# Scientific sources - Case44

## References

André Becker, Georgios M. Koutentakis, and Peter Schmelcher. 2025. *Two-Body Kapitza-Dirac Scattering of One-Dimensional Ultracold Atoms*. *arXiv*, 2512.15260. [Source](https://arxiv.org/abs/2512.15260v1). Version 1.

Exact short-pulse exchange-symmetric two-body diffraction, paired unit-normalized one-body observables and independent k4/k6 interaction families. Source higher-order suppression is a canonical source observation, not a compulsory direction under the effective task; long-time fidelity is excluded.

## Criterion-to-source crosswalk

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1, F2 | Becker, Koutentakis and Schmelcher (2025), article pp. 2–4, Eqs. (1)–(7), (13)–(22). Exchange-symmetric contact-interacting initial states, CM/relative coupling, quench and exact correlated propagation. Independent-particle or mean-field replacements are not equivalent. |
| F3 | Article pp. 4–5, Eqs. (23)–(25), (29)–(31); Appendix C. Laboratory one-body reduced density and its Fourier momentum distribution from the same state, each normalized to one particle. |
| O1 | Article pp. 10–11, Fig. 7(a,c), caption and interaction-state paragraphs. Smaller-wave-number real-space localization/extension across attractive, zero and repulsive interactions. |
| O2 | Article p. 11, Fig. 7(b,d), caption and momentum-sharpening paragraph. Smaller-wave-number momentum widths, contrast and integer orders; weak-order evidence is assessed only at credible resolution. |
| O3 | Article pp. 11–12, Fig. 8(a,c), caption and changed-lattice-spacing paragraph. Fresh larger-wave-number real-space family and denser spatial modulation, not a relabelled smaller-wave-number dataset. |
| O4 | Article pp. 11–12, Fig. 8(b,d), caption and larger-wave-number discussion. Fresh shifted first-order momentum family and actual resolved higher-order behavior. The source suppression statement is contextual; the effective task does not impose a predetermined second-order direction. |
| I1, M1, M2 | Article p. 11, Fig. 7(c,d), paragraphs on overlap, localization, momentum spread and phase gradients. Paired real/momentum evidence and interaction-dressed initial-state mechanism. Attractive localization/broad momentum and repulsive extension/sharp peaks are complementary, not independent arbitrary rescalings. |
| I2 | Article pp. 11–12, Fig. 8(c,d), interaction-ordering and lattice-spacing paragraphs. Interaction ordering and denser modulation in the independently recalculated larger-wave-number case, at the actual resolved scale. |
| I3, M4 | Article p. 4, momentum-transfer paragraph after Eq. (22); pp. 11–12, Figs. 7(d), 8(d). Physical entailment from the standing-lattice Fourier components gives momentum transfer in integer multiples of 2 hbar k_lat; changed wave number shifts the order locations. |
| M3, M7 | Article pp. 11–12, interaction-state and larger-wave-number mechanism paragraphs, Figs. 7(c,d), 8(c,d). Task-defined synthesis of independently established interaction-state and wave-number chains. A weak local modulation identification does not automatically erase the interaction mechanism. |
| M5, M6 | Article p. 8, Eq. (39) and recoil discussion; pp. 11–12, Figs. 7(d), 8(d); p. 5, Eq. (33). Recoil spacing, pulse area and finite-time access explain the candidate-established order behavior. The source direction is not a hidden mandatory prediction, and no fixed suppression factor is required. |
| M8, V1 | Article pp. 2–4, model/sector/quench; pp. 12–14, sudden-comparison discussion. Six-case exact-two-body short-pulse domain and distinction between direct evidence and conditional interpretation; long-time fidelity is outside scope. |
| V2 | Article p. 15, Appendix A, Figs. 10–11; pp. 4–5, same-state observable definitions. Bounded short-time basis/observable credibility. Conditional testing does not mandate a basis size or use unresolved long-time fidelity as gold. |

## Formula and convention notes

The laboratory Hamiltonian is $H=\sum_{i=1}^2[p_i^2/(2m)+m\omega^2x_i^2/2+U_0\cos^2(k_{\rm lat}x_i)]+\sqrt2g\delta(x_1-x_2)$ (Eq. (1)). With $R=(x_1+x_2)/\sqrt2$, $r=(x_1-x_2)/\sqrt2$ and $\ell=\sqrt{\hbar/(m\omega)}$, $H_{\rm rel}=p_r^2/(2m)+m\omega^2r^2/2+g\delta(r)$ and $H_{\rm lat}=U_0[1+\cos(\sqrt2k_{\rm lat}r)\cos(\sqrt2k_{\rm lat}R)]$. The latter couples CM and relative motion; post-quench factorization is not generally valid.

The initial state for each $g$ is $|\Psi_g(0)\rangle=|0_{\rm CM}\rangle|0_{{\rm rel},g}\rangle$, and $|\Psi_g(t_0)\rangle=e^{-iH(g,k_{\rm lat})t_0/\hbar}|\Psi_g(0)\rangle$. An equivalent Busch or numerical exact correlated representation is accepted in the exchange-even sector; it must not reuse the same interacting ground state for all three couplings.

For a unit-normalized two-body wavefunction, $\rho^{(1)}(x,x\prime)=\int dx_2\,\Psi(x,x_2)\Psi^*(x\prime,x_2)$, $n(x)=\rho^{(1)}(x,x)$ and $n(k)=(2\pi)^{-1}\int dx\,dx\prime\,e^{-ik(x-x\prime)}\rho^{(1)}(x,x\prime)$ (Eqs. (24)–(25), (29)). Thus $\int n(x)dx=\int n(k)dk=1$; physical momentum is $p=\hbar k$. A total-density normalization of two is equivalent only with an explicit common conversion; the unrelated source two-body-density prefactor is not imported.

The lattice contains Fourier transfers $\Delta p=\pm2\hbar k_{\rm lat}$, with order locations $p_n=2n\hbar k_{\rm lat}$ and free recoil scale $E_n=p_n^2/(2m)=4n^2E_R$, $E_R=\hbar^2k_{\rm lat}^2/(2m)$ (physical entailment, recoil discussion and Eq. (39)). Pulse area $K=\hbar^{-1}\int U_0(t)dt$ is Eq. (33). Larger recoil spacing alone does not license imposing an unreported numerical second-order suppression direction; judge the actual short-time calculation and supported scope.


## Scope of the source anchors

Source examples verify physical identity and robust structure; their numerical samples, plotting conventions and panel layout are not compulsory candidate deliverables. Reporting, reconstructibility, evidence attribution, conditional validation and Research Closure are examination-defined obligations, not additional results asserted by the article. A derivation or physical entailment is identified as such rather than presented as a directly printed source formula.