a foliation of space-time Σ t ֒→ M. In dS, t is time-like, and this corresponds to the usual Hamiltonian foliation; in AdS on the other hand, we will take t to be the (space-like) radial coordinate. We will keep track of the resulting signs by a parameter σ ⊥ , equal to ±1 in dS(AdS).\\n\\nProceeding as usual then, we get a vector field t that satisfies ∇ t t = 1 ≡ t(t) (so t = ∂ ∂t ) and a 1-form dt. Given a 4-metric, we can introduce the normal 1-form n as\\n\\nn = σ ⊥ Ndt , (2.9)\\n\\nwhich is normalized as (n, n) = σ ⊥ . The dual vector field n can be expanded as\\n\\nn = 1 N t - 1 N N , (2.10)\\n\\nwhere the shift N satisfies (N, n) = 0, and thus (t, n) = σ ⊥ N.\\n\\nNext, we will locally choose a basis of 1-forms\\n\\ne 0 = σ ⊥ n = Ndt , (2.11)\\n\\ne α = ẽα + N α dt .\\n\\n(2.12)\\n\\nThe ẽα span T * Σ t , and correspond to a 3-metric h ij = ẽα i ẽβ j η αβ . The quantities N α are the components of N: N α = e α i N i . These basis 1-forms are dual to {e 0 = n, e α = ẽα }, with e a (e b ) = δ b a . We expand the spin connection in the same basis 5\\n\\nω a b = q a b dt + ωa b , (2.13)\\n\\nwhich leads to\\n\\nR a b = Ra b + dt ∧ r a b , (2.14)\\n\\nwhere R is formed from ω and d only, and\\n\\nr a b = ωa b -dq a b -ωa c q c b + q a c ωc b . (2.15)\\n\\nNote that these quantities are merely decompositions along T * Σ t in the 4-geometry; we will introduce the intrinsically defined objects shortly.\\n\\nWe then find\\n\\nI HP = 2ǫ αβγ dt ∧ N( Rαβ + Λẽ α ∧ ẽβ ) ∧ ẽγ -2N α ( R0β ) ∧ ẽγ + r 0α ∧ ẽβ ∧ ẽγ .\\n\\n(2.16) 5 We have\\n\\nq a b = N ω 0 a b + N α ω α a b and ωa b ≡ ω α a b ẽα .\\n\\nAs is familiar, the lapse and shift appear as Lagrange multipliers. The constraints that they multiply are of course zero in any background (i.e. vacuum solution), such as (A)dS 4 . The final term in the action contains the real dynamics -r 0α depends on the components R 0α 0β of the Riemann tensor.\\n\\nNote though that the tensors used here are 4-dimensional. Let us define the \\\"electric field\\\"\\n\\nK α = σ ⊥ ω0 α = K βα ẽβ .\\n\\n(2.17)\\n\\nIn the case that ω is the torsion-free Levi-Civita connection, this agrees with the standard definition for extrinsic curvature, regarded as a vector-valued one-form. We then find\\n\\nRα β = (3) R α β -σ ⊥ K α ∧ K β , (2.18)\\n\\nand\\n\\nR0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ ( DK) α . (2.19)\\n\\nThese equations amount to the Gauss-Codazzi relations. Furthermore, r 0α contains time derivatives of ω0α as well as terms linear in components of q. We find\\n\\n2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = 2ǫ αβγ σ ⊥ Kα -( Dq) 0α ∧ ẽβ ∧ ẽγ , (2.20)\\n\\n= 2σ ⊥ ǫ αβγ Kα + q αδ K δ ∧ ẽβ ∧ ẽγ + 4q 0α ǫ αβγ T β ∧ ẽγ up to a total 3-derivative. We have defined the intrinsic 3-torsion T α = dẽ α + ωα β ∧ ẽβ . Since we wish to regard the ẽ as coordinate variables, 6 we integrate the first term by parts to obtain (up to the total time-derivative\\n\\n∂ ∂t 2σ ⊥ K α ∧ ẽβ ∧ ẽγ ǫ αβγ ) 2ǫ αβγ r 0α ∧ ẽβ ∧ ẽγ = Π α ∧ ėα + 4q 0α ǫ αβγ T β ∧ ẽγ + 2σ ⊥ q αδ ǫ αβγ K δ ∧ ẽβ ∧ ẽγ . (2.21)\\n\\nwhere we have defined the momentum 2-form\\n\\nΠ α = -4σ ⊥ ǫ αβγ K β ∧ ẽγ . (2.22)\\n\\nThe q ab appear as Lagrange multipliers. In particular, the q αβ constraint precisely sets the antisymmetric (torsional) part of the extrinsic curvature tensor K [αβ] to zero. Next, we define the \\\"magnetic field\\\"\\n\\nB α = 1 2 σ ⊥ ǫ αβγ ωβγ , ω αβ = -ǫ αβγ B γ . (2.23)\\n\\nand we find that the q 0α constraint\\n\\nǫ αβγ T β ∧ ẽγ = ǫ αβγ dẽ β ∧ ẽγ -σ ⊥ B β ∧ ẽβ ∧ ẽα = 0 , (2.24)\\n\\ninvolves only the antisymmetric part B [α,β] of the magnetic field B α = B αβ ẽβ . The antisymmetric part of B α spoils the gauge covariance of the constraint (2.24) under an SO(3) rotation of the dreibein ẽα , hence it represents degrees of freedom that can be gauged fixed to zero by an SO(3) rotation. On the other hand, an algebraic equation of motion connects the symmetric part of B αβ to derivatives of ẽα as dẽ α + ǫ αβγ B β ∧ ẽγ = 0 (2.25)\\n\\nAt the end, one is left with the canonically conjugate variables ẽα and Π α . These results are familiar from the metric formalism.\\n\\nDropping the torsional terms, we then arrive at the action\\n\\nI HP = dt ∧ ėα ∧ Π α + 2Nǫ αβγ ( (3) R αβ -σ ⊥ K α ∧ K β + Λẽ α ∧ ẽβ ) ∧ ẽγ -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ . (2.26) Furthermore, using * 3 ẽα ∧ ẽβ = 1 2 η γδ ǫ αβγ ẽδ , we have Πα = * 3 Π α = -2(K αβ -η αβ trK)ẽ β , (2.27)\\n\\nwhere trK = η αβ K αβ . We can solve the above equation to get\\n\\nK α = - 1 2 ( Παβ - 1 2 η αβ tr Π)ẽ β .\\n\\n(2.28)\\n\\nAs stated above, K αβ (and Παβ ) is symmetric when the torsion vanishes. Finally, with the definition (2.23) we find foot_3\\n\\nǫ αβγ (3) R αβ ∧ ẽγ = ǫ αβγ dω αβ + ωα δ ∧ ωδβ ∧ ẽγ = σ ⊥ 2 dB γ + ǫ αβγ B α ∧ B β ∧ ẽγ . (2.29)\\n\\nIntroducing B α is an unusual thing to do but it will play a role in duality: in this form, the Hamiltonian contains terms which are reminiscent of those of the Maxwell theory. The full HP action is of the form\\n\\nI HP = dt ∧ ėα ∧ Π α -4σ ⊥ N α ǫ αβγ ( DK) β ∧ ẽγ +2σ ⊥ N(2 dB γ + ǫ αβγ B α ∧ B β -ǫ αβγ K α ∧ K β + σ ⊥ Λǫ αβγ ẽα ∧ ẽβ ) ∧ ẽγ . (2.30)\\n\\nNote that the entire contribution of the cosmological constant appears in the last term of the Hamiltonian constraint.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Shifted Variables\",\n \"text\": \"It is possible to make a transformation of the canonical variables in order to absorb the cosmological constant term in (2.30 ). This can be achieved by introducing the new variables\\n\\nKα = K α -ρẽ α , (2.31)\\n\\nand requiring that\\n\\nρ 2 = σ ⊥ Λ . (2.32)\\n\\nThis is positive only when σ ⊥ and Λ are simultaneously positive or negative, as it is the case for both AdS 4 (Λ > 0) and dS 4 (Λ < 0). We will often write Λ = σ ⊥ /L 2 where L is a length scale. Under (2.31) the momentum 2-form becomes\\n\\nΠ α → P α -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ . (2.33)\\n\\nThe last term in (2.33) contributes a total time derivative to the action (of the form of a boundary cosmological term). We have introduced a new momentum variable\\n\\nP α = -4σ ⊥ ǫ αβγ Kβ ∧ ẽγ .\\n\\nThen, we get the action\\n\\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K + ρ T ) β ∧ ẽγ - 4 3 σ ⊥ ρǫ αβγ ∂ ∂t (ẽ α ∧ ẽβ ∧ ẽγ ) +2σ ⊥ N 2 d(B α ∧ ẽα ) + 2B γ ∧ T γ -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ .(2.34)\\n\\nNote that the shift constraint is still written in terms of the ordinary covariant derivative, and thus involves a non-linear term coupling B to K. Consistent with our previous discussion, we drop the terms involving the torsion T , and disregard the boundary term to obtain\\n\\nI HP = dt ∧ ėα ∧ P α -4σ ⊥ N α ǫ αβγ ( D K) β ∧ ẽγ +2σ ⊥ N 2 d(B α ∧ ẽα ) -ǫ αβγ B α ∧ B β + Kα ∧ Kβ + 2ρ Kα ∧ ẽβ ∧ ẽγ . (2.35)\\n\\nWe note that the parameter ρ can be of either sign (although, this sign does not appear in the second order equations of motion).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Linearization\",\n \"text\": \"Next, we linearize the above action around an appropriate fixed background. We expand as\\n\\nẽα = ẽα + E α , N = 1 + n, N α = n α , B α = B α + b α , Kα = Kα + k α . (2.36)\\n\\nThe background values should satisfy the constraints. The simplest choice is the background where Kα = 0 = B α .\\n\\n(2.37)\\n\\nIn fact, reaching this simple form was a motivation for the shift (2.31). Then, to quadratic order in the fluctuating fields the Hamiltonian gives\\n\\nI HP = dt ∧ Ėα ∧ p α -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + 4σ ⊥ n d(b α ∧ ẽα ) -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β + 2ρk α ∧ E β ∧ ẽγ , (2.38)\\n\\nwhere\\n\\np α = -4σ ⊥ ǫ αβγ k β ∧ ẽγ (2.39)\\n\\nare the linearized momentum variables conjugate to E α . In order to reach the form (2.38) the linear terms in the fluctuations must vanish. For this to happen we find the relationships ėα + ρẽ α = 0 .\\n\\n(2.40)\\n\\nNotice that we can also write the linearized action in the form\\n\\nI HP = dt ∧ ( Ėα + ρE α ) ∧ p α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ -4σ ⊥ n α ǫ αβγ dk β ∧ ẽγ + n 4σ ⊥ db γ + ρp γ ∧ ẽγ . (2.41)\\n\\nThe form of the first term, involving the momentum, makes clear that longitudinal fluctuations are non-dynamical. The natural time dependence of E α is of the form e -ρt (correspondingly, the natural time dependence of p α is e +ρt ). Other than that, we see that in comparing to the flat space action, in these variables, the only change is that the Hamiltonian constraint is modified. The solutions of (2.40) and (2.37) are components of (A)dS 4 spacetimes. We can solve (2.40) to obtain e 0 = dt, e α = e -ρt dx α .\\n\\n(2.42)\\n\\nWith these we construct the usual Poincaré metric on (A)dS which, however, covers only half of the space even though the parameter t runs from -∞ to +∞. The conformal boundary in these coordinates is at t = +∞. Then we derive\\n\\nω α 0 = -ρe -ρt dx α = -ρe α , (2.43)\\n\\nand so\\n\\nR α β = -σ ⊥ L 2 e α ∧ e β R α 0 = -1 L 2 e α ∧ e 0 R a b = - σ ⊥ L 2 e a ∧ e b . ( 2\\n\\nHence Ric ab = -3σ ⊥ L 2 η ab and R = -12σ ⊥ /L 2 = -12Λ. We also evaluate Π α = -4σ ⊥ ρǫ αβγ ẽβ ∧ ẽγ , Πα = 4ρẽ α , tr Π = 12ρ (2.45)\\n\\nB α = 0, K α = ρẽ α ⇒ Kα = 0 (2.46)\\n\\nNote that in this gauge, ( DK) α = 1 L T α = 0, which solves the shift constraint, while the Hamiltonian constraint is satisfied through a cancellation between the K 2 term and the cosmological term.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Linearized Gravitational Duality and Holography\",\n \"text\": \"Let us summarize what we have obtained so far. In the presence of a cosmological constant we have defined variables such that the action resembles most closely the action without the cosmological constant. This was done in order to look for a suitable background around which linear fluctuations are as simple as possible. Requiring that K (the \\\"electric field\\\") and B (the \\\"magnetic field\\\") vanish in such a background -as they do around flat space -we found that the background should be (A)dS 4 . Quite satisfactorily, both sign choices for ρ in the change of variables (2.31) lead to (A)dS 4 spacetimes.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Duality and Holography\",\n \"text\": \"This is the appropriate point to recall some salient features of duality rotations. In simple Hamiltonian systems the effect of the canonical transformation p → q and q → -p to the action is (see e.g. [19] )\\n\\nI = t 2 t 1 dt[p q -H(p, q)] → I D = t 2 t 1 dt[-q ṗ -H(q, -p)] . (3.1)\\n\\nNotice that I D involves the dual variables, for which we have however kept the same notation for simplicity. The transformed Hamiltonian H(q, -p) is in general not related to H(p, q). However, if H(q, -p) = H(p, q) we call the above transformation a duality. It then holds\\n\\nI D = I -qp t 2 t 1 . (3.2)\\n\\nThe dual action describes exactly the same dynamics as the initial one, up to a modification of the boundary conditions. For example, if I is stationary on the e.o.m for fixed q in the boundary, I D is stationary on the same e.o.m. for fixed p in the boundary. This simple example illustrates the role of duality in holography; a bulk duality transformation corresponds to a particular modification of the boundary conditions. This property of duality transformations is behind the remarkable holographic properties of electormagnetism in (A)dS 4 [2, 3] .\\n\\nClearly, the crucial properties of a duality transformation are to be canonical and to leave the Hamiltonian unchanged. However, consider a slight generalization\\n\\nS = t 2 t 1 dt[p q - 1 2 (p 2 + q 2 + 2λpq)] (3.3)\\n\\nwhere λ is an arbitrary parameter. The Hamiltonian now is not invariant under the canonical transformation p → q and q → -p -the pq term changes sign. Consequently, the first order form of the equations of motion are also not duality invariant. Nevertheless, the second order equation of motion is invariant. We will find that gravity in the presence of a cosmological constant follows precisely this model. Of course, gravity is a much more complicated constrained system, but as we will show, the constraints and Bianchi identities transform appropriately.\\n\\nWe also note that the canonical transformation (implemented by a generating functional of the first kind) p → q + 2λp , q → -p . (3.4) is of interest here. The above does not change the Hamiltonian and the transformed action differs from the initial one by total time derivative terms foot_4 S → S D = S -pq\\n\\nt 2 t 1 -λp 2 t 2 t 1 . (3.5)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Linearized gravitational duality\",\n \"text\": \"As a preamble to gravity we recall the duality properties of Maxwell theory\\n\\nI M ax = 1 2g 2 dt ∧ Ȧ ∧ * 3 E - 1 2 (E ∧ * 3 E + B ∧ * 3 B) -A 0 d * 3 E , (3.6)\\n\\nUnder the duality E → - * 3 B, B → * 3 E, Ã → ÃD , we find\\n\\nI M ax → I M ax,D = 1 2g 2 dt ∧ -ȦD ∧ B - 1 2 (E ∧ * 3 E + B ∧ * 3 B) + A 0 dB . (3.7)\\n\\nE and B in (3.7) should be expressed through ÃD . We observe that the kinetic term has changed sign, while the Hamiltonian remains invariant. In addition, the (Gauss) constraint is dualized to the trivial 'Bianchi' identity dB = 0 for the dual magnetic field.\\n\\nNext we try to apply a Maxwell-type duality map in gravity. We consider the following transformation around the fixed background (2.40)\\n\\nk α → -b α , b α → k α . (3.8)\\n\\nTo implement the map (3.8) we need to specify the mapping of E α to a 'dual 3-bein' E α . We do that using the linearized form of (2.25) as\\n\\nǫ αβγ b β ∧ ẽγ + dE α = 0 → ǫ αβγ k β ∧ ẽγ + dE α = 0 = dE α - 1 4σ ⊥ p α (3.9) Since p α = 4σ ⊥ dE α , it is natural to define p D,α = 4σ ⊥ dE α = -4σ ⊥ ǫ αβγ b β ∧ ẽγ , (3.10)\\n\\nand thus the mapping (3.8) is supplemented by\\n\\nE → E , E → -E , p → -p D , p D → p (3.11)\\n\\nNow, let us see the effects of the above duality mapping. The action transforms to\\n\\nI HP → I HP,D = dt ∧ -Ėα ∧ p D,α -ρE α ∧ p D,α -2σ ⊥ ǫ αβγ b α ∧ b β + k α ∧ k β ∧ ẽγ (3.12) +4σ ⊥ n α ǫ αβγ db β ∧ ẽγ + n 4σ ⊥ dk γ + ρp D,α ∧ ẽγ\\n\\nwhere now k α and b α should be expressed in terms of the dual variables E α and p D,α via (3.9) and (3.10). We notice that the 'kinetic' part Ė ∧ p of the action changes sign under the duality map, in direct analogy with the Maxwell case. However, the Hamiltonian is not invariant due to the change of sign of the second term in the first line of (3.12). We will discuss this further in a later section. For now, we note that this sign change would not show up in the equations of motion, written in second order form. It is important to also note that the constraints are transformed into quantities which in the next subsection we will recognize as the linearized Bianchi identities. This is to be expected since the duality transformations are canonical. We also note that it may be possible to choose an alternative canonical transformation, designed to leave the Hamiltonian invariant. The latter is presumably related to the work of Julia et. al. [15] and is considered in the Appendix.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Linearized Constraints and Bianchi Identities\",\n \"text\": \"By virtue of the discussion above we may now demonstrate that under the duality mapping (3.8) the linearized constraints transform to the linearized Bianchi identities as\\n\\nC α ≡ ǫ αβγ dk β ∧ ẽγ → -ǫ αβγ db β ∧ ẽγ (3.13) C 0 ≡ -σ ⊥ db γ -ρǫ αβγ k α ∧ ẽβ ∧ ẽγ → -σ ⊥ dk γ + ρǫ αβγ b α ∧ ẽβ ∧ ẽγ (3.14)\\n\\nTo identify the right hand sides, we first note that the Bianchi identities are\\n\\nB R a b = dR a b -R a c ∧ ω c b + ω a c ∧ R c b = 0 (3.15) B a T = dT a -R a b ∧ e b + ω a b ∧ T b = 0 (3.16)\\n\\nwhich are obtained from the definitions of R a b and T a by exterior differentiation. The first equation is satisfied identically. Since the torsion vanishes, the second equation tells us only that R a b ∧ e b = 0. If we do the 3+1 split, we find two equations. The first is\\n\\nB T α = -( (3) R α β -σ ⊥ K α ∧ K β ) ∧ ẽβ = 0 (3.17)\\n\\nwhich upon using the symmetry of K α linearizes to\\n\\nB T α = -ǫ αβγ db β ∧ ẽγ + . . . (3.18)\\n\\nNote that this is the image under duality of the shift constraint as in (3.13) .