cases/case17/task.md

# Gap and drive dependence of high-harmonic emission in two quantum models

## Background

The excitation gap and light-matter coupling are both central scales in
off-resonant high-harmonic generation. Their joint influence can be studied in
a dissipative two-level model and then compared with a two-band semiconductor
model that includes dispersion and momentum aggregation.

## Research problem

Determine how the excitation gap and peak light-matter coupling jointly
organize the polarization-based high-harmonic response of the two model
families below. Establish the physical origin of the gap dependence found in
the two-level model. Then assess which parts of that
explanation transfer to the semiconductor calculation and state the supported
scope of the conclusion.

How you construct and substantiate the investigation is for you to decide.
Ground every material claim in calculated evidence, and do not assume the
direction of the gap dependence or the outcome of the transfer test in advance.

## Physical models and inputs

For the two-level model, use units with \(e=\hbar=d=1\), where \(d\) is the
constant transition dipole. Take the energy unit to be
\(\hbar\omega_u=1~\mathrm{eV}\).

For the two-level branch, use

$$
H(t)=
\begin{pmatrix}
-\Delta/2 & -\Omega(t)\\
-\Omega(t) & \Delta/2
\end{pmatrix},
\qquad
\Omega(t)=\Omega_0\cos(\omega_0 t)\exp(-t^2/\tau^2).
$$

Begin in the lower field-free level. Include phenomenological longitudinal
relaxation toward that level at rate \(\gamma_L\) and transverse decay of the
coherence at rate \(\gamma_T\). Use

| Quantity | Value or condition |
|---|---|
| Carrier frequency | \(\omega_0=0.3\) |
| Pulse width | \(\tau=8.5\pi\) |
| Reference relaxation rates | \(\gamma_L=2.0\), \(\gamma_T=1.0\) |
| Excitation regime | Keep the study and its claims in the off-resonant regime, \(\omega_0\ll\Delta\) |
| Gap and coupling study | Sample a domain broad enough to compare coupling below, near, and above the gap and to support every global claim made |

The population difference and coherence may be derived from the density-matrix
equation. The radiating polarization is the expectation value of the dipole,
which is proportional to the real part of the coherence under this convention.

For the transfer study, use units with \(e=\hbar=a=1\), where \(a\) is the
lattice constant. Work at low doping on the two-dimensional square Brillouin
zone \(-\pi\leq k_x,k_y\leq\pi\), and use equal electron and hole dispersions

$$
E_{\mathbf k}^{e/h}=\frac{\Delta}{2}+\frac{k^2}{2m},
\qquad m=2.0.
$$

Let \(\theta_{\mathbf k}=\operatorname{atan2}(k_y,k_x)\), and introduce the
Nambu spinor
\(\Psi_{\mathbf k}=(\alpha_{\mathbf k},\beta^\dagger_{-\mathbf k})^T\).
Up to an irrelevant scalar term, use the single-\(\mathbf k\) Hamiltonian

$$
h_{\mathbf k}(t)=
\begin{pmatrix}
E_{\mathbf k}^{e}+\Omega(t)\cos\theta_{\mathbf k}
&
i\Omega(t)\sin\theta_{\mathbf k}
\\
-i\Omega(t)\sin\theta_{\mathbf k}
&
-E_{\mathbf k}^{h}-\Omega(t)\cos\theta_{\mathbf k}
\end{pmatrix}.
$$

Here \(\alpha_{\mathbf k}\) annihilates a conduction-band electron and
\(\beta_{-\mathbf k}\) annihilates a hole. Begin with an empty conduction band,
no holes, and zero interband polarization. Derive the corresponding standard
semiconductor Bloch equations for the electron and hole populations and one
complex interband polarization \(P_{\mathbf k}\). Use \(P_{\mathbf k}\)
consistently in the dynamics and observable. Add longitudinal population decay
and transverse polarization decay with \(\gamma_L=2.0\) and \(\gamma_T=1.0\).
Define the macroscopic polarization by

$$
p(t)=-\sum_{\mathbf k}
\left\langle\frac{\partial h_{\mathbf k}(t)}
{\partial\Omega(t)}\right\rangle.
$$

For this branch use the same Gaussian carrier form with \(\tau=16\pi\) and
\(\omega_0=0.3\). State all basis, phase, and momentum-sum conventions used.

## Observable and study boundary

Construct the HHG spectrum from the Fourier transform of the corresponding
polarization. Use the fifth-harmonic spectral intensity as the common
quantitative reference between models. Other evidence may be introduced when
it has a clear discriminating function.

The overall intensity scale is arbitrary. Declare the transform,
normalization, and comparison conventions, and keep them common wherever
relative magnitudes support a claim. Absolute source-scale intensity is not a
target.

All momentum quadratures, time records, grids, propagators, transform details,
windowing choices, parameter sampling, plotting conventions, and convergence
or stability tests are yours. Document them and establish that the reported
physical conclusions are reliable. No particular solver, grid, diagnostic,
number of figures, or validation procedure is prescribed.

## Submission

Submit one self-contained scientific report and an auditable offline
calculation package containing the code, 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 evidence package is yours.

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 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.