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

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

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

$$
f(t)=\sin^2\!\left(\frac{\pi t}{T}\right),
\qquad 0\le t\le T.
$$

Use the conventional reference field

$$
\mathbf E_{\mathrm{ref}}(t)=F_0 f(t)\sin(\omega_0 t)\,\hat{\mathbf x}
$$

and the delay-controlled shaped field

$$
\mathbf E_{\mathrm{sh}}(t;t_d)=F_0 f(t)
\left[
\frac{\sin\!\left(\omega_0(t-t_0-t_d)\right)}
     {\omega_0(t-t_0-t_d)}
-
\frac{\sin\!\left(\omega_0(t-t_0)\right)}
     {\omega_0(t-t_0)}
\right]\hat{\mathbf x},
\qquad t_0=2.14\tau,
$$

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](../../submission_requirements.md) for the common submission entry points.