\\n\\nThe second identity is\\n\\nB T 0 = -R0 α ∧ ẽα = -σ ⊥ ( DK) α = -σ ⊥ dk α + ρǫ αβγ b β ∧ ẽγ ∧ ẽα = 0 (3.19)\\n\\nwhere to arrive in the second line we used (2.46) . This is the image of the Hamiltonian constraint as in (3.14) . Summarizing, the duality transformations between linearized constraints and Bianchi identities are\\n\\nC α → B T,α C 0 → B 0 T (3.20) B T,α → -C α B 0 T → -C 0 (3.21)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Connection with other known dualities\",\n \"text\": \"The Maxwell-type duality operation (3. We begin with the spatial 2-forms when we have\\n\\nRαβ = -ǫ αβγ dB γ + σ ⊥ (B α ∧ B β -K α ∧ K β ) (3.23) R0 α = σ ⊥ ( dK α + K β ∧ ωβ α ) ≡ σ ⊥ (\\n\\nDK) α (3.24) and S0 γ = 1 2 σ ⊥ ǫ αβγ Rαβ (3.25) Sαβ = ǫ αβγ R0γ (3.26) If we linearize these expressions, we find under the duality transformation (3.8) Rab → -σ ⊥ Sab (3.27) Because the expressions (3.24) involve derivatives of B and K, the duality (3.8) is an 'integrated form' of the usual Riemann tensor duality, but implies it. Similarly, if we investigate the spatial 1-forms, we find r ab → -σ ⊥ s ab (3.28)\\n\\nTo arrive at this result we have set to zero the Lagrange multiplier field q.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The Effect on the Boundary Theory\",\n \"text\": \"It is well known that AdS is holographic. We may well ask, in the context of AdS/CFT, how the duality transformation that we have defined here acts in the boundary. We are instructed to consider the on-shell bulk action as a function of bulk fields. So, we evaluate the action on a solution to the equation of motion, resulting in a pure boundary term which is of the form\\n\\nS bdy = ∂M p α ∧ E α (4.1)\\n\\nApplying the duality transformation to the bulk theory, although the bulk action is not invariant as we have discussed above, nevertheless it may be easily shown that it induces a simple transformation on the (linearized) boundary term: it simply changes its sign.\\n\\nS dual bdy = - ∂M p D,α ∧ E α (4.2)\\n\\nThis transformation is exactly analogous to what happens in the Maxwell case: it amounts to the result [21] .\\n\\n10 G 2 G dual 2 = -1 . (4.3)\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"Conclusions and Outlook\",\n \"text\": \"Motivated by possible application in holography and in higher-spin gauge theory we have studied the duality properties of gravity in the Hamiltonian formulation. We have presented the gravity action in terms of suitable variables that closely resemble the electric and magnetic fields in Maxwell theory. We have found suitable \\\"electric\\\" and \\\"magnetic\\\" field variables, such that at the linearized level first order gravity most closely resembles electromagnetism. This can be done only around Minkowksi and (A)dS 4 backgrounds. We have implemented duality transformations in the linearized gravity fluctuations around these backgrounds. In the presence of a cosmological constant, the Hamiltonian changes, nevertheless the bulk dynamics remains unaltered, while the linearized lapse and shift constraints are mapped into the linearized Bianchi identities. Moreover, the duality transformations induce boundary terms whose relevance in holography we have briefly discussed. Finally, we have exhibited a modified duality rotation that leaves the bulk Hamiltonian invariant, while it induces boundary terms corresponding to relevant deformations.\\n\\nThe main implication of our results is that certain properties of correlations functions in three-dimensional CFTs mimic the duality of gravity. It would be interesting to extend our results to black-hole backgrounds and also when topological terms are present in the bulk. We also expect that one can analyze the duality of higher-spin gauge theories based on our first-order approach.\\n\\nThe work of A. C. P. was partially supported by the research program \\\"PYTHAGORAS II\\\" of the Greek Ministry of Education. RGL was supported in part by the U.S. Department of Energy under contract DE-FG02-91ER40709.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Appendix: other duality mappings\",\n \"text\": \"It is possible to find a transformation that leaves the Hamiltonian unchanged. Consider the following transformation in the fixed background (2.40)\\n\\nk α → -b α -2ρE α , b α → k α . (6.1)\\n\\nThe mapping to the dual dreibein is still specified by (3.9). A straightforward calculation reveals that the action transforms as The transformations (6.1) leaves unchanged the Hamiltonian and changes the action by the total \\\"time\\\" derivative terms shown in the first line of (6.2). Moreover, the linearized constraints transform into the linearized Bianchi identities. Let us see that in some detail. The second term in the shift constraint is zero since k α is a symmetric one form k α = k αβ ẽβ with k αβ = k βα ; see (2.21) . The term proportional to Λ in the lapse constraint is also zero. This is slightly more involved to see and it is based on the possibility of solving (3.9) for E α after gauge fixing. 11 One way to see this is in components. Write E α = E α β ẽβ and (3.9) becomes\\n\\nI\\n\\n∂ α E β γ -∂ γ E β α = ǫ β δα k δ γ -ǫ β δγ k δ α (6.3)\\n\\nIn the \\\"Lorentz gauge\\\" where ∂ α E β α = 0 = ∂ α k β α the above can be inverted as\\n\\nE α β = 1 ∂ 2 ǫ α δγ ∂ γ k δ β (6.4)\\n\\nUsing (6.4) one verifies that the last term in the lapse constraint vanishes. This modified duality transformation is probably related to the one considered by Julia et. al. in [15] .\"\n }\n]","isConversational":false}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":0,"numItemsPerPage":100,"numTotalItems":343056,"offset":0,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NjYzMjI3OCwianRpIjoiMzFjMzEzNmEtNWE1OS00MmZiLTlkMzAtOTAzMzYzM2M3MjRlIiwic3ViIjoiL2RhdGFzZXRzL2tvbnNtYW4vcGh5c2ljcy1jb3JwdXMiLCJleHAiOjE3ODY2MzU4NzgsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.eqitVDMP1NJGD8nVyYSHaDEJAAkUNwJ8Ie422wOAnAdByY1dV5qVJ7l1D6zmw3TKiLLvP-P4Oi3uzbq3u0WhDw","displayUrls":true,"splitSizeSummaries":[{"config":"quantum-physics","split":"train","numRows":343056,"numBytesParquet":15067827363}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
paper_id stringlengths 15 15 | arxiv_id stringlengths 9 9 | arxiv_version int32 1 1 | doi stringlengths 0 94 | content_hash stringlengths 64 64 | title stringlengths 5 243 | abstract stringlengths 13 2.11k | authors listlengths 1 816 | arxiv_categories listlengths 1 13 | primary_category stringclasses 116
values | journal stringclasses 1
value | keywords listlengths 0 0 | submission_date stringdate 2007-03-31 00:00:00 2012-12-06 00:00:00 | processed_date stringdate 2026-02-26 00:00:00 2026-02-26 00:00:00 | full_text stringlengths 29 1.72M | sections listlengths 1 507 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
arxiv:0704.0007 | 0704.0007 | 1 | 10.1103/PhysRevD.76.044016 | 8ae0e0f35013725cd790673ceb8a5ca5fa7caa17c3f9774fc6aa4adfcfb62edd | Polymer Quantum Mechanics and its Continuum Limit | A rather non-standard quantum representation of the canonical commutation
relations of quantum mechanics systems, known as the polymer representation has
gained some attention in recent years, due to its possible relation with Planck
scale physics. In particular, this approach has been followed in a symmetric
sector of... | [
"Alejandro Corichi",
"Tatjana Vukasinac and Jose A. Zapata"
] | [
"gr-qc"
] | gr-qc | [] | 2007-03-31 | 2026-02-26 | The so-called polymer quantum mechanics, a nonregular and somewhat 'exotic' representation of the canonical commutation relations (CCR) [1], has been used to explore both mathematical and physical issues in background independent theories such as quantum gravity [2, 3] . A notable example of this type of quantization, ... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The so-called polymer quantum mechanics, a nonregular and somewhat 'exotic' representation of the canonical commutation relations (CCR) [1], has been used to explore both mathematical and physical issues in background independe... | |
arxiv:0704.0015 | 0704.0015 | 1 | 10.1088/1126-6708/2007/05/034 | 736f71b146882dce29dd074d1d2d87f98333dd269e5d47f089d31b0d7812e8e5 | Fermionic superstring loop amplitudes in the pure spinor formalism | The pure spinor formulation of the ten-dimensional superstring leads to
manifestly supersymmetric loop amplitudes, expressed as integrals in pure
spinor superspace. This paper explores different methods to evaluate these
integrals and then uses them to calculate the kinematic factors of the one-loop
and two-loop massle... | [
"Christian Stahn"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | The pure spinor formulation of the ten-dimensional superstring leads to manifestly supersymmetric loop amplitudes, expressed as integrals in pure spinor superspace. This paper explores different methods to evaluate these integrals and then uses them to calculate the kinematic factors of the one-loop and two-loop massle... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The pure spinor formulation of the ten-dimensional superstring leads to manifestly supersymmetric loop amplitudes, expressed as integrals in pure spinor superspace. This paper explores different methods to evaluate these integrals and t... | |
arxiv:0704.0018 | 0704.0018 | 1 | ee51631f7ef4614c82d5dfcb38c0d10d13a798d2b8a12e1d77f585492414ebce | In quest of a generalized Callias index theorem | We give a prescription for how to compute the Callias index, using as
regulator an exponential function. We find agreement with old results in all
odd dimensions. We show that the problem of computing the dimension of the
moduli space of self-dual strings can be formulated as an index problem in
even-dimensional (loop-... | [
"Andreas Gustavsson"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | We give a prescription for how to compute the Callias index, using as regulator an exponential function. We find agreement with old results in all odd dimensions. We show that the problem of computing the dimension of the moduli space of self-dual strings can be formulated as an index problem in even-dimensional (loop-... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We give a prescription for how to compute the Callias index, using as regulator an exponential function. We find agreement with old results in all odd dimensions. We show that the problem of computing the dimension of the moduli space o... | ||
arxiv:0704.0034 | 0704.0034 | 1 | a05703051fdf4cf4df48610e330d72278db2f90f7aef16c061572996b8a9c0c0 | Origin of adaptive mutants: a quantum measurement? | This is a supplement to the paper arXiv:q-bio/0701050, containing the text of
correspondence sent to Nature in 1990. | [
"Vasily Ogryzko"
] | [
"q-bio.PE",
"q-bio.CB",
"quant-ph"
] | q-bio.PE | [] | 2007-03-31 | 2026-02-26 | This is a supplement to the paper arXiv:q-bio/0701050, containing the text of correspondence sent to Nature in 1990. Origin of adaptive mutants: a quantum measurement? Sir, -Several recent works described non-random induction of adaptive mutations by environmental stimuli 1-3 . The most obvious explanation of this str... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "This is a supplement to the paper arXiv:q-bio/0701050, containing the text of correspondence sent to Nature in 1990. Origin of adaptive mutants: a quantum measurement? Sir, -Several recent works described non-random induction of adapti... | ||
arxiv:0704.0042 | 0704.0042 | 1 | ff68e0616e33788a1cc39c55250b83169d82f127c9ade78202d572a62a8993c2 | General System theory, Like-Quantum Semantics and Fuzzy Sets | It is outlined the possibility to extend the quantum formalism in relation to
the requirements of the general systems theory. It can be done by using a
quantum semantics arising from the deep logical structure of quantum theory. It
is so possible taking into account the logical openness relationship between
observer an... | [
"Ignazio Licata"
] | [
"physics.gen-ph",
"quant-ph"
] | physics.gen-ph | [] | 2007-03-31 | 2026-02-26 | It is outlined the possibility to extend the quantum formalism in relation to the requirements of the general systems theory. It can be done by using a quantum semantics arising from the deep logical structure of quantum theory. It is so possible taking into account the logical openness relationship between observer an... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "It is outlined the possibility to extend the quantum formalism in relation to the requirements of the general systems theory. It can be done by using a quantum semantics arising from the deep logical structure of quantum theory. It is s... | ||
arxiv:0704.0046 | 0704.0046 | 1 | 10.1063/1.2779138 | f01371fa48c30b75baa9584d06be7c51746e1c39eccfa73e742b5f5b675b7ccb | A limit relation for entropy and channel capacity per unit cost | In a quantum mechanical model, Diosi, Feldmann and Kosloff arrived at a
conjecture stating that the limit of the entropy of certain mixtures is the
relative entropy as system size goes to infinity. The conjecture is proven in
this paper for density matrices. The first proof is analytic and uses the
quantum law of large... | [
"I. Csiszar",
"F. Hiai and D. Petz"
] | [
"quant-ph",
"cs.IT",
"math.IT"
] | quant-ph | [] | 2007-04-01 | 2026-02-26 | In a quantum mechanical model, Diósi, Feldmann and Kosloff arrived at a conjecture stating that the limit of the entropy of certain mixtures is the relative entropy as system size goes to infinity. The conjecture is proven in this paper for density matrices. The first proof is analytic and uses the quantum law of large... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "In a quantum mechanical model, Diósi, Feldmann and Kosloff arrived at a conjecture stating that the limit of the entropy of certain mixtures is the relative entropy as system size goes to infinity. The conjecture is proven in this paper... | |
arxiv:0704.0048 | 0704.0048 | 1 | 10.1088/0264-9381/24/19/S17 | 5831c84409a92a3da91bce6c44bb910afdfbe6594b6b49eb409254e409cc6c92 | Inference on white dwarf binary systems using the first round Mock LISA
Data Challenges data sets | We report on the analysis of selected single source data sets from the first
round of the Mock LISA Data Challenges (MLDC) for white dwarf binaries. We
implemented an end-to-end pipeline consisting of a grid-based coherent
pre-processing unit for signal detection, and an automatic Markov Chain Monte
Carlo post-processi... | [
"Alexander Stroeer",
"John Veitch",
"Christian Roever",
"Ed Bloomer",
"James\n Clark",
"Nelson Christensen",
"Martin Hendry",
"Chris Messenger",
"Renate Meyer",
"Matthew Pitkin",
"Jennifer Toher",
"Richard Umstaetter",
"Alberto Vecchio and\n Graham Woan"
] | [
"gr-qc",
"astro-ph"
] | gr-qc | [] | 2007-03-31 | 2026-02-26 | The data obtained from LISA [1] will contain a large number of white dwarf binary systems across the whole observational window [2] . At frequencies below ∼ 3 mHz the sources are so abundant that they produce a stochastic foreground whose intensity dominates the instrumental noise [3] . The closer (and louder) sources ... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The data obtained from LISA [1] will contain a large number of white dwarf binary systems across the whole observational window [2] . At frequencies below ∼ 3 mHz the sources are so abundant that they produce a stochastic foregrou... | |
arxiv:0704.0051 | 0704.0051 | 1 | 89bfe7bf5b6e413c61845825ba2507e7c5d1bbea46c828b96f384fd5323b3afc | Visualizing Teleportation | A novel way of picturing the processing of quantum information is described,
allowing a direct visualization of teleportation of quantum states and
providing a simple and intuitive understanding of this fascinating phenomenon.
