Historical research guide · 0938 · 2017

Manipulating Twisted Electron Beams

Alexander J. Silenko · Pengming Zhang · Liping Zou

In one minute

Alexander Silenko, Pengming Zhang and Liping Zou work out how to steer an electron that is twisting as it travels. A twisted, or vortex, electron beam carries orbital angular momentum — the wave winds around a hollow core the way water winds around a drain — and because electrons are charged, that winding gives the beam a magnetic moment as well. Earlier treatments were done wave-mechanically and mostly for magnetic fields alone, which the authors point out is not enough to steer such a beam at all. They take a different route: model the electron as a small rotating cloud of charge, apply Lorentz transformations, and derive a general relativistic equation for how the intrinsic orbital angular momentum precesses in any combination of electric and magnetic fields. The equation resembles the one for spin but carries no Thomas correction, and that difference is what their recipes exploit — separating the two twist directions, deflecting a beam while freezing its twist, rotating it, and flipping it on resonance.

Connection

Chapter 10 asks what the structure of a field can do when no ordinary force is doing the work, and this paper turns that structure into something a laboratory can aim: the first general equation of motion for a free electron’s own orbital angular momentum in arbitrary electric and magnetic fields, with four concrete devices built on it. Chapter 5 cares because quantised circulation around a phase singularity is the defining signature of a quantum fluid — and here it is a quantity you can sort, rotate and flip on demand.

From the source

A twisted electron can be treated classically as a rotating cloud of charge. The rotation that does not depend on the electron’s momentum defines an intrinsic orbital angular momentum, non-zero even for a particle at rest, while the motion of the centre of charge defines an extrinsic one — and the classical route is legitimate because relativistic equations of motion for momentum and spin are already known to agree exactly with their quantum-mechanical counterparts.Opening pages, the rotating-charged-cloud model and the intrinsic-extrinsic decomposition
The orbital helicity of a vortex electron — the projection of its orbital angular momentum on its direction of travel — is frame dependent, so the paraxial description can fail after a Lorentz transformation and the Bessel-wave description is the one to use, since it lets the axis be chosen along any direction and not only along the momentum.Eq. (1) for the Bessel wave, Eq. (2) for the boosted orbital angular momentum, Eq. (3) for the orbital helicity
The central result is a general relativistic equation for the Larmor precession of the intrinsic orbital angular momentum in arbitrary fields: the precession vector is minus the electron charge times the magnetic field minus the velocity crossed into the electric field, divided by twice the mass, the speed of light and the Lorentz factor. It has the form of the spin equation but without the Thomas correction, because the orbital angular momentum is defined by an antisymmetric tensor rather than in the accelerated rest frame.Eq. (9) and Eq. (10); the comparison of Eq. (14) with the spin Hamiltonian of Eq. (15)
Beams of opposite twist can be separated with a longitudinal magnetic field. A non-uniform field pushes oppositely twisted electrons in opposite directions, and even a uniform longitudinal field makes the velocity depend on the direction of the orbital angular momentum, so a Wien filter can then extract a single orbital polarisation.Separation of beams with opposite directions of the OAM, Eq. (16)
A deflector can bend a vortex beam while holding its twist fixed relative to the beam direction. Setting the crossed electric and magnetic fields so that the Larmor precession rate equals the rotation rate of the momentum freezes the orbital helicity, and the beam turns at a rate of minus the charge times the magnetic field over the mass, the speed of light and the Lorentz factor times the Lorentz factor squared plus one — effective, the authors state, for standard beams of order one hundred kiloelectronvolts.Freezing the intrinsic OAM in electromagnetic fields, Eq. (17) through Eq. (19)
What to watch: the same machinery is proposed for other particles. A Wien filter acts as an orbital-angular-momentum rotator, magnetic resonance with a longitudinal oscillating field flips the twist at a frequency distinct from the spin resonance, and the authors expect twisted positron beams to follow the same rules — naming the testing of magnetic materials and the formation of twisted positronium as the applications to look for.Rotator of the intrinsic OAM and Flipping the intrinsic OAM, Eq. (20); Summary paragraph

The way in

https://doi.org/10.1103/physrevlett.119.243903

Historical source-access record. This guide links to the original; it does not reproduce the paper text.

The published Letter is under the APS default licence, which is not a Creative Commons licence. The author version is on arXiv as arXiv:1709.00065 version 2, dated 3 January 2018, posted under the arXiv non-exclusive distribution licence — also not Creative Commons — and no Creative Commons statement appears in either copy. So only the authors’ own abstract is reproduced here; the summary, the claims and every equation locator below were written from the complete author version. Affiliations as given on that copy: Silenko at the Institute of Modern Physics of the Chinese Academy of Sciences in Lanzhou, the Bogoliubov Laboratory of Theoretical Physics at the Joint Institute for Nuclear Research in Dubna, and the Research Institute for Nuclear Problems at Belarusian State University in Minsk; Zhang and Zou at the Institute of Modern Physics and the University of Chinese Academy of Sciences.

How to cite it

Alexander J. Silenko, Pengming Zhang, Liping Zou (2017) Manipulating Twisted Electron Beams doi:10.1103/physrevlett.119.243903

Where it sits in the curriculum

Scalar waves and the field behind the fields