cases/case31/task.md

Coulomb-sensitive semiclassical modelling of strong-field photoelectron interference

Physical system and cases

Work in atomic units with hydrogen initially in its $1s$ state. Use $I_p=0.5$, ionic charge $Z=1$, magnetic quantum number $m=0$, and static polarizabilities $\alpha_{\mathrm H}=4.5$ and $\alpha_{\mathrm{H}^+}=0$. The laser is linearly polarized along $z$ and is defined by

$$\mathbf{A}(t)=(-1{)}^{n+1}\frac{c{F}_0}{\omega }{\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{\omega t}{2n}\right)\sin (\omega t){\mathbf{e}}_z,0\le t\le \frac{2\pi n}{\omega },\backslash mathbf\; A(t)=(-1)^\{n+1\}\backslash frac\{cF\_0\}\{\backslash omega\}\; \backslash sin^2\backslash !\backslash left(\backslash frac\{\backslash omega\; t\}\{2n\}\backslash right)\; \backslash sin(\backslash omega\; t)\backslash ,\backslash mathbf\; e\_z,\; \backslash qquad\; 0\backslash le\; t\backslash le\; \backslash frac\{2\backslash pi\; n\}\{\backslash omega\},$$

with $\mathbf A(t)=0$ outside this interval and $\mathbf F(t)=-(1/c),d\mathbf A/dt$. Use $n=8$, peak intensity $9.0\times10^{13}\ \mathrm{W,cm^{-2}}$, and wavelengths 800, 1200 and 1600 nm.

Use the quasistatic launch density

$$w(t_0,v_{0,\perp })\propto \exp \text{\hspace{0.17em}⁣}[-\frac{2{\kappa }^3}{3F}-\frac{\kappa v_{0,\perp }^2}{F}],\kappa =\sqrt{2I_p(F)},w(t\_0,v\_\{0,\backslash perp\})\backslash propto\; \backslash exp\backslash !\backslash left[-\backslash frac\{2\backslash kappa^3\}\{3F\}\; -\backslash frac\{\backslash kappa\; v\_\{0,\backslash perp\}^2\}\{F\}\backslash right],\; \backslash qquad\; \backslash kappa=\backslash sqrt\{2I\_p(F)\},$$

where $F=|F_z(t_0)|$ is the local field magnitude and

$$I_p(F)=I_p+\frac{1}{2}({\alpha }_{\mathrm{H}}-{\alpha }_{{\mathrm{H}}^{+}})F^2\mathrm{.}I\_p(F)=I\_p+\backslash frac\{1\}\{2\}\; (\backslash alpha\_\{\backslash mathrm\; H\}-\backslash alpha\_\{\backslash mathrm\{H\}^+\})F^2.$$

The exit coordinate is fixed by

$$-\frac{{\beta }_2(F)}{2\eta }+\frac{m^2-1}{8{\eta }^2}-\frac{F\eta }{8}=-\frac{I_p(F)}{4},{\beta }_2(F)=Z-\frac{1+\mathrm{\mid }m\mathrm{\mid }}{2}\sqrt{2I_p(F)}\mathrm{.}-\backslash frac\{\backslash beta\_2(F)\}\{2\backslash eta\}\; +\backslash frac\{m^2-1\}\{8\backslash eta^2\}\; -\backslash frac\{F\backslash eta\}\{8\}\; =-\backslash frac\{I\_p(F)\}\{4\},\; \backslash qquad\; \backslash beta\_2(F)=Z-\backslash frac\{1+|m|\}\{2\}\backslash sqrt\{2I\_p(F)\}.$$

Choose the positive real root $\eta_0$ on the field-lowered, outgoing side. Equivalently, after orienting the local polarization axis so that the field points along $+z$, take $\mathbf r_0=-\eta_0\mathbf e_z/2$; in the fixed laboratory axis this is

$${\mathbf{r}}_0=-\mathrm{sgn}[F_z(t_0)]\frac{{\eta }_0}{2}{\mathbf{e}}_z\mathrm{.}\backslash mathbf\; r\_0=-\backslash operatorname\{sgn\}[F\_z(t\_0)]\; \backslash frac\{\backslash eta\_0\}\{2\}\backslash ,\backslash mathbf\; e\_z.$$

The baseline launch has $v_{0,z}=0$. For the 800-nm case, also examine the nonzero longitudinal launch prescription

$$v_{0,z}(t_0)=-\frac{A_z(t_0)}{c}[\sqrt{1+{\gamma }^2(t_0,v_{0,\perp })}-1],v\_\{0,z\}(t\_0)=-\backslash frac\{A\_z(t\_0)\}\{c\}\; \backslash left[\backslash sqrt\{1+\backslash gamma^2(t\_0,v\_\{0,\backslash perp\})\}-1\backslash right],$$

$$\gamma (t_0,v_{0,\perp })=\frac{\omega \sqrt{2I_p+v_{0,\perp }^2}}{F_0{\sin }^2(\omega t_0\mathrm{/}2n)\mathrm{\mid }\cos (\omega t_0)\mathrm{\mid }}\mathrm{.}\backslash gamma(t\_0,v\_\{0,\backslash perp\})=\; \backslash frac\{\backslash omega\backslash sqrt\{2I\_p+v\_\{0,\backslash perp\}^2\}\}\; \{F\_0\backslash sin^2(\backslash omega\; t\_0/2n)|\backslash cos(\backslash omega\; t\_0)|\}.$$

The same quasistatic weight family applies to both longitudinal-launch cases.