The discussion is aimed at providing physicists a method of explaining
teleportation to non-s... | [
"Scott M. Cohen"
] | [
"physics.ed-ph",
"quant-ph"
] | physics.ed-ph | [] | 2007-04-02 | 2026-02-26 | One of the most exciting and fastest-growing fields of physics today is quantum information. Especially since the discovery by Shor [1, 2] that there exist calculations for which a quantum computer is apparently far more efficient than a classical computer, interest in understanding quantum information has increased at... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "One of the most exciting and fastest-growing fields of physics today is quantum information. Especially since the discovery by Shor [1, 2] that there exist calculations for which a quantum computer is apparently far more effici... | ||
arxiv:0704.0052 | 0704.0052 | 1 | 67ed34caf79c65f4a9d124daf92524fdfa9e55ddc4435a33fc69e1973c007809 | Quantum Field Theory on Curved Backgrounds. II. Spacetime Symmetries | We study space-time symmetries in scalar quantum field theory (including
interacting theories) on static space-times. We first consider Euclidean
quantum field theory on a static Riemannian manifold, and show that the
isometry group is generated by one-parameter subgroups which have either
self-adjoint or unitary quant... | [
"Arthur Jaffe (1) and Gordon Ritter (1) ((1) Harvard University)"
] | [
"hep-th"
] | hep-th | [] | 2007-03-31 | 2026-02-26 | The extension of quantum field theory to curved space-times has led to the discovery of many qualitatively new phenomena which do not occur in the simpler theory on Minkowski space, such as Hawking radiation; for background and historical references, see [2, 6, 18] . The reconstruction of quantum field theory on a Lore... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The extension of quantum field theory to curved space-times has led to the discovery of many qualitatively new phenomena which do not occur in the simpler theory on Minkowski space, such as Hawking radiation; for background and hi... | ||
arxiv:0704.0053 | 0704.0053 | 1 | 39613cb738fb14c9fcd290d9a7ddb7c314a984abeab7f7e78d81855d1e36fb82 | A Global Approach to the Theory of Special Finsler Manifolds | The aim of the present paper is to provide a global presentation of the
theory of special Finsler manifolds. We introduce and investigate globally (or
intrinsically, free from local coordinates) many of the most important and most
commonly used special Finsler manifolds: locally Minkowskian, Berwald,
Landesberg, genera... | [
"Nabil L. Youssef",
"S. H. Abed and A. Soleiman"
] | [
"math.DG",
"gr-qc"
] | math.DG | [] | 2007-03-31 | 2026-02-26 | The aim of the present paper is to provide a global presentation of the theory of special Finsler manifolds. We introduce and investigate globally (or intrinsically, free from local coordinates) many of the most important and most commonly used special Finsler manifolds : locally Minkowskian, Berwald, Landesberg, gener... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The aim of the present paper is to provide a global presentation of the theory of special Finsler manifolds. We introduce and investigate globally (or intrinsically, free from local coordinates) many of the most important and most commo... | ||
arxiv:0704.0064 | 0704.0064 | 1 | 10.1142/S0217751X08041591 | 8f2258c6a1705a57c7308510858d8a12ea1d4ff794e5ecc2bdc0558696d75389 | Nilpotent symmetry invariance in the superfield formulation: the
(non-)Abelian 1-form gauge theories | We capture the off-shell as well as the on-shell nilpotent
Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry invariance of the
Lagrangian densities of the four (3 + 1)-dimensional (4D) (non-)Abelian 1-form
gauge theories within the framework of the superfield formalism. In particular,
we provide the geometrical i... | [
"R. P. Malik (Bhu)"
] | [
"hep-th"
] | hep-th | [] | 2007-04-01 | 2026-02-26 | The geometrical superfield approach [1] [2] [3] [4] [5] [6] [7] [8] to Becchi-Rouet-Stora-Tyutin (BRST) formalism is one of the most attractive and intuitive approaches which enables us to gain some physical insights into the beautiful (but abstract mathematical) structures that are associated with the nilpotent (anti-... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The geometrical superfield approach [1] [2] [3] [4] [5] [6] [7] [8] to Becchi-Rouet-Stora-Tyutin (BRST) formalism is one of the most attractive and intuitive approaches which enables us to gain some physical insights into the beau... | |
arxiv:0704.0066 | 0704.0066 | 1 | 5a791c216f06fd6b10882db3e3fbfbdb15a2951484e49be57da703a2d3733cbb | Lagrangian quantum field theory in momentum picture. IV. Commutation
relations for free fields | Possible (algebraic) commutation relations in the Lagrangian quantum theory
of free (scalar, spinor and vector) fields are considered from mathematical
view-point. As sources of these relations are employed the Heisenberg
equations/relations for the dynamical variables and a specific condition for
uniqueness of the ope... | [
"Bozhidar Z. Iliev (Institute for Nuclear Research and Nuclear Energy",
"Bulgarian Academy of Sciences",
"Sofia",
"Bulgaria)"
] | [
"hep-th"
] | hep-th | [] | 2007-04-01 | 2026-02-26 | Possible (algebraic) commutation relations in the Lagrangian quantum theory of free (scalar, spinor and vector) fields are considered from mathematical view-point. As sources of these relations are employed the Heisenberg equations/relations for the dynamical variables and a specific condition for uniqueness of the ope... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Possible (algebraic) commutation relations in the Lagrangian quantum theory of free (scalar, spinor and vector) fields are considered from mathematical view-point. As sources of these relations are employed the Heisenberg equations/rela... | ||
arxiv:0704.0078 | 0704.0078 | 1 | 10.1088/0264-9381/24/14/008 | e7905f135b445b554e49d9dccd83f4cd5695b8ade30e0b980cc4cc285a3a05ec | Linear perturbations of matched spacetimes: the gauge problem and
background symmetries | We present a critical review about the study of linear perturbations of
matched spacetimes including gauge problems. We analyse the freedom introduced
in the perturbed matching by the presence of background symmetries and revisit
the particular case of spherically symmetry in n-dimensions. This analysis
includes settin... | [
"Marc Mars",
"Filipe C. Mena",
"Raul Vera"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-01 | 2026-02-26 | We present a critical review about the study of linear perturbations of matched spacetimes including gauge problems. We analyse the freedom introduced in the perturbed matching by the presence of background symmetries and revisit the particular case of spherically symmetry in n-dimensions. This analysis includes settin... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We present a critical review about the study of linear perturbations of matched spacetimes including gauge problems. We analyse the freedom introduced in the perturbed matching by the presence of background symmetries and revisit the pa... | |
arxiv:0704.0081 | 0704.0081 | 1 | 778900414d1a1f08fb8e88b60222a7cb368a7f0c24b131c30158994a60c01f60 | Quantum Deformations of Relativistic Symmetries | We discussed quantum deformations of D=4 Lorentz and Poincare algebras. In
the case of Poincare algebra it is shown that almost all classical r-matrices
of S. Zakrzewski classification correspond to twisted deformations of Abelian
and Jordanian types. A part of twists corresponding to the r-matrices of
Zakrzewski class... | [
"V.N. Tolstoy (INP",
"Moscow State University)"
] | [
"math.QA",
"hep-th",
"math-ph",
"math.MP",
"math.RT"
] | math.QA | [] | 2007-04-02 | 2026-02-26 | We discussed quantum deformations of D = 4 Lorentz and Poincaré algebras. In the case of Poincaré algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification correspond to twisted deformations of Abelian and Jordanian types. A part of twists corresponding to the r-matrices of Zakrzewski cla... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We discussed quantum deformations of D = 4 Lorentz and Poincaré algebras. In the case of Poincaré algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification correspond to twisted deformations of Abelian and... | ||
arxiv:0704.0083 | 0704.0083 | 1 | 10.1103/PhysRevLett.99.071301 | 3f56a48bd29e65a018261e0299f2f784310472a5b7f577b43b2f2cc6a41bf6d7 | Why there is something rather than nothing (out of everything)? | The path integral over Euclidean geometries for the recently suggested
density matrix of the Universe is shown to describe a microcanonical ensemble
in quantum cosmology. This ensemble corresponds to a uniform (weight one)
distribution in phase space of true physical variables, but in terms of the
observable spacetime ... | [
"A.O.Barvinsky"
] | [
"hep-th"
] | hep-th | [] | 2007-04-01 | 2026-02-26 | The path integral over Euclidean geometries for the recently suggested density matrix of the Universe is shown to describe a microcanonical ensemble in quantum cosmology. This ensemble corresponds to a uniform (weight one) distribution in phase space of true physical variables, but in terms of the observable spacetime ... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The path integral over Euclidean geometries for the recently suggested density matrix of the Universe is shown to describe a microcanonical ensemble in quantum cosmology. This ensemble corresponds to a uniform (weight one) distribution ... | |
arxiv:0704.0100 | 0704.0100 | 1 | 10.1143/PTP.118.715 | 99f97029a358be0732fb87926175d30da9524c34fa9f6637aa15c218dc49eb70 | Topology Change of Black Holes | The topological structure of the event horizon has been investigated in terms
of the Morse theory. The elementary process of topological evolution can be
understood as a handle attachment. It has been found that there are certain
constraints on the nature of black hole topological evolution: (i) There are n
kinds of ha... | [
"Daisuke Ida and Masaru Siino"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-02 | 2026-02-26 | Black holes in space-times of greater than or equal to five dimensions have rich topological structure. According to the well-known results of Hawking concerning the topology of black holes in four-dimensional space-time, the apparent horizon or the spatial section of the stationary event horizon is necessarily diffeom... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Black holes in space-times of greater than or equal to five dimensions have rich topological structure. According to the well-known results of Hawking concerning the topology of black holes in four-dimensional space-time, the appa... | |
arxiv:0704.0101 | 0704.0101 | 1 | fe528113aa79866c45eb45e2b6d5f7f386f0359c1df414aa2148395b56badb6c | The birth of string theory | In this contribution we go through the developments that in the years 1968 to
1974 led from the Veneziano model to the bosonic string. | [
"Paolo Di Vecchia"
] | [
"hep-th"
] | hep-th | [] | 2007-04-01 | 2026-02-26 | The sixties was a period in which strong interacting processes were studied in detail using the newly constructed accelerators at Cern and other places. Many new hadronic states were found that appeared as resonant peaks in various cross sections and hadronic cross sections were measured with increasing accuracy. In ge... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The sixties was a period in which strong interacting processes were studied in detail using the newly constructed accelerators at Cern and other places. Many new hadronic states were found that appeared as resonant peaks in variou... | ||
arxiv:0704.0103 | 0704.0103 | 1 | 10.1007/s10773-007-9450-y | b5569420bcf9c3028ac0cd9a97fe5e5d178d687547076d0da25a18a46e491bbf | Generalized regularly discontinuous solutions of the Einstein equations | The physical consistency of the match of piecewise-$C^0$ metrics is
discussed. The mathematical theory of gravitational discontinuity hypersurfaces
is generalized to cover the match of regularly discontinuous metrics. The
mean-value differential geometry framework on a hypersurface is introduced, and
corresponding comp... | [
"Gianluca Gemelli"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-01 | 2026-02-26 | The physical consistency of the match of piecewise-C 0 metrics is discussed. The mathematical theory of gravitational discontinuity hypersurfaces is generalized to cover the match of regularly discontinuous metrics. The mean-value differential geometry framework on a hypersurface is introduced, and corresponding compat... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The physical consistency of the match of piecewise-C 0 metrics is discussed. The mathematical theory of gravitational discontinuity hypersurfaces is generalized to cover the match of regularly discontinuous metrics. The mean-value diffe... | |
arxiv:0704.0116 | 0704.0116 | 1 | 10.1103/PhysRevD.75.127902 | 9f85542fe308a852df10252f186431c073464294ec78899d772556fdbc3036a9 | Stringy Jacobi fields in Morse theory | We consider the variation of the surface spanned by closed strings in a
spacetime manifold. Using the Nambu-Goto string action, we induce the geodesic
surface equation, the geodesic surface deviation equation which yields a Jacobi
field, and we define the index form of a geodesic surface as in the case of
point particl... | [
"Yong Seung Cho",
"Soon-Tae Hong"
] | [
"math-ph",
"hep-th",
"math.MP"
] | math-ph | [] | 2007-04-02 | 2026-02-26 | We consider the variation of the surface spanned by closed strings in a spacetime manifold. Using the Nambu-Goto string action, we induce the geodesic surface equation, the geodesic surface deviation equation which yields a Jacobi field, and we define the index form of a geodesic surface as in the case of point particl... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We consider the variation of the surface spanned by closed strings in a spacetime manifold. Using the Nambu-Goto string action, we induce the geodesic surface equation, the geodesic surface deviation equation which yields a Jacobi field... | |
arxiv:0704.0117 | 0704.0117 | 1 | 10.1088/0953-4075/40/11/002 | ab49ae90fd17c08af50d23a7e91bdb7703f6901f0b714d66f75e6faf5bbb44ee | Lower ground state due to counter-rotating wave interaction in trapped
ion system | We consider a single ion confined in a trap under radiation of two traveling
waves of lasers. In the strong-excitation regime and without the restriction of
Lamb-Dicke limit, the Hamiltonian of the system is similar to a driving
Jaynes-Cummings model without rotating wave approximation (RWA). The approach
we developed ... | [
"T. Liu",
"K.L. Wang",
"M. Feng"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | Ultracold ions trapped as a line are considered as a promising system for quantum information processing [1] . Since the first quantum gate performed in the ion trap [2] , there have been a series of experiments with trapped ions to achieve nonclassical states [3], simple quantum algorithm [4] , and quantum communicati... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Ultracold ions trapped as a line are considered as a promising system for quantum information processing [1] . Since the first quantum gate performed in the ion trap [2] , there have been a series of experiments with trapped io... | |
arxiv:0704.0121 | 0704.0121 | 1 | 10.1088/0264-9381/25/7/075001 | d667e8a785fe3fccabc03f16e745f748f169faf7f75bfc37ac9c853839a39ce5 | Meta-Stable Brane Configuration of Product Gauge Groups | Starting from the N=1 SU(N_c) x SU(N_c') gauge theory with fundamental and
bifundamental flavors, we apply the Seiberg dual to the first gauge group and
obtain the N=1 dual gauge theory with dual matters including the gauge
singlets. By analyzing the F-term equations of the superpotential, we describe
the intersecting ... | [
"Changhyun Ahn"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | Starting from the N = 1 SU(N c ) × SU(N ′ c ) gauge theory with fundamental and bifundamental flavors, we apply the Seiberg dual to the first gauge group and obtain the N = 1 dual gauge theory with dual matters including the gauge singlets. By analyzing the F-term equations of the superpotential, we describe the inters... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Starting from the N = 1 SU(N c ) × SU(N ′ c ) gauge theory with fundamental and bifundamental flavors, we apply the Seiberg dual to the first gauge group and obtain the N = 1 dual gauge theory with dual matters including the gauge singl... | |
arxiv:0704.0135 | 0704.0135 | 1 | 10.1103/PhysRevA.76.052105 | 17b1b58b19c72b2e4c86cbc9c53a65e08eed391855e30408198cb0ce7a198ac5 | A Single Trapped Ion as a Time-Dependent Harmonic Oscillator | We show how a single trapped ion may be used to test a variety of important
physical models realized as time-dependent harmonic oscillators. The ion itself
functions as its own motional detector through laser-induced electronic
transitions. Alsing et al. [Phys. Rev. Lett. 94, 220401 (2005)] proposed that
an exponential... | [
"Nicolas C. Menicucci and G. J. Milburn"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | The time-dependent quantum harmonic oscillator has long served as a paradigm for nonadiabatic timedependent Hamiltonian systems and has been applied to a wide range of physical problems by choosing the mass, the frequency, or both, to be time-dependent. The earliest application is to squeezed state generation in quantu... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The time-dependent quantum harmonic oscillator has long served as a paradigm for nonadiabatic timedependent Hamiltonian systems and has been applied to a wide range of physical problems by choosing the mass, the frequency, or b... | |
arxiv:0704.0137 | 0704.0137 | 1 | f75d6ea4258f3061a36baa4625616d5304a5b4f7168f59319f746f367c8447aa | Topological defects, geometric phases, and the angular momentum of light | Recent reports on the intriguing features of vector vortex bearing beams are
analyzed using geometric phases in optics. It is argued that the spin
redirection phase induced circular birefringence is the origin of topological
phase singularities arising in the inhomogeneous polarization patterns. A
unified picture of re... | [
"S C Tiwari"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | Recent reports on the intriguing features of vector vortex bearing beams are analyzed using geometric phases in optics. It is argued that the spin redirection phase induced circular birefringence is the origin of topological phase singularities arising in the inhomogeneous polarization patterns. A unified picture of re... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Recent reports on the intriguing features of vector vortex bearing beams are analyzed using geometric phases in optics. It is argued that the spin redirection phase induced circular birefringence is the origin of topological phase singu... | ||
