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## Background and physical system
A free electron traverses an initially empty, effectively single-mode optical
cavity over an interaction length $L$. The cavity frequency is $\omega$, its
longitudinal wave number is $\kappa$, and the electron enters with total energy
$E_0$. Emission of $n$ photons correlates the cavity Fock state with an electron
energy-loss state,
$$
|n\rangle_{e\gamma}
=|E_0-n\hbar\omega\rangle_e\otimes|n\rangle_\gamma,
\qquad n=0,1,2,\ldots .
$$
Let
$$
E_n=E_0-n\hbar\omega,
\qquad
k_n=\frac{\sqrt{E_n^2-m_e^2c^4}}{\hbar c},
\qquad
\Delta k_n=k_n-k_{n+1}-\kappa .
$$
Choose the cavity or grating momentum so that the first emission step is phase
matched, $\Delta k_0=0$. In the vacuum-coupling regime, use the exact
recoil-resolved single-mode ladder
$$
\frac{dc_n}{dz}
=\tilde g\,e^{-i\Delta k_n z}\sqrt{n+1}\,c_{n+1}
-\tilde g^{*}e^{+i\Delta k_{n-1}z}\sqrt{n}\,c_{n-1},
$$
where the second term is absent for $n=0$, $c_n(0)=\delta_{n0}$, and the
dimensionless interaction strength is $g_{\mathrm{Qu}}=\tilde gL$. The
photon-number probability is $P(n)=|c_n(L)|^2$. You may propagate this
non-autonomous system directly or use an exactly equivalent autonomous
representation that you derive.
Use the phase-matching width $\sigma$ as the recoil control. It is defined by
$\Delta k_\sigma L/2=\pi$ for the first zero of the corresponding
phase-matching factor. For electron kinetic energies below about $150$ keV,
the estimate
$$
\sigma\simeq
155\,\frac{[E_{\mathrm{kin}}(\mathrm{keV})]^{3/2}}
{[E_{\mathrm{ph}}(\mathrm{eV})]^2L(\mu\mathrm{m})}
$$
may be used, with $E_{\mathrm{ph}}=\hbar\omega$. State when you use this
estimate rather than the exact relativistic mismatch.
## Research problem
Determine how electron recoil and interaction strength jointly govern the
photon-number statistics and accessible ladder dynamics of this system. Your
investigation must cover both the low-coupling single-photon regime
$|g_{\mathrm{Qu}}|\lesssim0.5$ and a nonperturbative higher-coupling domain,
and must establish the domain over which a low-coupling description remains
physically adequate. Decide what statistical and state-generation claims are
supported by your calculation, explain them from the recoil-resolved dynamics,
and delimit their evidential scope.
Relate the dimensionless recoil domain you study to physically realizable
electron, photon, and interaction-length controls. Material claims about
higher-coupling structure or state generation must be supported by
reconstructible state-resolved ladder populations or an equivalent underlying
signal; a summary correlation alone is not sufficient.
The exact ladder above must supply the production result. A first-order
phase-matching or weak-coherent approximation may be used as a comparison, but
not as a replacement for the finite-recoil dynamics. Include a weak-recoil
reference and choose recoil and coupling cases sufficient to support the
claims you make.
You choose the physical parameter instances, sampling, ladder truncation,
propagation method, statistical diagnostics, comparison controls, and evidence
presentation. Explain how your choices support your claims. Exact reproduction
of a prescribed panel, plotting range, feature coordinate, colour scale, or
absolute display normalization is not required.
The required problem is confined to the single-photon phase-matched branch.
Multiphoton phase matching, externally driven classical-field operation,
analogue-horizon transport, and state-engineering claims outside this branch
are out of scope.
## Submission and resources
Submit one self-contained scientific report and an auditable offline
calculation package containing the task-specific derivation, code, numerical
data underlying all material claims, 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, panel count, filename scheme,
solver, grid, 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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