cases/case47/task.md

Complete intermediate propagation in strong-field detachment

Physical setting

Work in atomic units. Consider a one-electron short-range negative-ion model in a linearly polarized monochromatic field,

$$\mathbf{A}(t)=A_0\cos (\omega t){\mathbf{e}}_z,\mathbf{E}(t)=-\frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t},U_p=\frac{A_0^2}{4}\mathrm{.}\backslash mathbf\; A(t)=A\_0\backslash cos(\backslash omega\; t)\backslash ,\backslash mathbf\; e\_z,\; \backslash qquad\; \backslash mathbf\; E(t)=-\backslash frac\{\backslash mathrm\; d\backslash mathbf\; A\}\{\backslash mathrm\; dt\},\; \backslash qquad\; U\_p=\backslash frac\{A\_0^2\}\{4\}.$$

Use a wavelength of $1800\ \mathrm{nm}$, an intensity of $1.3\times10^{13}\ \mathrm{W,cm^{-2}}$, and a binding energy $I_p=3.4\ \mathrm{eV}$. Study detachment along the polarization axis. The short-range scattering potential is

$$V(r)=-\sum _{j=1}^2{a}_j\frac{e^{-{\alpha }_{j}r}}{r},V(r)=-\backslash sum\_\{j=1\}^\{2\}a\_j\backslash frac\{e^\{-\backslash alpha\_j\; r\}\}\{r\},$$

with

$$(a_1,a_2)=(5.137,3.863),({\alpha }_1,{\alpha }_2)=(1.288,3.545)\mathrm{.}(a\_1,a\_2)=(5.137,3.863),\; \backslash qquad\; (\backslash alpha\_1,\backslash alpha\_2)=(1.288,3.545).$$

The initial $p_z$ state is specified through its momentum-space dipole. For $\mathbf q=(\mathbf q_\perp,q_z)$, use

$$d_z(\mathbf{q})=\sum _{a=1}^4\frac{32\pi {\zeta }_a{N}_a}{(2\pi {)}^{3\mathrm{/}2}}\frac{q_{\perp }^2+{\zeta }_a^2-5q_z^2}{(q_{\perp }^2+q_z^2+{\zeta }_a^2{)}^4},d\_z(\backslash mathbf\; q)=\; \backslash sum\_\{a=1\}^\{4\}\; \backslash frac\{32\backslash pi\backslash zeta\_a\; N\_a\}\{(2\backslash pi)^\{3/2\}\}\; \backslash frac\{q\_\backslash perp^2+\backslash zeta\_a^2-5q\_z^2\}\; \{(q\_\backslash perp^2+q\_z^2+\backslash zeta\_a^2)^4\},$$

where

$$N_a=C_a\frac{(2{\zeta }_a{)}^{5\mathrm{/}2}}{\sqrt{4!}}\sqrt{\frac{3}{4\pi }},N\_a=C\_a\backslash frac\{(2\backslash zeta\_a)^\{5/2\}\}\{\backslash sqrt\{4!\}\}\; \backslash sqrt\{\backslash frac\{3\}\{4\backslash pi\}\},$$

$$(C_1,C_2,C_3,C_4)=(0.4704,0.3084,0.0988,0.2470),(C\_1,C\_2,C\_3,C\_4)=(0.4704,0.3084,0.0988,0.2470),$$

$$({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4)=(2.0754,3.9334,1.4660,0.9568)\mathrm{.}(\backslash zeta\_1,\backslash zeta\_2,\backslash zeta\_3,\backslash zeta\_4)\; =(2.0754,3.9334,1.4660,0.9568).$$

Plane waves obey $\langle\mathbf r|\mathbf q\rangle= (2\pi)^{-3/2}e^{i\mathbf q\cdot\mathbf r}$, so that

$$⟨\mathbf{p}\mathrm{\mid }V\mathrm{\mid }\mathbf{k}⟩=-\frac{1}{2{\pi }^2}\sum _{j=1}^2\frac{a_j}{\mathrm{\mid }\mathbf{p}-\mathbf{k}{\mathrm{\mid }}^2+{\alpha }_j^2}\mathrm{.}\backslash langle\backslash mathbf\; p|V|\backslash mathbf\; k\backslash rangle\; =-\backslash frac\{1\}\{2\backslash pi^2\}\backslash sum\_\{j=1\}^\{2\}\; \backslash frac\{a\_j\}\{|\backslash mathbf\; p-\backslash mathbf\; k|^2+\backslash alpha\_j^2\}.$$

Energy-channel amplitude identity

For an absorbed-photon order $n$, define

$$E_p=n\omega -I_p-U_p,p=\sqrt{2E_p},T=\frac{2\pi }{\omega }\mathrm{.}E\_p=n\backslash omega-I\_p-U\_p,\; \backslash qquad\; p=\backslash sqrt\{2E\_p\},\; \backslash qquad\; T=\backslash frac\{2\backslash pi\}\{\backslash omega\}.$$

The following definitions fix the phase and normalization of the commissioned long-pulse, first-Born energy-channel branch. Let $\dot{\boldsymbol\alpha}(t)=\mathbf A(t)$ and choose the periodic primitive

$${\mathcal{U}}_1(t)={\int }^t[\frac{{\mathbf{A}}^2(\tau )}{2}-U_p]\mathrm{d}\tau \mathrm{.}\backslash mathcal\; U\_1(t)=\backslash int^t\; \backslash left[\backslash frac\{\backslash mathbf\; A^2(\backslash tau)\}\{2\}-U\_p\backslash right]\backslash mathrm\; d\backslash tau.$$

