cases/case20/task.md

Diagnosing correlated resonances in high-harmonic emission

Background and research question

Autoionizing states embedded in the continuum can influence high-harmonic generation through correlated excitation and emission pathways. Distinguishing the roles of bright and dark resonances requires their response to be assessed within the correlated dynamics rather than assumed from their labels.

For the correlated one-dimensional helium model below, determine how resonances with the specified bright and dark autoionizing states affect the high-harmonic response as the driving frequency is tuned through each resonance. Establish what the computed amplitude and phase information reveals about the relation between the two resonance classes and what temporal information, if any, can be inferred for the resonant XUV emission. State the strongest conclusion supported by your evidence and its physical scope.

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

Physical system and research boundary

Use atomic units unless stated otherwise. Treat two indistinguishable electrons in one spatial dimension with the correlated TDSE

$$i\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\Psi }(x,y,t)=[-\frac{1}{2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}-\frac{1}{2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}+V(x,y)+(x+y)E(t)]\mathrm{\Psi }(x,y,t),i\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\backslash Psi(x,y,t)=\; \backslash left[-\backslash frac\{1\}\{2\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}\; -\backslash frac\{1\}\{2\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; y^2\}\; +V(x,y)+(x+y)E(t)\backslash right]\backslash Psi(x,y,t),$$

where

$$V(x,y)=-\frac{2}{\sqrt{x^2+a^2}}-\frac{2}{\sqrt{y^2+a^2}}+\frac{1}{\sqrt{(x-y{)}^2+b^2}},a=\frac{1}{\sqrt{2}},b=\frac{1}{\sqrt{3}}\mathrm{.}V(x,y)=-\backslash frac\{2\}\{\backslash sqrt\{x^2+a^2\}\}\; -\backslash frac\{2\}\{\backslash sqrt\{y^2+a^2\}\}\; +\backslash frac\{1\}\{\backslash sqrt\{(x-y)^2+b^2\}\},\; \backslash qquad\; a=\backslash frac\{1\}\{\backslash sqrt\{2\}\},\backslash quad\; b=\backslash frac\{1\}\{\backslash sqrt\{3\}\}.$$

Start from the symmetric field-free ground state. The target resonances are the even-parity dark $(1,1)$ autoionizing state near a 14-photon condition and the odd-parity bright $(1,2)$ autoionizing state near a 17-photon condition.

Use a linearly polarized field with peak intensity $8.0\times10^{14}\ \mathrm{W,cm^{-2}}$ and a fundamental photon energy near $3\ \mathrm{eV}$. The pulse has a four-cycle turn-on, an eight-cycle plateau and a four-cycle turn-off. As a task convention resolving the pulse wording, use a $\sin^2$ field-amplitude envelope on each ramp and a cosine carrier. Tune the fundamental frequency through both stated multiphoton-resonance neighbourhoods over domains adequate for the conclusions you make.

The correlated two-electron dynamics and both resonance classes are part of the research commission. Standard choices for the bound-state preparation, propagation, absorbing boundary, construction of the complex harmonic response, phase convention, sampling, convergence assessment and evidence representation are yours. Define the conventions needed to make your results reconstructible. No grid, box, absorber shape, time step, frequency spacing, transform window, phase-unwrapping algorithm, plotting range or fixed figure set is prescribed. Absolute harmonic strength is not graded when an overall normalization remains unspecified; relative structures used in a claim must be compared under a consistent normalization.

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.

The available CPU allocation, candidate active deadline, node, network, memory and scratch limits are specified in the accompanying resource_limits.json. Complete the derivation, implementation, calculation, validation, interpretation and final-file preparation within these resource limits. More expensive modelling 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.