Electron vortex beams in non-uniform magnetic fields

TL;DR

This paper develops a quantum theory for electron vortex beams in non-uniform magnetic fields, deriving their wave functions, conserved quantities, and dynamic behavior, with implications for quantum Hall systems.

Contribution

It introduces a comprehensive quantum model for electron vortex beams in non-uniform magnetic fields, including wave functions and conserved operators, extending to quantum Hall systems.

Findings

Wave functions are joint eigenstates of two gauge-independent operators.

Magnetic field variations can split a mode into superpositions.

Analysis applies to quantum Hall systems with time-dependent fields.

Abstract

We consider the quantum theory of paraxial non-relativistic electron beams in non-uniform magnetic fields, such as the Glaser field. We find the wave function of an electron from such a beam and show that it is a joint eigenstate of two ($z$-dependent) commuting gauge-independent operators. This generalized Laguerre-Gaussian vortex beam has a phase that is shown to consist of two parts, each being proportional to the eigenvalue of one of the two conserved operators and each having different symmetries. We also describe the dynamics of the angular momentum and cross-sectional area of any mode and how a varying magnetic field can split a mode into a superposition of modes. By a suitable change in frame of reference all of our analysis also applies to an electron in a quantum Hall system with a time-dependent magnetic field.