# Scientific sources - Case37

## References

Maxim Sirotin, Andrei Rasputnyi, Tomáš Chlouba, Roy Shiloh, and Peter Hommelhoff. 2026. *Quantum computing and quantum optics with recoiled free electrons*. *arXiv*, 2405.06560. [Source](https://arxiv.org/abs/2405.06560v2). Version 2; first submitted in 2024, revised in 2026.

Single-photon vacuum-cavity exact recoil ladder across low and nonperturbative higher coupling. Source version is arXiv:2405.06560v2 (19 May 2026); driven control, horizon and two-photon sections are excluded except to identify the scope boundary.

## Criterion-to-source crosswalk

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1, F2 | Sirotin et al. (2026), main text p. 6, joint-state paragraph; SI pp. 20–22, Eqs. (S34)–(S44). Vacuum joint ladder, link-dependent recoil, bosonic factors and exact direct/autonomous evolution. Populations are invariant under exact autonomization; fidelity-sensitive phases need not be. |
| F3, M5 | Main text p. 6, Eqs. (5)–(6); SI pp. 30–31, Eqs. (S64)–(S65), Fig. S4(a). Physical energy/photon/length mapping and the approximate phase-matching width; the physical example is not a compulsory parameter tuple. |
| O1, I1, M1, M2 | Main text pp. 7–8, Eq. (7), Fig. 4(a–c), low-coupling single-photon paragraphs. Weak-recoil Poisson reference, restricted ladder and antibunching/sub-Poissonian statistics. Joint populations or equivalent electron sidebands with reconstructible statistics are accepted. |
| O2, I2, M3, M4 | Main text pp. 7–8, Fig. 4(d–f), high-coupling paragraphs; SI p. 31, Fig. S5(a,b). Coherent redistribution, returns, few-level oscillations and large factorial correlation at low mean. A capped correlation map alone cannot establish occupation dynamics or high yield. |
| O3, I3, M7, V1 | SI pp. 21–22, Eqs. (S38)–(S44), limitation paragraph after Eq. (S41); main text p. 8, Fig. 4(c). Exact production versus first-order sinc/Magnus approximation; alternative exact representations and informative tail/phase discrimination remain acceptable without a prescribed solver. |
| M6, M8 | Main text pp. 6–8, Eqs. (5)–(7), Fig. 4(c–f); SI p. 22, Eqs. (S41)–(S44). Physical synthesis: mismatch limits accessible levels, coupling governs coherent dynamics. Integrated state-generation conclusions must come from the candidate rather than source labels. |
| V2 | SI pp. 21–22, Eqs. (S39)–(S44); main text p. 7, low-mean correlation paragraph; Fig. S5(b), SI p. 31. Concrete representation, truncation and low-mean interpretation threats; conditional risk handling is task-defined, not a universal required check. |
| V3 | Main text pp. 6–7, joint vacuum state and Sec. III A; Sec. III B, pp. 7–9, as an excluded contrast. Scope is the initially empty single-mode first-link single-photon branch; two-photon/QPM, driven PINEM and horizon cases are not commissioned outputs. |

## Formula and convention notes

Let $E_n=mc^2+K_0-n\hbar\omega$ be total electron energy after $n$ emitted photons, $k_n=\sqrt{E_n^2-m^2c^4}/(\hbar c)$, and $\Delta_n=k_n-k_{n+1}-\kappa$, with $\kappa=k_0-k_1$ for first-link phase matching. This relativistic dispersion is exact in the stated ladder model; a nonrelativistic $\sigma$ estimate does not replace exact propagation.

For $|\psi(z)\rangle=\sum_{n\ge0}c_n(z)|E_n,n\rangle$, a convention equivalent to SI Eq. (S38) is $\partial_zc_n=\widetilde g\sqrt{n+1}e^{-i\Delta_n z}c_{n+1}-\widetilde g^*\sqrt n e^{i\Delta_{n-1}z}c_{n-1}$, $c_n(0)=\delta_{n0}$, with the negative-index term absent. Here $z\in[0,L]$, $g_{\rm Qu}=\widetilde gL$; the generator is anti-Hermitian. Consistent conjugation/sign redefinitions are equivalent, but deleting successive-link mismatch is not.

$P_n=|c_n(L)|^2$, $\sum_nP_n=1$, $\bar n=\sum_nnP_n$, $g^{(2)}(0)=\sum_nn(n-1)P_n/\bar n^2$. The factorial second moment is not $\langle n^2\rangle$. For $\bar n>0$, the Fano factor is $F=\operatorname{Var}(n)/\bar n=1+\bar n[g^{(2)}-1]$; this is a derived statistically equivalent classification when the underlying populations/mean are available. At zero mean the normalized correlation is undefined, and large $g^{(2)}$ at tiny mean does not imply high yield.

For the stated small fractional energy-loss approximation, $\sigma\simeq155[K_0(\mathrm{keV})]^{3/2}/\{[\hbar\omega(\mathrm{eV})]^2L(\mu\mathrm m)\}$ for electron kinetic energy below about 150 keV (SI Eq. (S65)); Eq. (S64) covers the broader relativistic estimate. $\sigma$ estimates the first mismatch zero, not a sharp exact population cutoff; $g_{\rm Qu}$ and $\sigma$ jointly govern higher-coupling dynamics.


## 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.