# What can complex saddle structure establish about a short-pulse attoclock?

## Background

An attoclock uses the angular structure of a photoelectron momentum distribution produced by a short, nearly circularly polarized pulse. A complex-time strong-field description offers a comparatively inexpensive way to examine how ionization pathways are encoded in that signal, but its physical scope must be kept distinct from a full calculation with the residual Coulomb field.

## Research problem

Within a strong-field saddle-point model, determine which features of a short-pulse atomic attoclock signal can be accounted for by its complex saddle structure, and determine how the corresponding saddle topology changes when atomic hydrogen is replaced by fixed-nuclei molecular hydrogen at different orientations. Support an evidence-bounded conclusion with a computed in-plane atomic photoelectron signal and quantitative complex-saddle evidence for the atomic and molecular cases.

## Physical inputs

Use atomic units unless a quantity is stated otherwise. The vector potential is

\[
\mathbf A(t)=\frac{A_0}{\sqrt{1+\epsilon^2}}
\cos^4\!\left(\frac{\omega t}{2N}\right)
\left[
\cos(\omega t+\phi)\,\mathbf e_x
+\epsilon\sin(\omega t+\phi)\,\mathbf e_y
\right]
\]

for \(|t|<N\pi/\omega\), and is zero outside that interval. Take

\[
\epsilon=1,\qquad N=2,\qquad \phi=0,
\]

with \(\omega\) corresponding to an \(800\ \mathrm{nm}\) wavelength. Use the intensity convention \(I=(\omega A_0)^2\) in atomic units and include the reference case \(I=8\times10^{13}\ \mathrm{W\,cm^{-2}}\). You may examine additional intensities if they materially strengthen your argument.

For atomic hydrogen use \(I_p=0.5\) a.u. For fixed-nuclei \(\mathrm{H}_2\), use the equilibrium separation \(R=1.4\) a.u. and the frozen-core Hartree-Fock ionization potential \(I_p=15.6\ \mathrm{eV}\). Treat the molecular axis in each of these orientations:

- parallel to \(\mathbf e_y\), the peak-electric-field direction;
- parallel to \(\mathbf e_x\), the peak-vector-potential direction;
- parallel to \(\mathbf e_z\), perpendicular to the polarization plane.

## Method and scope boundary

Use the strong-field approximation with complex-time saddle-point evaluation. For \(\mathrm{H}_2\), use a two-centre LCAO description and derive the site-resolved stationary conditions by applying stationary phase to the centre-translated LCAO phase. Standard formulas may be derived or cited from eligible references; the root solver, discretization, branch-tracking strategy, representation and normalization are your choices and must be declared.

The required molecular result is the complex-saddle topology, not a molecular photoelectron distribution, molecular angular width or molecular offset angle. A TDSE, surface-flux calculation, Coulomb-corrected trajectory model and author-specific fitting prescription are outside the required scope. Claims must remain within the physical content of the commissioned Coulomb-free model.

## Research report and submission

Submit one self-contained scientific report and an auditable offline calculation package. Define the mathematical objects and conventions used, support every material claim with reconstructible numerical evidence, and state what the calculation does and does not establish. Include a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, together with concise environment and run instructions. The organization of the argument and the number and form of supporting figures are yours.

Resource limits are supplied exclusively by the accompanying
`resource_limits.json`. Its fixed CPU allocation, candidate-active
wall time, network boundary and temporary-storage allowance are binding.
No separate aggregate CPU-hour limit applies.
Evidence should be sufficiently reliable and reproducible for the claims made,
but no particular validation procedure, grid, plotting range or fitting rule is
prescribed.

## 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.