cases/case31/task.md

# Coulomb-sensitive semiclassical modelling of strong-field photoelectron interference

## Physical system and cases

Work in atomic units with hydrogen initially in its $1s$ state. Use
$I_p=0.5$, ionic charge $Z=1$, magnetic quantum number $m=0$, and static
polarizabilities $\alpha_{\mathrm H}=4.5$ and
$\alpha_{\mathrm{H}^+}=0$. The laser is linearly polarized along $z$ and is
defined by

$$
\mathbf A(t)=(-1)^{n+1}\frac{cF_0}{\omega}
\sin^2\!\left(\frac{\omega t}{2n}\right)
\sin(\omega t)\,\mathbf e_z,
\qquad 0\le t\le \frac{2\pi n}{\omega},
$$

with $\mathbf A(t)=0$ outside this interval and
$\mathbf F(t)=-(1/c)\,d\mathbf A/dt$. Use $n=8$, peak intensity
$9.0\times10^{13}\ \mathrm{W\,cm^{-2}}$, and wavelengths 800, 1200 and
1600 nm.

Use the quasistatic launch density

$$
w(t_0,v_{0,\perp})\propto
\exp\!\left[-\frac{2\kappa^3}{3F}
-\frac{\kappa v_{0,\perp}^2}{F}\right],
\qquad
\kappa=\sqrt{2I_p(F)},
$$

where $F=|F_z(t_0)|$ is the local field magnitude and

$$
I_p(F)=I_p+\frac{1}{2}
(\alpha_{\mathrm H}-\alpha_{\mathrm{H}^+})F^2.
$$

The exit coordinate is fixed by

$$
-\frac{\beta_2(F)}{2\eta}
+\frac{m^2-1}{8\eta^2}
-\frac{F\eta}{8}
=-\frac{I_p(F)}{4},
\qquad
\beta_2(F)=Z-\frac{1+|m|}{2}\sqrt{2I_p(F)}.
$$

Choose the positive real root $\eta_0$ on the field-lowered, outgoing side.
Equivalently, after orienting the local polarization axis so that the field
points along $+z$, take $\mathbf r_0=-\eta_0\mathbf e_z/2$; in the fixed
laboratory axis this is

$$
\mathbf r_0=-\operatorname{sgn}[F_z(t_0)]
\frac{\eta_0}{2}\,\mathbf e_z.
$$

The baseline launch has $v_{0,z}=0$. For the 800-nm case, also examine the
nonzero longitudinal launch prescription

$$
v_{0,z}(t_0)=-\frac{A_z(t_0)}{c}
\left[\sqrt{1+\gamma^2(t_0,v_{0,\perp})}-1\right],
$$

$$
\gamma(t_0,v_{0,\perp})=
\frac{\omega\sqrt{2I_p+v_{0,\perp}^2}}
{F_0\sin^2(\omega t_0/2n)|\cos(\omega t_0)|}.
$$

The same quasistatic weight family applies to both longitudinal-launch cases.

## Semiclassical response identity

Let $V(\mathbf r)=-Z/r$. The commissioned SCTS branch uses the mixed
initial-position-to-final-momentum generating function. In length gauge,
$\mathbf p_0=\mathbf v_0$, and its trajectory phase is defined, up to terms
common to all trajectories in the same final-momentum cell, by

$$
\Phi_{\mathrm{SCTS}}
=-\mathbf p_0\!\cdot\!\mathbf r_0+I_p t_0
-\operatorname{FP}\!\int_{t_0}^{\infty}
\left[
\frac{p^2(t)}{2}+V(\mathbf r(t))
-\mathbf r(t)\!\cdot\!\nabla V(\mathbf r(t))
\right]dt.
$$

Here $\operatorname{FP}$ means that, after the pulse, only the divergent phase
common to trajectories arriving in the same asymptotic-momentum cell is
removed. The finite trajectory-dependent Coulomb remainder must be retained.
The reference QTMC comparator is defined at the same level by

$$
\Phi_{\mathrm{QTMC}}
=I_p t_0
-\operatorname{FP}\!\int_{t_0}^{\infty}
\left[\frac{p^2(t)}{2}+V(\mathbf r(t))\right]dt.
$$

Equivalent action conventions may move endpoint or common asymptotic terms
between expressions, but they must reproduce the same relative phase in each
final-momentum cell for the stated SCTS or QTMC branch.

For an unbound trajectory, the final momentum $\mathbf k$ is its outgoing
asymptotic Coulomb momentum obtained from the post-pulse Kepler orbit, not its
instantaneous momentum at the end of the laser pulse. The graded observable is
the coherent asymptotic photoelectron momentum distribution obtained with an
unbiased treatment of the stated launch density. Its sampling, propagation,
coherent accumulation, variance reduction, asymptotic mapping and
regularization algorithms are student choices; an incoherent CTMC distribution
is not a substitute for this observable.

## Research problem

Assess the physical and quantitative credibility of the commissioned SCTS
description for the asymptotic photoelectron interference generated by the
stated fields. Use the defined QTMC branch and a converged single-active-
electron TDSE calculation at 800 nm as comparators, and determine the domain in
which any conclusions remain supported across the wavelength and launch cases.
Your report should explain what the calculated evidence does and does not
establish about the roles of the tunnel-launch input and the continuum phase.

The material derivation, numerical realization, evidence architecture,
parameter sampling, claim-bearing reductions and organization of the report
are for you to decide. No particular panel, crop, node count, energy or angular
reduction, intervention sequence, or expected model ordering is prescribed.
Do not assume that visual similarity by itself establishes model validity.

Root solver, sampling strategy, trajectory count, momentum binning, trajectory
integrator, tolerances, Coulomb finite-part implementation, asymptotic matching,
quantum-reference discretization, transforms, normalization, display scale and
plotting domain are student choices. State the choices that affect your claims
and provide evidence sufficiently reliable and reconstructible to support them.
A common overall normalization may be chosen and declared; absolute yield is
not a grading target unless independently established by your calculation.

Do not add multielectron, relativistic, focal-volume, macroscopic-propagation or
detector-response physics to the required production model. Clearly separated
exploratory extensions are permitted but cannot replace the stated commission.

## Submission and resources

Submit one self-contained scientific report and an auditable offline
calculation package containing the task-specific derivation, code, 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. No fixed file tree, filename scheme, panel count 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.