Scientific Sources — Case16
References
M. Eldar, Z. Chen, Y. Pan, and M. Krüger, “Self-Trapping of Slow Electrons in the Energy Domain,” Physical Review Letters 132, 035001 (2024), doi:10.1103/PhysRevLett.132.035001.
Quadratic electron dispersion, phase-matched sideband dynamics, spectral confinement, and the Raman–Nath/Bragg regime distinction.
Relevant locations
Eq. (1), pp. 2–3: full sideband dynamics.
Fig. 1(d), p. 2; discussion pp. 3–4: plane-wave confinement.
Eq. (2), definition of rho and regime discussion pp. 3–4.
Fig. 2(a)–(b), p. 4: regime map and Bragg exchange.
Discussion/conclusions pp. 4–5: tunability and model interpretation.
Supplemental Material to M. Eldar et al., Physical Review Letters 132, 035001 (2024).
Derivation of the sideband equation and the regime parameter, including its effective-interaction-length interpretation.
Relevant locations
Derivation through Eq. (S9): full sideband model.
Eqs. (S25)–(S26): regime discrimination.
Criterion-to-source crosswalk
| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | Eldar et al., Phys. Rev. Lett. 132, 035001 (2024), Eq. (1), pp. 2–3; Supplemental Material derivation through Eq. (S9). |
| F2, O2, I2, M3 | Eldar et al. (2024), Eq. (2), definition of $\rho$ and regime discussion, pp. 3–4; Supplemental Material Eqs. (S25)–(S26). |
| F3 | Eldar et al., Phys. Rev. Lett. 132, 035001 (2024), Eq. (1), pp. 2–3; Supplemental Material derivation through Eq. (S9). Eldar et al. (2024), Fig. 1(d), p. 2, and plane-wave confinement discussion, pp. 3–4. Eldar et al. (2024), Fig. 2(a)–(b), p. 4, and Bragg-regime discussion. |
| O1, I1, M2 | Eldar et al. (2024), Fig. 1(d), p. 2, and plane-wave confinement discussion, pp. 3–4. |
| O3, O4, I3, I4, M7 | Eldar et al. (2024), Fig. 2(a)–(b), p. 4, and Bragg-regime discussion. |
| M1 | Eldar et al., Phys. Rev. Lett. 132, 035001 (2024), Eq. (1), pp. 2–3; Supplemental Material derivation through Eq. (S9). Eldar et al. (2024), Fig. 1(d), p. 2, and plane-wave confinement discussion, pp. 3–4. |
| M4, M6 | Eldar et al., Phys. Rev. Lett. 132, 035001 (2024), Eq. (1), pp. 2–3; Supplemental Material derivation through Eq. (S9). Eldar et al. (2024), Fig. 2(a)–(b), p. 4, and Bragg-regime discussion. |
| M8, V1, V2 | Eldar et al. (2024), regime and confinement discussion/conclusions, pp. 4–5. |
Formula and convention notes
One accepted nonrelativistic convention for the full sideband equation is $$ \begin{aligned}i\hbar\dot a_n={}&-\left[n\hbar(\omega-v_0k_z)-n^2\frac{(\hbar k_z)^2}{2m}+U_p\right]a_n\&+\delta\left(k_{n+1}-\frac{k_z}{2}\right)a_{n+1}e^{i\phi}+\delta\left(k_{n-1}+\frac{k_z}{2}\right)a_{n-1}e^{-i\phi}\&-\frac{U_p}{2}\left(a_{n+2}e^{2i\phi_0}+a_{n-2}e^{-2i\phi_0}\right),\end{aligned} with $k_n=k_0+n k_z$, $U_p=e^2E_f^2/(4m\omega^2)$, $\delta=e\hbar E_f/(2m\omega)$ and $\phi=\phi_0+\pi/2$. Phase matching $k_z=\omega/v_0$ removes the linear diagonal term. A removable global phase, consistent convention transformation, or consistently retained relativistic factors does not change the model identity. For interpretation, neglecting recoil variation in the nearest-neighbour coefficient and small ponderomotive coupling gives i\hbar\dot a_n=\beta_d n^2a_n+\kappa a_{n+1}+\kappa^*a_{n-1},\qquad\beta_d=\frac{(\hbar k_z)^2}{2m},\quad\kappa=\frac{eE_f\hbar k_0}{2m\omega}e^{i(\phi_0+\pi/2)},\quad\rho=\frac{\beta_d}{|\kappa|}. $$ This is an analytical interpretation, not a replacement production model. $\rho\ll1$ is hopping-dominated Raman–Nath behaviour; $\rho\gg1$ is on-site-dominated Bragg behaviour; crossover is near $\rho\sim1$. Equivalent discriminators must show the same physics and ordering.
The observable is $P_n(t)=|a_n(t)|^2$ with absolute state normalization and a declared energy mapping. Under the stated sideband-energy convention, $E_n=E_{\rm ref}+n\hbar\omega$; a correctly transformed equivalent representation is accepted. Independent normalization of a selected state subset cannot establish full-ladder probability or suppression.
Case I uses the 100 eV reference, initial $n=0$, 1.54 eV photon, 0.5 V nm$^{-1}$ field and $\phi_0=0$. The characteristic full-model evolution is initial multi-order transport followed by slowing, bounded turning and recurrent reconcentration near the initial energy. Other functionally equivalent confinement evidence is accepted for the confinement output and integrated conclusion; early broadening alone or a smooth finite-basis distribution is insufficient.
Case II uses the 16 eV reference, initial 12 eV state $n=-1$, 4 eV photon, 0.1 V nm$^{-1}$ field and $\phi_0=0$. The task's unit convention is $10^8$ V m$^{-1}=0.1$ V nm$^{-1}$; the inconsistent source-caption unit must not replace it. Exchange is concentrated in $n=-1$ (12 eV) and $n=+1$ (20 eV), not the 16 eV reference, while the rest of the full ladder remains a small correction.
Broad quantitative anchors are Case-I $\rho$ of order $10^{-2}$ and Case-II $\rho$ of order $10^1$. Source times near 9 and 19 fs, a reference transfer near 680 fs and reference pair probability near 0.975 are comparison anchors, not exact timing, decimal, probability-threshold or display obligations.
Scope of the source anchors
Quadratic electron dispersion, phase-matched sideband dynamics, spectral confinement, and the Raman–Nath/Bragg regime distinction.
Derivation of the sideband equation and the regime parameter, including its effective-interaction-length interpretation.