# 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,

$$
\mathbf A(t)=A_0\cos(\omega t)\,\mathbf e_z,
\qquad
\mathbf E(t)=-\frac{\mathrm d\mathbf A}{\mathrm dt},
\qquad
U_p=\frac{A_0^2}{4}.
$$

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

$$
V(r)=-\sum_{j=1}^{2}a_j\frac{e^{-\alpha_j r}}{r},
$$

with

$$
(a_1,a_2)=(5.137,3.863),
\qquad
(\alpha_1,\alpha_2)=(1.288,3.545).
$$

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

$$
d_z(\mathbf q)=
\sum_{a=1}^{4}
\frac{32\pi\zeta_a N_a}{(2\pi)^{3/2}}
\frac{q_\perp^2+\zeta_a^2-5q_z^2}
{(q_\perp^2+q_z^2+\zeta_a^2)^4},
$$

where

$$
N_a=C_a\frac{(2\zeta_a)^{5/2}}{\sqrt{4!}}
\sqrt{\frac{3}{4\pi}},
$$

$$
(C_1,C_2,C_3,C_4)=(0.4704,0.3084,0.0988,0.2470),
$$

$$
(\zeta_1,\zeta_2,\zeta_3,\zeta_4)
=(2.0754,3.9334,1.4660,0.9568).
$$

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

$$
\langle\mathbf p|V|\mathbf k\rangle
=-\frac{1}{2\pi^2}\sum_{j=1}^{2}
\frac{a_j}{|\mathbf p-\mathbf k|^2+\alpha_j^2}.
$$

## Energy-channel amplitude identity

For an absorbed-photon order $n$, define

$$
E_p=n\omega-I_p-U_p,
\qquad
p=\sqrt{2E_p},
\qquad
T=\frac{2\pi}{\omega}.
$$

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

$$
\mathcal U_1(t)=\int^t
\left[\frac{\mathbf A^2(\tau)}{2}-U_p\right]\mathrm d\tau.
$$

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

$$
T_{\mathbf k i}^{(0)}(n')=
\frac{1}{T}\int_0^T\!\mathrm dt\,
\langle\mathbf k+\mathbf A(t)|\mathbf r\!\cdot\!\mathbf E(t)|\psi_i\rangle
\exp\!\left\{i\left[\mathbf k\!\cdot\!\boldsymbol\alpha(t)
+\mathcal U_1(t)+n'\omega t\right]\right\},
$$

$$
T_{\mathbf p\mathbf k}(m)=
J_m\!\left(\frac{\mathbf A_0\!\cdot(\mathbf k-\mathbf p)}{\omega}\right)
\langle\mathbf p|V|\mathbf k\rangle,
$$

and

$$
f_{n'}(x)=-k\int\!\mathrm d\Omega_{\mathbf k}\,
T_{\mathbf p\mathbf k}(n-n')T_{\mathbf k i}^{(0)}(n').
$$

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

$$
T_n^{\mathrm R}=\sum_{n'}\int_0^\infty
\frac{f_{n'}(x)}{x-x_{n'}-i0}\,\mathrm dx,
\qquad
x_{n'}=n'\omega-I_p-U_p,
$$

with

$$
\frac{1}{x-x_{n'}-i0}
=\mathcal P\frac{1}{x-x_{n'}}+i\pi\delta(x-x_{n'}).
$$

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](../../submission_requirements.md) for the common submission entry points.