Electron Vortex Beams in a Magnetic Field: A New Twist on Landau Levels and Aharonov-Bohm States

Abstract

We examine the propagation of the recently discovered electron vortex beams in a longitudinal magnetic field. We consider both the Aharonov-Bohm configuration with a single flux line and the Landau case of a uniform magnetic field. While stationary Aharonov-Bohm modes represent Bessel beams with flux- and vortex-dependent probability distributions, stationary Landau states manifest themselves as nondiffracting Laguerre-Gaussian beams. Furthermore, the Landau-state beams possess field- and vortex-dependent phases: (i) the Zeeman phase from coupling the quantized angular momentum to the magnetic field and (ii) the Gouy phase, known from optical Laguerre-Gaussian beams. Remarkably, together these phases determine the structure of Landau energy levels. This unified Zeeman-Landau-Gouy phase manifests itself in a nontrivial evolution of images formed by various superpositions of modes. We demonstrate that, depending on the chosen superposition, the image can rotate in a magnetic field with either (i) Larmor, (ii) cyclotron (double-Larmor), or (iii) zero frequency. At the same time, its centroid always follows the classical cyclotron trajectory, in agreement with the Ehrenfest theorem. Interestingly, the nonrotating superpositions reproduce stable multivortex configurations that appear in rotating superfluids. Our results open an avenue for the direct electron-microscopy observation of fundamental properties of free quantum-electron states in magnetic fields.

Alternate abstract:

Plain Language Summary

The simplest type of traveling light wave is a plane wave, whose wave fronts are planes perpendicular to the direction of the wave propagation. By comparison, the wave fronts of an optical vortex beam are spiral sheets with their common axis aligned along the direction of the beam propagation. Depending on the number of spiral sheets, the phase of the wave changes by a multiple of 2π along a loop enclosing the vortex axis. An orbital angular momentum (OAM) proportional to the phase change is carried by the beam. With their fascinating properties, vortex beams have been produced for acoustic, x-ray, visible-light, and radio waves, and have found a wide range of applications in optics, quantum information, and astronomy. Recently, vortex beams for free-electron matter waves were generated in electron microscopes. What has not yet been explored much is the fundamental physics contained in the fact that electron vortex beams can directly interact with applied magnetic fields. In this paper, we present some of the first theoretical explorations of how the interaction of an electron vortex beam with an external magnetic field gives rise to many fundamental quantum phenomena known only in other contexts.

We have explored two configurations of the magnetic-field–vortex-beam interaction. In the first, the so-called Aharonov-Bohm (infinitely thin shielded solenoid) configuration, a magnetic flux line is applied along the axis of the beam. The fundamental vortex-beam modes acquire flux-dependent radii. This can be regarded as a radial Aharonov-Bohm effect. In the second configuration, which corresponds to the so-called Landau configuration known in condensed matter physics, a uniform magnetic field is applied. The fundamental vortex-beam modes acquire field- and OAM-dependent phases. Importantly, we find that these phases describe known Landau energy levels and consist of two contributions: (i) the Zeeman interaction between the vortex OAM and the magnetic field as well as (ii) the optical Gouy phase caused by the transverse confinement of the beam. This finding leads to a unified Landau-Zeeman-Gouy description of these previously unrelated phenomena and the prediction of nontrivial behavior of quantum electron states in a magnetic field. In particular, we show that, in sharp contrast to classical electron-cyclotron motion, different superpositions of quantum electron-vortex modes can either rotate with distinct angular velocities or even reproduce nonrotating multiple-vortex configurations that also appear in rotating superfluids.

Our results allow the direct observation, using electron microscopy, of fundamental properties of quantum states of electrons in external fields.