Relativistic vortex electrons: paraxial versus non-paraxial regimes

Dmitry Karlovets

doi:10.1103/PhysRevA.98.012137

paper

Relativistic vortex electrons: paraxial versus non-paraxial regimes

arXiv:1803.10166 · doi:10.1103/PhysRevA.98.012137

Abstract

A plane-wave approximation in particle physics implies that a width of a massive wave packet $σ_{⊥}$ is much larger than its Compton wavelength $λ_{c} = ℏ/ m c$ . For Gaussian beams or for packets with the non-singular phases (say, the Airy beams), corrections to this approximation are attenuated as $λ_{c}^{2} / σ_{⊥}^{2} ≪ 1$ and usually negligible. Here we show that this situation drastically changes for particles with the phase vortices associated with an orbital angular momentum $ℓ ℏ$ . For highly twisted beams with $∣ ℓ ∣ ≫ 1$ , the non-paraxial corrections get $∣ ℓ ∣$ times enhanced and $∣ ℓ ∣$ can already be as large as $1 0^{3}$ . We describe the relativistic wave packets, both for vortex bosons and fermions, which transform correctly under the Lorentz boosts, are localized in a 3D space, and represent a non-paraxial generalization of the massive Laguerre-Gaussian beams. We compare such states with their paraxial counterpart paying specific attention to the relativistic effects and to the differences from the twisted photons. In particular, a Gouy phase is found to be Lorentz invariant and it generally depends on time rather than on a distance $z$ . By calculating the electron packet's mean invariant mass, magnetic moment, etc., we demonstrate that the non-paraxial corrections can already reach the relative values of $1 0^{- 3}$ . These states and the non-paraxial effects can be relevant for the proper description of the spin-orbit phenomena in relativistic vortex beams, of scattering of the focused packets by atomic targets, of collision processes in particle and nuclear physics, and so forth.

Minor changes compared to v2

Relativistic vortex electrons: paraxial versus non-paraxial regimes

Abstract

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