# Scientific sources - Case36

## References

Nivash R, Amol R. Holkundkar, and Jayendra N. Bandyopadhyay. 2023. *Temporal control of high-order harmonic cutoffs in periodic crystals*. *Physical Review A*, 108, 063503. [Source](https://doi.org/10.1103/PhysRevA.108.063503).

Coherent filled-valence full-BZ total-current HHG: matched driver support, delay-controlled cutoff energy/time and yield, and band-access mechanisms. Dephasing-on, subcycle transfer and alternate-lattice studies are contextual only.

## Criterion-to-source crosswalk

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | R, Holkundkar and Bandyopadhyay (2023), article p. 063503-2, Eqs. (1)–(3), Fig. 2(a,b), p. 063503-3. Periodic model, reference/shaped fields and vector potentials; matched case and delay binding. |
| F2, F3 | Article pp. 063503-2–3, Fig. 1, Eqs. (4)–(11). Bloch bands, driven coefficients and coherent occupied-valence full-BZ current. The initial-VB1 population diagnostic in Fig. 4 does not replace the main VB0+VB1 current sum. |
| F4, V1 | Article p. 063503-3, text following Eq. (11); Figs. 7(a–d), p. 063503-6, and 9(a–c), p. 063503-7. Current-derived spectrum, yield integral and claim-relevant timing/cutoff definitions. Reconstructibility and common-scale reporting are task-defined; no source threshold/window is compulsory. |
| O1, I1, M1 | Article p. 063503-4, Fig. 3(a–d) and Sec. III A. Matched spectral support, gap-organized plateaus and timing structure distinguish temporal-driver change from scalar rescaling. |
| O2 | Article p. 063503-7, Fig. 9(a,b) and Sec. III D. Delay-dependent cutoff-energy family and finite-domain optimum; no unique sample mesh or optimum coordinate. |
| O3 | Article p. 063503-7, Fig. 9(a,c) and Sec. III D. Band-integrated yield versus delay; plotted high-energy yield is rescaled for visibility, not a required absolute ratio. |
| O4, I3, M3 | Article pp. 063503-4–6, Fig. 3(a,b), Fig. 4(a–c), Fig. 6(a,c–e); Eqs. (12)–(14), p. 063503-4. Successive band access and interband gap identity. Gauge-correct populations, trajectories, gaps or equivalent discriminating evidence are accepted; no fixed band-population plot is required. |
| I2 | Article p. 063503-6, Fig. 7(a–d), caption and timing paragraphs. Cutoff-emission-time feature identity and delay dependence, independently of energy/yield credit. The printed exponent is a finite-model example, not a universal fitting target. |
| M2 | Article pp. 063503-1, 063503-4–6, crystal-momentum paragraph, Fig. 2(b), Fig. 6(b,c). Physical entailment of $k(t)=k_0+A(t)$ linking vector potential to band traversal. |
| M4 | Article p. 063503-7, Fig. 9(b,c), competition paragraph referring to Fig. 6(b–e). Enhanced access competes with overdrive, longer trajectories and dispersion; stronger vector potential alone need not increase cutoff or yield. |
| M5 | Article pp. 063503-6–8, Fig. 7(d), Fig. 9(b,c), Appendix Fig. 11(d). Distinct energy, timing and yield dependences; changed lattice gives a different timing exponent. Alternate lattice/driver calculations are context, not compulsory transfer outputs. |
| M6, V2 | Article p. 063503-4, Fig. 3(a–d); pp. 063503-5–6, Fig. 6(a–e), Fig. 7(a–d); Eq. (11), p. 063503-3. Band/timing cross-representation checks can expose wrong ridges, incomplete currents or spectral artefacts. Conditional validation does not prescribe a particular FFT or Gabor procedure. |
| M7 | Article pp. 063503-2–3, Sec. II; p. 063503-5, Sec. III B; p. 063503-8, Sec. V. Coherent thin-crystal single-electron model scope. Dephasing-on, subcycle-driver and alternate-lattice examples are not required main outputs. |

## Formula and convention notes

In atomic units $H_0=p^2/2+V(x)$, $V(x)=-V_0[1+\cos(2\pi x/d)]$; $H_0\phi_k^n=E_k^n\phi_k^n$ and $i\dot\alpha_k^s=E_k^s\alpha_k^s+A(t)\sum_u p_k^{su}\alpha_k^u$ (Eqs. (1), (4), (8)). $n,s,u$ label bands, $k$ is crystal momentum and $p_k^{su}$ the momentum matrix element.

$E=-\dot A$, with a consistently declared integration constant; $j_{ks}=-\operatorname{Re}\langle\psi_{ks}|p+A|\psi_{ks}\rangle$ and $j(t)=\sum_{s\in\mathrm{VB}}\int_{\mathrm{BZ}}j_{ks}(t)dk$ (Eqs. (10)–(11)). The coherent main object includes VB0+VB1; Fourier-transforming separate channel intensities and then summing is not equivalent to transforming the total current.

$S(\Omega)=|\int j(t)e^{i\Omega t}dt|^2$, $Y_{[\Omega_1,\Omega_2]}=T^{-1}\int_{\Omega_1}^{\Omega_2}S(\Omega)d\Omega$; $T$ is the analyzed pulse duration and the integration band and window must be declared. Cutoff emission time belongs to the same resolved high-energy feature, not the overall time maximum. Equivalent radiation/timing estimators are accepted with consistent mappings.

The semiclassical relation $k(t)=k_0+A(t)$ and emitted interband energy $\Omega_{mn}(t)=E_m[k(t)]-E_n[k(t)]$ (with $\hbar=1$) explain the band-access/cutoff mapping; a trajectory diagnostic is not a substitute for the filled-valence quantum current. Timing exponents and optimum delays are domain- and band-structure-dependent.


## 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.