cases/case01/task.md

# Phase characterization in a weak-pulse/strong-field ionization model

## Background

Directly characterizing the carrier-envelope phase (CEP) of a weak circularly
polarized isolated attosecond pulse (IAP) is difficult because the pulse does
not by itself provide the strongly nonlinear response used in established
infrared CEP measurements. Consider ground-state hydrogen exposed to the weak
IAP together with a much stronger, phase-stable circularly polarized infrared
(IR) field, with the emitted photoelectrons observed in the laser-polarization
plane.

## Research problem

Develop and assess a reproducible angular-streaking strategy for characterizing
the absolute IAP CEP, within the stated laboratory phase convention, from the
final photoelectron momentum or angular distribution in this two-pulse
configuration. Treat the known IR phase as the phase reference.
Demonstrate, by varying the IAP CEP, whether a usable phase-to-observable
relation can be established and whether the same diagnostic remains applicable
at both specified IAP wavelengths. Determine the level of CEP characterization
supported by the stated model and your calculated evidence.

How the strategy is constructed and substantiated is for you to decide. Assess
its success from the calculated evidence rather than assuming it in advance.

## Physical inputs

Use atomic units in the field and strong-field calculations. For each
individual pulse $i\in\{\mathrm{IAP},\mathrm{IR}\}$, use the source-stated
finite-support field

$$
\mathbf E_i(t)=E_{0,i}
\left[
\cos(\omega_i t+\theta_i)\,\hat{\mathbf x}
-\sin(\omega_i t+\theta_i)\,\hat{\mathbf y}
\right]
\cos^2\!\left(\frac{\pi t}{\tau_i}\right),
\qquad |t|<\frac{\tau_i}{2},
$$

and $\mathbf E_i(t)=0$ outside this support. The two pulse envelopes share the
same time origin. The fields rotate clockwise in the $x$-$y$ plane under this
convention.

| Quantity | Value or condition |
|---|---|
| Target | Ground-state hydrogen ($1s$) |
| Ionization potential | $I_p=0.5$ a.u. |
| IAP wavelengths | 67 nm and 100 nm |
| IR wavelength | 800 nm |
| Peak intensities | $I_{\mathrm{IAP}}=2\times10^{11}\,\mathrm{W/cm^2}$ and $I_{\mathrm{IR}}=2\times10^{14}\,\mathrm{W/cm^2}$ |
| Field amplitudes | $E_{0,i}=\sqrt{I_i/(3.51\times10^{16})}$ a.u. |
| Pulse durations | Four optical cycles, $\tau_i=4(2\pi/\omega_i)$ |
| Carrier frequencies | $\omega_i=2\pi c/\lambda_i$, converted consistently to atomic units |
| Phase reference | $\theta_{\mathrm{IR}}=0$; $\theta_{\mathrm{IAP}}$ is the characterization variable |
| Momentum domain of interest | Laser-polarization plane, $p_z=0$ |

State any conversion constants and phase, vector-potential, momentum-axis, and
angular conventions used in your work.

## Model scope

Use a standard ground-state-hydrogen ($1s$) ADK treatment as the simplified
tunnelling baseline and a standard strong-field approximation (SFA) for the
coherent final photoelectron calculation. Select and cite the precise standard
formulations you use. In the ADK-level model, neglect ground-state depletion
and post-release Coulomb action. Do not add a full TDSE calculation to the
required task.

All numerical and diagnostic choices not fixed above are yours. The same
physical and phase conventions must be used consistently, and the same
diagnostic strategy must be tested at both IAP wavelengths. Any normalization
or display scaling must be declared and kept common wherever relative
magnitudes are used as scientific evidence. Absolute photoelectron yield is
not a grading target unless it is fixed by the formulation you adopt.

You are responsible for checking that your numerical results are sufficiently
reliable and reproducible to support the conclusions you report. No particular
grid, solver, convergence sequence, error metric, or validation procedure is
prescribed.

## Submission

Submit a self-contained scientific report and an offline reproduction package
containing the code, the numerical data underlying the reported evidence, the
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. The
scientific organization of the report and supporting evidence, as well as
filenames below the common top-level collection entry points, are yours. No
full exploratory history or examiner-selected numerical method is required.

The active examination resources and limits are defined in the companion
`resource_limits.json`. All retained production calculations must run
within that fixed allocation and active wall time. More expensive modelling
does not receive 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.