# Near-resonant ionization beyond a resonant two-state description

## Physical setting

Consider fixed-nucleus hydrogen initially in the $1s$ state and driven by a
linearly polarized pulse near the field-free $1s$-$2p$ transition. Use atomic
units unless stated otherwise. Let the pulse have a sine-squared envelope

$$
g(t)=\sin^2\!\left(\frac{\pi t}{T}\right),
\qquad T=\frac{2\pi N}{\omega},
$$

and state the carrier and field conventions used in your calculation. Here
$E_0$ is the peak electric-field amplitude corresponding to the stated peak
intensity.

Use an energy-normalized active-sector model. Denote the initial state by
$|I\rangle=|1s\rangle$, the linearly accessible resonant state by
$|R\rangle=|2p,m=0\rangle$, and an energy-normalized continuum state by
$|\varepsilon\rangle$. The model must retain the pulse envelope, dynamic
Stark shifts and local ionization losses of the active sectors, and coherent
resonant and direct nonresonant contributions to the final continuum
amplitude. In this notation, $D_{IR}=\langle I|z|R\rangle$ is the bound-bound
dipole, $D_{R\varepsilon}=\langle R|z|\varepsilon\rangle$ is the resonant
bound-continuum dipole, and $\widetilde M_{I\varepsilon}$ is the effective
direct two-photon coupling from the initial state to the continuum. Use

$$
D_{IR}=\frac{256}{243\sqrt{2}},
\qquad
\widetilde M_{I\varepsilon}=-7.5,
$$

and

$$
S_I(t)=-0.7E_0^2g^2(t),\qquad
S_R(t)=2.6E_0^2g^2(t),\qquad
S_\varepsilon(t)=\frac{E_0^2}{4\omega^2}g^2(t).
$$

Here $S_I(t)$, $S_R(t)$, and $S_\varepsilon(t)$ are the instantaneous
ac-Stark energy shifts of the initial, resonant, and continuum sectors,
respectively. In this model $S_\varepsilon(t)=U_p(t)$ is the instantaneous
ponderomotive shift. These quantities enter the corresponding diagonal
energy or phase evolution; an equivalent rotating-frame convention may absorb
them into accumulated phases if the mapping is stated explicitly.

Construct a source-equivalent rotating-wave amplitude model from these
ingredients. Determine the required hydrogenic continuum matrix elements in
your declared energy-normalization convention, and derive the corresponding
detunings and local loss terms. Treat all dipole-allowed orthogonal continuum
channels consistently in the angle-integrated spectrum. A channel-resolved
sum or a declared equivalent-channel reduction is acceptable. Algebraically
equivalent rotating-frame, phase, gauge, and continuum-normalization
conventions are acceptable when their mapping to the stated physical objects
is explicit.

The physical family has the field-free transition frequency
$\omega_{IR}=E_R-E_I=0.375$, peak intensities $5\times10^{13}$ and
$10^{14}\ \mathrm{W\,cm^{-2}}$, and pulse lengths $N=60$ and $N=300$ cycles.
The final observable is the energy-normalized photoelectron spectrum. For the
lower- and higher-energy principal peaks of its near-resonant doublet, define

$$
W=|E_H-E_L|,
\qquad
A=\frac{|S_L-S_H|}{\max(S_L,S_H)},
$$

where $E_L,E_H$ are the peak energies and $S_L,S_H$ are their heights in the
same declared spectrum.

An independent three-dimensional single-active-electron TDSE calculation is
also required for the $N=60$, $5\times10^{13}\ \mathrm{W\,cm^{-2}}$ case in a
bounded near-resonant domain. It may use a partial-wave representation or a
physically equivalent full-dimensional hydrogen solver. Its role is to test a
material spectral claim from the active-sector calculation at an independent
model level. The evidence may use the central spectrum, a bounded
frequency-dependent response, or another same-domain comparison with
equivalent discriminating force. A complete frequency-duration map, an
$N=300$ TDSE calculation, angular distributions, partial-wave percentages,
and absolute ionization yield are outside the required scope.

## Research problem

Determine what physically material changes arise when the stated
nonresonant-state terms are included in the near-resonant ionization doublet
across the specified intensity and pulse-length family. Develop a causal
account that is supported by your calculated spectra and peak observables,
and assess which material claim is supported by the independent bounded TDSE
calculation.

Design evidence that can genuinely distinguish the contributions relevant to
your causal account. The intervention, comparison, or other discriminating
construction is your choice; no particular term switch or scan is required.
Do not assume the direction or magnitude of any effect, and limit every
conclusion to the model, cases, and response domain established by your data.

The task-specific derivation, numerical representation, frequency sampling,
peak estimator, evidence design, reliability assessment, plotting choices,
and report organization are for you to decide. Your implementation must be
sufficiently reliable for the claims you make. No particular grid, solver,
window, range, control procedure, number of figures, or panel arrangement is
prescribed.

## Submission and resources

Submit one self-contained scientific report and an auditable offline
calculation package. Include the task-specific derivation, code, numerical data
underlying all 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. No fixed file tree, filename scheme, complete
exploration history, or examiner-specific diagnostic package 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.