Propagation of vortex electron wave functions in a magnetic field

Abstract

The physics of coherent beams of photons carrying axial orbital angular momentum (OAM) is well understood and such beams, sometimes known as vortex beams, have found applications in optics and microscopy. Recently electron beams carrying very large values of axial OAM have been generated. In the absence of coupling to an external electromagnetic field the propagation of such vortex electron beams is virtually identical mathematically to that of vortex photon beams propagating in a medium with a homogeneous index of refraction. But when coupled to an external electromagnetic field the propagation of vortex electron beams is distinctly different from photons. Here we use the exact path integral solution to Schrodingers equation to examine the time evolution of an electron wave function carrying axial OAM. Interestingly we find that the nonzero OAM wave function can be obtained from the zero OAM wave function, in the case considered here, simply by multipling it by an appropriate time and position dependent prefactor. Hence adding OAM and propagating can in this case be replaced by first propagating then adding OAM. Also, the results shown provide an explicit illustration of the fact that the gyromagnetic ratio for OAM is unity. We also propose a novel version of the Bohm-Aharonov effect using vortex electron beams.

Electron vortices: Beams with orbital angular momentum

“…Thus, on traversing a cyclotron orbit, the resulting orbital angular momentum vector now points in the direction opposite to the initial orientation, as shown in Fig. 11 (Gallatin and McMorran, 2012).…”

“…The Landau configuration involves a constant magnetic field in a fixed direction (Bliokh and Nori, 2012a;Gallatin and McMorran, 2012;Greenshields et al, 2012;Landau and Lifshitz, 1977), whilst the AharanovBohm configuration can involve a single line of flux (Bliokh and Nori, 2012a). The vortex propagation direction may be transverse (Gallatin and McMorran, 2012), parallel (Bliokh et al, 2012;Greenshields et al, 2012) to the direction of the field or in an arbitrary orientation (Greenshields et al, 2014), and the interaction between the magnetic moment with the external field leads to interesting dynamics, as we now explain.…”

“…Solutions of the Schrödinger equation in the presence of several di↵erent field configurations have been investigated (Bliokh et al, 2012;Gallatin and McMorran, 2012;Greenshields et al, 2012Greenshields et al, , 2014Schattschneider et al, 2017;Velasco-Martínez et al, 2016). The Landau configuration involves a constant magnetic field in a fixed direction (Bliokh and Nori, 2012a;Gallatin and McMorran, 2012;Greenshields et al, 2012;Landau and Lifshitz, 1977), whilst the AharanovBohm configuration can involve a single line of flux (Bliokh and Nori, 2012a).…”

“…Substituting Eq.56 into Eq.55 and choosing LaguerreGaussian solutions propagating perpendicular to the magnetic field reveals interesting dynamics relating to the orientation of the OAM vector projected onto the electron trajectory -or the helicity of the beam (Gallatin and McMorran, 2012)…”

“…Note that the scale of the plots of <( 1) is roughly five orders of magnitude smaller than that of the | 1| 2 plots, so that the phase can be seen. Image from Gallatin and McMorran (2012).…”