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

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

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

$${\mathbf{p}}_{\perp }^e={\mathbf{v}}_{\perp }-{\mathbf{A}}_{\perp }(\eta ),p_{-}^e={\tilde{p}}_z+\frac{I_p}{c}\mathrm{.}\backslash mathbf\; p\_\backslash perp^e=\backslash mathbf\; v\_\backslash perp-\backslash mathbf\; A\_\backslash perp(\backslash eta),\; \backslash qquad\; p\_-^e=\backslash widetilde\; p\_z+\backslash frac\{I\_p\}\{c\}.$$

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

$$H=\frac{({\mathbf{p}}_{\perp }+{\mathbf{A}}_{\perp }{)}^2+(p_{-}+V\mathrm{/}c{)}^2}{2[1-(p_{-}+V\mathrm{/}c)\mathrm{/}c]}+V,H=\; \backslash frac\{(\backslash mathbf\; p\_\backslash perp+\backslash mathbf\; A\_\backslash perp)^2+(p\_-+V/c)^2\}\; \{2[1-(p\_-+V/c)/c]\}+V,$$

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 for the common submission entry points.