Non-linear quantum effects in electromagnetic radiation of a vortex electron

There is a controversy of how to interpret interactions of electrons with a large spatial coherence with light and matter. When such an electron emits a photon, it can do so either as if its charge were confined to a point within a coherence length, the region where a square modulus of a wave function $|ψ|^2$ is localized, or as a continuous cloud of space charge spread over it. This problem was addressed in a recent study R.~Remez, et al., Phys. Rev. Lett. {\bf 123}, 060401 (2019) where a conclusion was drawn in favor of the first (point) interpretation. Here we argue that there is an alternative explanation for the measurements reported in that paper, which relies on purely classical arguments and does not allow one to refute the second interpretation. We propose an experiment of Smith-Purcell radiation from a non-relativistic vortex electron carrying orbital angular momentum, which can unambiguously lead to the opposite conclusion. Beyond the paraxial approximation, the vortex packet has a non-point electric quadrupole moment, which grows as the packet spreads and results in a non-linear $L^3$-growth of the radiation intensity with the length $L$ of the grating when $L$ is much larger than the packet's Rayleigh length. Such a non-linear effect has never been observed for single electrons and, if detected, it would be a hallmark of the non-point nature of charge in a wave packet. Thus, two views on $|ψ|^2$ are complementary to each other and an electron radiates either as a point charge or as a continuous charge flow depending on the experimental conditions and on its quantum state. Our conclusions hold for a large class of non-Gaussian packets and emission processes for which the radiation formation length can exceed the Rayleigh length, such as Cherenkov radiation, transition radiation, diffraction radiation, and so forth.