cases/case36/task.md

Temporal control of solid-state high-harmonic emission with a shaped driver

Physical setting

Consider a one-dimensional crystal described in atomic units by the periodic single-electron potential

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

The crystal is driven along its periodic direction by a linearly polarized $3.2\ \mu\mathrm{m}$ pulse of duration $T=5\tau$, where $\tau=2\pi/\omega_0$. The peak-field scale $F_0$ corresponds to a peak intensity of $6.0\times10^{11}\ \mathrm{W,cm^{-2}}$, and

$$f(t)={\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{\pi t}{T}\right),0\le t\le T\mathrm{.}f(t)=\backslash sin^2\backslash !\backslash left(\backslash frac\{\backslash pi\; t\}\{T\}\backslash right),\; \backslash qquad\; 0\backslash le\; t\backslash le\; T.$$

Use the conventional reference field

$${\mathbf{E}}_{\mathrm{r}\mathrm{e}\mathrm{f}}(t)=F_{0}f(t)\sin ({\omega }_{0}t)\widehat{\mathbf{x}}\backslash mathbf\; E\_\{\backslash mathrm\{ref\}\}(t)=F\_0\; f(t)\backslash sin(\backslash omega\_0\; t)\backslash ,\backslash hat\{\backslash mathbf\; x\}$$

and the delay-controlled shaped field

$${\mathbf{E}}_{\mathrm{s}\mathrm{h}}(t;t_d)=F_{0}f(t)[\frac{\sin \text{\hspace{0.17em}⁣}({\omega }_0(t-t_0-t_d))}{{\omega }_0(t-t_0-t_d)}-\frac{\sin \text{\hspace{0.17em}⁣}({\omega }_0(t-t_0))}{{\omega }_0(t-t_0)}]\widehat{\mathbf{x}},t_0=2.14\tau ,\backslash mathbf\; E\_\{\backslash mathrm\{sh\}\}(t;t\_d)=F\_0\; f(t)\; \backslash left[\; \backslash frac\{\backslash sin\backslash !\backslash left(\backslash omega\_0(t-t\_0-t\_d)\backslash right)\}\; \{\backslash omega\_0(t-t\_0-t\_d)\}\; -\; \backslash frac\{\backslash sin\backslash !\backslash left(\backslash omega\_0(t-t\_0)\backslash right)\}\; \{\backslash omega\_0(t-t\_0)\}\; \backslash right]\backslash hat\{\backslash mathbf\; x\},\; \backslash qquad\; t\_0=2.14\backslash tau,$$

with each removable singularity understood by continuity. Use one consistent field-vector-potential convention throughout all comparisons.

The central study is the coherent independent-electron response with the two lowest bands initially filled; these bands form the valence manifold for this task. Quantitative dephasing models are outside the required scope. A velocity- gauge Bloch-basis propagation or a demonstrably equivalent gauge representation may be used. The primary observable is the emitted high-harmonic response constructed from the total current over the occupied initial states and the Brillouin zone. A common normalization is acceptable; absolute spectral intensity is not a grading target.

Research problem

Determine whether the delay-controlled shaped driver provides genuine temporal control of the high-harmonic cutoff response of this crystal, rather than merely changing an overall signal scale. Establish the comparison with the conventional reference pulse, determine how the cutoff energy, its emission time, and a defensibly defined harmonic yield depend on $t_d$, and explain the observed behavior in terms of the field-driven band dynamics.

Choose and justify a delay domain broad enough to support the dependence and scope you claim. Your evidence must make the physical cases and observable definitions auditable, but the derivation, numerical representation, cutoff and yield definitions, time-frequency or equivalent timing diagnostic, controls, reliability strategy, plotting choices, and report organization are for you to decide. If you infer a functional law, state how it was obtained and where the evidence supports or fails to support it.

No particular number, order, or layout of figures is prescribed. Crystal- momentum sampling, basis or plane-wave truncation, time step, propagator, Fourier estimator or taper, time-frequency resolution, visibility threshold, interpolation, normalization, and plot range are student choices. State the choices material to your claims and submit reconstructible numerical data.

Submission and resources

Submit one self-contained scientific report and an auditable offline calculation package containing the task-specific 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 are binding. No separate aggregate CPU-hour limit applies. Greater computational expense 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.