cases/case37/task.md

Recoil and coupling in free-electron photon statistics

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,

$$\mathrm{\mid }n{⟩}_{e\gamma }=\mathrm{\mid }{E}_0-n\mathrm{\hslash }\omega {⟩}_e\otimes \mathrm{\mid }n{⟩}_{\gamma },n=0,1,2,\dots \mathrm{.}|n\backslash rangle\_\{e\backslash gamma\}\; =|E\_0-n\backslash hbar\backslash omega\backslash rangle\_e\backslash otimes|n\backslash rangle\_\backslash gamma,\; \backslash qquad\; n=0,1,2,\backslash ldots\; .$$

Let

$$E_n=E_0-n\mathrm{\hslash }\omega ,k_n=\frac{\sqrt{E_n^2-m_e^2{c}^4}}{\mathrm{\hslash }c},\mathrm{\Delta }{k}_n=k_n-k_{n+1}-\kappa \mathrm{.}E\_n=E\_0-n\backslash hbar\backslash omega,\; \backslash qquad\; k\_n=\backslash frac\{\backslash sqrt\{E\_n^2-m\_e^2c^4\}\}\{\backslash hbar\; c\},\; \backslash qquad\; \backslash Delta\; k\_n=k\_n-k\_\{n+1\}-\backslash kappa\; .$$

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

$$\frac{d{c}_n}{dz}=\tilde{g}{e}^{-i\mathrm{\Delta }{k}_{n}z}\sqrt{n+1}{c}_{n+1}-{\tilde{g}}^{\ast }{e}^{+i\mathrm{\Delta }{k}_{n-1}z}\sqrt{n}{c}_{n-1},\backslash frac\{dc\_n\}\{dz\}\; =\backslash tilde\; g\backslash ,e^\{-i\backslash Delta\; k\_n\; z\}\backslash sqrt\{n+1\}\backslash ,c\_\{n+1\}\; -\backslash tilde\; g^\{*\}e^\{+i\backslash Delta\; k\_\{n-1\}z\}\backslash sqrt\{n\}\backslash ,c\_\{n-1\},$$

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

$$\sigma \simeq 155\frac{[E_{\mathrm{k}\mathrm{i}\mathrm{n}}(\mathrm{k}\mathrm{e}\mathrm{V}){]}^{3\mathrm{/}2}}{[E_{\mathrm{p}\mathrm{h}}(\mathrm{e}\mathrm{V}){]}^{2}L(\mu \mathrm{m})}\backslash sigma\backslash simeq\; 155\backslash ,\backslash frac\{[E\_\{\backslash mathrm\{kin\}\}(\backslash mathrm\{keV\})]^\{3/2\}\}\; \{[E\_\{\backslash mathrm\{ph\}\}(\backslash mathrm\{eV\})]^2L(\backslash mu\backslash mathrm\{m\})\}$$

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 for the common submission entry points.