# Scientific sources - Case20

## References

V. V. Strelkov. 2023. *Dark and bright autoionizing states in resonant high-order harmonic generation: Simulation via a one-dimensional helium model*. *Physical Review A*, 107, 053506. [Source](https://doi.org/10.1103/PhysRevA.107.053506).

Complex HHG amplitude/phase near bright and dark autoionizing resonances, interference pathways and dark-branch temporal inference. Source figures support physical structure, not fixed displays, exact widths or arbitrary-unit absolute ratios.

## 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, F2 | article p. 053506-2, Eqs. (1)-(2), Fig. 1 and Sec. III parity discussion; p. 053506-4, Sec. V pulse paragraph The correlated exchange-symmetric two-electron target distinguishes even dark (1,1) from odd bright (1,2) and their 14-/17-photon resonance neighborhoods. |
| F3, O3, M7 | article pp. 053506-4-6, Figs. 4(a,b), 6(a,b) and Sec. V A common complex-response and relative-scale convention is needed to compare resonance classes. Reconstruction and absolute-scale restrictions are task obligations, not source file requirements. |
| F4, I4, M5 | article p. 053506-6, Eq. (4), unnumbered emission-time relation and Eqs. (5)-(6) The H15-H13 phase difference determines relative emission timing and is not the resonant-minus-background phase of one harmonic. |
| O1, I1, I2, M2 | article pp. 053506-4-5, Fig. 4(a,b) and following bright-resonance paragraphs Bright H17 is locally enhanced, H19 responds and H15 is much weaker in intensity; H17/H19 have approximately one phase-cycle variation, distinct from H15. |
| O2, I3, M3 | article pp. 053506-5-6, Figs. 5, 6(a,b) and adjacent dark-resonance discussion The dark state acts through laser-dressed pathways; H13/H15 maxima are displaced and H13 has a destructive minimum. Fig. 5 is contextual, not an extra required output. |
| M1, M4 | article p. 053506-6, paragraph preceding Eq. (4), Eq. (4) and following phase analysis; pp. 053506-4-5, Figs. 4 and 6 Complex resonant and nonresonant amplitudes interfere; their different detuning phases link amplitude and phase and distinguish bright/direct from dark/dressed response. |
| M6 | article p. 053506-6, Eqs. (5)-(6), lifetime paragraph and Fig. 7 The source obtains a delay near 300 as, comparable with but not equal to the inverse-width scale near 470 as. These are scale anchors, not exact mandatory values. |
| V1, V2 | article pp. 053506-2, 053506-4 and 053506-6, model definition, pulse paragraph and Sec. VI The reduced correlated model and finite resonance domain bound inference. Phase-origin/unwrapping, normalization and finite-detuning risks are judged only when material. |

## 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.

Use a reconstructible complex harmonic response $D_q=\int dt\,w(t)d(t)e^{iq\omega_l t}$, or a consistently converted velocity/acceleration equivalent; $I_q\propto|D_q|^2$ and $\phi_q=\arg D_q$. Write $D_q=D_q^{\rm nr}+D_q^{\rm r}$ so interference contributes $2\operatorname{Re}(D_q^{\rm nr}D_q^{{\rm r}*})$. This is the complex-amplitude entailment of Sec. V, not an extra source-numbered equation.

With the stated Fourier/time convention, $\Delta\phi=\phi_{15}-\phi_{13}$ gives $t_e=\Delta\phi/(2\omega_l)$, while $\delta\phi_q=\phi_q^{\rm r}-\phi_q^{\rm nr}$ compares pathways of the same harmonic. The delay is $\tau=(\Delta\phi^{\rm r}-\Delta\phi^{\rm nr})/(2\omega_l)$ (Eqs. (4)-(6)). Convert phase branches and time origin consistently; compare the evidence-supported scale to $\hbar/\Gamma$, not an exact equality.

## 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.