cases/case35/task.md

# Testing phase-space descriptions of quantum-light high-harmonic generation

## Background and physical inputs

High-harmonic generation (HHG) driven by a nonclassical optical state need not
be characterized by a classical field with one fixed amplitude. The validity of
a phase-space treatment depends on whether its representation of the optical
state preserves the physically relevant coherence.

Work in atomic units with a one-dimensional single-active-electron model

$$
H_0=\frac{p^2}{2}+V(x),
\qquad
V(x)=-\frac{1}{\sqrt{x^2+1}}.
$$

For every quantum-light component and matched classical control, initialize the
electron in the normalized field-free ground state $|\psi_g\rangle$ of $H_0$.

Represent each quantum optical input on the square generalized von Neumann
lattice

$$
|\alpha_{mn}\rangle,
\qquad
\alpha_{mn}=\frac{\sqrt{\pi}}{2}(m+in),
\qquad m,n\in\mathbb Z.
$$

On the retained finite lattice, write the specified optical state as

$$
|\chi\rangle=\sum_i C_i|\alpha_i\rangle,
\qquad
G_{ij}=\langle\alpha_i|\alpha_j\rangle,
\qquad
C^\dagger G C=1,
$$

where the single index $i$ labels retained pairs $(m,n)$. The coefficients must
represent the stated optical state to the accuracy needed for the claims. Their
construction, lattice extent, and numerical solution are part of the work.

For each coherent component, propagate the same initial electronic state and
let $j_i(\Omega)$ denote its complex HHG response amplitude at angular
frequency $\Omega$, in a consistently declared dipole, current, or acceleration
convention. Define

$$
u_i(\Omega)=C_i j_i^*(\Omega).
$$

The required production response is the full nonorthogonal quadratic form

$$
S_{\mathrm{full}}(\Omega)\propto
u^\dagger(\Omega)G\,u(\Omega).
$$

The corresponding diagonal-only comparator is obtained by deleting every
$i\ne j$ term from the same form,

$$
S_{\mathrm{diag}}(\Omega)\propto
\sum_i G_{ii}|u_i(\Omega)|^2,
$$

with $G_{ii}=1$ for normalized coherent states. An equivalent nonorthogonal
coordinate convention is acceptable if it preserves the same metric quadratic
form. Any other positive-weight diagonal phase-space mixture is an additional
approximation: state its weights and keep it distinct from
$S_{\mathrm{full}}$.

For a coherent component with carrier frequency $\omega$, bind its classical
field amplitude and phase to its lattice amplitude by

$$
E_0=2g_0\sqrt{\omega}\,|\alpha|,
\qquad
\phi=-\arg\alpha.
$$

For a squeezed vacuum use $\alpha_r=\sinh r$, and for a Fock state use
$\alpha_N=\sqrt N$, when calibrating $g_0$ to the stated effective intensity.
For the single-mode cases, obtain the electronic response from the standard
one-dimensional TDSE driven by a ten-cycle trapezoidal electric field with a
two-cycle turn-on, six-cycle plateau, and two-cycle turn-off. For uniqueness in
the two-colour case, use the same common temporal envelope, with its total
duration defined by ten cycles of the 800 nm fundamental; this common-envelope
choice is task policy.

Investigate the following physical cases together with classical fields having
the same carrier frequencies, intensities, phase settings, pulse envelope, and
electronic initial state:

| case | quantum optical input | carrier and effective intensity |
|---|---|---|
| single mode | squeezed vacuum with real $r=4$ | $800\ \mathrm{nm}$, $1.0\times10^{13}\ \mathrm{W\,cm^{-2}}$ |
| single mode | Fock state $\lvert N=10000\rangle$ | $800\ \mathrm{nm}$, $1.0\times10^{14}\ \mathrm{W\,cm^{-2}}$ |
| two colour | an $800\ \mathrm{nm}$ coherent state with real amplitude and a $400\ \mathrm{nm}$ squeezed vacuum with $r=4$ whose extended quadrature lies on the imaginary-amplitude axis | $1.0\times10^{14}$ and $1.0\times10^{13}\ \mathrm{W\,cm^{-2}}$, respectively |

The primary observable is the emitted single-atom HHG spectrum. A common
normalization may be used for comparisons; absolute spectral intensity is not
a grading target.

## Research problem

Determine whether a diagonal phase-space description is sufficient to
characterize the HHG response of the stated nonclassical optical fields, and
establish how the conclusion depends on photon-state structure and on the
two-colour configuration. Base the conclusion on your own calculated radiation
signals, matched classical controls, and evidence that connects the
optical-state representation to the observed response. State the physical scope
within which your conclusion is supported.

The task-specific derivation, numerical implementation, evidence architecture,
spectral domain, diagnostics, and report organization are for you to decide. No
particular number or layout of figures is prescribed. Finite lattice extent,
coefficient cutoff, overlap storage, linear solver, TDSE grid and propagator,
absorbing boundary, time step, transform convention, spectral estimator,
normalization, display range, and reliability strategy are student choices.
State the choices material to your claims and provide sufficiently reliable and
reconstructible evidence.

The required production calculation is $S_{\mathrm{full}}$. Clearly separated
diagonal approximations are permitted and are needed only insofar as they help
answer the research question.

## Submission and resources

Submit one self-contained scientific report and an auditable offline
calculation package containing the task-specific 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
and environment information. No fixed file tree, filename scheme, panel 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.