# Scientific sources - Case48

## References

Christian Saugbjerg Lange and Lars Bojer Madsen. 2025. *Hierarchy of approximations for describing quantum light from high-harmonic generation: A Fermi-Hubbard-model study*. *Physical Review A*, 111, 013113. [Source](https://doi.org/10.1103/PhysRevA.111.013113).

Three retained photonic reductions share one eight-site Hubbard current engine and the same spectrum/Q/optimized-squeezing definitions. Two pulse cases form six members; output coverage milestones do not replace named-member evidence, and independent per-mode normalization is not equivalent to the adopted multimode trace.

## Criterion-to-source crosswalk

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | Lange and Madsen (2025), author manuscript pp. 2–3, Eqs. (9)–(12); pp. 7–8, Eqs. (42)–(44). Half-filled eight-site Hubbard dynamics, common propagated diagonal/transition currents and two pulse durations. Source Eq. (20) full decoupled branch is background, not a fourth required route. |
| F2 | Author manuscript pp. 4–5, Eq. (19), Eqs. (22a,b). Ground-transition branch product and initial vacuum. Same-branch other-mode norm product and one global trace are physical entailments of the task-adopted multimode state; fixed odd H1–H59 modes are task-defined, not a source display grid. |
| F3 | Author manuscript p. 5, Eqs. (23)–(26); p. 3, Eqs. (13), (15)–(16). Local-time elimination and Fock-state moments preserve the Markov-type route identity rather than silently using an independent branch. |
| F4 | Author manuscript pp. 5–7, Eqs. (28)–(41), especially (30)–(34), (38). MSA displaced current-correlation state, electronic expectation, discarded higher commutator and second-order coupling content; moment derivation must retain the declared order. |
| O1, O2, O3 | Author manuscript p. 8, Fig. 3(a–c); p. 9, Fig. 4(a–c); observable definitions p. 3, Eqs. (13), (15)–(16). Exactly six pulse×observable members, each compared across the three retained routes. First, broad and full coverage are task-defined milestones, not a source figure-count requirement or proof of named-member inference. |
| I1 | Author manuscript pp. 8–10, Fig. 3(a), Fig. 4(a), spectrum paragraphs. Both pulse spectra: main harmonic train, higher-frequency structure and route relation. Spectrum agreement alone does not identify full emitted-state fidelity. |
| I2 | Author manuscript pp. 8–10, Fig. 3(b), Fig. 4(b), denominator explanation after Eq. (40). Both pulse Mandel-Q members, with photon-number validity and defensible resolution. Source raw spikes or clipped plotting range are not targets. |
| I3 | Author manuscript pp. 8–11, Fig. 3(c), Fig. 4(c), low-harmonic squeezing discussion; Eq. (41), p. 7. Both pulse optimized-squeezing members and observable/region-dependent duration relation; no fixed inset/peak coordinate. |
| M1, M2 | Author manuscript pp. 6–7, Eqs. (32)–(41); pp. 9–11, Figs. 3–4 and spectral/nonclassical comparison paragraphs. Positive mechanism for why main spectral agreement need not preserve photon-number fluctuations or quadrature noise; no universal winning observable is imposed. |
| M3, M4 | Author manuscript pp. 9–11, Figs. 3–5, longer-pulse and averaged-error limitations paragraphs. Matched duration/frequency-region validity synthesis. Fig. 5 additional pulse-length scan is contextual, not a required new sweep. |
| M5, M6 | Author manuscript pp. 5–7, Eqs. (25)–(41); pp. 10–11, discarded-commutator explanation. Diagonal current controls leading spectra while transition-current correlations control nonclassical moments; local-time and higher-commutator reductions discard different information. |
| M7 | Author manuscript pp. 4–7, Eqs. (19), (22), (26), (30); observable definitions Eqs. (13), (15)–(16). Task-defined representation mapping to the same branch/state/order/global trace. Correct executable normalization is distinct from explaining it; no fourth route or continuum limit is inferred. |
| M8, V2 | Author manuscript pp. 9–11, Q denominator, narrow-frequency structure and MSA limitations; Eqs. (15)–(16), p. 3. Falsifiable candidate-arising limitations and conditional risks. Routine numerical checks are not detachable points or a prescribed method list. |
| V1 | Author manuscript pp. 7–8, finite Hubbard/pulse setting; p. 11, weak-coupling conclusion. Finite eight-site, weak-coupling, two-pulse three-route scope, not all approximations or an exact continuum limit. |

