| # Atomic high-harmonic generation in a quantized optical cavity |
|
|
| ## Physical setting |
|
|
| Work in atomic units. Consider one electron moving along coordinate $z$ in the |
| soft-Coulomb potential |
|
|
| $$ |
| V(z)=-\frac{1}{|z|+\eta},\qquad \eta=0.9871. |
| $$ |
|
|
| The nucleus is fixed at $z=0$. A single cavity mode of frequency $\omega_c$ is |
| represented by displacement coordinate $x_c$. In the length gauge, use the |
| two-coordinate Hamiltonian |
|
|
| $$ |
| \begin{aligned} |
| H(t)={}&-\frac{1}{2}\frac{\partial^2}{\partial z^2} |
| -\frac{1}{|z|+\eta} |
| -\frac{1}{2}\frac{\partial^2}{\partial x_c^2} |
| +\frac{\omega_c^2x_c^2}{2}\\ |
| &-\sqrt{2\omega_c}\,g_czx_c |
| +\frac{g_c^2}{\omega_c}z^2+zF(t). |
| \end{aligned} |
| $$ |
|
|
| The last term drives only the electron. The bilinear interaction and the |
| dipole self-energy are both part of the physical model. Use |
|
|
| $$ |
| F(t)=F_0\cos[\omega_0(t-T/2)] |
| \cos^2\!\left[\frac{\pi(t-T/2)}{T}\right], |
| \qquad 0\le t\le T, |
| $$ |
|
|
| with $F(t)=0$ outside the pulse and |
|
|
| $$ |
| F_0=0.09,\qquad \omega_0=0.05,\qquad |
| T=\frac{20\pi}{\omega_0}. |
| $$ |
| |
| Compare the uncoupled reference, $g_c=0$, with the three cavity cases |
|
|
| $$ |
| (\omega_c,g_c)=(0.05,0.01),\quad(0.10,0.01),\quad(0.3185,0.06). |
| $$ |
|
|
| For each coupled case, start from the field-free polaritonic ground state and |
| treat the dynamics by direct propagation of the two-coordinate |
| time-dependent Schrödinger equation. The propagated state must retain physical |
| norm loss rather than being renormalized at every step. Any conditional |
| coordinate or energy expectation values must use a declared, consistent norm |
| convention, while the emitted response must retain ionization loss. |
|
|
| ## Research problem |
|
|
| Determine whether and how the selected cavity configurations change the |
| high-harmonic response and its high-energy extent relative to the uncoupled |
| atom. Establish whether the coupled dynamics of the electronic and cavity |
| degrees of freedom provide a physically consistent explanation of the |
| spectral differences. |
|
|
| Your conclusion must distinguish effects supported by the computed cases from |
| broader claims that the evidence does not establish. Limit all conclusions to |
| the specified one-dimensional, single-electron, single-mode model. |
|
|
| The derivation, spatial representation, boundary treatment, initial-state |
| solver, propagator, time step, spectral transform, high-order or endpoint |
| estimator, evidence design, validation strategy, normalization display, |
| plotting choices, and report organization are for you to choose and justify. |
| No particular number, sequence, range, grid, window, solver, or layout of |
| figures is prescribed. |
|
|
| ## Submission and resources |
|
|
| Submit one self-contained scientific report and an auditable offline |
| calculation package containing 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. |
|
|
