On the Fast Electrons-Optical Modes Interaction Problem Error

Abstract

A recently developed relativistic quantum optics approach for a fast electron beam-optical modes interaction problem (Optica 6, 1524--1534 (2019)) contains suggestion about a smallness of the error induced by replacement ${\Psi }_{\vec{k}}\approx {\Psi }_{{\vec{k}}_0}$ for noninteracting free electron Dirac spinors at such wave vector $\vec{k}$ that $\left|\vec{k}-{\vec{k}}_0\right|\ll \left|{\vec{k}}_0\right|$ with ${\vec{k}}_0$ as a peaked wave vector of a focused electron beam. Here, based on quantum information approach we strictly investigated a degree of coincidence of spinors ${\Psi }_{\vec{k}}$ and ${\Psi }_{{\vec{k}}_0}$ by calculating fidelity for them. The analytical expression is obtained. It is shown that the fidelity inevitably depends on not only wave vectors $\vec{k}$ and ${\vec{k}}_0$ but also on mutual orientation of $[{\vec{k}}_0,\vec{k}]$ and ${\hat{\vec{s}}}^{†}\vec{\sigma }\hat{\vec{s}}$ with $\vec{s}$ is a two-component spin unit vector and $\vec{\sigma }$ is a vector of Pauli matrices. This result can be conceived as a chirality-like effect in the fidelity and constraints spin arbitrariness when applying the developed relativistic quantum optics approach to the problems of interest. A geometrical configuration of ${\vec{k}}_0$, $\vec{k}-{\vec{k}}_0$, and ${\hat{\vec{s}}}^{†}\vec{\sigma }\hat{\vec{s}}$ that is optimal in terms of fidelity is established.