arxiv:0704.0138 | 0704.0138 | 1 | 10.1103/PhysRevD.76.044007 | 3ba301ce5e70e09a4143cd5f8753dd327f2ff619fe397cd40de8fcf206841f95 | Circular and non-circular nearly horizon-skimming orbits in Kerr
spacetimes | We have performed a detailed analysis of orbital motion in the vicinity of a
nearly extremal Kerr black hole. For very rapidly rotating black holes (spin
a=J/M>0.9524M) we have found a class of very strong field eccentric orbits
whose angular momentum L_z increases with the orbit's inclination with respect
to the equat... | [
"Enrico Barausse",
"Scott A. Hughes",
"Luciano Rezzolla"
] | [
"gr-qc",
"astro-ph"
] | gr-qc | [] | 2007-04-02 | 2026-02-26 | The space-based gravitational-wave detector LISA [1] will be a unique tool to probe the nature of supermassive black holes (SMBHs), making it possible to map in detail their spacetimes. This goal is expected to be achieved by observing gravitational waves emitted by compact stars or black holes with masses µ ≈ 1 -100 M... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The space-based gravitational-wave detector LISA [1] will be a unique tool to probe the nature of supermassive black holes (SMBHs), making it possible to map in detail their spacetimes. This goal is expected to be achieved by o... | |
arxiv:0704.0140 | 0704.0140 | 1 | 10.1016/j.physletb.2007.08.026 | 666aaac79baf722a17a45b8f72128f29e25f1b9f014cf65105319dc3ee7806f0 | Entanglement entropy of two-dimensional Anti-de Sitter black holes | Using the AdS/CFT correspondence we derive a formula for the entanglement
entropy of the anti-de Sitter black hole in two spacetime dimensions. The
leading term in the large black hole mass expansion of our formula reproduces
exactly the Bekenstein-Hawking entropy S_{BH}, whereas the subleading term
behaves as ln S_{BH... | [
"Mariano Cadoni"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | Using the AdS/CFT correspondence we derive a formula for the entanglement entropy of the anti-de Sitter black hole in two spacetime dimensions. The leading term in the large black hole mass expansion of our formula reproduces exactly the Bekenstein-Hawking entropy SBH , whereas the subleading term behaves as ln SBH . T... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Using the AdS/CFT correspondence we derive a formula for the entanglement entropy of the anti-de Sitter black hole in two spacetime dimensions. The leading term in the large black hole mass expansion of our formula reproduces exactly th... | |
arxiv:0704.0144 | 0704.0144 | 1 | 10.1088/1475-7516/2009/02/007 | 502f365bbcbc68e856ad4bde1d13bfe2bd6767db20ac4b4d59c7d8c9f51fa3d9 | Eternal inflation and localization on the landscape | We model the essential features of eternal inflation on the landscape of a
dense discretuum of vacua by the potential $V(\phi)=V_{0}+\delta V(\phi)$,
where $|\delta V(\phi)|\ll V_{0}$ is random. We find that the diffusion of the
distribution function $\rho(\phi,t)$ of the inflaton expectation value in
different Hubble ... | [
"D. Podolsky",
"K. Enqvist"
] | [
"hep-th",
"astro-ph",
"gr-qc"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | We model the essential features of eternal inflation on the landscape of a dense discretuum of vacua by the potential V (φ) = V0 + δV (φ), where |δV (φ)| ≪ V0 is random. We find that the diffusion of the distribution function ρ(φ, t) of the inflaton expectation value in different Hubble patches may be suppressed due to... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We model the essential features of eternal inflation on the landscape of a dense discretuum of vacua by the potential V (φ) = V0 + δV (φ), where |δV (φ)| ≪ V0 is random. We find that the diffusion of the distribution function ρ(φ, t) of... | |
arxiv:0704.0145 | 0704.0145 | 1 | bb697fd8f846fc1c08d49ce99684bc4e78b602b4ac1a88ae22206ad50b4209aa | Singularity Resolution in Isotropic Loop Quantum Cosmology: Recent
Developments | Since the past Iagrg meeting in December 2004, new developments in loop
quantum cosmology have taken place, especially with regards to the resolution
of the Big Bang singularity in the isotropic models. The singularity resolution
issue has been discussed in terms of physical quantities (expectation values of
Dirac obse... | [
"Ghanashyam Date"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-02 | 2026-02-26 | Since the past Iagrg meeting in December 2004, new developments in loop quantum cosmology have taken place, especially with regards to the resolution of the Big Bang singularity in the isotropic models. The singularity resolution issue has been discussed in terms of physical quantities (expectation values of Dirac obse... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Since the past Iagrg meeting in December 2004, new developments in loop quantum cosmology have taken place, especially with regards to the resolution of the Big Bang singularity in the isotropic models. The singularity resolution issue ... | ||
arxiv:0704.0147 | 0704.0147 | 1 | f4e3ecd27fdf2db85fc052451f1350a72df99a3fcaae6faa64830b02aeba16fe | A POVM view of the ensemble approach to polarization optics | Statistical ensemble formalism of Kim, Mandel and Wolf (J. Opt. Soc. Am. A 4,
433 (1987)) offers a realistic model for characterizing the effect of
stochastic non-image forming optical media on the state of polarization of
transmittedlight. With suitable choice of the Jones ensemble, various Mueller
transformations - s... | [
"Sudha",
"A. V. Gopala Rao",
"A. R. Usha Devi and A. K. Rajagopal"
] | [
"physics.optics",
"physics.class-ph",
"quant-ph"
] | physics.optics | [] | 2007-04-02 | 2026-02-26 | The intensity and polarization of a beam of light passing through an isolated optical device undergoes a linear transformation. But this is an ideal situation because, in general, the optical system is embedded in some media such as atmosphere or other ambient material, which further modifies the polarization propertie... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The intensity and polarization of a beam of light passing through an isolated optical device undergoes a linear transformation. But this is an ideal situation because, in general, the optical system is embedded in some media such ... | ||
arxiv:0704.0149 | 0704.0149 | 1 | 10.1088/1742-6596/91/1/012001 | babd34aae0c4bd479db9f42c5c013c84b85c0036741df9993f8ea22f00ffc55a | Construction of initial data for 3+1 numerical relativity | This lecture is devoted to the problem of computing initial data for the
Cauchy problem of 3+1 general relativity. The main task is to solve the
constraint equations. The conformal technique, introduced by Lichnerowicz and
enhanced by York, is presented. Two standard methods, the conformal
transverse-traceless one and ... | [
"Eric Gourgoulhon (LUTH",
"CNRS / Observatoire de Paris / Univ. Paris 7)"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-02 | 2026-02-26 | The 3+1 formalism is the basis of most modern numerical relativity and has lead, along with alternative approaches [82] , to the recent successes in the binary black hole merger problem [6, 7, 99, 25, 26, 27, 28] (see [24, 69, 86] for a review). Thanks to the 3+1 formalism, the resolution of Einstein equation amounts ... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The 3+1 formalism is the basis of most modern numerical relativity and has lead, along with alternative approaches [82] , to the recent successes in the binary black hole merger problem [6, 7, 99, 25, 26, 27, 28] (see [24, 69, 86... | |
arxiv:0704.0166 | 0704.0166 | 1 | 10.1088/1126-6708/2007/05/047 | 969f036fa8b412543185b0b02e81a1874555556c0056a68af5f5029d7028cad8 | Supersymmetry breaking metastable vacua in runaway quiver gauge theories | In this paper we consider quiver gauge theories with fractional branes whose
infrared dynamics removes the classical supersymmetric vacua (DSB branes). We
show that addition of flavors to these theories (via additional non-compact
branes) leads to local meta-stable supersymmetry breaking minima, closely
related to thos... | [
"Inaki Garcia-Etxebarria",
"Fouad Saad",
"Angel M. Uranga"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | Systems of D-branes at singularities provide a very interesting setup to realize and study diverse non-perturbative gauge dynamics phenomena in string theory. In the context of N = 1 supersymmetric gauge field theories, systems of D3-branes at Calabi-Yau singularities lead to interesting families of tractable 4d strong... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Systems of D-branes at singularities provide a very interesting setup to realize and study diverse non-perturbative gauge dynamics phenomena in string theory. In the context of N = 1 supersymmetric gauge field theories, systems of... | |
arxiv:0704.0175 | 0704.0175 | 1 | 10.1088/1475-7516/2007/10/004 | ddc73a689f5b1abc0a2024a40159971b1b2edd52d945335f605bb7ffa6400bf3 | Solar System Constraints on Gauss-Bonnet Mediated Dark Energy | Although the Gauss-Bonnet term is a topological invariant for general
relativity, it couples naturally to a quintessence scalar field, modifying
gravity at solar system scales. We determine the solar system constraints due
to this term by evaluating the post-Newtonian metric for a distributional
source. We find a mass ... | [
"Luca Amendola",
"Christos Charmousis and Stephen C. Davis"
] | [
"astro-ph",
"gr-qc",
"hep-th"
] | astro-ph | [] | 2007-04-02 | 2026-02-26 | Although the Gauss-Bonnet term is a topological invariant for general relativity, it couples naturally to a quintessence scalar field, modifying gravity at solar system scales. We determine the solar system constraints due to this term by evaluating the post-Newtonian metric for a distributional source. We find a mass ... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Although the Gauss-Bonnet term is a topological invariant for general relativity, it couples naturally to a quintessence scalar field, modifying gravity at solar system scales. We determine the solar system constraints due to this term ... | |
arxiv:0704.0179 | 0704.0179 | 1 | 10.1103/PhysRevA.75.052106 | d39eb69b08b240bbbd706fc5d1cd420d2aa1e7fda46b24b7ff1f6682613fd534 | Experimental nonclassicality of single-photon-added thermal light states | We report the experimental realization and tomographic analysis of novel
quantum light states obtained by exciting a classical thermal field by a single
photon. Such states, although completely incoherent, possess a tunable degree
of quantumness which is here exploited to put to a stringent experimental test
some of th... | [
"Alessandro Zavatta",
"Valentina Parigi",
"Marco Bellini"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | The definition and the measurement of the nonclassicality of a quantum light state is a hot and widely discussed topic in the physics community; nonclassical light is the starting point for generating even more nonclassical states [1, 2] or producing the entanglement which is essential to implement quantum information ... | [
{
"section_type": "BACKGROUND",
"section_title": "INTRODUCTION",
"text": "The definition and the measurement of the nonclassicality of a quantum light state is a hot and widely discussed topic in the physics community; nonclassical light is the starting point for generating even more nonclassical states... | |
arxiv:0704.0194 | 0704.0194 | 1 | 10.1103/PhysRevB.77.075325 | 02598565965bfa7af91a44c7c2b2e5ee6dda1cbe143ab6d535bfb77e73fe21eb | Quantum mechanical approach to decoherence and relaxation generated by
fluctuating environment | We consider an electrostatic qubit, interacting with a fluctuating charge of
single electron transistor (SET) in the framework of exactly solvable model.
The SET plays a role of the fluctuating environment affecting the qubit's
parameters in a controllable way. We derive the rate equations describing
dynamics of the en... | [
"S.A. Gurvitz and D. Mozyrsky"
] | [
"cond-mat.mes-hall",
"quant-ph"
] | cond-mat.mes-hall | [] | 2007-04-02 | 2026-02-26 | The influence of environment on a single quantum system is the issue of crucial importance in quantum information science. It is mainly associated with decoherence, or dephasing, which transforms any pure state of a quantum system into a statistical mixture. Despite a large body of theoretical work devoted to decoheren... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The influence of environment on a single quantum system is the issue of crucial importance in quantum information science. It is mainly associated with decoherence, or dephasing, which transforms any pure state of a quantum sys... | |
arxiv:0704.0202 | 0704.0202 | 1 | 10.1088/1367-2630/9/6/206 | aec1b49ed6698e9bc7096ed01dbf802374bfe48e10ea829f1e872a3c24923292 | Towards Minimal Resources of Measurement-based Quantum Computation | We improve the upper bound on the minimal resources required for
measurement-based quantum computation. Minimizing the resources required for
this model is a key issue for experimental realization of a quantum computer
based on projective measurements. This new upper bound allows also to reply in
the negative to the op... | [
"Simon Perdrix"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | The discovery of new models of quantum computation (QC), such as the one-way quantum computer [7] and the projective measurement-based model [4] , have opened up new experimental avenues toward the realisation of a quantum computer in laboratories. At the same time they have challenged the traditional view about the na... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The discovery of new models of quantum computation (QC), such as the one-way quantum computer [7] and the projective measurement-based model [4] , have opened up new experimental avenues toward the realisation of a quantum compute... | |
arxiv:0704.0207 | 0704.0207 | 1 | 10.1088/0954-3899/34/8/S10 | e619f6f2484f53a0c62c1275bf154e2d99f7cc13d034107653ef12838f5e51c6 | Quark matter and the astrophysics of neutron stars | Some of the means through which the possible presence of nearly deconfined
quarks in neutron stars can be detected by astrophysical observations of
neutron stars from their birth to old age are highlighted. | [
"M Prakash"
] | [
"astro-ph",
"gr-qc",
"nucl-th"
] | astro-ph | [] | 2007-04-02 | 2026-02-26 | Utilizing the asymptotic freedom of QCD, Collins and Perry [1] first noted that the dense cores of neutron stars may consist of deconfined quarks instead of hadrons. The crucial question is whether observations of neutron stars from their birth to death through neutrino, photon and gravity-wave emissions can unequivoca... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Utilizing the asymptotic freedom of QCD, Collins and Perry [1] first noted that the dense cores of neutron stars may consist of deconfined quarks instead of hadrons. The crucial question is whether observations of neutron stars fr... | |
arxiv:0704.0214 | 0704.0214 | 1 | 060470b4dbeb72f041d1d9f0eb467fa26eb61cf66c63cb9dae16a7a3a8c0e5bc | A schematic model of scattering in PT-symmetric Quantum Mechanics | One-dimensional scattering problem admitting a complex, PT-symmetric
short-range potential V(x) is considered. Using a Runge-Kutta-discretized
version of Schroedinger equation we derive the formulae for the reflection and
transmission coefficients and emphasize that the only innovation emerges in
fact via a complexific... | [
"Miloslav Znojil"
] | [
"quant-ph",
"hep-th"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | One-dimensional scattering problem admitting a complex, PT -symmetric shortrange potential V (x) is considered. Using a Runge-Kutta-discretized version of Schrödinger equation we derive the formulae for the reflection and transmission coefficients and emphasize that the only innovation emerges in fact via a complexific... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "One-dimensional scattering problem admitting a complex, PT -symmetric shortrange potential V (x) is considered. Using a Runge-Kutta-discretized version of Schrödinger equation we derive the formulae for the reflection and transmission c... | ||
arxiv:0704.0221 | 0704.0221 | 1 | 10.1007/s10714-007-0472-9 10.1142/S0218271808012449 | a130963edee5fcad5bf3d395b233f9ef589cee0efc87491104a122e6065970a1 | The Return of a Static Universe and the End of Cosmology | We demonstrate that as we extrapolate the current $\Lambda$CDM universe
forward in time, all evidence of the Hubble expansion will disappear, so that
observers in our "island universe" will be fundamentally incapable of
determining the true nature of the universe, including the existence of the
highly dominant vacuum e... | [
"Lawrence M. Krauss (1",
"2) and Robert J. Scherrer (2) ((1) Case Western\n Reserve University",
"(2) Vanderbilt University)"
] | [
"astro-ph",
"gr-qc",
"hep-ph",
"hep-th"
] | astro-ph | [] | 2007-04-02 | 2026-02-26 | We demonstrate that as we extrapolate the current ΛCDM universe forward in time, all evidence of the Hubble expansion will disappear, so that observers in our "island universe" will be fundamentally incapable of determining the true nature of the universe, including the existence of the highly dominant vacuum energy, t... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We demonstrate that as we extrapolate the current ΛCDM universe forward in time, all evidence of the Hubble expansion will disappear, so that observers in our \"island universe\" will be fundamentally incapable of determining the true n... | |
arxiv:0704.0228 | 0704.0228 | 1 | 10.1209/0295-5075/78/10010 | 22a164d3e9d3ebb9f8267fe76e523e592d71abee6945767ca79ae21df8db21a5 | Einstein vs Maxwell: Is gravitation a curvature of space, a field in
flat space, or both? | Starting with a field theoretic approach in Minkowski space, the
gravitational energy momentum tensor is derived from the Einstein equations in
a straightforward manner. This allows to present them as {\it acceleration
tensor} = const. $\times$ {\it total energy momentum tensor}. For flat space
cosmology the gravitatio... | [
"Theo M. Nieuwenhuizen"
] | [
"gr-qc",
"astro-ph",
"quant-ph"
] | gr-qc | [] | 2007-04-02 | 2026-02-26 | Starting with a field theoretic approach in Minkowski space, the gravitational energy momentum tensor is derived from the Einstein equations in a straightforward manner. This allows to present them as acceleration tensor = const. × total energy momentum tensor. For flat space cosmology the gravitational energy is negat... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Starting with a field theoretic approach in Minkowski space, the gravitational energy momentum tensor is derived from the Einstein equations in a straightforward manner. This allows to present them as acceleration tensor = const. × tota... | |
