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We examine the propagation of the recently-discovered electron vortex beamsin a longitudinal magnetic field. We consider both the Aharonov-Bohmconfiguration with a single flux line and the Landau case of a uniform magneticfield. While stationary Aharonov-Bohm modes represent Bessel beams with flux-and vortex-dependent probability distributions, stationary Landau statesmanifest themselves as non-diffracting Laguerre-Gaussian beams. Furthermore,the Landau-state beams possess field- and vortex-dependent phases: (i) theZeeman phase from coupling the quantized angular momentum to the magnetic fieldand (ii) the Gouy phase, known from optical Laguerre-Gaussian beams.Remarkably, together these phases determine the structure of Landau energylevels. This unified Zeeman-Landau-Gouy phase manifests itself in a nontrivialevolution of images formed by various superpositions of modes. We demonstratethat, depending on the chosen superposition, the image can rotate in a magneticfield with either (i) Larmor, (ii) cyclotron (double-Larmor), or (iii) zerofrequency. At the same time, its centroid always follows the classicalcyclotron trajectory, in agreement with the Ehrenfest theorem. Remarkably, thenon-rotating superpositions reproduce stable multi-vortex configurations thatappear in rotating superfluids. Our results open up an avenue for the directelectron-microscopy observation of fundamental properties of free quantumelectron states in magnetic fields.

Electron vortex beams in a magnetic field: A new twist on Landau levels and Aharonov-Bohm states