Comment on: "Nonlinear quantum effects in electromagnetic radiation of a vortex electron"

Technical Summary: Comment on "Nonlinear quantum effects in electromagnetic radiation of a vortex electron"

Problem Statement
This paper addresses a scientific controversy regarding the interpretation of free-electron wavefunctions in spontaneous emission processes, specifically within the context of the Smith-Purcell effect. The dispute centers on a critique by Karlovets and Pupasov-Maximov (Ref1) of a prior experimental and theoretical study by Remez et al. (Ref2). Ref1 argued that the experimental parameters in Ref2 were invalid for observing quantum wave nature, claiming the measurements were conducted in the near-field rather than the far-field. Consequently, Ref1 suggested that the observed radiation patterns could be explained by a semiclassical interpretation (where the electron acts as a smeared-out charge density, $\rho(\mathbf{r}) = e|\psi(\mathbf{r})|^2$) rather than the quantum interpretation (where the electron effectively "collapses" to a point upon emission). The authors of this comment aim to refute the claims made in Ref1 regarding the experimental regime and to clarify the theoretical conditions under which the derivations in Ref1 hold.

Methodology
The authors employ two primary methods to address the criticisms:

1. Analytical Optical Arguments: The authors re-evaluate the Fraunhofer diffraction condition ($r \gg d^2/\lambda$) applied to the experiment. They argue that Ref1 incorrectly used the total incoherent beam width ($w_{beam}$) as the effective aperture $d$. Instead, they posit that for partially coherent sources, the relevant parameter is the transverse coherence length ($l_c$). They perform a quantitative comparison of the experimental observation distance ($r \approx 0.25$ m) against the Fraunhofer distances calculated using both $w_{beam}$ and $l_c$.
2. Numerical Simulations (FDTD): To validate their optical arguments, the authors conducted Finite-Difference Time-Domain (FDTD) simulations of Smith-Purcell emission. They scaled the problem to SEM energies ($E=30$ keV) with a grating periodicity of $\Lambda = 200$ nm. The simulations modeled an incoherent beam width ($w_{beam} = 20$ $\mu$m) while varying the coherence length ($l_c$) between 0.2 $\mu$m, 1 $\mu$m, and 4 $\mu$m. The near-field was propagated to a distance ($r = 100$ $\mu$m) that satisfies the far-field condition relative to $l_c$ ($r \gg l_c^2/\lambda$) but remains in the near-field relative to $w_{beam}$ ($r \ll w_{beam}^2/\lambda$), directly testing the regime claimed by Ref1.

Key Contributions and Results

• Correction of the Far-Field Regime: The authors demonstrate that the criticism regarding the experimental regime is based on a misunderstanding of coherence. By correctly identifying the transverse coherence length ($l_c$) as the effective aperture for interference, they show that the Fraunhofer condition is easily met in the Ref2 experiment ($r \gg l_c^2/\lambda$). Their simulations confirm that even at distances where the beam width suggests a near-field regime, the radiation divergence is determined by $l_c$, not $w_{beam}$.
• Validation of the Quantum Interpretation: The simulations reveal that under the semiclassical interpretation, varying $l_c$ should result in a dramatic change in angular divergence ($\Delta\phi \sim \lambda/l_c$). However, the experimental data from Ref2 showed no such change in divergence despite a six-fold variation in $l_c$. This lack of sensitivity to $l_c$ allows for the rejection of the semiclassical model in favor of the quantum interpretation, where emission is incoherent and point-like.
• Clarification of Theoretical Post-Selection: The authors identify a critical distinction in the theoretical derivations of Ref1. They argue that Ref1's results, which suggest a dependence of emission rates on the initial electron wavefunction, apply only when the final electron state is post-selected (observed in coincidence with the emitted photon). In standard cathodoluminescence experiments (like Ref2), where only the light is measured, the electron final state is "traced out." This tracing leads to decoherence, rendering the radiation independent of the initial electron wavefunction. The authors assert that Ref1's failure to account for this lack of post-selection in the experimental context led to the conflicting theoretical predictions.

Significance and Claims
The paper concludes that the concerns raised by Karlovets and Pupasov-Maximov regarding the experimental validity of Ref2 are unfounded. By establishing that the experiment was conducted in the far-field relative to the coherence length, the authors reaffirm that the experimental evidence supports the quantum wave nature of free electrons over the semiclassical charge-density model.

Furthermore, the authors suggest that the theoretical work in Ref1 remains a valuable contribution to the field, provided its scope is correctly understood. They posit that Ref1 offers novel expressions for emission rates beyond the paraxial approximation, but these expressions are specifically applicable to scenarios involving electron post-selection or coincidence measurements. By clarifying the necessity of electron post-selection for wavefunction-dependent emission, the authors believe the two studies can be reconciled, with Ref1 providing a theoretical framework for specific, advanced experimental configurations while Ref2 correctly describes the standard spontaneous emission regime.