cases/case28/task.md

Field-dependent high-harmonic generation in a periodic crystal

Background

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.

Consider a one-dimensional independent-electron crystal in atomic units, with

$$V(x)=-V_0[1+\cos \left(\frac{2\pi x}{a}\right)],V_0=0.37,a=8.V(x)=-V\_0\backslash left[1+\backslash cos\backslash left(\backslash frac\{2\backslash pi\; x\}\{a\}\backslash right)\backslash right],\; \backslash qquad\; V\_0=0.37,\backslash quad\; a=8.$$

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,

$$E(t)=E_0{\sin }^4\left(\frac{\pi t}{T_p}\right)\sin \left[\omega (t-\frac{T_p}{2})\right],0\le t\le T_p,E(t)=E\_0\backslash sin^4\backslash left(\backslash frac\{\backslash pi\; t\}\{T\_p\}\backslash right)\; \backslash sin\backslash left[\backslash omega\backslash left(t-\backslash frac\{T\_p\}\{2\}\backslash right)\backslash right],\; \backslash qquad\; 0\backslash leq\; t\backslash leq\; T\_p,$$

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.

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.

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.

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.

Submission

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.

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.

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.