Vacuum Faraday effect for electrons

Abstract

The optical Faraday effect describes the rotation of linear polarization upon propagation through a medium in the presence of a longitudinal magnetic field. The effect arises from a different phase delay between the right and left handed polarization components of the light. In this paper we predict a Faraday effect for a completely different system: electron vortices. Free electron vortex states were recently observed in transmission electron microscopy experiments, and they introduce new degrees of freedom into the probing of matter with electron beams. We associate a rotation of a vortex superposition with the fact that different phases are acquired by oppositely handed vortices propagating in a magnetic field. We show that, in contrast to the optical Faraday effect, the rotation of the electron beam occurs in vacuum and arises from the intrinsic chirality of the constituent vortex states.

Electron vortices: Beams with orbital angular momentum

“…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).…”

“…For the case when the electron vortex is propagating in the same direction as the magnetic field (B = B zẑ ) the non-relativistic Hamiltonian of the system takes the form (Greenshields et al, 2012) …”

“…55 for a vector potential as in Eq. 56 have the form of non-di↵racting LaguerreGaussian modes, with a fixed 'magnetic' width (w B ) and magnetic Rayleigh range (z B ) given by the strength of the field (Bliokh and Nori, 2012a;Greenshields et al, 2012).…”

Electron Singularities, Matter Wave Catastrophes, and Vortex Lattice Singularimetry

“…The rich diffraction physics of singular optics is vast and many concepts can be directly translated to electron microscopy. Interesting physics has been discovered by highlighting differences between electron and optical vortex beams, such as a remarkable interplay between Zeeman coupling, Landau levels and Gouy phases in the TEM [4] or the Faraday effect for electron vortex beams propagating in a magnetic field, which interestingly occurs in vacuum [5].Our singular electron optics experiments began by considering ways in which to spontaneously nucleate vortices in the TEM using aberrations. Using insights from Berry et al…”