Imaging the dynamics of free-electron Landau states

TL;DR

This paper presents the first real-space imaging of Landau states and their internal rotational dynamics of free electrons, revealing distinct classes of Landau modes with different rotational behaviors in a magnetic field.

Contribution

It demonstrates the real-space observation of Landau wave functions and their rotational dynamics using nanometer-scale electron vortex beams in a magnetic field.

Findings

Landau states with different quantum numbers exhibit three classes of rotational behavior.

The observed rotational frequencies include zero, Larmor, and cyclotron frequencies.

Results agree with recent theoretical predictions on Landau state dynamics.

Abstract

Landau levels and states of electrons in a magnetic field are fundamental quantum entities underlying the quantum-Hall and related effects in condensed matter physics. However, the real-space properties and observation of Landau wave functions remain elusive. Here we report the first real-space observation of Landau states and the internal rotational dynamics of free electrons. States with different quantum numbers are produced using nanometer-size electron vortex beams, with a radius chosen to match the waist of the Landau states, in a quasi-uniform magnetic field. Scanning the beams along the propagation direction, we reconstruct the rotational dynamics of the Landau wave functions with angular frequency ~100 GHz. We observe that Landau modes with different azimuthal quantum numbers belong to three classes, which are characterized by rotations with zero, Larmor, and cyclotron frequencies, respectively. This is in sharp contrast to the uniform cyclotron rotation of classical electrons, and in perfect agreement with recent theoretical predictions.