# Carrier-envelope control of directional above-threshold ionization

## Physical setting

Work in atomic units. Consider a linearly polarized few-cycle field along
$z$,

$$
\mathbf E(t)=E_0\sin^2\!\left(\frac{\omega t}{2N_c}\right)
\sin(\omega t+\phi)\,\mathbf e_z,
\qquad 0\leq t\leq \frac{2\pi N_c}{\omega},
$$

with $N_c=4$, $\omega=0.057$, intensity $10^{14}\ \mathrm{W\,cm^{-2}}$,
and $I_p=0.5$. This expression defines the electric field. Use

$$
\mathbf A(t)=-\int_0^t\mathbf E(s)\,\mathrm ds
$$

with zero endpoint vector potential. Let
$U_p=E_0^2/(4\omega^2)$.

Use the following separable short-range benchmark in the first-order
rescattering strong-field approximation. Set $\Gamma=1$, $\gamma=38$, and
$\epsilon=0.2$ as independent fixed inputs. Define

$$
\mathcal N^2=
\frac{\sqrt{2I_p}\left(\Gamma+\sqrt{2I_p}\right)^2}{4\pi^2},
$$

$$
\mathbf d(\mathbf q)=
i\mathcal N\mathbf q\,
\frac{(q^2+\Gamma^2)+(q^2/2+I_p)}
{(q^2+\Gamma^2)^{3/2}(q^2/2+I_p)^2},
$$

and

$$
B(\mathbf q)=
-\frac{2\gamma}{\sqrt{q^2+\Gamma^2}}
\left(1-\frac{4\pi^2 i\gamma}{|\mathbf q|+i\Gamma}\right)^{-1}.
$$

The continuum-continuum element is

$$
\begin{aligned}
\mathbf g(\mathbf q_1,\mathbf q_2)
={}&iB(\mathbf q_2)\mathbf q_1
\frac{3q_1^2-q_2^2+2\Gamma^2+i\epsilon}
{(q_1^2+\Gamma^2)^{3/2}(q_2^2-q_1^2-i\epsilon)^2}\\
&-iB^*(\mathbf q_1)\mathbf q_2
\frac{3q_2^2-q_1^2+2\Gamma^2-i\epsilon}
{(q_2^2+\Gamma^2)^{3/2}(q_1^2-q_2^2+i\epsilon)^2}.
\end{aligned}
$$

Use a standard first-order rescattering SFA, with a saddle treatment of the
intermediate momentum, to construct the complex direct amplitude
$b_0(\mathbf p)$ and the single-rescattering amplitude $b_1(\mathbf p)$ from
the supplied matrix elements. State the action, saddle-momentum, phase, and
normalization conventions used. Physically equivalent gauge and phase
formulations are acceptable when their mapping preserves the same target,
field, and observable identities.

The observable amplitude is $b=b_0+b_1$. The direct, rescattered, signed
interference, and coherent total contributions are $|b_0|^2$, $|b_1|^2$,
$2\operatorname{Re}(b_0b_1^*)$, and $|b_0+b_1|^2$, respectively. For opposite
longitudinal momenta at the same energy, define

$$
\mathcal A(E,\phi)=
\frac{P_-(E,\phi)-P_+(E,\phi)}
{P_-(E,\phi)+P_+(E,\phi)},
$$

where $P_-$ and $P_+$ denote emission with $p_z<0$ and $p_z>0$ under this
fixed convention.

## Research problem

Determine how the carrier-envelope phase controls directional
above-threshold ionization in this model. Establish whether one coherent
direct-rescattered account explains both the longitudinal energy asymmetry
and the structure of the polarization-plane momentum distribution.

Support the conclusion with your own physically discriminating numerical
evidence and a causal account tied to the model. The conclusion must be
bounded to the field and target model established by your calculation.

The numerical representation, integration strategy, sampling, evidence
architecture, diagnostic controls, display choices, and report organization
are for you to choose 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.