cases/case44/task.md

Interaction-shaped diffraction of two trapped atoms

Physical setting

Consider two equal-mass atoms in an exchange-symmetric state, confined to one dimension by a harmonic trap of frequency $\omega$. Their Hamiltonian is

$$H(t)=\sum _{\sigma =1}^2[-\frac{{\mathrm{\hslash }}^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_{\sigma }^2}+\frac{1}{2}m{\omega }^2{x}_{\sigma }^2+U(t){\cos }^2(k_{\mathrm{l}\mathrm{a}\mathrm{t}}{x}_{\sigma })]+\sqrt{2}g\delta (x_1-x_2)\mathrm{.}H(t)=\backslash sum\_\{\backslash sigma=1\}^\{2\}\backslash left[\; -\backslash frac\{\backslash hbar^2\}\{2m\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; x\_\backslash sigma^2\}\; +\backslash frac\{1\}\{2\}m\backslash omega^2x\_\backslash sigma^2\; +U(t)\backslash cos^2(k\_\{\backslash mathrm\{lat\}\}x\_\backslash sigma)\; \backslash right]\; +\backslash sqrt\{2\}\backslash ,g\backslash ,\backslash delta(x\_1-x\_2).$$

Use the oscillator length $\ell=\sqrt{\hbar/(m\omega)}$. The system begins in the ground state of the lattice-free Hamiltonian with the stated value of $g$. At $t=0$, a standing lattice of depth $U_0=-1000\hbar\omega$ is applied and remains constant until

$$t_0=2.5\times {10}^{-3}\mathrm{/}\omega \mathrm{.}t\_0=2.5\backslash times10^\{-3\}/\backslash omega.$$

Investigate the interaction cases

$$g\in \{-5,0,+5\}\mathrm{\hslash }\omega \mathrm{\ell }g\backslash in\backslash \{-5,0,+5\backslash \}\backslash ,\backslash hbar\backslash omega\backslash ell$$

for each of the two lattice wave numbers

$$k_{\mathrm{l}\mathrm{a}\mathrm{t}}\in \{4\mathrm{/}\mathrm{\ell },6\mathrm{/}\mathrm{\ell }\}\mathrm{.}k\_\{\backslash mathrm\{lat\}\}\backslash in\backslash \{4/\backslash ell,6/\backslash ell\backslash \}.$$

The required method level is an exact two-body quantum treatment that retains the contact-interaction correlations and the lattice-induced coupling of centre-of-mass and relative motion. Any converged basis, coordinate representation and time propagator that preserve this physical problem are acceptable. Independent-particle and mean-field substitutions are not equivalent treatments.

Research problem

Determine how and why the contact interaction and lattice wave number reshape the short-pulse two-body diffraction response. Base the assessment on the normalized one-body density in real space and the corresponding normalized momentum distribution at $t_0$.

Support a bounded physical conclusion with auditable, reconstructible evidence of your own design. The evidence should distinguish interaction-dependent state effects from changes associated with the lattice wave number. State which claims are established by the specified benchmark and which broader claims are not.

The basis cutoff, solver, time step, quadrature, coordinate grid, plotting domain, display convention, number of figures and order of presentation are not prescribed. Declare any convention that materially affects the physical interpretation.

Submission and resources

Submit one self-contained scientific report and an auditable offline calculation package. Include the task-specific derivation, code, numerical data underlying every material claim, a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, and concise run and environment information. No fixed file tree, filename scheme, panel count or complete exploration history is required.

Resource limits are supplied exclusively by the accompanying resource_limits.json. Its fixed CPU allocation, candidate-active wall time, network boundary are binding. No separate aggregate CPU-hour limit applies. Greater computational expense receives no additional credit.

Reference and integrity policy

This examination is conducted without network access or live literature retrieval. You may rely on your existing general scientific knowledge and, if any are explicitly supplied with the examination, on those offline materials or software documentation. Record every such source actually consulted. For formulas, methods, constants, or conventions used from general knowledge rather than a consulted source, state enough of the method basis, definition, value, units, and convention to make the calculation auditable; do not invent a citation. You are not required to identify or prove a publication date or to retrieve literature during the examination.

The hidden target article and its identifying metadata, figures or captions, supplementary material, author code or data, target-specific reproduction code, solution or scoring material, online solutions, and worked reproductions of this case must not be used. All submitted scientific figures and numerical data must be generated by your own implementation of the stated problem.

See submission requirements for the common submission entry points.