arxiv:0704.0232 | 0704.0232 | 1 | 4a56b2e7ec34ce8c00d29a9d0df3c84e728944549eb85e55ec38a4d0b73e8f38 | New algebraic aspects of perturbative and non-perturbative Quantum Field
Theory | In this expository article we review recent advances in our understanding of
the combinatorial and algebraic structure of perturbation theory in terms of
Feynman graphs, and Dyson-Schwinger equations. Starting from Lie and Hopf
algebras of Feynman graphs, perturbative renormalization is rephrased
algebraically. The Hoc... | [
"Christoph Bergbauer",
"Dirk Kreimer"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | As elements of perturbative expansions of Quantum field theories, Feynman graphs have been playing and still play a key role both for our conceptual understanding and for state-of-the-art computations in particle physics. This 1 article is concerned with several aspects of Feynman graphs: First, the combinatorics of pe... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "As elements of perturbative expansions of Quantum field theories, Feynman graphs have been playing and still play a key role both for our conceptual understanding and for state-of-the-art computations in particle physics. This 1 a... | ||
arxiv:0704.0240 | 0704.0240 | 1 | 10.1146/annurev.nucl.57.090506.123120 | 8f74bfa02fac7122abfb05d8312aa2a50217ef778782bdd953a27059735c323b | Viscosity, Black Holes, and Quantum Field Theory | We review recent progress in applying the AdS/CFT correspondence to
finite-temperature field theory. In particular, we show how the hydrodynamic
behavior of field theory is reflected in the low-momentum limit of correlation
functions computed through a real-time AdS/CFT prescription, which we
formulate. We also show ho... | [
"D. T. Son",
"A. O. Starinets"
] | [
"hep-th"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | 1 2 Son, Starinets
This review is about the recently emerging connection, through the gauge/gravity correspondence, between hydrodynamics and black hole physics. The study of quantum field theory at high temperature has a long history. It was first motivated by the Big Bang cosmology when it was hoped that early phase... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "1 2 Son, Starinets"
},
{
"section_type": "BACKGROUND",
"section_title": "INTRODUCTION",
"text": "This review is about the recently emerging connection, through the gauge/gravity correspondence, between hydrodynamics... | |
arxiv:0704.0245 | 0704.0245 | 1 | 10.1088/1126-6708/2007/07/002 | 2a173fa7f0bb0e54db35e723c0ed83458928b85329dcea45e520d5d198a792a5 | One-loop MHV Rules and Pure Yang-Mills | It has been known for some time that the standard MHV diagram formulation of
perturbative Yang-Mills theory is incomplete, as it misses rational terms in
one-loop scattering amplitudes of pure Yang-Mills. We propose that certain
Lorentz violating counterterms, when expressed in the field variables which
give rise to st... | [
"Andreas Brandhuber",
"Bill Spence",
"Gabriele Travaglini",
"Konstantinos\n Zoubos"
] | [
"hep-th",
"hep-ph"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | 1 {a.brandhuber, w.j.spence, g.travaglini, k.zoubos}@qmul.ac.uk Contents 1 Introduction 1 2 Background 3 2.1 The classical MHV Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 A four-dimensional regulator for lightcone Yang-Mills . . . . . . . . . . . . . 6 2.3 The one-loop (++++) amplitude . . . . . ... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "1 {a.brandhuber, w.j.spence, g.travaglini, k.zoubos}@qmul.ac.uk Contents 1 Introduction 1 2 Background 3 2.1 The classical MHV Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 A four-dimensional regulator for ligh... | |
arxiv:0704.0247 | 0704.0247 | 1 | 10.1088/1126-6708/2007/07/046 | bf8e7c8ad3d0d7319400cec91eff26db7fcb257f5a326c8e6755831de4e3278a | Geometry of four-dimensional Killing spinors | The supersymmetric solutions of N=2, D=4 minimal ungauged and gauged
supergravity are classified according to the fraction of preserved
supersymmetry using spinorial geometry techniques. Subject to a reasonable
assumption in the 1/2-supersymmetric time-like case of the gauged theory, we
derive the complete form of all ... | [
"S. L. Cacciatori",
"M. M. Caldarelli",
"D. Klemm",
"D. S. Mansi",
"D. Roest"
] | [
"hep-th",
"gr-qc"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | The supersymmetric solutions of N = 2, D = 4 minimal ungauged and gauged supergravity are classified according to the fraction of preserved supersymmetry using spinorial geometry techniques. Subject to a reasonable assumption in the 1/2supersymmetric time-like case of the gauged theory, we derive the complete form of a... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The supersymmetric solutions of N = 2, D = 4 minimal ungauged and gauged supergravity are classified according to the fraction of preserved supersymmetry using spinorial geometry techniques. Subject to a reasonable assumption in the 1/2... | |
arxiv:0704.0251 | 0704.0251 | 1 | 10.1103/PhysRevA.76.042309 | 377bbca34ff04927ba11a6c996fae271f96407730fe7a625810d51b06e081bb2 | Entanglement of Subspaces and Error Correcting Codes | We introduce the notion of entanglement of subspaces as a measure that
quantify the entanglement of bipartite states in a randomly selected subspace.
We discuss its properties and in particular we show that for maximally
entangled subspaces it is additive. Furthermore, we show that maximally
entangled subspaces can pla... | [
"Gilad Gour",
"Nolan R. Wallach"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | We introduce the notion of entanglement of subspaces as a measure that quantify the entanglement of bipartite states in a randomly selected subspace. We discuss its properties and in particular we show that for maximally entangled subspaces it is additive. Furthermore, we show that maximally entangled subspaces can pla... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We introduce the notion of entanglement of subspaces as a measure that quantify the entanglement of bipartite states in a randomly selected subspace. We discuss its properties and in particular we show that for maximally entangled subsp... | |
arxiv:0704.0258 | 0704.0258 | 1 | 10.1140/epjb/e2007-00296-x | 96b172a7d5bf4dfce0e35c8b9e85024619374d97db6d0892a0a15eee3ce76b46 | Correlation functions in the Non Perturbative Renormalization Group and
field expansion | The usual procedure of including a finite number of vertices in Non
Perturbative Renormalization Group equations in order to obtain $n$-point
correlation functions at finite momenta is analyzed. This is done by exploiting
a general method recently introduced which includes simultaneously all vertices
although approxima... | [
"Diego Guerra",
"Ramon Mendez-Galain and Nicolas Wschebor"
] | [
"hep-th",
"cond-mat.stat-mech"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | In nearly all fields in physics, there are systems having a large number of strongly correlated constituents. These cannot be treated with usual perturbative methods. Phase transitions and critical phenomena, disordered systems, strongly correlated electrons, quantum chromodynamics at large distances, are just a few ex... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In nearly all fields in physics, there are systems having a large number of strongly correlated constituents. These cannot be treated with usual perturbative methods. Phase transitions and critical phenomena, disordered systems... | |
arxiv:0704.0262 | 0704.0262 | 1 | 10.1088/1126-6708/2007/06/067 | 19cd9dce875df2ee7b54bc584bd0d87527c95d622d0c83b18bce087e2c3afc07 | Stringy Instantons at Orbifold Singularities | We study the effects produced by D-brane instantons on the holomorphic
quantities of a D-brane gauge theory at an orbifold singularity. These effects
are not limited to reproducing the well known contributions of the gauge theory
instantons but also generate extra terms in the superpotential or the
prepotential. On the... | [
"Riccardo Argurio",
"Matteo Bertolini",
"Gabriele Ferretti",
"Alberto Lerda\n and Christoffer Petersson"
] | [
"hep-th"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | Contents 1. Introduction 1 2. Preliminaries 3 3. The N = 1 Z 2 × Z 2 orbifold 7 3.1 Instanton sector 10 3.2 Recovery of the ADS superpotential 11 3.3 Absence of exotic contributions 14 3.4 Study of the back-reaction 16
4. The N = 1 Z 2 × Z 2 orientifold 17 4.1 Instanton sector 19 5. An N = 2 example: the Z 3 orientifo... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Contents 1. Introduction 1 2. Preliminaries 3 3. The N = 1 Z 2 × Z 2 orbifold 7 3.1 Instanton sector 10 3.2 Recovery of the ADS superpotential 11 3.3 Absence of exotic contributions 14 3.4 Study of the back-reaction 16\n\n4. The N ... | |
arxiv:0704.0264 | 0704.0264 | 1 | 0ab569bfe706c57e3dc4b25b50ee03583e2c1873f6ddb49382cb12928fa8323e | Gluon Radiation of an Expanding Color Skyrmion in the Quark-Gluon Plasma | The density of states and energy spectrum of the gluon radiation are
calculated for the color current of an expanding hydrodynamic skyrmion in the
quark gluon plasma with a semiclassical method. Results are compared with those
in literatures. | [
"Jian Dai"
] | [
"hep-ph",
"hep-th",
"nucl-th"
] | hep-ph | [] | 2007-04-02 | 2026-02-26 | 1 E-mail: jdai@sci.ccny.cuny.edu
In this letter, we address the issue of gluon radiation during the hydrodynamic stage in the evolution of the deconfined hot QCD matter or quark gluon plasma (QGP) [1] (for review see for example [2] ).
The medium induced gluon radiation has been thoroughly explored in the context of ... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "1 E-mail: jdai@sci.ccny.cuny.edu"
},
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "In this letter, we address the issue of gluon radiation during the hydrodynamic stage in the evolution o... | ||
arxiv:0704.0268 | 0704.0268 | 1 | 18e9df294e9c7831bbe67f56a5262e1041a2ed0cc8162203d75ad72592c837f1 | Automated Generation of Layout and Control for Quantum Circuits | We present a computer-aided design flow for quantum circuits, complete with
automatic layout and control logic extraction. To motivate automated layout for
quantum circuits, we investigate grid-based layouts and show a performance
variance of four times as we vary grid structure and initial qubit placement.
We then pro... | [
"Mark Whitney",
"Nemanja Isailovic",
"Yatish Patel",
"John Kubiatowicz"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-02 | 2026-02-26 | We present a computer-aided design flow for quantum circuits, complete with automatic layout and control logic extraction. To motivate automated layout for quantum circuits, we investigate grid-based layouts and show a performance variance of four times as we vary grid structure and initial qubit placement. We then pro... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We present a computer-aided design flow for quantum circuits, complete with automatic layout and control logic extraction. To motivate automated layout for quantum circuits, we investigate grid-based layouts and show a performance varia... | ||
arxiv:0704.0278 | 0704.0278 | 1 | 10.1088/0264-9381/24/13/009 | fed9bb61e28a64d27ff273594f1325a8e6ee8168c1801dc30de7af1494379996 | q-Deformed spin foam models of quantum gravity | We numerically study Barrett-Crane models of Riemannian quantum gravity. We
have extended the existing numerical techniques to handle q-deformed models and
arbitrary space-time triangulations. We present and interpret expectation
values of a few selected observables for each model, including a spin-spin
correlation fun... | [
"Igor Khavkine",
"J. Daniel Christensen"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-02 | 2026-02-26 | Spin foam models were first introduced as a space-time alternative to the spin network description of states in loop quantum gravity [3] . The most studied spin foam models are due to Barrett and Crane [8, 9] . A spin foam is a discretization of space-time where the fundamental degrees of freedom are the areas labellin... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Spin foam models were first introduced as a space-time alternative to the spin network description of states in loop quantum gravity [3] . The most studied spin foam models are due to Barrett and Crane [8, 9] . A spin foam is a di... | |
arxiv:0704.0284 | 0704.0284 | 1 | 10.1103/PhysRevD.75.124008 | 050e5d6a0fc0de02938b89b75e31009b008c3dde3207e3d2b4c1e1bcc6ef1b80 | Second Order Perturbative Calculation of Quasinormal Modes of
Schwarzschild Black Holes | We analytically calculate to second order the correction to the asymptotic
form of quasinormal frequencies of four dimensional Schwarzschild black holes
based on the monodromy analysis proposed by Motl and Neitzke. Our results are
in good agreement with those obtained from numerical calculation. | [
"Hsien-chung Kao"
] | [
"hep-th",
"gr-qc"
] | hep-th | [] | 2007-04-02 | 2026-02-26 | Quasinormal modes (QNMs) were originally observed in considering the scattering or emission of gravitational waves by Schwarzschild black holes [1] . It was found that a characteristic damped oscillation, which only depends on the black hole mass, dominated the time evolution in a certain period of time. Since then QNM... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Quasinormal modes (QNMs) were originally observed in considering the scattering or emission of gravitational waves by Schwarzschild black holes [1] . It was found that a characteristic damped oscillation, which only depends on the... | |
arxiv:0704.0296 | 0704.0296 | 1 | 10.1103/PhysRevD.75.104015 | ef23e0c2953d4682cf77d3e5947190e55072c520e3502c8369ae74ec26c34517 | Generalized Twistor Transform And Dualities, With A New Description of
Particles With Spin, Beyond Free and Massless | A generalized twistor transform for spinning particles in 3+1 dimensions is
constructed that beautifully unifies many types of spinning systems by mapping
them to the same twistor, thus predicting an infinite set of duality relations
among spinning systems with different Hamiltonians. Usual 1T-physics is not
equipped t... | [
"Itzhak Bars and Bora Orcal"
] | [
"hep-th"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | A generalized twistor transform for spinning particles in 3+1 dimensions is constructed that beautifully unifies many types of spinning systems by mapping them to the same twistor , thus predicting an infinite set of duality relations among spinning systems with different Hamiltonians. Usual 1T-physics is not equipped ... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "A generalized twistor transform for spinning particles in 3+1 dimensions is constructed that beautifully unifies many types of spinning systems by mapping them to the same twistor , thus predicting an infinite set of duality relations a... | |
arxiv:0704.0299 | 0704.0299 | 1 | 10.1103/PhysRevD.75.124022 | ea0edaa46fecaf2fdba24d6d9faf816a86beb0298436e6e342bffaadcce1f210 | Parametrized Post-Newtonian Expansion of Chern-Simons Gravity | We investigate the weak-field, post-Newtonian expansion to the solution of
the field equations in Chern-Simons gravity with a perfect fluid source. In
particular, we study the mapping of this solution to the parameterized
post-Newtonian formalism to 1 PN order in the metric. We find that the PPN
parameters of Chern-Sim... | [
"Stephon Alexander and Nicolas Yunes"
] | [
"hep-th",
"astro-ph",
"gr-qc"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | We investigate the weak-field, post-Newtonian expansion to the solution of the field equations in Chern-Simons gravity with a perfect fluid source. In particular, we study the mapping of this solution to the parameterized post-Newtonian formalism to 1 PN order in the metric. We find that the PPN parameters of Chern-Sim... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We investigate the weak-field, post-Newtonian expansion to the solution of the field equations in Chern-Simons gravity with a perfect fluid source. In particular, we study the mapping of this solution to the parameterized post-Newtonian... | |
arxiv:0704.0314 | 0704.0314 | 1 | 10.1103/PhysRevD.76.027502 | 1b38d11269562682f9f9bac6c878b017939e7d85962ba39ba8426a667a4df365 | Extra dimensions and Lorentz invariance violation | We consider effective model where photons interact with scalar field
corresponding to conformal excitations of the internal space (geometrical
moduli/gravexcitons). We demonstrate that this interaction results in a
modified dispersion relation for photons, and consequently, the photon group
velocity depends on the ener... | [
"Viktor Baukh",
"Alexander Zhuk",
"Tina Kahniashvili"
] | [
"hep-ph",
"astro-ph",
"hep-th"
] | hep-ph | [] | 2007-04-03 | 2026-02-26 | We consider effective model where photons interact with scalar field corresponding to conformal excitations of the internal space (geometrical moduli/gravexcitons). We demonstrate that this interaction results in a modified dispersion relation for photons, and consequently, the photon group velocity depends on the ener... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We consider effective model where photons interact with scalar field corresponding to conformal excitations of the internal space (geometrical moduli/gravexcitons). We demonstrate that this interaction results in a modified dispersion r... | |
arxiv:0704.0323 | 0704.0323 | 1 | 10.1088/1751-8113/41/15/155303 | a843256b0bab16e58310af032903a5199db900cbb46a890de67a9f4205ca8ae9 | General sequential quantum cloning | Some multipartite quantum states can be generated in a sequential manner
which may be implemented by various physical setups like microwave and optical
cavity QED, trapped ions, and quantum dots etc. We analyze the general N to M
qubits Universal Quantum Cloning Machine (UQCM) within a sequential generation
scheme. We ... | [
"Gui-Fang Dang",
"Heng Fan"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Some multipartite quantum states can be generated in a sequential manner which may be implemented by various physical setups like microwave and optical cavity QED, trapped ions, and quantum dots etc. We analyze the general N to M (N ≤ M ) qubits Universal Quantum Cloning Machine (UQCM) within a sequential generation sc... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Some multipartite quantum states can be generated in a sequential manner which may be implemented by various physical setups like microwave and optical cavity QED, trapped ions, and quantum dots etc. We analyze the general N to M (N ≤ M... | |
arxiv:0704.0332 | 0704.0332 | 1 | ab1b535c10971747aea9297bf8482333825895fbc83801d4337ff9487811ca54 | The effect of a fifth large-scale space-time dimension on the
conservation of energy in a four dimensional Universe | The effect of introducing a fifth large-scale space-time dimension to the
equations of orbital dynamics was analysed in an earlier paper by the authors.
