cases/case33/task.md

# Single-atom recollision beyond the dipole approximation

## Background and physical inputs

At sufficiently long wavelength or high intensity, the magnetic part of a
laser field can displace an ionized electron during its continuum excursion
and thereby alter recollision-based high-harmonic generation (HHG). Consider
whether a pair of non-collinear, counter-rotating, equal-frequency circularly
polarized beams can turn this normally weak nondipole response into a useful
single-atom spectroscopic observable and a controllable recollision process.

Work in atomic units. The beams propagate in the $x,z$ plane with

$$
\mathbf{k}_{\pm}=k(\pm\sin\theta,0,\cos\theta),
\qquad k=\frac{\omega}{c}.
$$

At a nuclear position $\mathbf r_0=(x_0,0,z_0)$, use the monochromatic vector
potential

$$
\mathbf A(\mathbf r_0,t)=\frac{F}{\omega}
\begin{pmatrix}
\cos\theta\,\cos(kz_0\cos\theta-\omega t)
\cos(kx_0\sin\theta)\\
\cos(kz_0\cos\theta-\omega t)
\sin(kx_0\sin\theta)\\
\sin\theta\,\sin(kz_0\cos\theta-\omega t)
\sin(kx_0\sin\theta)
\end{pmatrix},
\qquad
\mathbf F(t)=-\partial_t\mathbf A(\mathbf r_0,t).
$$

The two-beam cases below are evaluated at $z_0=0$ and
$kx_0\sin\theta=\pi/2$. Use a single-active-electron strong-field
approximation (SFA) including the spatial dependence of the field to first
order in $1/c$. A suitable length-gauge Hamiltonian family is

$$
H_L(t)=\frac{\mathbf p^2}{2}
+\mathbf r\cdot\mathbf F(t)
+\mathbf r\cdot\nabla\mathbf A(t)\cdot\mathbf p
+V_0.
$$

The required continuum construction must retain the ionization-time
dependence introduced by the nondipole term. Standard saddle-point and uniform
approximations may be used, but their numerical realization and the evidence
used to establish the final claim are your choices.

Investigate the following physical cases:

| target | wavelength | intensity | field case |
|---|---:|---:|---|
| He | $800\ \mathrm{nm}$ | $1.0\times10^{15}\ \mathrm{W\,cm^{-2}}$ | two-beam field, $\theta=4^\circ$ |
| He | $1.6\ \mu\mathrm m$ | $3.2\times10^{14}\ \mathrm{W\,cm^{-2}}$ | two-beam field, $\theta=4^\circ$ |
| He | $1.6\ \mu\mathrm m$ | $1.0\times10^{15}\ \mathrm{W\,cm^{-2}}$ | two-beam field, $\theta=4^\circ$ |
| $\mathrm{Ne}^{6+}$ | $800\ \mathrm{nm}$ | $1.0\times10^{17}\ \mathrm{W\,cm^{-2}}$ | dipole reference; nondipole linear reference; two-beam field with $\theta=2^\circ$ |

Use the following ionization potentials:

$$
I_p(\mathrm{He})=0.90357\ \mathrm{a.u.},
\qquad
I_p(\mathrm{Ne}^{6+}\!\to\!\mathrm{Ne}^{7+})=7.61707\ \mathrm{a.u.}
$$

The charge-state label in the second value identifies removal of one electron
from the stated $\mathrm{Ne}^{6+}$ target. Use a consistently documented,
physically defensible single-active-electron bound-continuum transition dipole
and apply it consistently within every comparison for a given target. The
primary graded observable is the transverse single-atom harmonic emission; do
not include the longitudinal
$z$-polarized component in that observable. Absolute spectral intensity is
not a grading target.

## Research problem

Determine whether, and within what evidence-supported scope, the stated
two-beam geometry provides a sensitive HHG probe and control of nondipole
recollision dynamics. Establish the conclusion from your own calculation of
the specified cases, identify the physical origin of the decisive spectral
evidence, and distinguish what that evidence demonstrates from what remains
dependent on the first-order single-atom model.

The task-specific derivation, numerical realization, evidence architecture,
controls, spectral domain, and organization of the report are for you to
decide. No particular number or layout of outputs is prescribed. Quadrature,
root finding, orbit handling, sampling, regularization, transform convention,
normalization, display range, and convergence strategy are student choices.
State the choices that affect your claims and provide evidence sufficiently
reliable and reconstructible to support them.

Do not add macroscopic propagation, phase matching, relativistic corrections,
or a different target model to the required production calculation. Clearly
separated exploratory extensions are permitted but cannot replace the stated
model and cases.

## Submission

Submit one self-contained scientific report and an offline reproduction
package containing the derivation, code, numerical data underlying every
material claim, 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.