# Quantum-statistical radiation pressure in strong-field ionization

## Physical setting

Work in atomic units. The laser propagates along $+z$ and is circularly
polarized in the $x$-$y$ plane. The measured longitudinal observable is the
photoelectron momentum $p_z$. Treat direct ionization with a leading-order
light-front nondipole strong-field approximation; rescattering is outside this
problem.

Compare two measurement regimes.

For the photon-traced regime, consider H and Na driven either by bright
squeezed vacuum (BSV) or by coherent light at the same mean incident
intensity. Use wavelength $1580\,\mathrm{nm}$, a ten-cycle pulse, and an
intensity domain within $10^{12}$--$10^{13}\,\mathrm{W\,cm^{-2}}$ broad
enough to resolve the state-dependent intensity response. For BSV, define the
amplitude width by

$$
I=2\omega^2 A_{\rm rms}^2,
$$

and use the bright-field limit. Standard, cited ionization potentials for H
and Na may be used.

For the photon-number-conditioned regime, consider Na at $400\,\mathrm{nm}$
with squeezing parameter $r=15$, mean incident intensity
$10^{13}\,\mathrm{W\,cm^{-2}}$, mode-amplitude scale
$\epsilon_V=3.2\times10^{-8}$ a.u., and a ten-cycle pulse. Study one
neighboring pair of final photon-number channels in the low-effective-intensity
regime near $10^{13}\,\mathrm{W\,cm^{-2}}$. The channel-conditioned field
amplitude is to be evaluated by a bounded two-coherent-point approximation:
the two points have phases separated by $\pi$ and use a common radius

$$
\rho=\sqrt{\frac{n}{2-\tanh r}}
$$

for the neighboring pair. At each point, evaluate the complete finite-pulse
direct nondipole-SFA amplitude. Use the exact squeezed-vacuum/Fock overlaps and
sum the two complex contributions before taking the modulus squared.

If you use a matched incoherent control, it must differ only by removal of the
cross term. Such a control is one optional diagnostic, not a required evidence
route. Any alternative diagnostic used to distinguish coherent and
incoherent explanations must preserve the same field points, overlaps,
physical parameters, numerical domain, normalization relation, and final
electron observable, so that it isolates the amplitude-level cross term rather
than changing the physical problem.

For finite-pulse calculations, the supplied reference convention is a
$\cos^4$ vector-potential envelope with total duration $10T$. Handedness,
axis, overall phase, normalization, and smooth-envelope conventions that are
physically equivalent are acceptable when declared and when they do not alter
the material conclusions. No particular momentum grid, time sampling,
quadrature, binning, transform window, or display convention is prescribed.

## Research problem

Determine how longitudinal radiation-pressure transfer depends on the quantum
state of the driving field and on whether the outgoing photon state is traced
or resolved into neighboring final-number channels. For the bounded
conditioned case, determine what information, if any, the channel-resolved
electron signal supports about the relation between alternative field
pathways. Establish which similarities and differences between the two
measurement regimes are supported by your own calculation, without assuming
in advance that conditioning must reveal, preserve, or remove a particular
kind of information.

Distinguish conclusions supported by the bounded low-field conditioned case
from claims requiring a larger channel or intensity survey. The detailed
derivation, numerical realization, evidence package, controls,
representations, plotting choices, and report organization are for you to
choose and justify. The conclusion must remain within the direct-ionization,
leading-order nondipole and stated target/field scope.

## Submission and resources

Submit one self-contained scientific report and an auditable offline
calculation package containing the task-specific derivation, code, underlying
numerical data for every material claim, 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, figure
count, output order, 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.