# Scientific sources - Case35

## References

Yi-Ben Wang and Xue-Bin Bian. 2025. *High-order harmonic generation in quantum light by a generalized von Neumann lattice method*. *Physical Review A*, 111, 043111. [Source](https://doi.org/10.1103/PhysRevA.111.043111).

Full nonorthogonal quantum-light HHG, state-specific single-mode comparisons and coherent-plus-squeezed two-colour transfer. Source-only two-mode and broader-state examples delimit approximations rather than add commissioned calculations.

## Criterion-to-source crosswalk

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | Wang and Bian (2025), article pp. 043111-6–7, Fig. 3(a,b), Fig. 4(a), Eq. (26); Appendices E–F, pp. 043111-10–11, Eqs. (E1)–(E2), (F1)–(F4). Matter ground state, field-amplitude calibration, single-mode state cases and two-colour frequency/phase controls; Fig. 4(b) is a different, unrequired two-mode squeezed case. |
| F2, M1 | Article pp. 043111-2–4, Eqs. (7)–(11), (18)–(22); Fig. 2(c,d), p. 043111-5. Nonorthogonal coefficients, overlap metric, normalization and full complex response. The task uses the explicitly declared equivalent quadratic-form convention. |
| F3, O4, M5 | Article pp. 043111-4–6, Eqs. (22)–(25), Fig. 3(a,b); Appendix G, Fig. 7(a,b), p. 043111-11. Full versus coefficient-diagonal and Husimi approximations; overlap smallness is state-dependent. The task comparator deletes off-diagonal terms from the same form, not from an independently normalized mixture. |
| O1, I1, M2 | Article p. 043111-6, Fig. 3(a) and squeezed-vacuum paragraph; Fig. 2(a,c), p. 043111-5. Extended squeezed harmonic support and altered envelope linked to longitudinal field-amplitude support; spectra are vertically offset, so absolute inter-curve ordinate ratios are not targets. |
| O2, I2, M3 | Article p. 043111-6, Fig. 3(b), Fock-state paragraph; Fig. 2(b,d), p. 043111-5. Closer Fock/coherent broad cutoff and structure, not pointwise identity or an all-state theorem. |
| M4 | Article pp. 043111-5–6, Figs. 2(a–d), 3(a,b), discussion after Fig. 3. Synthesis of the two separately established single-mode state-response chains; static state distributions alone are insufficient dynamic evidence. |
| O3, I3, M6, M7 | Article pp. 043111-6–7, Eq. (26), Fig. 4(a) and its caption and adjacent first-scenario paragraph. 800-nm coherent plus 400-nm squeezed case, phase alignment, even-harmonic family and broad quantum/classical spectral relation. Fig. 4(b) is not this comparison. |
| M8 | Article p. 043111-6, Eq. (25), Figs. 3(a,b), 4(a); Appendix G, Fig. 7(a,b), p. 043111-11. Evidence-based synthesis of state dependence, diagonal adequacy and two-colour transfer; the final integrated judgment is task-defined. |
| V1 | Article p. 043111-7, Sec. V; Appendices E–F, pp. 043111-10–11. Single-atom 1D model and theoretical coupling-domain limitations, not macroscopic or universal validation. |
| V2 | Article p. 043111-5, coefficient residual/normalization paragraph; p. 043111-6, Eq. (25); Appendices F–G, pp. 043111-10–11. Potential finite-lattice, response and approximation failure modes; conditional testing is task-defined, with no compulsory residual threshold or source grid. |

## Formula and convention notes

Nonorthogonal identity: $|\chi\rangle=\sum_i C_i|\alpha_i\rangle$, $G_{ij}=\langle\alpha_i|\alpha_j\rangle=\exp[-(|\alpha_i|^2+|\alpha_j|^2)/2+\alpha_i^*\alpha_j]$, $C^\dagger GC=1$. The projection system is $GC=Q$, $Q_i=\langle\alpha_i|\chi\rangle$; $Q_i$ is not itself an expansion coefficient (Eqs. (8)–(11)).

In the task convention $u_i(\Omega)=C_i j_i^*(\Omega)$, the full response is $S_{\rm full}(\Omega)\propto u^\dagger Gu$, and $S_{\rm diag}(\Omega)\propto\sum_i G_{ii}|u_i|^2$ deletes only off-diagonal terms. Here $j_i$ is the component complex HHG response at emitted frequency $\Omega$; equivalent conjugation/index conventions must preserve the complete quadratic form. This is not an arbitrary positive Husimi mixture.

The common electronic initial state is the normalized ground state of $H_0=p^2/2-1/\sqrt{x^2+a^2}$, with the stated $a=1$ a.u.; component propagation may use $H_i=H_0+xE_i(t)$ or the gauge-equivalent velocity form. Optical calibration is $E_{0i}=2g_0\sqrt{\omega}|\alpha_i|$, $\phi_i=-\arg\alpha_i$; $\langle n\rangle=\sinh^2r$ for squeezed vacuum and $N$ for Fock states (Appendices E–F). These relations bind the declared finite-pulse single-atom cases, not realistic macroscopic yields.


## Scope of the source anchors

Source examples verify physical identity and robust structure; their numerical samples, plotting conventions and panel layout are not compulsory candidate deliverables. Reporting, reconstructibility, evidence attribution, conditional validation and Research Closure are examination-defined obligations, not additional results asserted by the article. A derivation or physical entailment is identified as such rather than presented as a directly printed source formula.