The results showed good agreement with the observed flat rotation curves of
galaxies and the Pioneer Anomaly. This analysis did not require the
modification of Newtoni... | [
"M.B. Gerrard and T.J. Sumner"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | Conservation of energy with 5 dimensions 2 1. Introduction In an earlier paper [1] we introduced a fifth large-scale space-time dimension, r to Newton's Second Law, as applied to systems with angular velocity. The resulting analysis of the orbital motion of galaxies, which considered only the role of baryonic matter, i... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Conservation of energy with 5 dimensions 2 1. Introduction In an earlier paper [1] we introduced a fifth large-scale space-time dimension, r to Newton's Second Law, as applied to systems with angular velocity. The resulting analysi... | ||
arxiv:0704.0340 | 0704.0340 | 1 | cd1ccc64371eab104adafb8415e5f5fbdd051067c946e8a0113e4f759380cc2f | Phonon-mediated decay of an atom in a surface-induced potential | We study phonon-mediated transitions between translational levels of an atom
in a surface-induced potential. We present a general master equation governing
the dynamics of the translational states of the atom. In the framework of the
Debye model, we derive compact expressions for the rates for both upward and
downward ... | [
"Fam Le Kien",
"S. Dutta Gupta",
"K. Hakuta"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Over the past few years, tight confinement of cold atoms has drawn considerable attention. The interest in this area is motivated not only by the fundamental nature of the problem, but also by its potential applications in atom optics and quantum information. A method for microscopic trapping and guiding of individual ... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Over the past few years, tight confinement of cold atoms has drawn considerable attention. The interest in this area is motivated not only by the fundamental nature of the problem, but also by its potential applications in atom... | ||
arxiv:0704.0353 | 0704.0353 | 1 | 10.1103/PhysRevC.75.047303 | 28deb216dbe5c3af9126240a6e9997c54fd309608bb5238cec5b8dbbafdf9e2a | Spin and pseudospin symmetries and the equivalent spectra of
relativistic spin-1/2 and spin-0 particles | We show that the conditions which originate the spin and pseudospin
symmetries in the Dirac equation are the same that produce equivalent energy
spectra of relativistic spin-1/2 and spin-0 particles in the presence of vector
and scalar potentials. The conclusions do not depend on the particular shapes
of the potentials... | [
"P. Alberto",
"A. S. de Castro",
"M. Malheiro"
] | [
"nucl-th",
"quant-ph"
] | nucl-th | [] | 2007-04-03 | 2026-02-26 | We show that the conditions which originate the spin and pseudospin symmetries in the Dirac equation are the same that produce equivalent energy spectra of relativistic spin-1/2 and spin-0 particles in the presence of vector and scalar potentials. The conclusions do not depend on the particular shapes of the potentials... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We show that the conditions which originate the spin and pseudospin symmetries in the Dirac equation are the same that produce equivalent energy spectra of relativistic spin-1/2 and spin-0 particles in the presence of vector and scalar ... | |
arxiv:0704.0362 | 0704.0362 | 1 | 0d2f7b5e699452837dc706c96b5b46bce67719bb41f8c12fe46dfcc9004589f4 | The Arctic Circle Revisited | The problem of limit shapes in the six-vertex model with domain wall boundary
conditions is addressed by considering a specially tailored bulk correlation
function, the emptiness formation probability. A closed expression of this
correlation function is given, both in terms of certain determinant and
multiple integral,... | [
"F. Colomo and A.G. Pronko"
] | [
"math-ph",
"hep-th",
"math.MP"
] | math-ph | [] | 2007-04-03 | 2026-02-26 | The Arctic Circle has first appeared in the study of domino tilings of large Aztec diamonds [EKLP, JPS]. The name originates from the fact that in most configurations the dominoes are 'frozen' outside the circle inscribed into the diamond, while the interior of the circle is a disordered, or 'temperate', zone. Further ... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The Arctic Circle has first appeared in the study of domino tilings of large Aztec diamonds [EKLP, JPS]. The name originates from the fact that in most configurations the dominoes are 'frozen' outside the circle inscribed into the... | ||
arxiv:0704.0366 | 0704.0366 | 1 | 10.1103/PhysRevD.76.064033 | 5157bc79c20f18512af060924b37c7f790332765bba987a088039c64f82c758e | Generalized Nariai Solutions for Yang-type Monopoles | A detailed study of the geometries that emerge by a gravitating generalized
Yang monopole in even dimensions is carried out. In particular, those which
present black hole and cosmological horizons. This two-horizon system is
thermally unstable. The process of thermalization will drive both horizons to
coalesce. This li... | [
"Pablo Diaz",
"Antonio Segui"
] | [
"gr-qc",
"hep-th"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | 1 Introduction 2 2 The gravitational coupling. Some geometrical features 3 3 The horizon coalecence geometry 4 3.1 Case m = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Case m = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Conclusions 11 A Proof of the finite ... | [
{
"section_type": "OTHER",
"section_title": "Contents",
"text": "1 Introduction 2 2 The gravitational coupling. Some geometrical features 3 3 The horizon coalecence geometry 4 3.1 Case m = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Case m = 0 . . . . . . . . . . . . . . ... | |
arxiv:0704.0367 | 0704.0367 | 1 | d1ea2d5ebc8bc890820988399fa841935bb6bc94bd400b7160c0b0b2e3602f9e | Instanton representation of Plebanski gravity. Consistency of the
initital value constraints under time evolution | The instanton representation of Plebanski gravity provides as equations of
motion a Hodge self-duality condition and a set of `generalized' Maxwell's
equations, subject to gravitational degrees of freedom encoded in the initial
value constraints of general relativity. The main result of the present paper
will be to pro... | [
"Eyo Eyo Ita III"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | In [1] a new formulation of general relativity was presented, named the instanton representation of Plebanski gravity. The basic dynamical variables are an SO(3, C) gauge connection A a µ and a matrix Ψ ae taking its values in two copies of SO(3, C). 1 The consequences of the associated action I Inst were determined vi... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "In [1] a new formulation of general relativity was presented, named the instanton representation of Plebanski gravity. The basic dynamical variables are an SO(3, C) gauge connection A a µ and a matrix Ψ ae taking its values in two... | ||
arxiv:0704.0371 | 0704.0371 | 1 | 10.1007/s10714-007-0428-0 | dd1f8cc3e5015dd43a6b13bb28a43a7df9d6d3060cda78a0bc035ea5a9ff314c | Dark energy interacting with neutrinos and dark matter: a
phenomenological theory | A model for a flat homogeneous and isotropic Universe composed of dark
energy, dark matter, neutrinos, radiation and baryons is analyzed. The fields
of dark matter and neutrinos are supposed to interact with the dark energy. The
dark energy is considered to obey either the van der Waals or the Chaplygin
equations of st... | [
"G. M. Kremer"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | A model for a flat homogeneous and isotropic Universe composed of dark energy, dark matter, neutrinos, radiation and baryons is analyzed. The fields of dark matter and neutrinos are supposed to interact with the dark energy. The dark energy is considered to obey either the van der Waals or the Chaplygin equations of st... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "A model for a flat homogeneous and isotropic Universe composed of dark energy, dark matter, neutrinos, radiation and baryons is analyzed. The fields of dark matter and neutrinos are supposed to interact with the dark energy. The dark en... | |
arxiv:0704.0372 | 0704.0372 | 1 | 10.1088/1751-8113/40/11/013 | 0e304d6a4369de20e39f410749114d5070c26242ff3a56ae76d8c25256102174 | Levy-Lieb constrained-search formulation as a minimization of the
correlation functional | The constrained-search formulation of Levy and Lieb, which formally defines
the exact Hohenberg-Kohn functional for any N-representable electron density,
is here shown to be equivalent to the minimization of the correlation
functional with respect to the N-1 conditional probability density, where N is
number of electro... | [
"L.Delle Site"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | The constrained-search formulation of Levy and Lieb, which formally defines the exact Hohenberg-Kohn functional for any N -representable electron density, is here shown to be equivalent to the minimization of the correlation functional with respect to the N -1 conditional probability density, where N is number of elect... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The constrained-search formulation of Levy and Lieb, which formally defines the exact Hohenberg-Kohn functional for any N -representable electron density, is here shown to be equivalent to the minimization of the correlation functional ... | |
arxiv:0704.0373 | 0704.0373 | 1 | 73b49bc09e35b75ceb60fe75a9c900fcfb3c49ccef47c52d7651e63b64d3f27e | Reality of linear and angular momentum expectation values in bound
states | In quantum mechanics textbooks the momentum operator is defined in the
Cartesian coordinates and rarely the form of the momentum operator in spherical
polar coordinates is discussed. Consequently one always generalizes the
Cartesian prescription to other coordinates and falls in a trap. In this work
we introduce the di... | [
"Utpal Roy",
"Suranjana Ghosh",
"T. Shreecharan",
"Kaushik Bhattacharya"
] | [
"quant-ph",
"hep-th"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Quantum mechanics is a treasure house of peculiar and interesting things. Elementary textbooks of quantum mechanics [1, 2, 3] generally start with the postulates which are required to define the nature of the dynamical variables in the theory and their commutation relations. The choice of the dynamical variables is not... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Quantum mechanics is a treasure house of peculiar and interesting things. Elementary textbooks of quantum mechanics [1, 2, 3] generally start with the postulates which are required to define the nature of the dynamical variable... | ||
arxiv:0704.0376 | 0704.0376 | 1 | 10.1103/PhysRevA.76.012337 | 2b23c349f79ec23643ab7e201b461b290f8bcb9bf9938ef0f1cc65aefa49574b | Environmental noise reduction for holonomic quantum gates | We study the performance of holonomic quantum gates, driven by lasers, under
the effect of a dissipative environment modeled as a thermal bath of
oscillators. We show how to enhance the performance of the gates by suitable
choice of the loop in the manifold of the controllable parameters of the laser.
For a simplified,... | [
"Daniele Parodi",
"Maura Sassetti",
"Paolo Solinas",
"Nino Zangh\\`i"
] | [
"quant-ph",
"cond-mat.mes-hall"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | The major challenge for quantum computation is posed by the fact that generically quantum states are very delicate objects quite difficult to control with the required accuracy-typically, by means of external driving fields, e.g., a laser. The interaction with the many degrees of freedom of the environment causes decoh... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The major challenge for quantum computation is posed by the fact that generically quantum states are very delicate objects quite difficult to control with the required accuracy-typically, by means of external driving fields, e.... | |
arxiv:0704.0377 | 0704.0377 | 1 | 10.1088/0954-3899/36/4/045001 | c9b267f312751f7fb7b5e1c66f7f4a0de81e94dad51fcf4ab1923542757529fc | The lifetime of unstable particles in electromagnetic fields | We show that the electromagnetic moments of unstable particles (resonances)
have an absorptive contribution which quantifies the change of the particle's
lifetime in an external electromagnetic field. To give an example we compute
here the imaginary part of the magnetic moment for the cases of the muon and
the neutron ... | [
"Daniele Binosi (ECT*",
"Trento)",
"Vladimir Pascalutsa (ECT*",
"Trento &\n Mainz U.)"
] | [
"hep-ph",
"hep-th",
"nucl-th"
] | hep-ph | [] | 2007-04-03 | 2026-02-26 | The electromagnetic (e.m.) moments of a particle are among the few fundamental quantities which describe the particle properties and as such have thoroughly been studied. The most renowned examples are the magnetic moments of the electron and the muon which have been measured to unprecendented accuracy and yielded a nu... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The electromagnetic (e.m.) moments of a particle are among the few fundamental quantities which describe the particle properties and as such have thoroughly been studied. The most renowned examples are the magnetic moments of t... | |
arxiv:0704.0386 | 0704.0386 | 1 | 10.1103/PhysRevLett.99.150401 | 4cdf211d453d201045c8a8892026ddb0e2d51231bc554fbbf685f8b762731c33 | Quantum non-local effects with Bose-Einstein condensates | We study theoretically the properties of two Bose-Einstein condensates in
different spin states, represented by a double Fock state. Individual
measurements of the spins of the particles are performed in transverse
directions, giving access to the relative phase of the condensates. Initially,
this phase is completely u... | [
"Franck Lalo\\\"e (LKB - Lhomond)",
"William J. Mullin (UMASS)"
] | [
"quant-ph",
"cond-mat.other"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | We study theoretically the properties of two Bose-Einstein condensates in different spin states, represented by a double Fock state. Individual measurements of the spins of the particles are performed in transverse directions, giving access to the relative phase of the condensates. Initially, this phase is completely u... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We study theoretically the properties of two Bose-Einstein condensates in different spin states, represented by a double Fock state. Individual measurements of the spins of the particles are performed in transverse directions, giving ac... | |
arxiv:0704.0389 | 0704.0389 | 1 | 10.1103/PhysRevD.75.124007 10.1103/PhysRevD.82.029901
10.1103/PhysRevD.82.129903 | 8bbc0a5c277ac9087fae73f2351273d64758f1d12c30292eb64f6a3c3c9216cc | Evolution of the Carter constant for inspirals into a black hole: effect
of the black hole quadrupole | We analyze the effect of gravitational radiation reaction on generic orbits
around a body with an axisymmetric mass quadrupole moment Q to linear order in
Q, to the leading post-Newtonian order, and to linear order in the mass ratio.
