# Spectral confinement of phase-matched slow electrons

## Background

A free electron exchanging photons with a phase-matched optical mode occupies
an energy ladder separated by the photon energy. Many descriptions developed
for much faster electrons linearize the dispersion and treat this ladder as an
effectively extended multilevel system. Whether that approximation remains
adequate at low kinetic energy must be established rather than assumed.

## Research problem

Determine whether and how a phase-matched optical field can confine the
energy-domain evolution of a slow electron. Establish a quantitative physical
distinction between Raman-Nath and Bragg dynamics, and assess whether the
specified low-energy interaction can realize coherent control within a reduced
energy-state manifold.

The derivation, numerical construction, choice of discriminating evidence and
organization of the argument are yours. Your conclusions must follow from your
calculation rather than from an assumed regime label or a finite numerical
basis.

## Physical model and inputs

Use a one-dimensional relativistic minimal-coupling model along the electron
propagation direction. Expand the field-free dispersion through quadratic
order about the initial momentum $p_0$. Retain the symmetrized $Ap$ coupling
and the $A^2$ interaction in the production model. Let $e>0$ denote the
magnitude of the electron charge and use the monochromatic optical mode

$$
A(z,t)=-\frac{E_f}{\omega}
\sin(\omega t-k_z z+\phi_0).
$$

Impose phase matching, $k_z=\omega/v_0$. Use a Floquet-Bloch sideband
description with momenta $k_n=k_0+n k_z$ and energies separated by
$\hbar\omega$. Derive the coupled dynamics needed for the research question;
do not replace the production model by a linear nonrecoil model. A consistently
transformed charge, phase or sideband-index convention is acceptable.

Investigate both cases below with the same physical model:

| Quantity | Case I | Case II |
|---|---:|---:|
| Phase-matched reference kinetic energy | $100\ \mathrm{eV}$ | $16\ \mathrm{eV}$ |
| Initial state | single-energy $n=0$ state | $12\ \mathrm{eV}$ state, $n=-1$ |
| Photon energy $\hbar\omega$ | $1.54\ \mathrm{eV}$ | $4\ \mathrm{eV}$ |
| Electric-field amplitude $E_f$ | $0.5\ \mathrm{V\,nm^{-1}}$ | $0.1\ \mathrm{V\,nm^{-1}}$ |
| Initial optical phase $\phi_0$ | $0$ | $0$ |

Task unit-resolution convention for Case II: the field amplitude is
$10^8\ \mathrm{V\,m^{-1}}=0.1\ \mathrm{V\,nm^{-1}}$.

Construct the final electron-energy evidence from the sideband probabilities
$|a_n(t)|^2$ with a declared energy mapping and normalization convention.
Absolute probability is physically fixed by state normalization; colour,
interpolation and display layout are not grading targets.

## Research boundary

Use the stated one-dimensional Floquet-Bloch minimal-coupling model. A full
Maxwell-field simulation, multidimensional electron propagation or an
experimental device model is outside the required task. Analytical reductions
may support the argument, but they do not replace the production calculation
when omitted couplings change the physical object.

All basis truncations, time sampling, propagation algorithms, plotting domains
and diagnostic constructions not fixed above are yours. You are responsible
for ensuring that the evidence is sufficiently reliable and reproducible for
the claims made, but no grid, solver, truncation size, survey range or
validation procedure is prescribed.

## Submission and resources

Submit one self-contained scientific report and an offline reproduction
package containing the producing code, reconstructible numerical data
underlying 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. The organization of the report and the choice and
number of figures are yours. No fixed file tree, filename scheme, complete
exploratory history or diagnostic prescribed by the evaluator is required.

The available CPU allocation, candidate active deadline, node, network,
memory and scratch limits are specified in the accompanying
`resource_limits.json`. Complete the derivation, implementation,
calculation, validation, interpretation and final-file preparation within these
resource limits. More expensive modelling receives no additional credit.

## Reference and integrity policy

Complete this task without network access or live literature
retrieval. You may rely on your existing general scientific knowledge and, if any are
explicitly supplied with this task, 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 while completing this task.

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.