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## Background and research question
Autoionizing states embedded in the continuum can influence high-harmonic
generation through correlated excitation and emission pathways. Distinguishing
the roles of bright and dark resonances requires their response to be assessed
within the correlated dynamics rather than assumed from their labels.
For the correlated one-dimensional helium model below, determine how
resonances with the specified bright and dark autoionizing states affect the
high-harmonic response as the driving frequency is tuned through each
resonance. Establish what the computed amplitude and phase information reveals
about the relation between the two resonance classes and what temporal
information, if any, can be inferred for the resonant XUV emission. State the
strongest conclusion supported by your evidence and its physical scope.
Design the task-specific derivation, calculation, evidence and scientific
argument needed to answer this question. Do not assume the outcome in advance.
## Physical system and research boundary
Use atomic units unless stated otherwise. Treat two indistinguishable
electrons in one spatial dimension with the correlated TDSE
$$
i\frac{\partial}{\partial t}\Psi(x,y,t)=
\left[-\frac{1}{2}\frac{\partial^2}{\partial x^2}
-\frac{1}{2}\frac{\partial^2}{\partial y^2}
+V(x,y)+(x+y)E(t)\right]\Psi(x,y,t),
$$
where
$$
V(x,y)=-\frac{2}{\sqrt{x^2+a^2}}
-\frac{2}{\sqrt{y^2+a^2}}
+\frac{1}{\sqrt{(x-y)^2+b^2}},
\qquad
a=\frac{1}{\sqrt{2}},\quad b=\frac{1}{\sqrt{3}}.
$$
Start from the symmetric field-free ground state. The target resonances are
the even-parity dark $(1,1)$ autoionizing state near a 14-photon condition and
the odd-parity bright $(1,2)$ autoionizing state near a 17-photon condition.
Use a linearly polarized field with peak intensity
$8.0\times10^{14}\ \mathrm{W\,cm^{-2}}$ and a fundamental photon energy near
$3\ \mathrm{eV}$. The pulse has a four-cycle turn-on, an eight-cycle plateau
and a four-cycle turn-off. As a task convention resolving the pulse wording,
use a $\sin^2$ field-amplitude envelope on each ramp and a cosine carrier.
Tune the fundamental frequency through both stated multiphoton-resonance
neighbourhoods over domains adequate for the conclusions you make.
The correlated two-electron dynamics and both resonance classes are part of
the research commission. Standard choices for the bound-state preparation,
propagation, absorbing boundary, construction of the complex harmonic
response, phase convention, sampling, convergence assessment and evidence
representation are yours. Define the conventions needed to make your results
reconstructible. No grid, box, absorber shape, time step, frequency spacing,
transform window, phase-unwrapping algorithm, plotting range or fixed figure
set is prescribed. Absolute harmonic strength is not graded when an overall
normalization remains unspecified; relative structures used in a claim must
be compared under a consistent normalization.
## Submission and resources
Submit one self-contained scientific report and an auditable offline
calculation package containing the producing code, reconstructible numerical
data for 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. The organization of the argument and the choice and
number of figures are yours. Evidence must be sufficiently reliable and
reproducible for the claims made, but no particular validation procedure is
required.
The available CPU allocation, candidate active deadline, node, network,
memory and scratch limits are specified in the accompanying
`resource_limits.json`. Complete the derivation, implementation,
calculation, validation, interpretation and final-file preparation within these
resource limits. More expensive modelling 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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