| # 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. |
|
|