## Formula and convention notes

For the stated Hubbard ring, $H=-t_0\sum_{j,\mu}(e^{iaA(t)}c^\dagger_{j\mu}c_{j+1,\mu}+\mathrm{h.c.})+U\sum_jn_{j\uparrow}n_{j\downarrow}$ and $\hat j=-iat_0\sum_{j,\mu}(e^{iaA}c^\dagger_{j\mu}c_{j+1,\mu}-\mathrm{h.c.})$ (Eqs. (42)–(44)). $j_{mn}(t)=\langle\phi_m(t)|\hat j(t)|\phi_n(t)\rangle$ comes from the common electronic propagation, not separate fitted currents for each photonic route.

For the adopted ground-transition state $|\Psi\rangle=\sum_m|\phi_m\rangle\bigotimes_{\ell\in\mathcal K}|\chi_\ell^{(m)}\rangle$, define $N_{m\ell}=\langle\chi_\ell^{(m)}|\chi_\ell^{(m)}\rangle$ and $\mathcal Z=\sum_m\prod_{\ell\in\mathcal K}N_{m\ell}$. Then $\langle O_k\rangle=\mathcal Z^{-1}\sum_m\langle\chi_k^{(m)}|O_k|\chi_k^{(m)}\rangle\prod_{\ell\ne k}N_{m\ell}$. This is a trace derivation from the orthogonal electronic branches and product ansatz, not an independently printed source moment formula. $\mathcal K=\{H1,H3,\ldots,H59\}$ is the fixed task physical-mode set; modewise renormalization destroys this object.

The common observables are $S(\omega_k)=\omega_k^3\sum_\sigma\langle n_{k\sigma}\rangle/[g_0^2(2\pi)^2c^3]$, $Q=\operatorname{Var}(n)/\langle n\rangle-1$, and $\eta=-10\log_{10}[4\min_\theta\operatorname{Var}(X_\theta)]$ with $X_\theta=(ae^{-i\theta}+a^\dagger e^{i\theta})/2$ (Eqs. (13), (15)–(16)). For normalized moments, $\min_\theta\operatorname{Var}(X_\theta)=\{1+2[\langle n\rangle-|\langle a\rangle|^2]-2|\langle a^2\rangle-\langle a\rangle^2|\}/4$ is an equivalent derived minimization. Q is undefined at zero mean; no hidden photon-number cutoff is imposed.

The MSA state is $|\chi_k^{(i)}\rangle=D[\beta_{ii}^{(k)}]\exp[-\langle W_k^2\rangle_{\rm el}/2]|0\rangle$, $W_k=\int dt\,A_{Q,k}(t)[j_H(t)-\langle j_H(t)\rangle]$ and $\beta_{mn}^{(k)}=-ig_0\omega_k^{-1/2}\int dt\,j_{mn}(t)e^{i\omega_kt}$ for a fixed real polarization (Eqs. (28)–(31)). The approximation neglects $[\dot W,W]$ and retains second-order $g_0$ state content. The covariance is $\langle j_H(t)j_H(t\prime)\rangle-\langle j_H(t)\rangle\langle j_H(t\prime)\rangle=\sum_{m\ne i}j_{im}(t)j_{mi}(t\prime)$ (Eq. (33)); it is not reducible to the diagonal-current spectrum.


## 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.