cases/case39/task.md

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

$$V(z)=-\frac{1}{\mathrm{\mid }z\mathrm{\mid }+\eta },\eta =\mathrm{0.9871.}V(z)=-\backslash frac\{1\}\{|z|+\backslash eta\},\backslash qquad\; \backslash eta=0.9871.$$

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

The last term drives only the electron. The bilinear interaction and the dipole self-energy are both part of the physical model. Use

$$F(t)=F_0\cos [{\omega }_0(t-T\mathrm{/}2)]{\cos }^2\text{\hspace{0.17em}⁣}\left[\frac{\pi (t-T\mathrm{/}2)}{T}\right],0\le t\le T,F(t)=F\_0\backslash cos[\backslash omega\_0(t-T/2)]\; \backslash cos^2\backslash !\backslash left[\backslash frac\{\backslash pi(t-T/2)\}\{T\}\backslash right],\; \backslash qquad\; 0\backslash le\; t\backslash le\; T,$$

with $F(t)=0$ outside the pulse and

$$F_0=0.09,{\omega }_0=0.05,T=\frac{20\pi }{{\omega }_0}\mathrm{.}F\_0=0.09,\backslash qquad\; \backslash omega\_0=0.05,\backslash qquad\; T=\backslash frac\{20\backslash pi\}\{\backslash omega\_0\}.$$

Compare the uncoupled reference, $g_c=0$, with the three cavity cases

$$({\omega }_c,g_c)=(0.05,0.01),(0.10,0.01),(0.3185,0.06)\mathrm{.}(\backslash omega\_c,g\_c)=(0.05,0.01),\backslash quad(0.10,0.01),\backslash quad(0.3185,0.06).$$

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 for the common submission entry points.