# Photon-momentum partition in direct strong-field ionization

## Physical setting

Consider fixed-nucleus hydrogen exposed to a finite, right-handed circularly
polarized plane-wave pulse that propagates along the laboratory $z$ direction.
Use atomic units unless stated otherwise. Work to leading observable order in
$1/c$ within a direct, length-gauge nondipole strong-field approximation.
Rescattering and spin dynamics are outside the required scope.

For the state-dependent study, use a carrier wavelength of $1400\ \mathrm{nm}$
and compare the $1s$, $2s$, $2p_{m=0}$, and $2p_{m=\pm1}$ initial states. For
the photon-order study, use the $1s$ state at a peak intensity of
$10^{14}\ \mathrm{W\,cm^{-2}}$ and vary the carrier frequency over a justified
sequence that extends from few-photon to substantially larger absorbed-photon
orders. Choose a smooth finite pulse envelope and duration that resolve the
claims you make, and state the convention used.

The primary observable is the normalized mean photoelectron momentum along
the propagation direction. Overall ionization yield is not a grading target.

## Research problem

Determine how photon linear momentum is partitioned into longitudinal
photoelectron momentum during direct strong-field ionization, including its
dependence on the initial orbital and its evolution from finite to large
absorbed-photon order. Decide from moments calculated within the same physical
branch whether a well-defined asymptotic partition law is established.

Your report must distinguish a relation established by computed observables
from one assumed or inserted analytically. Explain the physical origin and
scope of the state dependence and of any asymptotic law you infer. Limit every
conclusion to the direct circular-polarization branch actually studied.

The task-specific derivation, numerical representation, integration strategy,
frequency and intensity sampling, evidence design, reliability assessment,
plotting choices, and report organization are for you to decide. Your evidence
must make the physical states, pulse
convention, observable definitions, and case-to-data mapping auditable. No
particular number, order, or layout of figures is prescribed.

## Submission and resources

Submit one self-contained scientific report and an auditable offline
calculation package. Include 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, filename scheme, numerical grid,
solver, window, plot range, 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.