Electron interferometer

Abstract: Double-slit electron interferometers fabricated in high mobility two-dimensional electron gases are powerful tools for studying coherent wave-like phenomena in mesoscopic systems. However, they suffer from low visibility of the interference patterns due to the many channels present in each slit, and from poor sensitivity to small currents due to their open geometry. Moreover, these interferometers do not function in high magnetic fields--such as those required to enter the quantum Hall effect regime--as the field destroys the symmetry between left and right slits. Here we report the fabrication and operation of a single-channel, two-path electron interferometer that functions in a high magnetic field. This device is the first electronic analogue of the optical Mach-Zehnder interferometer, and opens the way to measuring interference of quasiparticles with fractional charges. On the basis of measurements of single edge state and closed geometry transport in the quantum Hall effect regime, we find that the interferometer is highly sensitive and exhibits very high visibility (62%). However, the interference pattern decays precipitously with increasing electron temperature or energy. Although the origin of this dephasing is unclear, we show, via shot-noise measurements, that it is not a decoherence process that results from inelastic scattering events.

Abstract: The diffraction of electrons through a nanoscale hologram that imprints a certain phase modulation on the electrons’ wavefunction produces a non-spreading electron Airy beam that follows a parabolic trajectory and can reconstruct its original shape after passing an obstacle. Light, as is widely known, travels in straight lines. Yet a few years ago it was shown that specially tailored light beams can follow a curved trajectory, without spreading. Such beams follow a waveform known from quantum mechanics, called the Airy function, a concept originally developed by the astronomer Sir George Biddell Airy in work on the trajectories of light in rainbows. Now, with the demonstration of Airy beams consisting of free electrons, new possibilities for manipulating electrons are in prospect. Airy electron beam arcs were generated by the diffraction of electrons through a nanoscale hologram, which imprints a specific phase modulation on the electrons' wavefunction. These beams can bend in space without any external force, stay localized over distances of up to 100 metres and self-heal after passing an obstacle. Possible applications include use in high-performance electron microscopes and as a basis for a new type of electron interferometer. Within the framework of quantum mechanics, a unique particle wave packet exists1 in the form of the Airy function2,3. Its counterintuitive properties are revealed as it propagates in time or space: the quantum probability wave packet preserves its shape despite dispersion or diffraction and propagates along a parabolic caustic trajectory, even though no force is applied. This does not contradict Newton’s laws of motion, because the wave packet centroid propagates along a straight line. Nearly 30 years later, this wave packet, known as an accelerating Airy beam, was realized4 in the optical domain; later it was generalized to an orthogonal and complete family of beams5 that propagate along parabolic trajectories, as well as to beams that propagate along arbitrary convex trajectories6. Here we report the experimental generation and observation of the Airy beams of free electrons. These electron Airy beams were generated by diffraction of electrons through a nanoscale hologram7,8,9, which imprinted on the electrons’ wavefunction a cubic phase modulation in the transverse plane. The highest-intensity lobes of the generated beams indeed followed parabolic trajectories. We directly observed a non-spreading electron wavefunction that self-heals10, restoring its original shape after passing an obstacle. This holographic generation of electron Airy beams opens up new avenues for steering electronic wave packets like their photonic counterparts, because the wave packets can be imprinted with arbitrary shapes5 or trajectories6.

Abstract: We present a measurement-based scheme for performing braiding operations on Majorana zero modes in mesoscopic superconductor islands and for detecting their non-Abelian statistics without moving or hybridizing them. In our scheme for ``braiding without braiding'', the topological qubit encoded in any pair of well-separated Majorana zero modes is read out from the transmission phase shift in electron teleportation through the island in the Coulomb-blockade regime. We propose experimental setups to measure the teleportation phase shift via conductance in an electron interferometer or persistent current in a closed loop.

Abstract: In the 1970s the prominent goal was to overcome the limitations of electron microscopy caused by aberrations of electron lenses by the development of electron holography. In the meantime this problem has been solved, not only in the roundabout way of holography, but directly by correcting the aberrations of the lenses. Nevertheless, many quantitative electron microscopical measurement methods?e.g. mapping and visualization of electric and magnetic fields?were developed within the context of holography and have become fields of their own. In this review we focus on less popular electron interferometric experiments which complement the field of electron holography. The paper is organized as follows. After a short sketch of the development of electron biprism interferometry after its invention in 1954, recent advances in technology are discussed that made electron biprism interferometry an indispensable tool for solving fundamental and applied questions in physics: the development and preparation of conventional and single-atom field electron and field ion sources with their extraordinary properties. Single- and few-atom sources exhibit spectacular features: their brightness at 100?keV exceeds that of conventional field emitters by two orders in magnitude. Due to the extremely small aberrations of diode field emitter extraction optics, the virtual source size of single-atom tips is on the order of 0.2?nm. As a consequence it illuminates an area 7?cm in diameter on a screen at a distance of 15?cm coherently. Projection electron micrographs taken with these sources reach spatial resolutions of atomic dimensions and in-line holograms are?due to the absence of lenses with their aberrations?not blurred. Their reconstruction is straightforward. By addition of a carbon nanotube biprism into the beam path of a projection microscope a lensless electron interferometer has been realized. In extremely ultrahigh vacuum systems flicker noise is practically absent in the new sources. In the context of holography, methods have been developed to record holograms without modulation of the biprism fringes by waves diffracted at the edges of the biprism filament. This simplifies the reconstruction of holograms and the evaluation of interferograms (taken, e.g. to extract a spectrum by Fourier analysis of the fringe system) significantly. A major section is devoted to the influence of electromagnetic and gravito-inertial potentials and fields on the quantum mechanical phase of matter waves: the Aharonov?Bohm effect, the inertial Aharonov?Bohm effect and its realization, the Sagnac effect and Sagnac experiments with atoms, superfluid helium, Bose?Einstein condensates, electrons and ions and their potential as rotation sensors are discussed. M?llenstedt and Wohland discovered in a crossed beam analyzer (Wien filter) an optical element for charged particles that shifts wave packets longitudinally that transverse a Wien filter on laterally separated paths. This new optical element rendered it possible to measure coherence lengths and the spectrum of charged particle waves by visibility- and Fourier-spectroscopy, to perform a 'Welcher Weg' experiment, to re-establish seemingly lost longitudinal coherence in an interferometer for charged particles and to realize a decoherence free quantum eraser. A precision test of decoherence according to a proposal from Anglin and Zurek and biprism interferences with helium atoms close the section on first-order coherence experiments. The topics of the last section are Hanbury Brown?Twiss correlations and an antibuching experiment of free electrons.