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Microscopy

Two-electron interference in a coherent beam

Tetsuji Kodama, Nobuyuki Osakabe

Published Apr 2, 2013• 3 citations
Publisher

Abstract

The interference between quantum amplitude for two electrons, emitted from two source points, to be detected at two detection points, is a direct result of quantum exchange statistics. Such interference is observed in the coincidence probability, compared with that of statistically independent electrons, by computing the time correlation function from the arrival times of the electrons. When the two detectors are separated by a distance less than the coherence length, the coincidence probability is suppressed for electrons (antibunching) due to the Pauli principle, even though they do not interact with each other. However, electrons are charged particles. The Coulomb potential, which governs the scattering of one charged particle by another, is so long ranged. It is obvious that we must consider the Pauli principle and the Coulomb interactions simultaneously. This paper deals with basic experimental and theoretical investigations of the antibunching behavior of electrons in a free beam by considering the Pauli principle and the direct Coulomb interaction between two individual electrons. The experimentally found dependences are described in a model which considers the Coulomb scattering and theoretical values of correlation signals evaluated by analytical calculations agree with those determined by experiment. A study of the time correlation function from the arrival times of the electrons will lead to an understanding of the physical processes that take place in electron guns.

DOI
10.1093/jmicro/dft012

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The interference between quantum amplitude for two electrons, emitted from two source points, to be detected at two detection points, is a direct result of quantum exchange statistics. Such interference is observed in the coincidence probability, compared with that of statistically independent electrons, by computing the time correlation function from the arrival times of the electrons. When the two detectors are separated by a distance less than the coherence length, the coincidence probability is suppressed for electrons (antibunching) due to the Pauli principle, even though they do not interact with each other. However, electrons are charged particles. The Coulomb potential, which governs the scattering of one charged particle by another, is so long ranged. It is obvious that we must consider the Pauli principle and the Coulomb interactions simultaneously. This paper deals with basic experimental and theoretical investigations of the antibunching behavior of electrons in a free beam by considering the Pauli principle and the direct Coulomb interaction between two individual electrons. The experimentally found dependences are described in a model which considers the Coulomb scattering and theoretical values of correlation signals evaluated by analytical calculations agree with those determined by experiment. A study of the time correlation function from the arrival times of the electrons will lead to an understanding of the physical processes that take place in electron guns.
Correlations between successive detections in beams of free electrons are studied with a transmission electron microscope. For incoherent illumination of the detectors, a certain random coincidence probability is observed, indicative for uncorrelated arrival times of the electrons. When the illumination is changed from incoherent to coherent, a reduction of the random coincidence probability is observed, indicative for antibunching in the arrival times of the electrons. However, the amount of reduction is larger than the theoretically expected value calculated from the Pauli principle, forbidding more than one identical fermion to occupy the same quantum state. For a certain coherent illumination of the detectors, where we use magnetic lenses in electron microscopes for magnifications of the coherence length, we find an enhanced coincidence probability, indicative for bunching in the arrival times of the electrons. This originates from correlations in beams of free electrons due to Coulomb interactions.
For a continuous beam, particles that arrive at random times show a flat second-order correlation function, g(2), as measured by a flat coincidence spectrum. A reduction in the likelihood for two particles in such a continuous beam to arrive at the same time is called antibunching, observed as a dip in the otherwise flat coincidence spectrum. For a pulsed beam, the coincidence spectrum consists of a series of equal height peaks, where the "dip" manifests as a reduction in the height of the zero-delay time peak. For electrons, such a dip is an experimental signature of Coulomb repulsion and Pauli pressure. This paper discusses another effect that can produce a similar signature but that does not originate from the properties of the physical system under scrutiny. Instead, the detectors and electronics used to measure those coincidences suffer significantly even from weak crosstalk. A simple model that explains our experimental observations is given. Furthermore, we provide an experimental approach to correct this type of crosstalk.
Very much like the ubiquitous quantum interference of a single particle with itself, quantum interference of two independent, but indistinguishable, particles is also possible. This interference is a direct result of quantum exchange statistics, however, it is observed only in the joint probability to find the particles in two separated detectors. Here we report the first observation of such interference fringes between two independent and non-interacting electrons in an interferometer proposed by Yurke et al. and Samuelsson et al. Our experiment resembles the "Hanbury Brown and Twiss" (HBT) experiment, which was performed with classical waves. In the experiment, two independent and mutually incoherent electron beams were each partitioned into two trajectories. The combined four trajectories enclosed an Aharonov-Bohm (AB) flux (but not the two trajectories of a single electron). While individual currents were found to be independent of the AB flux, as expected, the cross-correlation between current fluctuations in two opposite points across the device exhibited strong AB oscillations. This is a direct signature of orbital entanglement between two electrons even though they never interact with each other.
We investigate the intensity interference between pairs of electrons using a spin-polarized electron beam having a high polarization and a narrow energy width. We observe spin-dependent antibunching on the basis of coincident counts of electron pairs performed with a spin-polarized transmission electron microscope, which could control the spin-polarization without any changes in the electron optics. The experimental results show that the time correlation was only affected by the spin polarization, demonstrating that the antibunching is associated with fermionic statistics. The coherent spin-polarized electron beam facilitates the extraction of intrinsic quantum interference.
Understanding how fermionic exchange and Coulomb repulsion jointly shape two-electron correlations is essential for identifying genuine quantum signatures in multi-electron interference experiments. To address this interplay, we investigate Hanbury Brown and Twiss interference of two electrons generated by two independent needle-tip emitters within a full quantum-mechanical framework. In the absence of Coulomb interaction, the approach reproduces the results previously obtained within a quantum path formalism. For Coulomb-interacting electrons, we predict characteristic features absent in a semiclassical description: a pronounced Coulomb-dominated suppression region as well as Coulomb-induced phase offsets and fringe shifts. At the same time, outside of the Coulomb-dominated region, the spatial oscillation frequency is essentially governed by fermionic exchange symmetry. Our results establish quantitative parameter regimes for disentangling Coulomb interaction from fermionic exchange symmetry in such experiments.
Interference experiments characterized by the degree of first-order coherence (Young’s fringes, Fresnel, and Fraunhofer diffraction, etc.) have been performed with massive particles, in particular electrons and neutrons, as well as with light. Optical intensity interference or correlation experiments characterized by the degree of second-order coherence have not yet been performed with particles. The theory for such experiments shows striking differences in the behavior of electrons compared with that of light deriving from the differences in quantum statistics and charge. Photons are neutral bosons. They can be prepared in states that exhibit bunching, no bunching, or antibunching; they are unaffected by the presence of electromagnetic potentials. Electrons are charged fermions. Theory indicates that they exhibit antibunching regardless of the coherence of the source, and that their degree of second-order coherence can be influenced by electromagnetic fields through which they do not pass. A remarkable example is the Hanbury Brown-Twiss experiment with electrons that have interacted locally with the vector potential of an inaccessible magnetic field. The electron intensity correlation at two detectors is shown to be sensitive to the confined magnetic flux; experimental conditions differ considerably from those of the well-known Aharonov-Bohm effect.