Vortex particles in axially symmetric fields and applications of the quantum Busch theorem

Abstract

The possibilities to accelerate vortex electrons with orbital angular momentum (OAM) to relativistic energies and to produce vortex ions, protons, and other charged particles crucially depend on whether the OAM is conserved during the acceleration and on how phase space of the wave packet evolves. We show that both the OAM and a mean emittance of the packet, the latter obeying the Schrödinger uncertainty relation, are conserved in axially symmetric fields of electric and magnetic lenses, typical for accelerators and electron microscopes, as well as in Penning traps. Moreover, a linear approximation of weakly inhomogeneous fields works much better for single packets than for classical beams. We analyze quantum dynamics of the packet’s rms radius ⟨ρ 2⟩, relate this dynamics to a generalized form of the van Cittert–Zernike theorem, applicable at arbitrary distances from a source and for non-Gaussian packets, and adapt the Courant–Snyder formalism to describe the evolution of the packet’s phase space. The vortex beams can therefore be accelerated, focused, steered, trapped, and even stored in azimuthally symmetric fields and traps, somewhat analogously to the classical angular-momentum-dominated beams. We also give a quantum version of the Busch theorem, which states how one can produce vortex electrons with a magnetized cathode during either field- or photoemission, as well as vortex ions and protons by using a magnetized stripping foil employed to change a charge state of ions. Spatial coherence of the packets plays a crucial role in these applications and we provide the necessary estimates for particles of different masses.

Coherent superposition of orthogonal modes

“…A related beam quality factor, the M-square parameter, which is also proportional to the phase space area and thus to the emittance, is employed in laser physics and light optics. The concept of an emittance as conserved quantity of motion, with a lower limit following the Heisenberg relation, and the Courant-Snyder formalism has recently also been applied to describe the manipulation of quantum mechanical vortex particles and the evolution of a wave packet in phase space [7]. Despite its solid foundation, the application of the rms envelope and the Courant-Snyder formalism is, however, not in all cases obvious.…”

Twisted particles in heavy-ion collisions

“…The following analysis based on precedent studies [35][36][37][38][39][40][41][42][43][44] allows us to predict that all charged particles produced in noncentral HIC also have significant OAMs. It is very important to determine what quantum states are suitable for a production of such particles.…”

“…It has been demonstrated in Refs. [43,44] on the basis of the quantum Busch theorem [52,53] that particle beams generated inside a magnetic field take the form of LG beams outside of this field. This important achievement predicts wide dissemination of twisted beams in nature.…”

“…Unlike Refs. [43,44], we condider the case of coinciding the trajectory symmetry axis with the solenoid one. It can be shown [58] that the average values L z do not depend on a shift of the former axis from the latter one.…”

“…It has been noted in Refs. [43,44] that twisted particles are produced in states with L z = 0 and L z = 0. This note is absolutely correct for specific experimental conditions [43,44] and does not contradict to our analysis.…”