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

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

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

$$\mathbf{A}({\mathbf{r}}_0,t)=\frac{F}{\omega }\left(\begin{array}{c}\cos \theta \cos (k{z}_0\cos \theta -\omega t)\cos (k{x}_0\sin \theta )\\ \cos (k{z}_0\cos \theta -\omega t)\sin (k{x}_0\sin \theta )\\ \sin \theta \sin (k{z}_0\cos \theta -\omega t)\sin (k{x}_0\sin \theta )\end{array}\right),\mathbf{F}(t)=-{\mathrm{\partial }}_t\mathbf{A}({\mathbf{r}}_0,t)\mathrm{.}\backslash mathbf\; A(\backslash mathbf\; r\_0,t)=\backslash frac\{F\}\{\backslash omega\}\; \backslash begin\{pmatrix\}\; \backslash cos\backslash theta\backslash ,\backslash cos(kz\_0\backslash cos\backslash theta-\backslash omega\; t)\; \backslash cos(kx\_0\backslash sin\backslash theta)\backslash \backslash \; \backslash cos(kz\_0\backslash cos\backslash theta-\backslash omega\; t)\; \backslash sin(kx\_0\backslash sin\backslash theta)\backslash \backslash \; \backslash sin\backslash theta\backslash ,\backslash sin(kz\_0\backslash cos\backslash theta-\backslash omega\; t)\; \backslash sin(kx\_0\backslash sin\backslash theta)\; \backslash end\{pmatrix\},\; \backslash qquad\; \backslash mathbf\; F(t)=-\backslash partial\_t\backslash mathbf\; A(\backslash mathbf\; r\_0,t).$$

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

$$H_L(t)=\frac{{\mathbf{p}}^2}{2}+\mathbf{r}\cdot \mathbf{F}(t)+\mathbf{r}\cdot \mathrm{\nabla }\mathbf{A}(t)\cdot \mathbf{p}+V_0\mathrm{.}H\_L(t)=\backslash frac\{\backslash mathbf\; p^2\}\{2\}\; +\backslash mathbf\; r\backslash cdot\backslash mathbf\; F(t)\; +\backslash mathbf\; r\backslash cdot\backslash nabla\backslash mathbf\; A(t)\backslash cdot\backslash mathbf\; p\; +V\_0.$$

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:

$$I_p(\mathrm{H}\mathrm{e})=0.90357\mathrm{a}\mathrm{.}\mathrm{u}\mathrm{.},I_p({\mathrm{N}\mathrm{e}}^{6+}\text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}{\mathrm{N}\mathrm{e}}^{7+})=7.61707\mathrm{a}\mathrm{.}\mathrm{u}\mathrm{.}I\_p(\backslash mathrm\{He\})=0.90357\backslash \; \backslash mathrm\{a.u.\},\; \backslash qquad\; I\_p(\backslash mathrm\{Ne\}^\{6+\}\backslash !\backslash to\backslash !\backslash mathrm\{Ne\}^\{7+\})=7.61707\backslash \; \backslash mathrm\{a.u.\}$$

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 for the common submission entry points.