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.
Electron vortex beams were only recently discovered and their potential as a probe for magnetism in materials was shown. Here we demonstrate a new method to produce electron vortex beams with a diameter of less than 1.2 AA. This unique way to prepare
free electrons to a state resembling atomic orbitals is fascinating from a fundamental physics point of view and opens the road for magnetic mapping with atomic resolution in an electron microscope.
The holographic mask technique is used to create freely moving electrons with quantized angular momentum. With electron optical elements they can be focused to vortices with diameters below the nanometer range. The understanding of these vortex beams
is important for many applications. Here we present a theory of focused free electron vortices. The agreement with experimental data is excellent. As an immediate application, fundamental experimental parameters like spherical aberration and partial coherence are determined.