This system admits three constants of the motion in absence of radiation
reaction: ene... | [
"Eanna E. Flanagan and Tanja Hinderer"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | We analyze the effect of gravitational radiation reaction on generic orbits around a body with an axisymmetric mass quadrupole moment Q to linear order in Q, to the leading post-Newtonian order, and to linear order in the mass ratio. This system admits three constants of the motion in absence of radiation reaction: ene... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We analyze the effect of gravitational radiation reaction on generic orbits around a body with an axisymmetric mass quadrupole moment Q to linear order in Q, to the leading post-Newtonian order, and to linear order in the mass ratio. Th... | |
arxiv:0704.0397 | 0704.0397 | 1 | 10.1103/PhysRevA.75.063803 | ee238a6a21e36cb923522ce06c9ab6af9261e7ac7d5d8cbd300f92fb188bc89f | Conditional generation of path-entangled optical NOON states | We propose a measurement protocol to generate path-entangled NOON states
conditionally from two pulsed type II optical parametric oscillators. We
calculate the fidelity of the produced states and the success probability of
the protocol. The trigger detectors are assumed to have finite dead time, and
for short pulse tri... | [
"Anne E. B. Nielsen and Klaus Molmer"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Nonclassical states of light have many applications, and a number of different protocols exist for the generation of various classes of states. The two-mode maximally entangled N -photon states
|NOON = 1 √ 2 |N, 0 + e iφ |0, N , ( 1
)
the so-called NOON states, are particularly interesting because a single-photon ph... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Nonclassical states of light have many applications, and a number of different protocols exist for the generation of various classes of states. The two-mode maximally entangled N -photon states\n\n|NOON = 1 √ 2 |N, 0 + e iφ |0,... | |
arxiv:0704.0399 | 0704.0399 | 1 | 10.1016/j.physletb.2007.05.052 | 4afd0ebd0871067862405df8d67c23ae9c96c7ba7cb0d4a02d6c7877a8d4ae34 | Hawking radiation of linear dilaton black holes | We compute exactly the semi-classical radiation spectrum for a class of
non-asymptotically flat charged dilaton black holes, the so-called linear
dilaton black holes. In the high frequency regime, the temperature for these
black holes generically agrees with the surface gravity result. In the special
case where the bla... | [
"G. Clement",
"J.C. Fabris and G.T. Marques"
] | [
"gr-qc",
"hep-th"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | We compute exactly the semi-classical radiation spectrum for a class of non-asymptotically flat charged dilaton black holes, the socalled linear dilaton black holes. In the high frequency regime, the temperature for these black holes generically agrees with the surface gravity result. In the special case where the blac... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We compute exactly the semi-classical radiation spectrum for a class of non-asymptotically flat charged dilaton black holes, the socalled linear dilaton black holes. In the high frequency regime, the temperature for these black holes ge... | |
arxiv:0704.0400 | 0704.0400 | 1 | 78c2458a58debec64016d66dde5a09da5009f228daa68bafc495157d156633b4 | The S-Matrix of AdS/CFT and Yangian Symmetry | We review the algebraic construction of the S-matrix of AdS/CFT. We also
present its symmetry algebra which turns out to be a Yangian of the centrally
extended su(2|2) superalgebra. | [
"Niklas Beisert"
] | [
"nlin.SI",
"cond-mat.stat-mech",
"hep-th"
] | nlin.SI | [] | 2007-04-03 | 2026-02-26 | Bethe's ansatz [1] for solving a one-dimensional integrable model was and remains a powerful tool in contemporary theoretical physics: 75 years ago it solved one of the first models of quantum mechanics, the Heisenberg spin chain [2] ; today it provides exact solutions for the spectra of certain gauge and string theori... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction and Overview",
"text": "Bethe's ansatz [1] for solving a one-dimensional integrable model was and remains a powerful tool in contemporary theoretical physics: 75 years ago it solved one of the first models of quantum mechanics, the Heisenber... | ||
arxiv:0704.0403 | 0704.0403 | 1 | 10.1038/nphoton.2007.46 | 3379021d8f8fb881332ac988f54b65201d8a992a55c2cb7511ddb58b81618ee8 | Review: Semiconductor Quantum Light Sources | Lasers and LEDs display a statistical distribution in the number of photons
emitted in a given time interval. New applications exploiting the quantum
properties of light require sources for which either individual photons, or
pairs, are generated in a regulated stream. Here we review recent research on
single-photon so... | [
"Andrew J Shields"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Applying quantum light states to photonic applications allows functionalities that are not possible using 'ordinary' classical light. For example, carrying information with single-photons provides a means to test the secrecy of optical communications, which could soon be applied to the problem of sharing digital crypto... | [
{
"section_type": "OTHER",
"section_title": "Applications of Quantum Photonics",
"text": "Applying quantum light states to photonic applications allows functionalities that are not possible using 'ordinary' classical light. For example, carrying information with single-photons provides a means to test t... | |
arxiv:0704.0409 | 0704.0409 | 1 | 10.1103/PhysRevA.76.032114 | 1272817a2a40683f12f237ad18c903107c8e6e73ef9bf6f87210091b7dfac177 | On the over-barrier reflection in quantum mechanics with multiple
degrees of freedom | We present an analytic example of two dimensional quantum mechanical system,
where the exponential suppression of the probability of over-barrier reflection
changes non-monotonically with energy. The suppression is minimal at certain
"optimal" energies where reflection occurs with exponentially larger
probability than ... | [
"D.G. Levkov",
"A.G. Panin",
"S.M. Sibiryakov"
] | [
"quant-ph",
"hep-th",
"nlin.CD",
"physics.atom-ph",
"physics.chem-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Tunneling and over-barrier reflection are the characteristic non-perturbative phenomena in quantum mechanics. They typically occur with exponentially small probabilities,
P ∝ e -F/ , ( 1
)
where F is the suppression exponent; still, the above phenomena are indispensable in understanding a wide variety of physical si... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Tunneling and over-barrier reflection are the characteristic non-perturbative phenomena in quantum mechanics. They typically occur with exponentially small probabilities,\n\nP ∝ e -F/ , ( 1\n\n)\n\nwhere F is the suppression expon... | |
arxiv:0704.0418 | 0704.0418 | 1 | 10.1103/PhysRevLett.99.147202 | 4841c2d087beb058c39c1425bfc13ff2ca5d9f6c1afcf6090f8283292d79ce39 | Entanglement entropy at infinite randomness fixed points in higher
dimensions | The entanglement entropy of the two-dimensional random transverse Ising model
is studied with a numerical implementation of the strong disorder
renormalization group. The asymptotic behavior of the entropy per surface area
diverges at, and only at, the quantum phase transition that is governed by an
infinite randomness... | [
"Yu-Cheng Lin",
"Ferenc Igloi and Heiko Rieger"
] | [
"cond-mat.dis-nn",
"quant-ph"
] | cond-mat.dis-nn | [] | 2007-04-03 | 2026-02-26 | The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, and only at, the quantum phase transition that is governed by an infinite randomness... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, ... | |
arxiv:0704.0420 | 0704.0420 | 1 | bdacf2eacc8ee140d5826823b0c6aa04e4cbe11cafca16852f4ed47d78546a24 | The Hourglass - Consequences of Pure Hamiltonian Evolution of a
Radiating System | Hourglass is the name given here to a formal isolated quantum system that can
radiate. Starting from a time when it defines the system it represents clearly
and no radiation is present, it is given straightforward Hamiltonian evolution.
The question of what significance hourglasses have is raised, and this question
is ... | [
"Donald McCartor"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | Hourglass is the name given here to a formal isolated quantum system that can radiate. Starting from a time when it defines the system it represents clearly and no radiation is present, it is given straightforward Hamiltonian evolution. The question of what significance hourglasses have is raised, and this question is ... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Hourglass is the name given here to a formal isolated quantum system that can radiate. Starting from a time when it defines the system it represents clearly and no radiation is present, it is given straightforward Hamiltonian evolution.... | ||
arxiv:0704.0425 | 0704.0425 | 1 | ef888e2c75769b2020de3d3b4cdef51d358f0413bb8eb3c4e2e378f7e6ec387c | QED for fields obeying a square root operator equation | Instead of using local field equations - like the Dirac equation for spin-1/2
and the Klein-Gordon equation for spin-0 particles - one could try to use
non-local field equations in order to describe scattering processes. The latter
equations can be obtained by means of the relativistic energy together with the
correspo... | [
"Tobias Gleim"
] | [
"hep-th"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | Instead of using local field equations -like the Dirac equation for spin-1/2 and the Klein-Gordon equation for spin-0 particles -one could try to use non-local field equations in order to describe scattering processes. The latter equations can be obtained by means of the relativistic energy together with the correspond... | [
{
"section_type": "OTHER",
"section_title": "Tobias Gleim",
"text": "Instead of using local field equations -like the Dirac equation for spin-1/2 and the Klein-Gordon equation for spin-0 particles -one could try to use non-local field equations in order to describe scattering processes. The latter equat... | ||
arxiv:0704.0431 | 0704.0431 | 1 | 10.1142/9789812834300_0199 | 4eb59b844cbe83ff4aab47b61143f1b80337823676b9269d898b997200243f54 | Fragmentation of general relativistic quasi-toroidal polytropes | We investigate the role of rotational instabilities in the context of black
hole formation in relativistic stars. In addition to the standard scenario - an
axially symmetric dynamical instability forming a horizon at the star's center
- the recently found low-$T/|W|$ instabilities are shown to lead to
fragmentation and... | [
"Burkhard Zink",
"Nikolaos Stergioulas",
"Ian Hawke",
"Christian D. Ott",
"Erik\n Schnetter",
"Ewald Mueller"
] | [
"gr-qc",
"astro-ph.HE"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | How do black holes form from relativistic stars? This question is of great fundamental and practical importance in gravitational physics and general relativistic astrophysics. On the fundamental level, black holes are genuinely relativistic objects, and thus the study of their production involves questions about horizo... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "How do black holes form from relativistic stars? This question is of great fundamental and practical importance in gravitational physics and general relativistic astrophysics. On the fundamental level, black holes are genuinely rel... | |
arxiv:0704.0435 | 0704.0435 | 1 | 10.1088/0264-9381/24/17/009 | 89e4dcf26ccedb9a759319a45998444094d94075b23c34890556082636a15cdf | Type D Einstein spacetimes in higher dimensions | We show that all static spacetimes in higher dimensions are of Weyl types G,
I_i, D or O. This applies also to stationary spacetimes if additional
conditions are fulfilled, as for most known black hole/ring solutions. (The
conclusions change when the Killing generator becomes null, such as at Killing
horizons.) Next we... | [
"Vojtech Pravda",
"Alena Pravdova",
"Marcello Ortaggio"
] | [
"gr-qc",
"hep-th"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | We show that all static spacetimes in higher dimensions n > 4 are necessarily of Weyl types G, I i , D or O. This applies also to stationary spacetimes provided additional conditions are fulfilled, as for most known black hole/ring solutions. (The conclusions change when the Killing generator becomes null, such as at K... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We show that all static spacetimes in higher dimensions n > 4 are necessarily of Weyl types G, I i , D or O. This applies also to stationary spacetimes provided additional conditions are fulfilled, as for most known black hole/ring solu... | |
arxiv:0704.0440 | 0704.0440 | 1 | 10.1103/PhysRevLett.99.130402 | 242e9f1047522986c26fd027b61b1fd1b1be0f7092c6f585d9fa2765a3a83eea | Dynamics of a quantum phase transition in a ferromagnetic Bose-Einstein
condensate | We discuss dynamics of a slow quantum phase transition in a spin-1
Bose-Einstein condensate. We determine analytically the scaling properties of
the system magnetization and verify them with numerical simulations in a one
dimensional model. | [
"Bogdan Damski and Wojciech H. Zurek"
] | [
"cond-mat.other",
"hep-th",
"quant-ph"
] | cond-mat.other | [] | 2007-04-03 | 2026-02-26 | We discuss dynamics of a slow quantum phase transition in a spin-1 Bose-Einstein condensate. We determine analytically the scaling properties of the system magnetization and verify them with numerical simulations in a one dimensional model.
Studies of phase transitions have traditionally focused on equilibrium scaling... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We discuss dynamics of a slow quantum phase transition in a spin-1 Bose-Einstein condensate. We determine analytically the scaling properties of the system magnetization and verify them with numerical simulations in a one dimensional mo... | |
arxiv:0704.0442 | 0704.0442 | 1 | 10.1364/JOSAB.24.002195 | 552e0178ea613901c96919ff19108b49213ee274a88c9ee7976deec1049e13b7 | Quantum electromagnetic X-waves | We show that two distinct quantum states of the electromagnetic field can be
associated to a classical vector X wave or a propagation-invariant solution of
Maxwell equations. The difference between the two states is of pure quantum
mechanical origin since they are internally entangled and disentangled,
respectively and... | [
"Alessandro Ciattoni",
"Claudio Conti"
] | [
"physics.optics",
"physics.gen-ph",
"quant-ph"
] | physics.optics | [] | 2007-04-03 | 2026-02-26 | We show that two distinct quantum states of the electromagnetic field can be associated to a classical vector X wave or a propagation-invariant solution of Maxwell equations. The difference between the two states is of pure quantum mechanical origin since they are internally entangled and disentangled, respectively and... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We show that two distinct quantum states of the electromagnetic field can be associated to a classical vector X wave or a propagation-invariant solution of Maxwell equations. The difference between the two states is of pure quantum mech... | |
arxiv:0704.0444 | 0704.0444 | 1 | 10.1103/PhysRevD.79.046005 | d1636fe8181ec4bdd12a40cdd36e0aa6bed81159e7b6428fc691bd17a1bc8fa3 | Multiple Unfoldings of Orbifold Singularities: Engineering Geometric
Analogies to Unification | Katz and Vafa showed how charged matter can arise geometrically by the
deformation of ADE-type orbifold singularities in type IIa, M-theory, and
F-theory compactifications. In this paper we use those same basic ingredients,
used there to geometrically engineer specific matter representations, here to
deform the compact... | [
"Jacob L. Bourjaily"
] | [
"hep-th"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | One of the ways in which a gauge theory with massless charged matter can arise in type IIa, Mtheory, and F-theory is known as geometrical engineering. In this framework, gauge theory at low energy arises from co-dimension four singular surfaces in the compactification manifold [2] and charged matter arises as isolated ... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "One of the ways in which a gauge theory with massless charged matter can arise in type IIa, Mtheory, and F-theory is known as geometrical engineering. In this framework, gauge theory at low energy arises from co-dimension four ... | |
arxiv:0704.0445 | 0704.0445 | 1 | 10.1103/PhysRevD.76.046004 | 4da8123da9f581af73bb6ab2b9302999a3c6bb3aaa17b17d9163f3be237d2596 | Geometrically Engineering the Standard Model: Locally Unfolding Three
Families out of E8 | This paper extends and builds upon the results of an earlier paper, in which
we described how to use the tools of geometrical engineering to deform
geometrically-engineered grand unified models into ones with lower symmetry.
This top-down unfolding has the advantage that the relative positions of
singularities giving r... | [
"Jacob L. Bourjaily"
] | [
"hep-th"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | In [2] , Katz and Vafa showed how to geometrically engineer matter representations in terms of the local singularity structure of type IIa, M-theory, and Ftheory compactifications. In that framework matter and gauge theory both have purely geometrical origins: SU n , SO 2n and E n gauge theories arise from the existenc... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In [2] , Katz and Vafa showed how to geometrically engineer matter representations in terms of the local singularity structure of type IIa, M-theory, and Ftheory compactifications. In that framework matter and gauge theory both... | |
arxiv:0704.0449 | 0704.0449 | 1 | 10.1088/1126-6708/2007/10/023 | 58e5b0bbc995ed67138e9dcf8a2726c11c1bccc96b640d73f50fa59508d16626 | Worldsheet Instantons and Torsion Curves, Part B: Mirror Symmetry | We apply mirror symmetry to the problem of counting holomorphic rational
curves in a Calabi-Yau threefold X with Z_3 x Z_3 Wilson lines. As we found in
Part A [hep-th/0703182], the integral homology group H_2(X,Z)=Z^3 + Z_3 + Z_3
contains torsion curves. Using the B-model on the mirror of X as well as its
covering spac... | [
"Volker Braun",
"Maximilian Kreuzer",
"Burt A. Ovrut",
"Emanuel Scheidegger"
] | [
"hep-th"
] | hep-th | [] | 2007-04-03 | 2026-02-26 | We apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z 3 ⊕ Z 3 Wilson lines. As we found in Part A [1], the integral homology group H 2 (X, Z) = Z 3 ⊕ Z 3 ⊕ Z 3 contains torsion curves. Using the B-model on the mirror of X as well as its covering spaces, we co... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z 3 ⊕ Z 3 Wilson lines. As we found in Part A [1], the integral homology group H 2 (X, Z) = Z 3 ⊕ Z 3 ⊕ Z 3 contains torsio... | |
arxiv:0704.0456 | 0704.0456 | 1 | 10.1088/1751-8113/40/25/S37 | 515eb130c1ea7f3255b1b6a9f4dae76d33c4ab7fd315a3c6938003a8c23328da | Particle propagation in cosmological backgrounds | We study the quantum propagation of particles in cosmological backgrounds, by
considering a doublet of massive scalar fields propagating in an expanding
universe, possibly filled with radiation. We focus on the dissipative effects
related to the expansion rate. At first order, we recover the expected result
that the de... | [
"Daniel Arteaga"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-03 | 2026-02-26 | We study the quantum propagation of particles in cosmological backgrounds, by considering a doublet of massive scalar fields propagating in an expanding universe, possibly filled with radiation. We focus on the dissipative effects related to the expansion rate. At first order, we recover the expected result that the de... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We study the quantum propagation of particles in cosmological backgrounds, by considering a doublet of massive scalar fields propagating in an expanding universe, possibly filled with radiation. We focus on the dissipative effects relat... | |
arxiv:0704.0457 | 0704.0457 | 1 | 10.1103/PhysRevB.76.014511 | 4e8ebebd01ba06ac30445d5b77bde13649dba9ea0e7dbcd07646f21bc4448016 | Quantum analysis of a linear DC SQUID mechanical displacement detector | We provide a quantum analysis of a DC SQUID mechanical displacement detector
within the sub-critical Josephson current regime. A segment of the SQUID loop
forms the mechanical resonator and motion of the latter is transduced
inductively through changes in the flux threading the loop. Expressions are
derived for the det... | [
"M. P. Blencowe",
"E. Buks"
] | [
"cond-mat.supr-con",
"cond-mat.mes-hall",
"quant-ph"
] | cond-mat.supr-con | [] | 2007-04-03 | 2026-02-26 | In a series of recent experiments 1, 2, 3 and related theoretical work, 4, 5, 6, 7, 8, 9 it was demonstrated that a displacement detector based on either a normal or superconducting single electronic transistor (SSET) can resolve the motion of a micron-scale mechanical resonator close to the quantum limit as set by Hei... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In a series of recent experiments 1, 2, 3 and related theoretical work, 4, 5, 6, 7, 8, 9 it was demonstrated that a displacement detector based on either a normal or superconducting single electronic transistor (SSET) can resol... | |
arxiv:0704.0461 | 0704.0461 | 1 | 10.1103/PhysRevA.76.022102 | 2030e896fe190178eff537073e190c2174761dc1a1cc7d97d92c6f814b29b5e0 | Entanglement increase from local interactions with
not-completely-positive maps | Simple examples are constructed that show the entanglement of two qubits
being both increased and decreased by interactions on just one of them. One of
the two qubits interacts with a third qubit, a control, that is never entangled
or correlated with either of the two entangled qubits and is never entangled,
but become... | [
"Thomas F. Jordan",
"Anil Shaji and E. C. G. Sudarshan"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-03 | 2026-02-26 | We construct simple examples here that show the entanglement of two qubits being both increased and decreased by interactions on just one of them. In our first and basic step, taken in Sec. II, we have one of the two qubits interact with a third qubit, a control, that is never entangled or correlated with either of the... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "We construct simple examples here that show the entanglement of two qubits being both increased and decreased by interactions on just one of them. In our first and basic step, taken in Sec. II, we have one of the two qubits int... | |
arxiv:0704.0476 | 0704.0476 | 1 | 10.1103/PhysRevA.76.033615 | 491c6baa121b1c5f43f439a50b386e310d14dba84727c26df92a3b0724362aaf | Geometric phase of an atom inside an adiabatic radio frequency potential | We investigate the geometric phase of an atom inside an adiabatic radio
frequency (rf) potential created from a static magnetic field (B-field) and a
time dependent rf field. The spatial motion of the atomic center of mass is
shown to give rise to a geometric phase, or Berry's phase, to the adiabatically
evolving atomi... | [
"P. Zhang and L. You"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | Magnetic trapping is an important enabling technology for the active research field of neutral atomic quantum gases. A variety of trap potentials can be developed using magnetic (B-) fields with different spatial distributions and time variations. For instance, the widely used quadrupole trap and the Ioffe-Pritchard tr... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Magnetic trapping is an important enabling technology for the active research field of neutral atomic quantum gases. A variety of trap potentials can be developed using magnetic (B-) fields with different spatial distributions ... | |
arxiv:0704.0482 | 0704.0482 | 1 | 10.1103/PhysRevA.76.062311 | 4caec5185f281f351422ba4711e16fc0b298b4789732d3ff316689474088ac07 | Implementation of holonomic quantum computation through engineering and
manipulating environment | We consider an atom-field coupled system, in which two pairs of four-level
atoms are respectively driven by laser fields and trapped in two distant
cavities that are connected by an optical fiber. First, we show that an
effective squeezing reservoir can be engineered under appropriate conditions.
