cases/case03/task.md

# Time-domain sampling of semiclassical photoelectron momentum distributions

Assessment issue date: 20 August 2026

## Background

Semiclassical strong-field models can represent an ionization amplitude either on a final-momentum grid or through complex ionization times followed by real-time trajectories. This project asks whether a change of sampling variables preserves a coherent two-dimensional photoelectron momentum distribution and how Coulomb-corrected and adiabatic trajectory models differ in the structures they produce.

Use atomic units unless a quantity is explicitly given in SI units.

## Research objective

For hydrogen in the pulse defined below, calculate six photoelectron momentum distributions (PMDs) in the signed $(p_x,p_z)$ plane:

1. a direct no-Coulomb SFA benchmark;
2. a no-Coulomb complex-time-sampled PMD without the time-to-momentum change-of-variables factor;
3. the corresponding no-Coulomb complex-time-sampled PMD with that factor;
4. a Coulomb-including QTMC PMD;
5. an ordinary momentum-sampled CCSFA PMD;
6. a complex-time-sampled CCSFA PMD.

The two CCSFA calculations must represent the same physical model and differ only in their sampling direction. The QTMC calculation uses its own standard adiabatic initialization, as described below.

## Physical system and pulse

Use hydrogen with $I_p=0.5$, a four-cycle linearly polarized pulse of wavelength $800\,\mathrm{nm}$ and peak intensity $1.0\times10^{14}\,\mathrm{W\,cm^{-2}}$, and polarization along $z$.

Define

$$
\mathbf A(t)=A_0\cos^2(\beta t)\sin(\omega t+\phi)\,\hat{\mathbf e}_z,
\qquad
\mathbf E(t)=-\frac{d\mathbf A}{dt},
$$

with $N=4$, $T=2\pi N/\omega$, and $\beta=\pi/T=\omega/(2N)$. Convert the stated wavelength and intensity to atomic units using a standard convention and state the conversion used.

Task convention: take $\phi=0$, use the pulse on the real interval $t_0=-T/2$ to $t_f=T/2$, and set it to zero outside that interval. At complex time, use the analytic continuation of the interior trigonometric expression; do not apply a real-time support test to a complex argument.

## Starting relations and derivation targets

### Direct no-Coulomb benchmark

Use the length-gauge SFA amplitude

$$
M(\mathbf p)=i\int_{t_0}^{t_f}dt'\,
e^{i\widetilde S_{\mathbf p}(t')}
\left\langle \mathbf p+\mathbf A(t')\middle|\mathbf r\cdot\mathbf E(t')\middle|1s\right\rangle,
$$

with the finite-endpoint phase convention

$$
\widetilde S_{\mathbf p}(t')=-\int_{t'}^{t_f}d\tau
\left\{\frac{[\mathbf p+\mathbf A(\tau)]^2}{2}+I_p\right\}.
$$

Derive or state the standard plane-wave hydrogen-$1s$ length-gauge dipole and its continuum normalization. If you take the expression from examiner-supplied offline material, cite that source. A momentum-independent common amplitude factor is immaterial, but the momentum dependence and phase convention must be retained.

The direct benchmark is $|M(\mathbf p)|^2$ evaluated on a converged signed $(p_x,p_z)$ grid with $p_y=0$.

### Complex saddles and time sampling

The complex ionization times $t_s=t_r+i t_i$ satisfy

$$
\frac{1}{2}[p_z+A_z(t_s)]^2+\frac{1}{2}p_x^2+I_p=0.
$$

Write $A_z(t_s)=A_r(t_r,t_i)+iA_i(t_r,t_i)$. Starting from this equation, derive the map from a sampled pair $(t_r,t_i)$ to the real momentum components and the absolute area Jacobian

$$
J_p=\left|\det\frac{\partial(p_x,p_z)}{\partial(t_r,t_i)}\right|.
$$

Retain both signed transverse solutions wherever the derived momentum is real; discard only inadmissible samples and document the criterion. Derive the expanded expression for $J_p$ rather than taking it as supplied. The sign of the oriented determinant is not an additional physical phase.

For a saddle contribution use

$$
a_s^{(\mathrm{nc})}=
\frac{(2I_p)^{5/4}}
{\sqrt{2}\,E_z(t_s)[p_z+A_z(t_s)]}
\exp\!\left[i\widetilde S_{\mathbf p}(t_s)\right]
$$

for the no-Coulomb sampled branches. The unweighted branch omits only $J_p$. The weighted branch multiplies this same contribution by $J_p$; neither branch may omit the saddle prefactor, sampling-cell measure or proposal-density correction appropriate to its estimator, or coherent phase.

