cases/case29/task.md

Resonant EUV high-harmonic spectroscopy of correlated helium

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:

$$H(t)=\sum _{j=1}^2\{\frac{1}{2}{[p_j+\frac{A(t)}{c}]}^2-\frac{2}{\sqrt{x_j^2+a_N}}\}+\frac{1}{\sqrt{(x_1-x_2{)}^2+a_{ee}}},H(t)=\backslash sum\_\{j=1\}^\{2\}\; \backslash left\backslash \{\backslash frac\{1\}\{2\}\backslash left[p\_j+\backslash frac\{A(t)\}\{c\}\backslash right]^2\; -\backslash frac\{2\}\{\backslash sqrt\{x\_j^2+a\_N\}\}\backslash right\backslash \}\; +\backslash frac\{1\}\{\backslash sqrt\{(x\_1-x\_2)^2+a\_\{ee\}\}\},$$

where $p_j=-i\partial_{x_j}$, $a_N=0.50$, $a_{ee}=0.32$, and

$$E(t)=-\frac{1}{c}\frac{dA(t)}{dt}\mathrm{.}E(t)=-\backslash frac\{1\}\{c\}\backslash frac\{dA(t)\}\{dt\}.$$

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

$$E(t)=E_0{\sin }^2\left(\frac{{\omega }_{0}t}{64}\right)\sin ({\omega }_{0}t),0\le t\le \frac{64\pi }{{\omega }_0},E(t)=E\_0\backslash sin^2\backslash left(\backslash frac\{\backslash omega\_0\; t\}\{64\}\backslash right)\; \backslash sin(\backslash omega\_0t),\; \backslash qquad\; 0\backslash le\; t\backslash le\; \backslash frac\{64\backslash pi\}\{\backslash omega\_0\},$$

and $E(t)=0$ outside this interval, with

$${\omega }_0=0.676,E_0=\mathrm{0.2154.}\backslash omega\_0=0.676,\; \backslash qquad\; E\_0=0.2154.$$

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 for the common submission entry points.