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## Background
An intense extreme-ultraviolet pulse can drive a bound-state resonance while
ionization is already strong. In this regime, coherent population transfer and
continuum dynamics may coexist in the emitted high-harmonic radiation.
Work in atomic units with the following one-dimensional, two-active-electron
soft-core helium model:
$$
H(t)=\sum_{j=1}^{2}
\left\{\frac{1}{2}\left[p_j+\frac{A(t)}{c}\right]^2
-\frac{2}{\sqrt{x_j^2+a_N}}\right\}
+\frac{1}{\sqrt{(x_1-x_2)^2+a_{ee}}},
$$
where $p_j=-i\partial_{x_j}$, $a_N=0.50$, $a_{ee}=0.32$, and
$$
E(t)=-\frac{1}{c}\frac{dA(t)}{dt}.
$$
The initial state is the normalized field-free ground state of this
Hamiltonian. A gauge-equivalent formulation is acceptable.
For the central case, use a 32-cycle pulse
$$
E(t)=E_0\sin^2\left(\frac{\omega_0 t}{64}\right)
\sin(\omega_0t),
\qquad 0\le t\le \frac{64\pi}{\omega_0},
$$
and $E(t)=0$ outside this interval, with
$$
\omega_0=0.676,
\qquad E_0=0.2154.
$$
This is the model resonance between the field-free ground state and the lowest
dipole-coupled odd state; it corresponds to a photon energy of about $18.4$
eV and a peak intensity of about $1.6\times10^{15}\ \mathrm{W\,cm^{-2}}$.
When varying the photon energy, retain the same cycle count and peak intensity
unless a clearly separated control is being studied.
## Research problem
Using the stated correlated two-electron TDSE as the required production
model, determine whether near-resonant intense EUV driving sustains coherent
population exchange on a subfemtosecond timescale, establish how the relevant
dynamics is encoded in the high-harmonic response, and decide which features
can and cannot be accounted for by an isolated bound two-state picture.
Support the conclusion with an auditable calculation of the central case and
a comparison domain of your design that is sufficient to distinguish a
resonance-specific response from a generic strong-field signal. The
derivation, numerical realization, evidence architecture, controls, scan
extent, diagnostics, and report organization are for you to decide. No fixed
number or layout of outputs is prescribed.
A reduced two-state model may be used as an interpretive or falsification
tool, but it is not a substitute for the correlated TDSE production
calculation. Clearly identify any phenomenological inputs and do not present a
fit as an independent prediction.
Spatial representation, box and boundary treatment, propagation algorithm,
grid, time step, field-free-state construction, spectral estimator, transform
sign, window, post-pulse propagation, normalization, display range, and
comparison sampling are student choices. State the choices that affect your
claims and provide evidence sufficiently reliable and reconstructible to
support them. Absolute spectral intensity and pointwise agreement with an
undisclosed transform convention are not grading targets.
## 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.
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