# Interpreting high-harmonic emission from a driven periodic system

## Background

The high-harmonic spectrum of a periodic system can encode electron dynamics
that are not directly visible in the driving field. Finite pulse shape and
duration can also alter how that information appears in the emitted response.
A useful physical account must therefore connect calculated emission to the
underlying dynamics without presupposing which structures or mechanisms will
be found.

## Research problem

Determine what the significant structures in the high-harmonic response of the
symmetric one-dimensional periodic system below reveal about its electron
dynamics. Establish whether one physically consistent account can explain the
spectral organization you find, its symmetry properties, and the differences
among the stated pulse cases.

Support the conclusion with your own calculation of the emitted response and
with whatever electron-dynamical evidence is needed to distinguish the
proposed explanation from plausible alternatives. Assess the scope of the
inference from the evidence rather than assuming the mechanism in advance. The
derivation, numerical realization, diagnostics, figures and organization of
the scientific argument are yours.

## Physical inputs

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

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

Prepare the electron initially in the field-free top valence-band state at
$k=0$, corresponding to the second Bloch band in ascending field-free energy
order for the stated potential, and use a linearly polarized field

$$
E(t)=E_0 f(t)\cos\!\left[\omega\left(t-\frac{\tau}{2}\right)\right].
$$

The wavelength is $3.2\,\mu\mathrm{m}$ and the peak intensity is
$8.77\times10^{11}\,\mathrm{W/cm^2}$. Determine $E_0$ and $\omega$ using a
consistent standard conversion to atomic units, and state the constants used.
Study all three pulse cases:

| Case | Envelope on $0\leq t\leq\tau$ | Total duration |
|---|---|---:|
| A | $f(t)=1$ | 15 optical cycles |
| B | $f(t)=\cos^2\!\left[\pi(t-\tau/2)/\tau\right]$ | 10 optical cycles |
| C | $f(t)=\cos^2\!\left[\pi(t-\tau/2)/\tau\right]$ | 15 optical cycles |

For this task, the field is zero outside the stated pulse interval. State your
gauge, band-index, Fourier-sign, harmonic-order and normalization conventions.
Any common arbitrary display scale must be declared; absolute emitted intensity
is not a grading target.

## Model and scope

Use a standard one-dimensional single-active-electron TDSE treatment of the
specified periodic system. You may work in coordinate space, a Bloch-state
representation or another demonstrably equivalent formulation. Construct the
high-harmonic response from a standard current, dipole-velocity or
dipole-acceleration observable, and account for any frequency weighting when it
matters to a scientific claim.

All numerical and diagnostic choices not fixed above are yours. In particular,
no spatial or momentum grid, box, band truncation, propagator, time step,
absorber, spectral window, frequency range, peak estimator or time-frequency
method is prescribed. You are responsible for making the reported evidence
sufficiently reliable and reproducible for the conclusions it supports, but no
particular convergence sequence, residual or validation procedure is required.

## Submission

Submit one self-contained scientific report and an auditable offline
calculation package. Include the task-specific derivation actually used, source
code, an executable command or script, machine-readable data underlying every
material claim, 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 choice and number of supporting plots and controls are yours;
no source-figure layout, fixed crop, annotation style, file tree 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. A more expensive implementation 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.