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 non-diffracting 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. Remarkably, the non-rotating superpositions reproduce stable multi-vortex configurations that appear in rotating superfluids. Our results open up an avenue for the direct electron-microscopy observation of fundamental properties of free quantum electron states in magnetic fields.
Highlights
Propagating waves carrying intrinsic orbital angular momentum (OAM), known as vortex beams, are widely explored and employed in optics [1,2,3]
2r22 w2m þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where wm 1⁄4 2 @=jeBj 1⁄4 2@=mjj is the magnetic length parameter, 1⁄4 eB=2m is the Larmor frequency corresponding to the g factor g 1⁄4 1 for OAM [4,10,14], and we introduce the parameter 1⁄4 sgnB 1⁄4 Æ1, which indicates the direction of the magnetic field
These Landau-Zeeman-Gouy relations bring about phases strongly dependent on the mode quantum numbers and the magnetic field; below, we argue that this finding can be employed in interferometry of electron vortex beams
Summary
Propagating waves carrying intrinsic orbital angular momentum (OAM), known as vortex beams, are widely explored and employed in optics [1,2,3]. Coupling the quantized OAM to a magnetic field and (ii) the Gouy phase, known from optical LG beams [1,23] Together, these phases determine the structure of Landau energy levels. The interference patterns rotate in a magnetic field with a rate that is strongly dependent on the chosen superposition of modes—The angular velocity can vary between the Larmor, cyclotron, and zero frequencies This fact allows direct experimental observations of different terms in the Landau-level structure, akin to optical pattern rotations in Gouy-phase diffraction experiments [24,25,26,27] and Berry-phase [28,29,30,31,32] and rotational-Doppler-effect [33,34,35] observations. This fact allows direct experimental observations of different terms in the Landau-level structure, akin to optical pattern rotations in Gouy-phase diffraction experiments [24,25,26,27] and Berry-phase [28,29,30,31,32] and rotational-Doppler-effect [33,34,35] observations. (It is worth noticing that the optical Berry-phase and rotational-Doppler (Coriolis) effects are caused by rotations of the reference frame and are similar to the electron Zeeman effect, according to Larmor’s theorem.)
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call