# Timing information in molecular photoionization

## Background

Photoelectron wave packets emitted from the two regions of an aligned
diatomic ion can interfere. When a short, high-energy pulse propagates across
the molecule, relative emission timing may be encoded in the final
molecule-frame signal. The same angular structure can also be reshaped by the
state-dependent photoionization response, so a numerical shift is not by
itself a timing measurement.

## Research problem

Using the physical model below, determine what quantitative information about
the relative birth time of the two emitted wave packets, if any, can be
reliably inferred from the final photoelectron signal. Apply the investigation
to the lowest symmetric and first antisymmetric initial states and determine
whether the evidence supports an initial-state dependence.

How the inference is constructed and substantiated is for you to decide.
Establish its physical meaning and identifiability from the computed evidence
rather than assuming in advance that a particular angular feature is a valid
clock.

## Physical inputs

Use atomic units and the two-dimensional soft-core $\mathrm H_2^+$ model below.
Work in coordinates $(z,x)$ with nuclei at $z=\pm R/2$, the molecular and
pulse-propagation axes along $\hat{\mathbf z}$, and pulse polarization along
$\hat{\mathbf x}$. The field-free potential is

$$
V(z,x)=-\frac{1}{\sqrt{(z-R/2)^2+x^2+a}}
-\frac{1}{\sqrt{(z+R/2)^2+x^2+a}}.
$$

Use $R=2$ and $a=4.33$. The two initial states are the lowest symmetric and
first antisymmetric eigenstates of this same field-free Hamiltonian.

Retain propagation of the pulse across the molecule. Its finite-support vector
potential is

$$
\begin{aligned}
\mathbf A(z,t)={}&A_0
\sin^2\!\left[\frac{\omega_e}{c}(z+a_0)-\omega_e t\right]
\sin\!\left[\frac{\omega}{c}(z-ct)\right]\hat{\mathbf x}\\
&\times\left[
\theta_H\!\left(z-ct+\frac{\pi c}{\omega_e}+a_0\right)
-\theta_H(z-ct+a_0)
\right],
\end{aligned}
$$

where $\omega_e=\omega/(2N_{\mathrm{cyc}})$. Use

| Quantity | Value |
|---|---:|
| Peak intensity | $1.0\times10^{14}\ \mathrm{W\,cm^{-2}}$ |
| Central wavelength | $5$ nm |
| Pulse duration | $N_{\mathrm{cyc}}=30$ cycles |
| Initial pulse displacement | $a_0=10$ a.u. |

Determine $\omega$ and $A_0$ using conventions consistent with a standard
velocity-gauge nondipole TDSE. State the Fourier, momentum-axis, angular and
delay-sign conventions used.

## Method boundary

A standard velocity-gauge nondipole TDSE treatment is required. All
task-specific analytical, numerical and evidential constructions are yours.
Any comparison between the two initial states must use the same operational
definition or demonstrate that the definitions are equivalent.

You are responsible for establishing that the computed evidence is
sufficiently reliable and physically discriminating for the conclusion made.
No numerical implementation, post-processing recipe, display convention or
validation procedure is prescribed. Absolute photoelectron yield is not a
grading target.

## Submission

Submit one self-contained scientific report and an offline reproduction
package containing the derivation, executable code, reconstructible final
photoelectron data for both initial states, the data underlying any
quantitative timing 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. The choice and number of figures are yours. No fixed
file tree, filename scheme, complete exploratory history or examiner-selected
diagnostic is required.

The active examination resources and limits are defined in the companion
`resource_limits.json`. All retained production calculations must run
within that fixed allocation and active wall time. More expensive modelling
does not receive 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.