cases/case26/task.md

# Ellipticity-dependent high-harmonic emission from graphene

## Background

Elliptically polarized light can drive electrons through a two-dimensional
crystal in ways that have no direct gas-phase analogue. In graphene, the
emitted high-harmonic signal may therefore contain information about the
driven band dynamics.

Consider undoped monolayer graphene in a nearest-neighbour, two-band
tight-binding model. Let

$$
\mathbf a_1=\frac{a}{2}(\sqrt{3},1),\qquad
\mathbf a_2=\frac{a}{2}(\sqrt{3},-1),
$$

with $a=2.46\ \mathring{\mathrm A}$. Use the standard nearest-neighbour
two-band graphene Hamiltonian with hopping energy $\gamma=3.03\ \mathrm{eV}$
and work in the Houston basis, with $\mathbf q=\mathbf k+\mathbf A(t)$ and
$\mathbf F(t)=-\partial_t\mathbf A(t)$. Work in atomic units after conversion
of the supplied dimensional quantities. Use an undoped 300 K Fermi-Dirac
initial state, include an interband dephasing time $T_2=35\ \mathrm{fs}$, and
neglect longitudinal population relaxation.

The driving field is

$$
F_x(t)=\frac{F_0}{\sqrt{1+\epsilon^2}}
e^{-2\ln2\,(t/t_d)^2}\cos(\omega_0t),
$$

$$
F_y(t)=\frac{\epsilon F_0}{\sqrt{1+\epsilon^2}}
e^{-2\ln2\,(t/t_d)^2}\sin(\omega_0t),
$$

with $0\leq\epsilon\leq1$, central wavelength $4.7\ \mu\mathrm m$,
$t_d=160\ \mathrm{fs}$, $F_0=8.2348\times10^6\ \mathrm{V\,cm^{-1}}$,
and zero carrier-envelope phase. Fix the major axis along $k_x$.

## Research problem

Determine how the fifth-, seventh-, and ninth-harmonic responses of this model
depend on the driver ellipticity. Establish, using a physically discriminating
evidence package of your own design, which microscopic part or parts of the
model account for any nontrivial ellipticity dependence you find. Explain the
physical and numerical evidence on which your conclusion rests and delimit the
scope supported by that evidence.

Your evidence design, number of plots, intermediate diagnostics, and order of
argument are for you to decide. No particular current decomposition,
momentum-space intervention, or source-style panel sequence is required.

## Method boundary

Use the stated two-band Houston-basis density-matrix model over a complete
reciprocal primitive cell. All band, optical-coupling, and emission quantities
must be mutually consistent with the same eigenstates and gauge.

Construct the physical total current coherently in the time domain. For a
two-component current, define the total spectral strength by

$$
I(\omega)\propto
|\omega\widetilde J_x(\omega)|^2+
|\omega\widetilde J_y(\omega)|^2.
$$

Obtain a harmonic yield by integrating a declared neighbourhood of the
relevant harmonic peak. For the principal ellipticity comparison, normalize
each harmonic order to its own value at $\epsilon=0$. If you compare resolved
current components, keep them on one common declared scale; do not normalize
each component independently in a way that erases their relative magnitude.
Absolute spectral normalization is not a grading target.

The reciprocal-space grid, time grid, propagator, finite-record interval, FFT
window, zero padding, harmonic integration bandwidth, convergence strategy,
and display convention are student choices. State them and show that the
physical claims are supported at the resolution used. Physically equivalent
choices are accepted; no reference-program setting is prescribed.

Do not add doping, additional bands, electron-hole interactions, macroscopic
propagation, or a fitted relaxation model to the required calculation. Such
extensions may be discussed only as clearly separated exploratory work.

## Submission and resources

Submit one self-contained scientific report and an auditable offline
calculation package. Include the 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/environment
information. No fixed file tree, filename scheme, figure 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.