cases/case46/task.md

Carrier-envelope control of directional above-threshold ionization

Physical setting

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

$$\mathbf{E}(t)=E_0{\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{\omega t}{2N_c}\right)\sin (\omega t+\varphi ){\mathbf{e}}_z,0\le t\le \frac{2\pi N_c}{\omega },\backslash mathbf\; E(t)=E\_0\backslash sin^2\backslash !\backslash left(\backslash frac\{\backslash omega\; t\}\{2N\_c\}\backslash right)\; \backslash sin(\backslash omega\; t+\backslash phi)\backslash ,\backslash mathbf\; e\_z,\; \backslash qquad\; 0\backslash leq\; t\backslash leq\; \backslash frac\{2\backslash pi\; N\_c\}\{\backslash omega\},$$

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

$$\mathbf{A}(t)=-{\int }_0^t\mathbf{E}(s)\mathrm{d}s\backslash mathbf\; A(t)=-\backslash int\_0^t\backslash mathbf\; E(s)\backslash ,\backslash mathrm\; ds$$

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

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

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

and

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

The continuum-continuum element is

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

$$\mathcal{A}(E,\varphi )=\frac{P_{-}(E,\varphi )-P_{+}(E,\varphi )}{P_{-}(E,\varphi )+P_{+}(E,\varphi )},\backslash mathcal\; A(E,\backslash phi)=\; \backslash frac\{P\_-(E,\backslash phi)-P\_+(E,\backslash phi)\}\; \{P\_-(E,\backslash phi)+P\_+(E,\backslash phi)\},$$

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 for the common submission entry points.