# Scientific sources - Case23

## References

Neda Boroumand, Adam Thorpe, Graeme Bart, Andrew Parks, Mohamad Toutounji, Giulio Vampa, Thomas Brabec, and Lu Wang. 2025. *Strong field physics in open quantum systems*. *arXiv*, 2502.10240. [Source](https://arxiv.org/abs/2502.10240v1). Version 1.

Bath-resolved second-order post-pulse population, spectral normalization, mandatory temperature/coupling/cutoff regimes, contribution-isolating mechanism and relaxation-time adequacy. No hidden full sweep or globally exact surrogate is implied.

## Criterion-to-source crosswalk

Locations refer to the article or named preprint/manuscript page, not a figure-file index. Source examples establish scientific identity and robust structure; the task defines required reporting, case coverage and risk-triggered validation. No original source images are reproduced.

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | preprint p. 2, Eqs. (1)-(2), Theory; p. 3, Results pulse/model paragraph Longitudinal spin-boson coupling modifies the driven two-band system through a thermal bosonic bath, not a fitted transverse transition rate. |
| F2 | preprint p. 3, Eq. (5); Supplement Sec. IV spectral-density convention; stated task spectral-density annex Positive-frequency normalization and the odd extension must preserve the bath influence. The visible normalization annex fixes the numerical convention; there is no interchangeable unconverted coupling strength. |
| F3 | preprint p. 3, Eqs. (3)-(5) Causally ordered second-order population contains the Stark-dressed action and complex thermal correlation before Brillouin-zone integration. |
| O1, O3, I1, I3, M1, M2 | preprint pp. 3-5, Eqs. (4)-(6), Fig. 3(a)(i-iii) and Fig. 4(a); stated task eight-case matrix The source establishes temperature/coupling/cutoff-dependent enhancement and suppression. The exact mandatory endpoints belong to the task; the source panels use illustrative cutoffs and are not the eight-case numerical answer. |
| O2, I2, M6 | preprint p. 4, Fig. 3(b,c), especially caption identifying removal of Im C; Eq. (5) A same-object change of the phase-bearing correlation isolates a causal contribution; the plotted source example is not a compulsory additional parameter case or unique control. |
| M3, M4, M5 | preprint pp. 3-5, Eq. (5), Fig. 3(b,c) and following physical-interpretation paragraphs The real correlation dephases while the imaginary finite-memory part modifies phase/action. A correct causal explanation must remain tied to the submitted intervention. |
| M7, V1 | preprint pp. 3-4, paragraph following Eq. (5), Fig. 2(b,c) and Debye-limit discussion The high-temperature Debye limit gives a relaxation-time form, not global equivalence for arbitrary bath spectra or low temperature. |
| M8, V3 | preprint pp. 2-3, equilibrium initial-state, second-order and high-temperature/multi-boson limitations; p. 5, discussion The evidence supports the stated equilibrium longitudinal bosonic-bath approximation; it does not establish heating, melting, fermionic scattering or experimental readout. |
| V2 | preprint p. 3, Eqs. (3)-(6); stated task observable and evidence commission Normalization, causal ordering and intervention identity provide the scientific basis for actual material risks. Validation is conditional; the source does not prescribe an extra numerical checklist. |

## Formula and convention notes

Source-numbered identities below are paraphrased mathematically. Explicit algebraic consequences and accepted model-conditioned representations are identified as such; they do not add a required solver, plot or validation recipe.

At accelerated momentum $K_t=K+eA(t)/\hbar$, $\Omega=2ed(K_t)E(t)/\hbar$ and $E_s=\sqrt{E(K_t)^2+\hbar^2|\Omega|^2}$. The population is $n_c(K,t)=\tfrac12\operatorname{Re}\int_{-\infty}^t dt_1\int_{-\infty}^{t_1}dt_2\,\Omega^*(t_1)\Omega(t_2)e^{iS(t_1,t_2)+C(t_1-t_2)}$, with $S=\hbar^{-1}\int_{t_2}^{t_1}E_s(K_\tau)d\tau$ (Eqs. (3)-(4)); keep causal order and the chosen normalization.

The correlation is $C(s)=\int_{-\infty}^{\infty}d\omega\,J(\omega)\{i\sin(\omega s)-[1-\cos(\omega s)]\coth[\hbar\omega/(2k_BT)]\}$ (Eq. (5)), using the stated odd all-frequency extension. Its real and imaginary parts have distinct decohering and phase-bearing roles. $\eta=\int_{\rm BZ}n_c(K,\infty;j_0)dK/\int_{\rm BZ}n_c(K,\infty;0)dK$ (Eq. (6)); compare all mandatory cases to that same no-bath object.

In the high-temperature Debye limit only, $C(s)\to-s/T_2$ for causal $s\ge0$, with $T_2=\hbar/(2\pi k_BTj_0)$ in the source convention. This does not replace the finite-memory correlation for the other spectral densities.

The task spectral annex fixes $J_+(\omega)=2\sum_q|g_q/(\hbar\omega_q)|^2\delta(\omega-\omega_q)$ for $\omega>0$, its odd all-frequency extension, and $J(\omega)=j_0\omega\exp(-|\omega|/\omega_c)/\omega_c^2$. Thus J has units of inverse angular frequency and j0 is dimensionless. This normalization is task-defined; the source Eq. (5) supplies the correlation structure, not permission to change the coupling without conversion.

## Scope of the source anchors

Figure panels, displayed source coordinates, curve multipliers, numerical grids and named controls are reference examples, not compulsory submission layouts or hidden pointwise targets. Equivalent evidence must preserve the actual physical observable, case, relative scale and discriminating function. Correct data, candidate identification, causal explanation and risk resolution are separate evidential acts. Existing branch-local dependencies, applicability/N/A and descriptive Research Closure rules remain in the case rubric.