# 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.