Radiation of "breathing" vortex electron packets in magnetic field

TL;DR

The paper addresses whether breathing NSLG vortex electrons radiate away their orbital angular momentum while propagating in a longitudinal magnetic field. Using a semiclassical Maxwell framework with NSLG-derived charge/current densities, it derives the period-averaged radiated power $\langle P\rangle_{T_c}$ and OAM loss rate $\langle dL/dt\rangle_{T_c}$, showing both scale as $(2n+|l|+1)^2$ and depend strongly on the field as $\langle P\rangle_{T_c}\propto H^6$ and $\langle dL/dt\rangle_{T_c}\propto H^5$. The results indicate that for realistic linac parameters the energy loss and OAM depletion are negligible, supporting the viability of accelerating vortex electrons while preserving their vorticity; only in extreme cases of very broad initial packets might measurable OAM loss occur, calling for careful consideration of the semiclassical regime. The work provides practical guidance for maintaining vortex-beam coherence in high-energy applications and suggests experimental tests to distinguish semiclassical from fully quantum radiation mechanisms.

Abstract

When a vortex electron with an orbital angular momentum (OAM) enters a magnetic field, its quantum state is described with a nonstationary Laguerre-Gaussian (NSLG) state rather than with a stationary Landau state. A key feature of these NSLG states is oscillations of the electron wave packet's root-mean-square (r.m.s.) radius, similar to betatron oscillations. Classically, such an oscillating charge distribution is expected to emit photons. This raises a critical question: does this radiation carry away OAM, leading to a loss of the electron's vorticity? To investigate this, we solve Maxwell's equations using the charge and current densities derived from an electron in the NSLG state. We calculate the total radiated power and the angular momentum of the emitted field, quantifying the rate at which a vortex electron loses its energy and OAM while propagating in a longitudinal magnetic field. We find both the radiated power and the angular momentum losses to be negligible indicating that linear accelerators (linacs) appear to be a prominent tool for maintaining vorticity of relativistic vortex electrons and other charged particles, at least in the quasi-classical approximation.

Figures (5)

• Figure 1: Schematic of the physical setup. A vortex electron wave packet enters a solenoid of length $d$. As it propagates along the $z$-axis, its wave function oscillates (the "breathing" NSLG mode). This non-stationary charge distribution emits radiation. The field emitted from a source point $\bm{r}$ at an earlier time $\tau$ is observed at a distant point $\bm{R}$.
• Figure 2: Angular distribution of the period-averaged radiation power $\langle \dd P / \dd \Omega \rangle_{T_{\text{c}}} \propto (1 + \cos^2{\theta}) \sin^2{\theta}$. (a) 3D spherical representation. (b) Polar plot with $\theta$ as the angular coordinate and the radial distance representing the distribution. Both plots share a common color scale indicating the distribution magnitude.
• Figure 3: Period-averaged radiated power as a function of the initial wave packet's width deviation $\sigma_0$. The value of $\sigma_0' = -3.1 \times 10^{-4} c$ was taken from Ref. schattschneider2014imaging that has been thoroughly considered in sec. V B of Sizykh2024Apr. The following parameters are used: $H = 1$ T (corresponding to $\sigma_{\text{L}} \approx 36$ nm and $\hbar \omega_{\text{c}} \approx 10^{-4}$ eV), $n = 0$, $l = 10$. The panels show the dependence over three distinct scales of $\sigma_0$: (a) nanometers, (b) micrometers, and (c) sub-millimeters.
• Figure 4: Period-averaged OAM loss per 1 km linac flight time $t = 3.5 \; \mu\text{s}$ as a function of the normalized initial wave packet's width deviation $\sigma_0/\sigma_\text{L}$. The following parameters are used: $H = 1$ T (corresponding to $\sigma_{\text{L}} \approx 36$ nm and $\hbar \omega_{\text{c}} \approx 10^{-4}$ eV), $\sigma'_0 = 0$, $n = 0$. The panels show the dependence over three distinct scales of $\sigma_0/\sigma_\text{L}$: (a) $\sigma_0 \approx \sigma_\text{L}$, (b) $\sigma_0 > \sigma_\text{L}$, and (c) $\sigma_0 \gg \sigma_\text{L}$. Horizontal line $\Delta L_z = 1 \hbar$ on the graph (c) shows the limitation of the utilized semiclassical approach that does not allow to predict angular momentum loss exceeding the single quantum of OAM.
• Figure 5: The ratio of the energy radiated for one cyclotron period to the energy of the electron's transverse motion, $E_{\text{rad}} / E_{\perp}$, as a function of the initial transversal size deviation $\sigma_0$ of the wave packet. The following parameters are used: $H = 1$ T (corresponding to $\sigma_{\text{L}} \approx 36$ nm and $\hbar \omega_{\text{c}} \approx 10^{-4}$ eV), $\sigma'_0 = 0$, $n = 0$. The panels show the dependence over three distinct scales of $\sigma_0$: (a) nanometers, (b) micrometers, and (c) sub-millimeters.