| # Interaction-shaped diffraction of two trapped atoms |
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|
| ## 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 |
|
|
| $$ |
| H(t)=\sum_{\sigma=1}^{2}\left[ |
| -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_\sigma^2} |
| +\frac{1}{2}m\omega^2x_\sigma^2 |
| +U(t)\cos^2(k_{\mathrm{lat}}x_\sigma) |
| \right] |
| +\sqrt{2}\,g\,\delta(x_1-x_2). |
| $$ |
| |
| 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 |
|
|
| $$ |
| t_0=2.5\times10^{-3}/\omega. |
| $$ |
| |
| Investigate the interaction cases |
| |
| $$ |
| g\in\{-5,0,+5\}\,\hbar\omega\ell |
| $$ |
| |
| for each of the two lattice wave numbers |
| |
| $$ |
| k_{\mathrm{lat}}\in\{4/\ell,6/\ell\}. |
| $$ |
|
|
| 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](../../submission_requirements.md) for the common submission entry points. |
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|
