# Scientific Sources — Case09

## References

Zhong Guan, Xiao-Xin Zhou, and Xue-Bin Bian. 2016. *High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses*. *Physical Review A*, 93, 033852. [Source](https://doi.org/10.1103/PhysRevA.93.033852)

High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses

Mengxi Wu, Shambhu Ghimire, David A. Reis, Kenneth J. Schafer, and Mette B. Gaarde. 2015. *High-harmonic generation from Bloch electrons in solids*. *Physical Review A*, 91, 043839. [Source](https://doi.org/10.1103/PhysRevA.91.043839)

Secondary periodic Bloch/Houston method and physical-equivalence boundary, not a required auxiliary implementation.

## 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, F2 | Guan et al. (2016), article pp.033852-1-2, Eqs.(1)-(7), Fig.1(a-d), paragraph choosing k=0 in band 2: B-spline generalized metric and physical initial-state identity, not finite-box ordinal. |
| F3 | Guan et al. (2016), article p.033852-2, Eq.(8), and p.033852-3 Fourier/window paragraph: laser-induced current and spectrum. Gauge/derivative equivalence requires actual observable mapping, not appearance alone. |
| F4, O4, M5, M6, V1 | Guan et al. (2016), article pp.033852-2-4, Fig.1(a,b), Fig.2 and Fig.4(a,b), Sec.III.A comparison to Bloch/Houston expansions: static bands plus driven response comparisons. Distinct failure-mode sensitivity and case/range-specific reliability conclusions are task-defined; no named auxiliary method is compulsory. Secondary standard-method support: Wu et al. (2015), article pp.043839-2,6, Secs.II-III, Fig.4(a-c), Eqs.(22),(23) and Fig.5: matched physical k=0 valence initial state, Houston/Bloch equivalence and band-truncation limitations. |
| O1, I1, M1 | Guan et al. (2016), article p.033852-3, Fig.2 and two-plateau paragraph: first plateau and roughly five-orders-weaker second plateau, with cutoff scales near harmonic orders 40 and 120; bands 2/3 and higher-band pathways differ. |
| O2, O3, I2, I3, M2, M3, M4 | Guan et al. (2016), article pp.033852-2-4, Eqs.(9)-(12), Fig.4(a,b): same-band/intraband low-order contribution and different-band/interband higher-first-plateau dominance at Case-B intensity. Current components combine coherently; their powers do not simply add. |
| M7, V2 | Guan et al. (2016), article pp.033852-1-4, SAE model boundary, Sec.III.A numerical-stability comparison and last p.033852-4 model-potential qualification. Reconciling material disagreements and bounded reliability claims are task-defined, not universal experimental equivalence. |

## Formula and convention notes

For F1-F3/O3/M2-M4, article p.033852-2, Eqs.(4),(6)-(12), a nonorthogonal B-spline representation obeys $iS\dot C=H(t)C$, with norm $C^\dagger SC$. For physical band projectors $P_b$,
$$j_{\rm intra}=-\frac em\sum_b\langle\psi|P_b\hat pP_b|\psi\rangle,\quad
j_{\rm inter}=-\frac em\sum_{b\ne b'}\langle\psi|P_b\hat pP_{b'}|\psi\rangle,\quad j=j_{\rm intra}+j_{\rm inter}.$$
Thus $|\widetilde j|^2=|\widetilde j_{\rm intra}|^2+|\widetilde j_{\rm inter}|^2+2\Re(\widetilde j_{\rm intra}\widetilde j_{\rm inter}^*)$. In a different gauge the current and projectors must be transformed consistently. Equivalent methods must preserve the same physical k=0 band-2 state and causal content; matching a finite-box ordinal, swapping component names, or comparing two implementations sharing the relevant error class does not establish equivalence or independent reliability.

## 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.