Choose and state a physically reasonable standard contour-deformation/steepest-descent and Stokes prescription. Completeness means the full contributing upper-half-plane set under that declared prescription within a demonstrated converged complex-time root domain. Count each distinct contribution once, apply the same saddle-selection and branch-continuation convention to the ordinary and time-sampled calculations, and report a root-count and stability audit as the domain is enlarged or refined. The root domain, seeds, root finder, sampling allocation, and branch-tracking implementation are your numerical choices.

### Coulomb-including trajectories

For the CCSFA branches, distinguish the real saddle/drift momentum parameter $\mathbf q=(q_x,0,q_z)$ from the asymptotic momentum $\mathbf p_\infty$ reached after propagation. The complex saddle relation is

$$
\frac{1}{2}[q_z+A_z(t_s)]^2+\frac{1}{2}q_x^2+I_p=0.
$$

Ordinary CCSFA samples $\mathbf q$ and solves this relation for the contributing saddles. Time-sampled CCSFA samples admissible $t_s$ and maps it to $\mathbf q$, with absolute area Jacobian

$$
J_q=\left|\det\frac{\partial(q_x,q_z)}{\partial(t_r,t_i)}\right|.
$$

Derive the expanded $\mathbf q$ map and $J_q$ from the saddle relation. Both branches then use the same nonadiabatic saddle launch at $t_r$. With $\sigma=\pm1$ for the two signed transverse branches,

$$
\mathbf r_0=\left(0,0,\operatorname{Re}\int_{t_s}^{t_r}A_z(\tau)\,d\tau\right),
$$

$$
\mathbf v_0=\left(
\sigma\sqrt{A_i^2(t_r,t_i)-2I_p},\,0,\,-A_r(t_r,t_i)+A_z(t_r)
\right).
$$

Propagate after release with

$$
\frac{d\mathbf r}{dt}=\mathbf v,
\qquad
\frac{d\mathbf v}{dt}=-\mathbf E(t)-\frac{\mathbf r}{r^3}.
$$

After the pulse has ended, obtain each outgoing trajectory's $\mathbf p_\infty$ from its end-of-pulse phase-space coordinates using a stated standard analytic Coulomb/Kepler asymptotic mapping, or a demonstrably equivalent converged continuation. Include the corresponding finite regularized post-pulse Coulomb phase consistently in all three Coulomb-including branches. This fixes the asymptotic observable and coherent phase, but does not prescribe a propagation time, matching radius, integrator, or regularization algorithm.

Use the under-barrier phase

$$
S_s=-\int_{t_s}^{t_r}d\tau
\left\{\frac{[\mathbf q+\mathbf A(\tau)]^2}{2}+I_p\right\}
$$

and the retained post-release Coulomb action family

$$
S_C=-\int_{t_r}^{\infty}d\tau
\left[\frac{\mathbf v^2(\tau)}{2}-\frac{2}{r(\tau)}+I_p\right].
$$

Only relative phases within a final-momentum cell affect its coherent yield. Give the infinite-limit expression a finite numerical meaning either by matching phases to a representative momentum for each finite cell or by using a trajectory-specific regularized action, with physically equivalent converged variants accepted. Apply one declared asymptotic-map and phase convention to all three Coulomb-including branches and jointly demonstrate stability as both its tail parameters and the final-momentum cell width are refined. No particular matching radius, cutoff, quadrature, regularization, or tolerance is prescribed.

For each CCSFA trajectory, combine the saddle prefactor with the phase $S_s+S_C$ (using your converged representative of $S_C$); in the time-sampled branch also apply $J_q$. Propagate both CCSFA branches and coherently bin them by the reached $\mathbf p_\infty$. The ordinary branch is not an asymptotic-momentum shooting or boundary-value calculation, and no $\mathbf q\!\to\!\mathbf p_\infty$ change-of-variables Jacobian is introduced. For QTMC, use the amplitude magnitude implied by the stated standard launch distribution and the retained post-release action as its coherent phase. These statements fix the physical combination, not an integration or binning algorithm.

All three Coulomb PMDs use an $x$-$z$ planar model with $y=v_y=p_y=0$. For the QTMC branch, choose and state a standard ADK ionization rate and the corresponding one-component signed transverse-$p_x$ launch distribution together with a standard Landau static tunnel exit. Do not substitute a cylindrical radial transverse distribution or a $p_y$-marginal for this planar model. The QTMC adiabatic launch is not to be replaced by the nonadiabatic CCSFA launch above. Use the same post-release equations, retained Coulomb action family, tail convention, final-momentum binning, and coherent-yield definition used for the CCSFA comparison.

