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 for the common submission entry points.