cases/case12/task.md

Selective high-energy harmonic response in a weakly inhomogeneous field

Background and research question

In a periodic solid-state model, the high-energy part of a high-harmonic response can be much weaker than its lower-energy part, while simply raising the incident intensity may be incompatible with a material damage threshold. A weak spatial variation of the driving field offers a possible alternative, but a brighter spectrum alone does not show whether the useful response has been selectively changed or merely rescaled.

For the model and field family below, determine whether weak spatial inhomogeneity can produce a selective and physically useful change of the high-energy harmonic response. If it can, establish the microscopic origin and decide whether the calculated evidence supports a change in the emission dynamics rather than only a common intensity rescaling. If it cannot, delimit the negative conclusion with equally strong evidence.

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

Physical system

Use atomic units unless stated otherwise. Treat one active electron in the one-dimensional periodic potential

$$V(x)=-V_0[1+\cos \left(\frac{2\pi x}{a_0}\right)],V_0=0.37,a_0=8.V(x)=-V\_0\backslash left[1+\backslash cos\backslash left(\backslash frac\{2\backslash pi\; x\}\{a\_0\}\backslash right)\backslash right],\; \backslash qquad\; V\_0=0.37,\; \backslash qquad\; a\_0=8.$$

The initial physical state is the field-free $k=0$ state at the top of the valence band, at the minimum band gap. Use the source-model length-gauge interaction

$$U_{ϵ}(x,t)=xE(t)+ϵx^{2}E(t),U\_\{\backslash epsilon\}(x,t)=xE(t)+\backslash epsilon\; x^2E(t),$$

where $E(t)$ is a linearly polarized pulse. To close the source-unspecified carrier convention without selecting a numerical method, use the following task convention:

$$E(t)=E_0{\cos }^2\text{\hspace{0.17em}⁣}\left(\frac{\pi t}{T}\right)\cos (\omega t),\mathrm{\mid }t\mathrm{\mid }\le \frac{T}{2},E(t)=E\_0\backslash cos^2\backslash !\backslash left(\backslash frac\{\backslash pi\; t\}\{T\}\backslash right)\backslash cos(\backslash omega\; t),\; \backslash qquad\; |t|\backslash leq\; \backslash frac\{T\}\{2\},$$

and $E(t)=0$ outside that interval, with peak intensity $8.77\times10^{11}\ \mathrm{W,cm^{-2}}$, wavelength $3200$ nm and $T=6(2\pi/\omega)$. Compare

$$ϵ=0,0.0002,0.0005\backslash epsilon=0,\backslash qquad\; 0.0002,\backslash qquad\; 0.0005$$

in the same atomic-unit coordinate convention. Use the same field, time origin and sign conventions in every compared case, and state the atomic-unit conversion used for $E_0$ and $\omega$.

Research boundary

Use a one-electron time-dependent Schrödinger treatment of the stated model. The representation, propagation method and scientific evidence package are your choices. Construct the emitted high-harmonic response from a physically consistent radiation observable. Any resolved or decomposed quantity used as evidence must be given a reconstructible mathematical definition; no particular decomposition is prescribed.

Absolute harmonic yield is not a grading target. Declare any common normalization and representation conventions used in a comparison. You are responsible for ensuring that the evidence is sufficiently reliable and reproducible to support the claims made, but no grid, box, solver, absorber, window, transform, survey range or validation procedure is prescribed.

Submission and resources

Submit one self-contained scientific report and an offline reproduction package containing the producing code, the numerical data underlying 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 report and the choice of evidence are yours. No fixed figure set, file tree, filename scheme or complete exploratory history 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.