Gouy phase for relativistic quantum particles

Abstract

Recently Gouy rotation was observed with focused non-relativistic electron vortex beams. If the electrons in vortex beams are very fast we have to take into account relativistic effects to completely describe the Gouy phase on them. Exact Hermite-Gaussian solutions to the Klein-Gordon equation for particle beams are obtained here that make explicit the 4-position of the focal point of the beam. These are Bateman-Hillion solutions with modified phase factors to take into account the rest mass of the particles. They enable a relativistic expression for the Gouy phase to be determined. It is in fact shown all the solutions are form invariant under Lorentz transformations. It is further shown for the exact solutions to correspond to those of the Schrödinger equation the relative time between the focal point and any point in the beam must be constrained to be a specific function of the relative spatial coordinates.

Lorentz-covariant quantum 4-potential and orbital angular momentum for the transverse confinement of matter waves

“…(1) for a beam is to use a Bateman inspired ansatz. In an earlier paper [14], the following trial form was taken as the starting point for the derivation of the positive-energy Hermite-Gaussian beam solutions…”

“…As in an earlier paper [14] the solution to be applied here is to use Dirac delta function notation to impose a relationship ξ 3 − v 3 τ = 0 between the relative position ξ i = x i − X i and relative time τ = t − T . This relates back to the idea that particles in continuous wave beams can be assigned a precise axial velocity v 3 .…”

“…In two recent papers, exact BatemanHillion solutions were obtained for the Hermite-Gaussian modes of both electromagnetic [13] and quantum particle [14] beams. These are detailed solutions for particle beams that include Gouy phase [15][16][17].…”

“…In this paper a transformation will be made to the Bateman-Hillion solutions of the Klein-Gordon equation for particle beams to account for an earlier finding [14] that the components of the 4-momentum of the particles must have a shift in them related to the complex shift in the 4-position coordinates needed for the accurate description of any wave beam. This will be shown to facilitate a calculation for the total energy of each particle in terms of the rest mass of the particle, the kinetic energy of the propagation of the particle along the axis of the beam and the kinetic energy locked up in the transverse mass flows.…”

Poisson's spot and Gouy phase

“…In the coherent matter wave context the Gouy phase has been explored in [19,[31][32][33]. Experimental realizations were made in different systems such as BoseEinstein condensates [24], electron vortex beams [25] and astigmatic electron matter waves using in-line holography [26].…”