| # 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, |
| |
| $$ |
| |n\rangle_{e\gamma} |
| =|E_0-n\hbar\omega\rangle_e\otimes|n\rangle_\gamma, |
| \qquad n=0,1,2,\ldots . |
| $$ |
| |
| 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. |
|
|