In the ordinary CCSFA branch, sample $\mathbf q$ and solve for the contributing saddles. In the time-sampled CCSFA branch, sample admissible complex times and map them to $\mathbf q$. Apart from this reversal of sampling direction and the required $J_q$ factor, the two branches must share the same physical launch, saddle set, propagation, action, tail, $\mathbf p_\infty$ binning, and phase conventions.

## Output construction and plotting

For every sampled or propagated branch, coherently add the complex contributions that reach the same final-momentum cell and then form

$$
P_b=\left|\sum_{j\in b} a_j\right|^2.
$$

Here $a_j$ contains the branch-appropriate initial magnitude, phase, absolute area Jacobian when required, and estimator weights. Use the same momentum-cell definition within each comparison group. Label every saved raw complex array as either a cell-integrated accumulation or an amplitude density/cell average. Record deterministic quadrature-cell measures or Monte Carlo proposal-density corrections separately from $J_p$ or $J_q$. If momentum-cell areas are unequal or change under refinement, convert the arrays to one consistently declared density convention before quantitative comparison. Report that complex array and the corresponding nonnegative $P_b$ array. Absolute normalization is not specified; state any one momentum-independent factor used for each branch. For display, unit-maximum normalization and either a linear or declared logarithmic color transform are acceptable. Absolute magnitude is not graded.

Use $p_z$ as the horizontal axis and $p_x$ as the vertical axis, both in atomic units. The following are reference display windows, not mandatory computation domains:

- no-Coulomb comparison: $-1\le p_z\le1$ and $-1\le p_x\le1$;
- Coulomb-including comparison: $-2\le p_z\le2$ and $-1.5\le p_x\le2.5$.

Submit the six PMDs with consistent orientation and readable scales. For the no-Coulomb group, also show quantitative slices or another data-derived comparison from the same arrays sufficient to assess agreement and sampling artifacts. For the Coulomb group, include a common data-derived comparison sufficient to assess global support, envelope, and resolved major ring or ridge positions; matched local regions or slices may be used where their structures are demonstrably resolved. Original annotations, colors, boxes, crops, or exact display ranges are not required.

## Numerical choices and evidence

You choose the momentum/time grids, sampling allocation, root solver, differentiation method, trajectory integrator, action quadrature and tail realization, threshold/caustic treatment, tolerances, refinement order, optimization, and parallelization. Document these choices and provide convergence or stability evidence for every conclusion that depends on them, including threshold/caustic and cell-width refinement where material. Claims about global support, envelope, or resolved major structures must remain stable at the resolution used for those claims. Finer structure that exceeds the resolution needed for a stated conclusion need not satisfy pointwise convergence; do not use an unresolved feature as evidence for a physical conclusion. Sampling branches should report random seeds, proposal/sample counts, propagated-root or trajectory counts, outgoing counts, and final per-cell coverage; deterministic branches should report grid and quadrature coverage. Use `not applicable` where a statistic does not belong to a branch.

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.

If independent replicas are used, combine compatible complex estimators before forming $P_b$; duplicate random seeds are not independent physical samples. Report the resources actually used.

## 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.

## Analysis questions

1. Using the three no-Coulomb PMDs and your quantitative comparisons, determine whether complex-time sampling reproduces the direct SFA benchmark and under what numerical conditions. Connect your derived mapping and change of measure to identifiable signals in the computed arrays. Distinguish a physical difference from a sampling, binning, or convergence artifact, and state the evidential basis and supported scope of your conclusion.

2. First determine whether the ordinary and time-sampled CCSFA calculations represent the same computed distribution within your demonstrated numerical accuracy. Then compare the QTMC result with the CCSFA family. Identify the specific global or local signals that distinguish the model families, interpret those signals using the stated differences in their initialization or contributing trajectories, and state the evidential basis and supported scope of each conclusion. Do not infer a mechanism from a feature that is not resolved in your submitted data.

## Submission requirements

Submit:

Include a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge.

- the derivation from the stated relations to the computable time-to-momentum map and change-of-variables factor;
- the six figures and the underlying numerical arrays, including the complex accumulations needed to reconstruct each PMD;
- code and exact production/plotting commands;
- branch-appropriate run metadata, numerical settings, normalization/display conventions, and resource use;
- convergence or stability evidence and the quantitative comparisons used in the analysis; and
- answers to both analysis questions, with references for any formula or convention actually taken from supplied offline material.

Report failed or superseded runs only when they materially affected parameter selection, convergence, numerical validity, or physical interpretation. No fixed directory tree is required, but the submitted material must be sufficient for an examiner to reconstruct every plotted result.

See [submission requirements](../../submission_requirements.md) for the common submission entry points.