Semiclassical response identity

Let $V(\mathbf r)=-Z/r$. The commissioned SCTS branch uses the mixed initial-position-to-final-momentum generating function. In length gauge, $\mathbf p_0=\mathbf v_0$, and its trajectory phase is defined, up to terms common to all trajectories in the same final-momentum cell, by

$${\mathrm{\Phi }}_{\mathrm{S}\mathrm{C}\mathrm{T}\mathrm{S}}=-{\mathbf{p}}_0\text{\hspace{0.17em}⁣}\cdot \text{\hspace{0.17em}⁣}{\mathbf{r}}_0+I_p{t}_0-\mathrm{FP}\text{\hspace{0.17em}⁣}{\int }_{t_0}^{\mathrm{\infty }}[\frac{p^2(t)}{2}+V(\mathbf{r}(t))-\mathbf{r}(t)\text{\hspace{0.17em}⁣}\cdot \text{\hspace{0.17em}⁣}\mathrm{\nabla }V(\mathbf{r}(t))]dt\mathrm{.}\backslash Phi\_\{\backslash mathrm\{SCTS\}\}\; =-\backslash mathbf\; p\_0\backslash !\backslash cdot\backslash !\backslash mathbf\; r\_0+I\_p\; t\_0\; -\backslash operatorname\{FP\}\backslash !\backslash int\_\{t\_0\}^\{\backslash infty\}\; \backslash left[\; \backslash frac\{p^2(t)\}\{2\}+V(\backslash mathbf\; r(t))\; -\backslash mathbf\; r(t)\backslash !\backslash cdot\backslash !\backslash nabla\; V(\backslash mathbf\; r(t))\; \backslash right]dt.$$

Here $\operatorname{FP}$ means that, after the pulse, only the divergent phase common to trajectories arriving in the same asymptotic-momentum cell is removed. The finite trajectory-dependent Coulomb remainder must be retained. The reference QTMC comparator is defined at the same level by

$${\mathrm{\Phi }}_{\mathrm{Q}\mathrm{T}\mathrm{M}\mathrm{C}}=I_p{t}_0-\mathrm{FP}\text{\hspace{0.17em}⁣}{\int }_{t_0}^{\mathrm{\infty }}[\frac{p^2(t)}{2}+V(\mathbf{r}(t))]dt\mathrm{.}\backslash Phi\_\{\backslash mathrm\{QTMC\}\}\; =I\_p\; t\_0\; -\backslash operatorname\{FP\}\backslash !\backslash int\_\{t\_0\}^\{\backslash infty\}\; \backslash left[\backslash frac\{p^2(t)\}\{2\}+V(\backslash mathbf\; r(t))\backslash right]dt.$$

Equivalent action conventions may move endpoint or common asymptotic terms between expressions, but they must reproduce the same relative phase in each final-momentum cell for the stated SCTS or QTMC branch.

For an unbound trajectory, the final momentum $\mathbf k$ is its outgoing asymptotic Coulomb momentum obtained from the post-pulse Kepler orbit, not its instantaneous momentum at the end of the laser pulse. The graded observable is the coherent asymptotic photoelectron momentum distribution obtained with an unbiased treatment of the stated launch density. Its sampling, propagation, coherent accumulation, variance reduction, asymptotic mapping and regularization algorithms are student choices; an incoherent CTMC distribution is not a substitute for this observable.

Research problem

Assess the physical and quantitative credibility of the commissioned SCTS description for the asymptotic photoelectron interference generated by the stated fields. Use the defined QTMC branch and a converged single-active- electron TDSE calculation at 800 nm as comparators, and determine the domain in which any conclusions remain supported across the wavelength and launch cases. Your report should explain what the calculated evidence does and does not establish about the roles of the tunnel-launch input and the continuum phase.

The material derivation, numerical realization, evidence architecture, parameter sampling, claim-bearing reductions and organization of the report are for you to decide. No particular panel, crop, node count, energy or angular reduction, intervention sequence, or expected model ordering is prescribed. Do not assume that visual similarity by itself establishes model validity.

Root solver, sampling strategy, trajectory count, momentum binning, trajectory integrator, tolerances, Coulomb finite-part implementation, asymptotic matching, quantum-reference discretization, transforms, normalization, display scale and plotting domain are student choices. State the choices that affect your claims and provide evidence sufficiently reliable and reconstructible to support them. A common overall normalization may be chosen and declared; absolute yield is not a grading target unless independently established by your calculation.

Do not add multielectron, relativistic, focal-volume, macroscopic-propagation or detector-response physics to the required production model. Clearly separated exploratory extensions are permitted but cannot replace the stated commission.

Submission and resources

Submit one self-contained scientific report and an auditable offline calculation package containing the task-specific derivation, code, numerical data underlying 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.