# Scientific Sources — Case07

## References

Ákos Gombkötő, Sándor Varró, Péter Mati, and Péter Földi. 2020. *High-order harmonic generation as induced by a quantized field: Phase-space picture*. *Physical Review A*, 101, 013418. [Source](https://doi.org/10.1103/PhysRevA.101.013418)

High-order harmonic generation as induced by a quantized field: Phase-space picture

## Criterion-to-source crosswalk

Article page identifiers below are journal pages, not PDF viewer indices. Supplemental pages are explicitly marked. The crosswalk distinguishes direct source facts, mathematical entailments and task-defined evidence duties. Original images and PDFs are referenced only; none are reproduced.

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1, M1, M2 | Gombkoto et al. (2020), article pp.013418-2-3, Eqs.(1),(2),(8),(14) and paragraph preceding Eq.(15): complete no-RWA Hamiltonian, branch motion under field-plus-interaction and constant dipole if the atomic term is absent. |
| F2 | Gombkoto et al. (2020), article pp.013418-4-5, Eqs.(17)-(24): nonorthogonal coherent-state initialization, moving-lattice metric and atomic cross-branch mixing. |
| F3 | Gombkoto et al. (2020), article pp.013418-5-6, Sec.III.B, Eqs.(25)-(31), explicit reduced density operator paragraph: partial trace over atom and normalized field phase-space construction. |
| F4, O2, O3, O4, I2, I3, M5 | Gombkoto et al. (2020), article pp.013418-7-8, Eq.(34), Fig.5 upper/middle/lower spectra and caption: 20-cycle material-dipole spectra at increasing coherent amplitudes, peaks, plateau and growing cutoff. Peak fine detail is model-specific, not a compulsory exact match. |
| O1, I1, M4 | Gombkoto et al. (2020), article pp.013418-6-7, Eqs.(19),(24),(29), Fig.4(top) first-cycle illustration. The full moving-lattice task uses the complete reduced state; the minimal two-state discussion of Eqs.(32),(33) is illustrative and its smooth montage is not a unique full-model target. |
| M3, M6, M7 | Gombkoto et al. (2020), article pp.013418-3,6-8, Eqs.(16),(22),(23),(34), Sec.IV and final Sec.VI paragraphs: atomic term mixes branches and enables changing material dipole/radiation. Term interventions can establish equivalent causal evidence, but a changed line position alone is insufficient. |
| V1, V2 | Gombkoto et al. (2020), article pp.013418-7-9, Sec.V peak-detail limitation and Sec.VI-VII generalization/scope; normalization and complete-lattice stability follow Eqs.(18),(24). Positive model scope and triggered validation are task duties, not a demand to reproduce one lattice size. |

## Formula and convention notes

For F2-F4/M1-M3, article pp.013418-3-5, Eqs.(14),(18),(22)-(26), one consistent branch convention is $\gamma=-\Omega/\omega$,
$$\alpha_\pm(t)=\pm\gamma+(\alpha\mp\gamma)e^{-i\omega t},\quad
i\hbar N^{++}\dot c^+=(\hbar\omega_0/2)N^{+-}c^-,\quad
i\hbar N^{--}\dot c^-=(\hbar\omega_0/2)N^{-+}c^+.$$
The static Gram inverse initializes coefficients; the metric norm is $(c^+)^\dagger N^{++}c^+ +(c^-)^\dagger N^{--}c^-=1$. The invariant field object is $\rho_f=\mathrm{Tr}_a|\Psi\rangle\langle\Psi|$ and $\int W(\alpha)d^2\alpha=1$. With $\widetilde H=H-H_a$, $[\widetilde H,\sigma_x]=0$, whereas $\dot{\langle\sigma_x\rangle}=-\omega_0\langle\sigma_y\rangle$ in the usual Pauli convention for the full Hamiltonian. These identities explain why coherent-label motion alone is not harmonic radiation; equivalent causal term controls are accepted.

## Scope of the source anchors

The locators identify scientific objects and mechanisms, not compulsory figure layouts, solver settings, exact grids, pointwise tolerances or a fixed evidence package. Canonical examples may be replaced by functionally equivalent evidence subject to the rubric. Source readback verifies the candidate's science; it cannot supply missing candidate presentation, identification or inference. Reporting, applicability/N/A and descriptive Research Closure duties remain those stated in the rubric and task.