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

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

and the vector potential

$$\mathbf{A}(\eta )=f(\eta )A_0[\cos (\omega \eta ){\mathbf{e}}_x+\sin (\omega \eta ){\mathbf{e}}_y]\mathrm{.}\backslash mathbf\; A(\backslash eta)=f(\backslash eta)A\_0\; \backslash left[\backslash cos(\backslash omega\backslash eta)\backslash ,\backslash mathbf\; e\_x+\; \backslash sin(\backslash omega\backslash eta)\backslash ,\backslash mathbf\; e\_y\backslash right].$$

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

$$f(\eta )={\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{\pi \eta }{T}\right),0\le \eta \le T,f(\backslash eta)=\backslash sin^2\backslash !\backslash left(\backslash frac\{\backslash pi\backslash eta\}\{T\}\backslash right),\; \backslash qquad\; 0\backslash leq\backslash eta\backslash leq\; T,$$

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

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

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 for the common submission entry points.