# Channel-resolved high-harmonic generation in driven two-band solids

## Background

High-harmonic generation (HHG) in solids can contain interband and intraband
responses produced by resonant and nonresonant electronic dynamics. In the
strong-field regime these contributions need not be independently observable,
because amplitudes from many excitation pathways contribute coherently to the
physical signal.

Consider an initially filled valence band and empty conduction band in a
one-dimensional, inversion-symmetric two-band solid. Work in atomic units,
neglect electron-hole interactions, and choose the Bloch-state gauge in which
the transition dipole is real. For an initial crystal momentum $K$, define
$K_t=K+A(t)$ and $F(t)=-\partial_t A(t)$. The exact density matrix satisfies

$$
i\partial_t\rho(K,t)=[H(K_t,t),\rho(K,t)],
\qquad
H(K_t,t)=\frac{1}{2}
\begin{pmatrix}
\varepsilon(K_t) & \Omega(K_t,t)\\
\Omega(K_t,t) & -\varepsilon(K_t)
\end{pmatrix},
$$

where $\Omega(K_t,t)=2F(t)d(K_t)$. Use

$$
\varepsilon(K)=E_g+\sum_n\Delta_n[1-\cos(nKa)],
\qquad
d(K)=d_0\frac{\varepsilon(0)}{\varepsilon(K)},
$$

and the finite pulse

$$
F(t)=F_0\exp[-(t/\tau_0)^2]\sin(\omega_0t),
\qquad T_0=\frac{2\pi}{\omega_0}.
$$

## Research problem

Determine whether a strong-field adiabatic-following description can give a
quantitatively and physically adequate account of HHG in the systems below,
and use the calculation to establish what can be inferred about the roles of
real and virtual electronic transitions as the material and driving field are
changed. Your conclusion must be supported by computed physical signals from
the stated cases and must distinguish claims established by the model from
claims that remain unresolved within its scope.

The derivation, numerical strategy, evidential representation, comparisons,
and order of argument are for you to design. Do not assume the validity of the
approximation or a channel mechanism in advance.

## Physical cases

| Case | Band and dipole parameters | Driving field |
|---|---|---|
| Model semiconductor | $E_g=0.129$, $\Delta_1=0.17$, $d_0=3.64$, $a=5.3$ | $F_0=0.002$, $\omega_0=0.015$, $\tau_0=6T_0$ |
| Model dielectric, long pulse | $E_g=0.32$, $(\Delta_1,\Delta_2,\Delta_3,\Delta_4)=(0.06,-0.0035,-0.001,-0.0007)$, $d_0=6.5$, $a=9.45$ | $F_0=0.012$, $\omega_0=0.06$, $\tau_0=6T_0$ |
| Model dielectric, field family | Same dielectric band and dipole parameters | $F_0=0.005$ and $0.012$, $\omega_0=0.06$, $\tau_0=3T_0$ |

## Physical and method boundary

The investigation is confined to direct propagation of the density-matrix
equation and a strong-field adiabatic-following (SFAF) treatment obtained in
the instantaneous adiabatic basis through second order in the corresponding
Dyson expansion. The task-specific derivation of computable amplitudes and
currents remains part of the work.

For the channel identity, let
$\lambda(K_t,t)=[\varepsilon^2(K_t)+|\Omega(K_t,t)|^2]^{1/2}$ and write the
SFAF transition amplitude as

$$
u(K,t)=\exp\!\left[-i\int_{-\infty}^{t}
\lambda(K_\tau,\tau)\,d\tau\right]v(K,t).
$$

The real/nonreal distinction is defined by filtering the complex amplitude
$v$, not a population or an HHG power spectrum. With tildes denoting a time
Fourier transform, use

$$
v_i(K,t)=\mathcal F^{-1}\!\left[G_i(\omega)\widetilde v(K,\omega)\right],
\qquad i\in\{r,nr\},
$$

where

$$
G_r(\omega)=
\begin{cases}
1, & |\omega|\leq\omega_0/2,\\
0, & |\omega|>\omega_0/2,
\end{cases}
\qquad
G_{nr}(\omega)=1-G_r(\omega).
$$

Any resolved currents must follow consistently from these amplitudes. The
physical HHG spectrum is formed from the coherently assembled total current,
not by adding independently squared component spectra. A frozen-valence-band
limit may be used as an optional control, but it is not required.

Time and crystal-momentum grids, propagation algorithms, finite-record
treatment, spectral windows, zero padding, spectral reduction, tolerances,
diagnostics, and display conventions are student choices. State the choices
needed to reconstruct your work and provide evidence sufficiently reliable for
the claims made. No particular grid, range, refinement sequence, validation
procedure, diagnostic, or figure organization is prescribed. Declare any
normalization or display scaling and preserve the relative information used in
the argument. Absolute spectral normalization is not a grading target.

Do not add dephasing, many-body reservoirs, propagation through a macroscopic
sample, or additional bands to the required calculation. Such extensions may
be discussed only as clearly separated exploratory work.

## Submission

Submit one self-contained scientific report and an auditable offline
calculation package. Include the task-specific derivation, 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. No fixed file tree, filename scheme, panel count, 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. No separate aggregate CPU-hour limit applies.

## 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.