| # Diagnosing correlated resonances in high-harmonic emission |
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| ## Background and research question |
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| 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. |
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| 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. |
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| Design the task-specific derivation, calculation, evidence and scientific |
| argument needed to answer this question. Do not assume the outcome in advance. |
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| ## Physical system and research boundary |
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| Use atomic units unless stated otherwise. Treat two indistinguishable |
| electrons in one spatial dimension with the correlated TDSE |
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|
| $$ |
| i\frac{\partial}{\partial t}\Psi(x,y,t)= |
| \left[-\frac{1}{2}\frac{\partial^2}{\partial x^2} |
| -\frac{1}{2}\frac{\partial^2}{\partial y^2} |
| +V(x,y)+(x+y)E(t)\right]\Psi(x,y,t), |
| $$ |
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|
| where |
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|
| $$ |
| V(x,y)=-\frac{2}{\sqrt{x^2+a^2}} |
| -\frac{2}{\sqrt{y^2+a^2}} |
| +\frac{1}{\sqrt{(x-y)^2+b^2}}, |
| \qquad |
| a=\frac{1}{\sqrt{2}},\quad b=\frac{1}{\sqrt{3}}. |
| $$ |
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|
| 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. |
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| 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. |
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| 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. |
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| ## Submission and resources |
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| 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. |
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| 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. |
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| ## Reference and integrity policy |
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| 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. |
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| 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. |
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| See [submission requirements](../../submission_requirements.md) for the common submission entry points. |
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