cases/case02/task.md

# Multiphoton ionization from driven scattering responses

## Background

Time-independent perturbation theory can avoid the long propagation times of
a pulsed calculation, but multiphoton ionization becomes difficult once an
intermediate absorption step is already in the continuum. A useful method
must then recover absolute channel amplitudes from a driven scattering
response without confusing open intermediate continua with the selected final
photoelectron channel.

## Research problem

Develop and assess a time-independent method for extracting absolute
generalized multiphoton ionization cross sections from driven responses of
ground-state hydrogen. Apply it to two-, three-, and four-photon ionization,
including photon-energy regions in which one or more intermediate steps are
open. Determine what the resulting partial-wave family establishes about the
opening of above-threshold pathways and whether the asymptotic content of a
representative complex response supports the extraction principle.

The derivation, numerical construction, choice of scientific evidence and
organization of the argument are yours. Establish the validity and limits of
the extraction from your calculated responses rather than assuming that an
outgoing-wave fit is automatically meaningful.

## Physical inputs

Use atomic units internally. The target is hydrogen in its normalized,
positive-real $1s$ ground state, with energy $E_0=-1/2$. The light is linearly
polarized along $z$, and the interaction operator for each absorbed photon is
the velocity-form dipole $D=p_z$.

Treat the following photon orders:

| Photon order | Conventional reported unit |
|---|---|
| $K=2$ | $\mathrm{cm^4\,s}$ |
| $K=3$ | $\mathrm{cm^6\,s^2}$ |
| $K=4$ | $\mathrm{cm^8\,s^3}$ |

Resolve every electric-dipole-allowed final partial wave reached from the
$1s$ state. For $K=2$, also report the arithmetic total of those partial
cross sections. The allowed angular channels and their order dependence are
part of your derivation.

Choose an energetically allowed photon-energy domain sufficient to establish
the channel-opening and high-energy claims you make. The primary graded
objects are the real-energy, unbroadened perturbative responses away from
exact bound-state poles. Exact pole samples may be omitted or explicitly
flagged; do not replace the primary result with a phenomenologically
broadened spectrum.

As a method-level audit, also treat at least one student-chosen complex radial
response for which an intermediate absorption step is open. Assess its
outgoing Coulomb-wave content in the unscaled asymptotic region. Any real-axis
continuation beyond the ECS onset must be distinguished from the computed
response on the ECS contour.

State the Coulomb-wave normalization, phase, radial-wave convention,
atomic-unit conversion constants and generalized-cross-section convention
used. Absolute cross sections, rather than independently normalized display
curves, are the grading target.

## Method boundary

Use ordered time-independent perturbative driven equations and ECS outgoing
boundary conditions. In the response convention for this task, every photon
step consists of one application of $D$; do not insert an additional
factorial, permutation, field-amplitude or order-dependent coefficient into
the successive response states.

You must derive the task-specific response hierarchy, the asymptotic
coefficient extraction and the conversion from the selected outgoing
coefficient to the reported cross section. Standard Coulomb functions, ECS,
linear algebra and atomic-unit conversions may be selected and cited. No
particular radial basis, knot sequence, ECS angle, box extent, sampling grid,
fit implementation, solver, tolerance or plotting layout is prescribed.

You are responsible for checking that the calculated evidence is sufficiently
reliable and reproducible for the conclusions made. Numerical reliability has
no separate deliverable or prescribed test sequence; it is reflected in the
correctness and quality of the reported physical evidence.

## Submission

Submit one self-contained scientific report and an offline reproduction
package containing the derivation, executable code, reconstructible absolute
partial-cross-section data, the complex radial data and asymptotic
reconstruction used in the method assessment, 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 and number of
figures are yours. No fixed file tree, filename scheme, complete exploratory
history or examiner-selected diagnostic is required.

The active examination resources and limits are defined in the companion
`resource_limits.json`. All retained production calculations must run
within that fixed allocation and active wall time. More expensive modelling
does not receive 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.