Then, we show that a t... | [
"Zhang-qi Yin",
"Fu-li Li",
"Peng Peng"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | Quantum computation, attracting much current interest since Shor's algorithm [1] was proposed, depends on two key factors: quantum entanglement and precision control of quantum systems. Unfortunately, quantum systems are inevitably coupled to their environment so that entanglement is too fragile to be retained. This ma... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Quantum computation, attracting much current interest since Shor's algorithm [1] was proposed, depends on two key factors: quantum entanglement and precision control of quantum systems. Unfortunately, quantum systems are inevit... | |
arxiv:0704.0488 | 0704.0488 | 1 | 10.1007/s10714-007-0431-5 | 5164ad34ca4de631f8a4c37f1aba38127c537f07d91eec4dfc4322512c2ad2cd | Teleparallel Version of the Stationary Axisymmetric Solutions and their
Energy Contents | This work contains the teleparallel version of the stationary axisymmetric
solutions. We obtain the tetrad and the torsion fields representing these
solutions. The tensor, vector and axial-vector parts of the torsion tensor are
evaluated. It is found that the axial-vector has component only along $\rho$
and $z$ directi... | [
"M. Sharif and M. Jamil Amir"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-04 | 2026-02-26 | This work contains the teleparallel version of the stationary axisymmetric solutions. We obtain the tetrad and the torsion fields representing these solutions. The tensor, vector and axial-vector parts of the torsion tensor are evaluated. It is found that the axial-vector has component only along ρ and z directions. Th... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "This work contains the teleparallel version of the stationary axisymmetric solutions. We obtain the tetrad and the torsion fields representing these solutions. The tensor, vector and axial-vector parts of the torsion tensor are evaluate... | |
arxiv:0704.0489 | 0704.0489 | 1 | 10.2478/s11534-008-0018-0 | 05a07c71e5ae114221137d0f5aa9063589e159ec0dabb4c42451345ca0534803 | Relativistic treatment in D-dimensions to a spin-zero particle with
noncentral equal scalar and vector ring-shaped Kratzer potential | The Klein-Gordon equation in D-dimensions for a recently proposed Kratzer
potential plus ring-shaped potential is solved analytically by means of the
conventional Nikiforov-Uvarov method. The exact energy bound-states and the
corresponding wave functions of the Klein-Gordon are obtained in the presence
of the noncentra... | [
"Sameer M. Ikhdair",
"Ramazan Sever"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | In various physical applications including those in nuclear physics and high energy physics [1, 2] , one of the interesting problems is to obtain exact solutions of the relativistic equations like Klein-Gordon and Dirac equations for mixed vector and scalar potential. The Klein-Gordon and Dirac wave equations are frequ... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In various physical applications including those in nuclear physics and high energy physics [1, 2] , one of the interesting problems is to obtain exact solutions of the relativistic equations like Klein-Gordon and Dirac equatio... | |
arxiv:0704.0490 | 0704.0490 | 1 | 10.1103/PhysRevD.76.111701 | 33a1d1b81936bb4d490d360820f2123e47c0f5994f09751e7382ad683a799b76 | Long Distance Signaling Using Axion-like Particles | The possible existence of axion-like particles could lead to a new type of
long distance communication. In this work, basic antenna concepts are defined
and a Friis-like equation is derived to facilitate long-distance link
calculations. An example calculation is presented showing that communication
over distances of 10... | [
"Daniel D. Stancil"
] | [
"hep-ph",
"astro-ph",
"hep-th"
] | hep-ph | [] | 2007-04-04 | 2026-02-26 | The possible existence of axion-like particles could lead to a new type of long distance communication. In this work, basic antenna concepts are defined and a Friis-like equation is derived to facilitate long-distance link calculations. An example calculation is presented showing that communication over distances of 10... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The possible existence of axion-like particles could lead to a new type of long distance communication. In this work, basic antenna concepts are defined and a Friis-like equation is derived to facilitate long-distance link calculations.... | |
arxiv:0704.0495 | 0704.0495 | 1 | 10.3842/SIGMA.2007.075 | 1901032a0ef23b90804048b2aef0d82a6e8e98bc6f56ec72e49cd08936aac7b0 | The Veldkamp Space of Two-Qubits | Given a remarkable representation of the generalized Pauli operators of
two-qubits in terms of the points of the generalized quadrangle of order two,
W(2), it is shown that specific subsets of these operators can also be
associated with the points and lines of the four-dimensional projective space
over the Galois field... | [
"Metod Saniga (ASTRINSTSAV)",
"Michel Planat (FEMTO-ST)",
"Petr Pracna\n (JH-Inst)",
"Hans Havlicek (TUW)"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | A deeper understanding of the structure of Hilbert spaces of finite dimensions is of utmost importance for quantum information theory. Recently, we made an important step in this respect by demonstrating that the commutation algebra of the generalized Pauli operators on the 2 N -dimensional Hilbert spaces is embodied i... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "A deeper understanding of the structure of Hilbert spaces of finite dimensions is of utmost importance for quantum information theory. Recently, we made an important step in this respect by demonstrating that the commutation algeb... | |
arxiv:0704.0505 | 0704.0505 | 1 | 10.1103/PhysRevD.76.085001 | aeda426d1071e75e895afdc97caa1fa9e76af03cbfa583d2180a72529e4ba2a9 | Exact Solutions of Einstein-Yang-Mills Theory with Higher-Derivative
Coupling | We construct a classical solution of an Einstein-Yang-Mills system with a
fourth order term with respect to the field strength of the Yang-Mills field.
The solution provides a spontaneous compactification proposed by Cremmer and
Scherk; ten-dimensional space-time with a cosmological constant is compactified
to the four... | [
"Hironobu Kihara",
"Muneto Nitta"
] | [
"hep-th",
"gr-qc"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | We construct a classical solution of an Einstein-Yang-Mills system with a fourth order term with respect to the field strength of the Yang-Mills field. The solution provides a compactification proposed by Cremmer and Scherk; ten-dimensional space-time with a cosmological constant is compactified to the four-dimensional... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We construct a classical solution of an Einstein-Yang-Mills system with a fourth order term with respect to the field strength of the Yang-Mills field. The solution provides a compactification proposed by Cremmer and Scherk; ten-dimensi... | |
arxiv:0704.0507 | 0704.0507 | 1 | 10.1103/PhysRevD.76.124023 | 22f44bb72e223a65c9213c7d3684c85f57dc3b9bd321cac32466088966d0a47e | E_6 and the bipartite entanglement of three qutrits | Recent investigations have established an analogy between the entropy of
four-dimensional supersymmetric black holes in string theory and entanglement
in quantum information theory. Examples include: (1) N=2 STU black holes and
the tripartite entanglement of three qubits (2-state systems), where the common
symmetry is ... | [
"M. J. Duff and S. Ferrara"
] | [
"hep-th",
"quant-ph"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | 1 m.duff@imperial.ac.uk 2 Sergio.Ferrara@cern.ch Contents 1 D = 4 black holes and qubits 1.1 N = 2 black holes and the tripartite entanglement of three qubits . . . . . . 1.2 N = 2 black holes and the bipartite entanglement of two qubits . . . . . . . 1.3 N = 8 black holes and the tripartite entanglement of seven qubit... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "1 m.duff@imperial.ac.uk 2 Sergio.Ferrara@cern.ch Contents 1 D = 4 black holes and qubits 1.1 N = 2 black holes and the tripartite entanglement of three qubits . . . . . . 1.2 N = 2 black holes and the bipartite entanglement of two ... | |
arxiv:0704.0511 | 0704.0511 | 1 | 10.1007/s10946-007-0032-5 | be892cb8194712d11f25f536c9a91df2f945e518862e27506bb42e0662579c98 | A unified approach to SIC-POVMs and MUBs | A unified approach to (symmetric informationally complete) positive operator
valued measures and mutually unbiased bases is developed in this article. The
approach is based on the use of operator equivalents expanded in the enveloping
algebra of SU(2). Emphasis is put on similarities and differences between
SIC-POVMs a... | [
"O. Albouy (IPNL)",
"M. R. Kibler (IPNL)"
] | [
"quant-ph",
"math-ph",
"math.MP"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | The importance of finite-dimensional spaces for quantum mechanics is well recognized (see for instance [1]-[3]). In particular, such spaces play a major role in quantum information theory, especially for quantum cryptography and quantum state tomography [4]- [27] .
Along this vein, a symmetric informationally complete... | [
{
"section_type": "BACKGROUND",
"section_title": "INTRODUCTION",
"text": "The importance of finite-dimensional spaces for quantum mechanics is well recognized (see for instance [1]-[3]). In particular, such spaces play a major role in quantum information theory, especially for quantum cryptography and q... | |
arxiv:0704.0516 | 0704.0516 | 1 | 4a119e65b80170869ee3687296117b16f01ab82d1c97a18278fec0312c5f9da7 | Effects of Imperfect Gate Operations in Shor's Prime Factorization
Algorithm | The effects of imperfect gate operations in implementation of Shor's prime
factorization algorithm are investigated. The gate imperfections may be
classified into three categories: the systematic error, the random error, and
the one with combined errors. It is found that Shor's algorithm is robust
against the systemati... | [
"Hao Guo",
"Gui Lu Long and Yang Sun"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | Shor's factorization algorithm [1] is a very important quantum algorithm, through which one has demonstrated the power of quantum computers. It has greatly promoted the worldwide research in quantum computing over the past few years. In practice, however, quantum systems are subject to influence of environment, and in ... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Shor's factorization algorithm [1] is a very important quantum algorithm, through which one has demonstrated the power of quantum computers. It has greatly promoted the worldwide research in quantum computing over the past few ... | ||
arxiv:0704.0520 | 0704.0520 | 1 | e2ed075f4ade80918e070091647f48ec5d73e5a1179cb737dc87b8224a6ec3ab | A critical theory of quantum entanglement for the Hydrogen molecule | In this paper we investigate some entanglement properties for the Hydrogen
molecule considered as a two interacting spin 1/2 (qubit) model. The
entanglement related to the $H_{2}$ molecule is evaluated both using the von
Neumann entropy and the Concurrence and it is compared with the corresponding
quantities for the tw... | [
"Tina A.C. Maiolo",
"Luigi Martina",
"Giulio Soliani"
] | [
"quant-ph",
"cond-mat.other",
"physics.atom-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | Entanglement is a physical observable measured by the von Neumann entropy or, alternatively, by the Concurrence of the system under consideration.
The concept of entanglement gives a physical meaning to the electron correlation energy in structures of interacting electrons. The electron correlation is not directly obs... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction and the model",
"text": "Entanglement is a physical observable measured by the von Neumann entropy or, alternatively, by the Concurrence of the system under consideration.\n\nThe concept of entanglement gives a physical meaning to the electr... | ||
arxiv:0704.0523 | 0704.0523 | 1 | 10.1103/PhysRevA.76.042103 | 1ca28034a1181cbdbddecf9d042ad8b097ea11a943736a666791eb33df4d9ea3 | Quantum superpositions and entanglement of thermal states at high
temperatures and their applications to quantum information processing | We study characteristics of superpositions and entanglement of thermal states
at high temperatures and discuss their applications to quantum information
processing. We introduce thermal-state qubits and thermal-Bell states, which
are a generalization of pure-state qubits and Bell states to thermal mixtures.
A scheme is... | [
"Hyunseok Jeong and Timothy C. Ralph"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | We study characteristics of superpositions and entanglement of thermal states at high temperatures and discuss their applications to quantum information processing. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure-state qubits and Bell states to thermal mixtures. A scheme is... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We study characteristics of superpositions and entanglement of thermal states at high temperatures and discuss their applications to quantum information processing. We introduce thermal-state qubits and thermal-Bell states, which are a ... | |
arxiv:0704.0526 | 0704.0526 | 1 | a1a26ce33595a638cf1c21c15b33b4f6450ff8e123099f4f85a016d59c12c65c | Fractional WKB Approximation | Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is
investigated in this paper using the fractional calculus. In the fractional
case the wave function is constructed such that the phase factor is the same as
the Hamilton's principle function "S". To demonstrate our proposed approach two
examples a... | [
"Eqab M. Rabei",
"Ibrahim M. A. Altarazi",
"Sami I. Muslih",
"Dumitru Baleanu"
] | [
"math-ph",
"hep-th",
"math.MP"
] | math-ph | [] | 2007-04-04 | 2026-02-26 | Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case the wave function is constructed such that the phase factor is the same as the Hamilton's principle function "S". To demonstrate our proposed approach two examples a... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case the wave function is constructed such that the phase factor is the same as the Ha... | ||
arxiv:0704.0527 | 0704.0527 | 1 | 10.1103/PhysRevD.75.125023 | edd179bddd702d53493d85c4e4f40b62a69e7e5d63dd63196e931a133fab844c | Towards Skyrmion Stars: Large Baryon Configurations in the
Einstein-Skyrme Model | We investigate the large baryon number sector of the Einstein-Skyrme model as
a possible model for baryon stars. Gravitating hedgehog skyrmions have been
investigated previously and the existence of stable solitonic stars excluded
due to energy considerations. However, in this paper we demonstrate that by
generating gr... | [
"B.M.A.G. Piette and G.I. Probert"
] | [
"hep-th"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | The Skyrme model, in its initial form, was proposed and developed by T.H.R. Skyrme in a series of papers as a non-linear field theory of pions [2] , [3] . Skyrme's initial idea was to think of baryons (in particular the nucleons) as secondary structures arising from a more fundamental mesonic fluid. The key property of... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The Skyrme model, in its initial form, was proposed and developed by T.H.R. Skyrme in a series of papers as a non-linear field theory of pions [2] , [3] . Skyrme's initial idea was to think of baryons (in particular the nucleons) ... | |
arxiv:0704.0530 | 0704.0530 | 1 | 10.1088/1126-6708/2007/06/065 | e09094677ad52079a512d808ae2511bbf4a65c00c4c0f9afba29ba8ee99b8a1e | Noncommutative Solitons in a Supersymmetric Chiral Model in 2+1
Dimensions | We consider a supersymmetric Bogomolny-type model in 2+1 dimensions
originating from twistor string theory. By a gauge fixing this model is reduced
to a modified U(n) chiral model with N<=8 supersymmetries in 2+1 dimensions.
After a Moyal-type deformation of the model, we employ the dressing method to
explicitly constr... | [
"Olaf Lechtenfeld",
"Alexander D. Popov"
] | [
"hep-th"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | We consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from twistor string theory. By a gauge fixing this model is reduced to a modified U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions. After a Moyal-type deformation of the model, we employ the dressing method to explicitly cons... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from twistor string theory. By a gauge fixing this model is reduced to a modified U(n) chiral model with 2N ≤ 8 supersymmetries in 2+1 dimensions. After a M... | |
arxiv:0704.0531 | 0704.0531 | 1 | 10.1088/1126-6708/2007/11/079 | b0e695f3b0dd96c805ccd0cc547d6cced9bf6b91d8fab221fb029b978f81b498 | Gravitational Duality Transformations on (A)dS4 | We discuss the implementation of electric-magnetic duality transformations in
four-dimensional gravity linearized around Minkowski or (A)dS4 backgrounds. In
the presence of a cosmological constant duality generically modifies the
Hamiltonian, nevertheless the bulk dynamics is unchanged. We pay particular
attention to t... | [
"Robert G. Leigh",
"Anastasios C. Petkou"
] | [
"hep-th"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | We discuss the implementation of electric-magnetic duality transformations in fourdimensional gravity linearized around Minkowski or (A)dS 4 backgrounds. In the presence of a cosmological constant duality generically modifies the Hamiltonian, nevertheless the bulk dynamics is unchanged. We pay particular attention to t... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We discuss the implementation of electric-magnetic duality transformations in fourdimensional gravity linearized around Minkowski or (A)dS 4 backgrounds. In the presence of a cosmological constant duality generically modifies the Hamilt... |