cases/case21/task.md

# Can smooth single-lobe fields generate spectral interference?

## Background and research question

Momentum-space interference in strong-field particle production is often
associated with carrier oscillations, pulse trains or several distinct
real-time production events. It is not evident whether such temporal
substructure is necessary.

For the two smooth, carrier-free single-lobe fields specified below,
determine whether spectral interference can arise without subcycle structure,
under what nonadiabatic conditions it becomes physically relevant, and what
sets its origin. Assess whether the same explanation remains valid when the
problem is transferred from scalar QED to a gapped two-dimensional Dirac
material. State the strongest conclusion supported by your computed evidence
and its physical scope.

Design the task-specific derivation, calculation, evidence and scientific
argument needed to answer this question. Do not assume the outcome in advance.

## Physical systems and research boundary

### Driving fields

Write the spatially homogeneous, linearly polarized electric field as

$$
E(t)=E_0 e(t/\tau),
\qquad
A(t)=-\int_0^t E(t')\,dt'.
$$

Compare the Gaussian and weakly deformed profiles

$$
e_{\mathrm G}(z)=\exp(-z^2),
\qquad
e_{\mathrm D}(z)=\exp(-z^2-z^4).
$$

Use the same field and vector-potential conventions consistently in both
physical settings.

### Scalar-QED setting

Use $\hbar=c=1$, electron mass $m$ and electron charge $e<0$. The peak field
is

$$
E_0=0.1E_{\mathrm c},
\qquad E_{\mathrm c}=\frac{m^2}{|e|}.
$$

Consider the longitudinal asymptotic particle spectrum at zero transverse
momentum for

$$
\tau=5,\ 10,\ 20\ m^{-1},
\qquad
\gamma=\frac{m}{|e|E_0\tau}=2,\ 1,\ 0.5,
$$

respectively. Production spectra must be obtained from exact numerical scalar
mode evolution or a physically equivalent exact kinetic formulation.
Complex-time turning-point or WKB methods may be used to investigate the
mechanism, but an approximate semiclassical spectrum is not a substitute for
the exact production signal.

### Gapped-Dirac-material setting

Use the spatially uniform two-dimensional Hamiltonian

$$
H_{\mathbf k}(t)=v_{\mathrm F}\boldsymbol{\sigma}\cdot
[\hbar\mathbf k-e\mathbf A(t)]+\Delta\sigma_z,
$$

where $\Delta$ is the Dirac mass term, equal to half the band gap. Take

$$
v_{\mathrm F}=10^6\ \mathrm{m\,s^{-1}},\qquad
\Delta=0.1\ \mathrm{eV},\qquad
E_0=40\ \mathrm{kV\,cm^{-1}},
$$

and consider the post-pulse longitudinal conduction-band occupation at zero
transverse wave vector for

$$
\tau=20,\ 30,\ 50\ \mathrm{fs},
\qquad
\gamma_{\mathrm D}=\frac{\Delta}{|e|v_{\mathrm F}E_0\tau}
=1.25,\ 0.833,\ 0.5,
$$

respectively. Use exact numerical Dirac evolution, an adiabatic-basis
Bogoliubov formulation, a quantum-kinetic formulation, or another physically
equivalent exact evolution with the filled valence band as the incoming
state.

The final occupations are asymptotic post-pulse observables. The numerical
time boundary must therefore be far enough into the field-free region for the
definition used in your calculation to be valid; no particular finite cutoff
is prescribed.

The momentum domain, sampling, evolution algorithm, time step, asymptotic
truncation, complex-root method, diagnostic construction and evidence
representation are yours. Define the conventions needed to make the results
reconstructible. No fixed grid, solver, tolerance, plotting range, figure set
or author-specific decomposition is prescribed. Absolute occupations are
graded when they follow from the stated normalization; comparisons displayed
with an additional arbitrary scale must retain the underlying unscaled data.

## Submission and resources

Submit one self-contained scientific report and an auditable offline
calculation package containing the 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 organization of the argument and the choice and
number of figures are yours. Evidence must be sufficiently reliable and
reproducible for the claims made, but no particular validation procedure 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. More expensive modelling 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.