cases/case25/task.md

# How is the quantum state of a driving field encoded in crystal high-harmonic radiation?

## Background and central research question

The field driving a strong-field process need not be well represented by one
classical waveform. Its coherent-state amplitudes can instead be distributed
according to different quantum states, even when those states have the same
mean photon number.

Investigate how the quantum state of the driving field is encoded in
intraband high-harmonic radiation from a crystal. Determine which spectral
and time-domain properties of the generated radiation provide defensible
state-dependent evidence, explain their physical origin within the stated
model, and delimit the strongest inference supported by the calculation. Do
not assume the outcome in advance.

Design the task-specific derivation, numerical calculation, evidence and
scientific argument needed to answer this one research question. The form and
organization of the evidence are yours.

## Physical model and indispensable inputs

Use atomic units unless otherwise stated. Consider a one-dimensional cut of a
single conduction band along the field-polarization direction. Its dispersion
is

$$
E(q)=\sum_{l=0}^{5} b_l\cos(laq),
$$

with

$$
\begin{aligned}
a&=5.32,\\
b_1&=-0.0814, & b_2&=-0.0024, & b_3&=-0.0048,\\
b_4&=-0.0003, & b_5&=-0.0009.
\end{aligned}
$$

The constant $b_0$ does not affect the intraband current and may be set to
zero. The band is occupied by ten electrons: the five crystal momenta

$$
q_m=m\Delta q,\qquad m=-2,-1,0,1,2,
\qquad \Delta q=\frac{2\pi}{10a},
$$

are each occupied in both spin orientations.

For a coherent-state amplitude $\alpha=|\alpha|e^{i\phi}$, take the
semiclassical vector potential to be

$$
A_\alpha(t)=\frac{2g_0}{\sqrt{\omega_0}}
f(t)|\alpha|\sin(\omega_0t-\phi),
$$

where

$$
g_0=4.0\times10^{-8},\qquad \omega_0=0.005.
$$

The envelope $f(t)$ has a three-cycle $\sin^2$ turn-on, ten cycles at unit
amplitude, and a three-cycle $\sin^2$ turn-off. A shift of the time origin is
an equivalent carrier-phase convention when applied consistently.

Compare coherent, thermal, Fock and bright squeezed-vacuum driving states at
the common mean photon number

$$
\bar N=7.35\times10^{11}.
$$

Use the exact diagonal Glauber--Sudarshan weights

$$
P_{\rm coh}(\alpha)=\delta^{(2)}(\alpha-\alpha_L),
\qquad |\alpha_L|^2=\bar N,
$$

and

$$
P_{\rm th}(\alpha)=\frac{1}{\pi\bar N}
\exp\!\left(-\frac{|\alpha|^2}{\bar N}\right).
$$

Choose $\alpha_L>0$ to fix the common time-origin convention. For the Fock
state $|n\rangle$, use $n=\bar N$ and the large-photon-number approximative
positive-$P$ (APP) diagonal weight

$$
P_{\rm Fock}^{\rm APP}(\alpha)=
\frac{e^{-|\alpha|^2}|\alpha|^{2n}}{\pi n!}.
$$

For the bright squeezed vacuum $\hat S(r)|0\rangle$, take real $r$ with

$$
\sinh^2 r=\bar N,
$$

so that $r\simeq14.35$, and use

$$
P_{\rm BSV}^{\rm APP}(\alpha)=
\frac{1}{\pi\cosh r}
\exp\!\left[
-\frac{2[\operatorname{Re}\alpha]^2}{1+e^{-2r}}
-\frac{2[\operatorname{Im}\alpha]^2}{1+e^{2r}}
\right].
$$

The coherent and thermal averages are exact within this single-band model.
The Fock and bright-squeezed-vacuum weights define the APP approximation used
for this task. APP is prescribed only for the harmonic spectrum and the first
two time-domain moments of the generated electric field; stronger emitted-
field claims require independent support.

## Method boundary and numerical freedom

Obtain the four-state finite-pulse emitted-energy signal and the generated-
field expectation value and zero-point-subtracted uncertainty. Derive the
task-specific relations needed to define and calculate these observables from
the stated model; no final output formula is supplied.

The finite-pulse calculation is the spectrum-generating route. A periodic
Floquet treatment, Bessel-function analysis or a semiclassical cutoff estimate
may be used as supporting methods, but cannot replace the finite-pulse result.

The quadrature or sampling representation, time and frequency domains, grids,
steps, Fourier convention, differentiation method, truncations, tolerances,
diagnostics and reliability strategy are student choices. State the
conventions needed to reconstruct your observables. Equivalent normalizations
are acceptable when unscaled data and the comparison convention are retained;
absolute intensity is not graded when it is left arbitrary by the model.

## Submission and resources

Submit one self-contained scientific report and an auditable offline
calculation package containing producing code, reconstructible numerical data
for 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 evidence must be sufficiently reliable to support the claims
made, but no particular validation procedure, plot layout or number of figures
is prescribed.

Resource limits are supplied exclusively by the accompanying
`resource_limits.json`. Its fixed CPU allocation, candidate-active
wall time, network boundary and temporary-storage allowance are binding.
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.