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 ${\left|\psi \right|}^2$ 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. 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 a purely classical concept of a so-called prewave zone and does not allow one to refute the second interpretation. We propose an experiment of Smith-Purcell radiation from a nonrelativistic vortex electron carrying orbital angular momentum, which can unambiguously lead to the opposite conclusion. Beyond the paraxial approximation, the vortex packet has a nonpoint electric quadrupole moment, which grows as the packet spreads and results in a nonlinear $L^3$ 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 nonlinear effect has never been observed for single electrons and, if detected, it would be a hallmark of the nonpoint nature of charge in a wave packet. Thus, two views on ${\left|\psi \right|}^2$ 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.

中文翻译：

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涡旋电子电磁辐射中的非线性量子效应

关于如何解释与光和物质具有大空间相干性的电子的相互作用存在争议。当这样的电子发射光子时，它可以像将其电荷限制在相干长度内的点一样，该长度是波函数的平方模量的区域${\left|\psi \right|}^2$是局部的，还是空间电荷在其上连续散布的云。最近的研究[R. Remez等。，物理 牧师 123，060401（2019）]中得出的结论是对第一个（点）解释的支持。在这里，我们争辩说，该论文中报道的测量存在另一种解释，它依赖于所谓的预波带的纯粹经典概念，不允许人们反驳第二种解释。我们提出了一个来自非相对论涡旋电子携带轨道角动量的史密斯-珀塞尔辐射的实验，该实验可以明确地得出相反的结论。除了近轴近似之外，涡旋包还具有非点电四极矩，该矩随包扩散而增长，并导致非线性${\mathrm{大号}}^3$ 辐射强度随长度的增长 $\mathrm{大号}$ 时的光栅 $\mathrm{大号}$比数据包的瑞利长度大得多。对于单个电子从未观察到这种非线性效应，如果检测到，它将成为波包中电荷的非点性质的标志。因此，关于${\left|\psi \right|}^2$彼此互补，根据实验条件及其量子态，电子以点电荷或连续电荷流的形式辐射。我们的结论适用于辐射形成长度可能超过瑞利长度的一大类非高斯数据包和发射过程，例如切伦科夫辐射，跃迁辐射，衍射辐射等。