For $\mathbf k$ with $k=\sqrt{2x}$, define

$$T_{\mathbf{k}i}^{(0)}(n^{\mathrm{\prime }})=\frac{1}{T}{\int }_0^T\text{\hspace{0.17em}⁣}\mathrm{d}t⟨\mathbf{k}+\mathbf{A}(t)\mathrm{\mid }\mathbf{r}\text{\hspace{0.17em}⁣}\cdot \text{\hspace{0.17em}⁣}\mathbf{E}(t)\mathrm{\mid }{\psi }_i⟩\exp \text{\hspace{0.17em}⁣}\left\{i[\mathbf{k}\text{\hspace{0.17em}⁣}\cdot \text{\hspace{0.17em}⁣}\mathit{\alpha }(t)+{\mathcal{U}}_1(t)+n^{\mathrm{\prime }}\omega t]\right\},T\_\{\backslash mathbf\; k\; i\}^\{(0)\}(n\text{'})=\; \backslash frac\{1\}\{T\}\backslash int\_0^T\backslash !\backslash mathrm\; dt\backslash ,\; \backslash langle\backslash mathbf\; k+\backslash mathbf\; A(t)|\backslash mathbf\; r\backslash !\backslash cdot\backslash !\backslash mathbf\; E(t)|\backslash psi\_i\backslash rangle\; \backslash exp\backslash !\backslash left\backslash \{i\backslash left[\backslash mathbf\; k\backslash !\backslash cdot\backslash !\backslash boldsymbol\backslash alpha(t)\; +\backslash mathcal\; U\_1(t)+n\text{'}\backslash omega\; t\backslash right]\backslash right\backslash \},$$

$$T_{\mathbf{p}\mathbf{k}}(m)=J_m\text{\hspace{0.17em}⁣}\left(\frac{{\mathbf{A}}_0\text{\hspace{0.17em}⁣}\cdot (\mathbf{k}-\mathbf{p})}{\omega }\right)⟨\mathbf{p}\mathrm{\mid }V\mathrm{\mid }\mathbf{k}⟩,T\_\{\backslash mathbf\; p\backslash mathbf\; k\}(m)=\; J\_m\backslash !\backslash left(\backslash frac\{\backslash mathbf\; A\_0\backslash !\backslash cdot(\backslash mathbf\; k-\backslash mathbf\; p)\}\{\backslash omega\}\backslash right)\; \backslash langle\backslash mathbf\; p|V|\backslash mathbf\; k\backslash rangle,$$

and

$$f_{n^{\mathrm{\prime }}}(x)=-k\int \text{\hspace{0.17em}⁣}\mathrm{d}{\mathrm{\Omega }}_{\mathbf{k}}{T}_{\mathbf{p}\mathbf{k}}(n-n^{\mathrm{\prime }})T_{\mathbf{k}i}^{(0)}(n^{\mathrm{\prime }})\mathrm{.}f\_\{n\text{'}\}(x)=-k\backslash int\backslash !\backslash mathrm\; d\backslash Omega\_\{\backslash mathbf\; k\}\backslash ,\; T\_\{\backslash mathbf\; p\backslash mathbf\; k\}(n-n\text{'})T\_\{\backslash mathbf\; k\; i\}^\{(0)\}(n\text{'}).$$

The direct branch in this same convention is $T_n^{\mathrm D}=T_{\mathbf p i}^{(0)}(n)$. The rescattered branch is

$$T_n^{\mathrm{R}}=\sum _{n^{\mathrm{\prime }}}{\int }_0^{\mathrm{\infty }}\frac{f_{n^{\mathrm{\prime }}}(x)}{x-x_{n^{\mathrm{\prime }}}-i0}\mathrm{d}x,x_{n^{\mathrm{\prime }}}=n^{\mathrm{\prime }}\omega -I_p-U_p,T\_n^\{\backslash mathrm\; R\}=\backslash sum\_\{n\text{'}\}\backslash int\_0^\backslash infty\; \backslash frac\{f\_\{n\text{'}\}(x)\}\{x-x\_\{n\text{'}\}-i0\}\backslash ,\backslash mathrm\; dx,\; \backslash qquad\; x\_\{n\text{'}\}=n\text{'}\backslash omega-I\_p-U\_p,$$

with

$$\frac{1}{x-x_{n^{\mathrm{\prime }}}-i0}=\mathcal{P}\frac{1}{x-x_{n^{\mathrm{\prime }}}}+i\pi \delta (x-x_{n^{\mathrm{\prime }}})\mathrm{.}\backslash frac\{1\}\{x-x\_\{n\text{'}\}-i0\}\; =\backslash mathcal\; P\backslash frac\{1\}\{x-x\_\{n\text{'}\}\}+i\backslash pi\backslash delta(x-x\_\{n\text{'}\}).$$

An on-shell reduction retains only the delta-function part of this denominator. The physical direct-plus-rescattered spectrum is formed from the complex amplitude $T_n^{\mathrm D}+T_n^{\mathrm R}$ using one common rate prefactor and one declared common display normalization. Algebraically equivalent long-pulse conventions are acceptable only when their mapping preserves the photon-order energy, intermediate-channel measure, Cauchy branch, and direct--rescattered relative phase.

Research problem

Determine how the complete energy-channel treatment of intermediate propagation changes the forward detachment spectrum relative to its on-shell reduction, and whether rescattering remains physically consequential when it is combined coherently with direct detachment.

Develop a task-specific derivation and calculation capable of answering this question. Use your computed signal to establish the physical origin of any material difference, identify the limits of the claim that your evidence supports, and address a plausible alternative explanation when it could change the central conclusion. The choice of decomposition, diagnostic controls, numerical representation, quadrature, sampling, and evidence layout is yours. No particular number or arrangement 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, reconstructible numerical data underlying all material claims, references actually used, and concise run and environment information. No fixed file tree, filename scheme, or complete exploration history is required.

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

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.