| # Field-dependent high-harmonic generation in a periodic crystal |
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| ## Background |
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| An intense mid-infrared field can drive electrons through several bands of a |
| crystal. The resulting high-harmonic spectrum contains information about both |
| intraband motion and interband excitation and recombination. A central question |
| is whether changes in the spectrum as the field grows can be explained by |
| electron-hole trajectories in crystal momentum space. |
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|
| Consider a one-dimensional independent-electron crystal in atomic units, with |
|
|
| $$ |
| V(x)=-V_0\left[1+\cos\left(\frac{2\pi x}{a}\right)\right], |
| \qquad V_0=0.37,\quad a=8. |
| $$ |
|
|
| The two valence bands are initially filled throughout the first Brillouin zone. |
| Use an independent-electron TDSE in the velocity gauge for this periodic |
| crystal, with $E(t)=-\partial_t A(t)$. The spatial or basis representation is |
| your choice, provided it converges to the same Hamiltonian and initial state. |
| |
| Use a linearly polarized pulse of wavelength $\lambda=3200$ nm and total |
| support $T_p=99.66$ fs, |
|
|
| $$ |
| E(t)=E_0\sin^4\left(\frac{\pi t}{T_p}\right) |
| \sin\left[\omega\left(t-\frac{T_p}{2}\right)\right], |
| \qquad 0\leq t\leq T_p, |
| $$ |
|
|
| and $E(t)=0$ outside this interval, where $\omega=2\pi c/\lambda$. The stated |
| $T_p$ is the entire nonzero support of the pulse. |
| |
| ## Research problem |
| |
| Determine how the high-harmonic response of this crystal changes with the peak |
| field amplitude $E_0$, and establish whether a momentum-space trajectory model |
| can account for the material spectral regimes, their limiting photon energies, |
| and the timing of the emission. |
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|
| Establish the conclusion from your own auditable calculation of the stated |
| model, and distinguish what the evidence demonstrates from what remains |
| model-dependent. The derivation, computational route, evidence architecture, |
| controls, field domain, and organization of the report are for you to decide. |
| No particular number or layout of outputs is prescribed. |
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|
| Spatial and crystal-momentum representations, propagation algorithms, grids, |
| time steps, transforms, time-frequency windows, spectral estimators, cutoff |
| definitions, tolerances, and display ranges are student choices. State the |
| choices that affect your claims and provide evidence sufficiently reliable and |
| reconstructible to support them. Declare normalization and display scaling; |
| absolute spectral intensity is not a grading target. |
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| Do not add electron-electron interaction, dephasing, disorder, propagation |
| through a macroscopic sample, or phenomenological band modifications to the |
| required production calculation. Clearly separated exploratory extensions are |
| permitted, but they cannot replace the stated model. |
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| ## Submission |
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| Submit one self-contained scientific report and an offline reproduction |
| package containing the derivation, code, numerical data underlying all |
| 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. No fixed file tree, filename scheme, panel count, or complete |
| exploration history is required. |
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|
| Resource limits are supplied exclusively by the accompanying |
| `resource_limits.json`. Its fixed CPU allocation, candidate-active |
| wall time, network boundary and temporary-storage allowance are binding. |
| No separate aggregate CPU-hour limit applies. |
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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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