cases/case09/task.md

# High-harmonic emission from a one-dimensional periodic system

## Background and research question

A coordinate-space finite-basis calculation can generate a smooth radiation
spectrum without necessarily preserving all of the relevant physics of a
periodic system. For the model and driving conditions below, assess whether a
coordinate-space B-spline treatment provides a physically reliable account of
high-harmonic emission and determine what physical conclusions are supported
by your calculations.

Design the derivation, computation, tests, evidence and scientific argument
needed to answer this question. Do not assume in advance that the method is
reliable or that a single interpretation applies to the full response.

## Physical system

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

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

under the electric-dipole approximation. The initial physical state belongs
to band 2 at crystal momentum $k=0$.

Use a linearly polarized finite-support field

$$
E(t)=E_0\cos^2\!\left(\frac{\pi t}{T}\right)
\cos(\omega t+\varphi),
\qquad |t|<\frac{T}{2},
\qquad T=8\frac{2\pi}{\omega},
$$

and set it to zero outside the support. The carrier-phase and time-origin
convention is not fixed; declare one and use the same physical field in every
calculation being compared.

| Case | Peak intensity | Wavelength |
|---|---:|---:|
| A | $8.087\times10^{11}\ \mathrm{W\,cm^{-2}}$ | $3.2\ \mu\mathrm m$ |
| B | $4.5\times10^{11}\ \mathrm{W\,cm^{-2}}$ | $3.2\ \mu\mathrm m$ |

Determine the field amplitude and carrier frequency using standard
atomic-unit conversions and state the conventions used.

## Research boundary

Use a coordinate-space B-spline TDSE as the primary method. Standard
periodic-state, analytical or numerical methods may be used as needed. All
other modelling, numerical, diagnostic and presentation choices are yours.
Establish that the evidence supporting each retained claim is sufficiently
reliable and reproducible. Absolute spectral yield is not a grading target.

## Submission and resources

Submit one self-contained scientific report and an offline reproduction
package containing the producing code, numerical data underlying the retained
evidence, 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 scientific figure set, file tree or exploratory history
is required.

The active examination resources and limits are defined in the companion
`resource_limits.json`. All retained production calculations must run
within that fixed allocation and active wall time. More expensive modelling
does not receive 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.