cases/case15/task.md

# Resolving molecular photoemission timing from nondipole momentum signals

## Background and research question

In single-photon ionization of a diatomic target, a propagating light field
does not reach spatially separated parts of the molecule simultaneously. The
resulting phase information may be encoded in the photoelectron momentum
distribution, but the same distribution also contains structure arising from
the molecular bound state and from ordinary two-centre interference. A
momentum displacement alone is therefore not yet a timing measurement.

For the fixed-nuclei molecular ion and XUV field below, determine whether the
calculated photoelectron signal supports a physically meaningful inference of
the relative photoemission time associated with the two molecular centres.
Establish how the inference depends on internuclear separation and molecular
orientation, when an interference-based timing interpretation is adequate,
and what the agreement or disagreement of independent quantum descriptions
actually establishes. If the evidence does not support a unique timing claim,
state the strongest narrower conclusion that it does support.

Design the task-specific derivation, computation, evidence and scientific
argument needed to answer this question. Do not assume the outcome in advance.

## Physical system

Use atomic units unless stated otherwise. Treat fixed-nuclei $\mathrm{H}_2^+$
in its field-free gerade ground state. The laser propagates along $z$ and is
circularly polarized in the $x$-$y$ plane. If $\beta$ is the angle between the
molecular axis and $z$, use the light-cone time

$$
\eta=t-\frac{z}{c}
$$

and the vector potential

$$
\mathbf A(\eta)=f(\eta)A_0
\left[\cos(\omega\eta)\,\mathbf e_x+
\sin(\omega\eta)\,\mathbf e_y\right].
$$

Use $\omega=10$ and a pulse duration $T=20(2\pi/\omega)$. As a task
convention, use

$$
f(\eta)=\sin^2\!\left(\frac{\pi\eta}{T}\right),
\qquad 0\leq\eta\leq T,
$$

with $f=0$ outside that interval. Use $c=137.036$ in atomic units.

The weak-field nondipole SFA/NSFA branch uses $A_0=0.01$. Use at least the
compact case $R=2$ and the separated-centre case
$R=10$, and investigate enough additional separations and orientations in
$0\leq\beta\leq\pi/2$ to support the scope of your conclusion. For a
weak-field orientation study use $A_0=0.02$.

## Research boundary

Use a nondipole strong-field approximation that retains the light-cone phase,
together with an independent fixed-nuclei nondipole time-dependent
Schroedinger treatment of the central physical signal. For an aligned
representative TDSE calculation, a two-dimensional $(x,z)$ realization may
use the attractive two-centre soft-core potential

$$
V_R(x,z)=-\frac{1}{\sqrt{x^2+(z-R/2)^2+\alpha(R)}}
-\frac{1}{\sqrt{x^2+(z+R/2)^2+\alpha(R)}},
$$

where $\alpha(R)$ is determined by matching the reference field-free
$\mathrm{H}_2^+$ electronic energy at the same $R$. That reference energy may
be obtained from a qualified field-free molecular calculation or eligible
standard literature. If TDSE evidence is used away from $\beta=0$, rotate the
two-centre geometry consistently; the orientation study may instead be
carried by the weak-field nondipole branch. The representative TDSE comparison
uses $A_0=1.0$. This is a separate one-photon-regime calculation, not a
matched-intensity comparison with the weak-field approximation.

Determine the field-free state and $I_p(R)$ consistently within each declared
model. Standard LCAO, numerical bound-state and mathematically equivalent
routes are permitted if their approximations and comparison boundary are made
explicit. The observable definitions, extraction rule, numerical method,
sampling, convergence strategy and evidence representation are your choices.
No grid, box, absorber, propagation time, fit window, root solver, momentum
range or fixed figure set is prescribed. Absolute probability is not graded
when only a common or per-case normalization is defined; do not use such a
normalization to support a relative-yield claim.

## Submission and resources

Submit one self-contained scientific report and an auditable offline
calculation package containing the producing code, reconstructible numerical
data for 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 argument and the choice and
number of figures are yours. Evidence must be sufficiently reliable and
reproducible for the claims made, but no particular validation procedure 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.