Zitterbewegung of relativistic electrons in a magnetic field and its simulation by trapped ions

TL;DR

This paper investigates the motion of relativistic electrons in magnetic fields, analyzing Zitterbewegung through analytical calculations and proposing trapped ion simulations to observe these quantum effects.

Contribution

It provides exact analytical solutions for electron trajectories in magnetic fields and demonstrates how to simulate Zitterbewegung using trapped ions and laser excitations.

Findings

Magnetic fields induce intraband frequency components in Zitterbewegung.

Energy quantization affects the frequency and stationary nature of electron motion.

Simulations suggest observable effects of magnetic fields on Zitterbewegung in trapped ions.

Abstract

One-electron 3+1 and 2+1 Dirac equations are used to calculate the motion of a relativistic electron in a vacuum in the presence of an external magnetic field. First, calculations are carried on an operator level and exact analytical results are obtained for the electron trajectories which contain both intraband frequency components, identified as the cyclotron motion, as well as interband frequency components, identified as the trembling motion (Zitterbewegung, ZB). Next, time-dependent Heisenberg operators are used for the same problem to compute average values of electron position and velocity employing Gaussian wave packets. It is shown that the presence of a magnetic field and the resulting quantization of the energy spectrum has pronounced effects on the electron Zitterbewegung: it introduces intraband frequency components into the motion, influences all the frequencies and makes the motion stationary (not decaying in time) in case of the 2+1 Dirac equation. Finally, simulations of the 2+1 Dirac equation and the resulting electron ZB in the presence of a magnetic field are proposed and described employing trapped ions and laser excitations. Using simulation parameters achieved in recent experiments of Gerritsma and coworkers we show that the effects of the simulated magnetic field on ZB are considerable and can certainly be observed.