cases/case41/task.md

# State-resolved longitudinal momentum transfer in light-front tunnelling

## Physical setting

Work in atomic units and retain terms through first order in $1/c$. Consider
single-active-electron tunnelling in a circularly polarized, long-wavelength
laser field. The propagation direction defines $z$ and the magnetic quantum
number. Use the light-front time and longitudinal momentum

$$
\eta=t-z/c,
\qquad
p_-=p_z-\frac{\mathbf p^2}{2c}+O(c^{-2}).
$$

Study the following channels:

- hydrogen $1s$, with $I_p=0.5$ and residual charge $Z=1$;
- argon $3p$, with $I_p=15.7596\ \mathrm{eV}$ and residual charge $Z=1$, for
  $m=1$ and $m=0$.

### Case-specific launch and branch identity

Use the standard light-front nondipole Coulomb-corrected SFA/PPT differential
launch probability in $(\eta,v_\perp,\widetilde p_z)$. For this case, retain
the asymptotic Coulomb bound-state factor and the state factor
$|Y_{lm}(\widetilde{\mathbf p}(\eta_s))|^2$ evaluated at the complex saddle
momentum. The tunnel exit is the real endpoint of that same saddle trajectory,
and its momentum convention is bound by

$$
\mathbf p_\perp^e=\mathbf v_\perp-\mathbf A_\perp(\eta),
\qquad
p_-^e=\widetilde p_z+\frac{I_p}{c}.
$$

Carry the same differential launch probability as the trajectory weight and
normalize each state/policy marginal when constructing its $p_-$ distribution
or moment. Algebraically equivalent saddle, normalization, or coordinate
conventions are acceptable if they map explicitly to this same state factor,
exit, $p_-^e$, and weight. Deriving the standard PPT measure and saddle exit is
part of the task.

The **short-range/no-sub-barrier-Coulomb launch control** removes the
under-barrier Coulomb correction and Coulomb asymptotic enhancement while
retaining the same nondipole kinetic window, pulse, state/angular identity,
launch variables, and $p_-^e$ definition. It is distinct from the
**sub-barrier-only asymptotic branch**, which uses the Coulomb-corrected launch
ensemble and exit but omits the post-exit Coulomb interaction. The **full
asymptotic branch** uses that same Coulomb-corrected launch ensemble and
includes the post-exit potential $V=-Z/r$.

For the full branch, propagate the light-front Hamiltonian

$$
H=
\frac{(\mathbf p_\perp+\mathbf A_\perp)^2+(p_-+V/c)^2}
{2[1-(p_-+V/c)/c]}+V,
$$

and map outgoing trajectories to asymptotic momenta after the field is off.
The sub-barrier-only branch uses the corresponding field-only post-exit
evolution and the same final $p_-$ observable. Choose and justify a circular
pulse and a field domain within the tunnelling, leading-$1/c$ regime; the
conclusions must not depend on an isolated scan point.

## Research problem

Determine the initial-state dependence and physical origin of longitudinal
momentum transfer in the specified light-front tunnelling model, and assess
which birth-stage conclusions remain supportable from the asymptotic electron
momentum. Use all three launch channels and both argon channels in the
asymptotic comparison.

Build a causal account from reconstructible momentum evidence of your own
design. The distribution, weighted-component, moment-family, intervention, or
other representation is your choice provided that it distinguishes the
material state, launch-policy, and post-exit branch identities. Do not assume
any peak topology, sign, ordering, or field trend. Bound the conclusion to the
regime and evidence established by your calculation.

The derivation, field sampling, momentum representation, quadrature or
trajectory sampling, propagator, time step, field-off criterion, asymptotic
mapping implementation, convergence strategy, plotting choices, and report
organization are for you to decide and justify. No particular number, order,
range, or layout of figures is prescribed.

## Submission and resources

Submit one self-contained scientific report and an auditable offline
calculation package containing 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, 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.