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

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

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

$$
U_{\epsilon}(x,t)=xE(t)+\epsilon x^2E(t),
$$

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:

$$
E(t)=E_0\cos^2\!\left(\frac{\pi t}{T}\right)\cos(\omega t),
\qquad |t|\leq \frac{T}{2},
$$

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

$$
\epsilon=0,\qquad 0.0002,\qquad 0.0005
$$

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