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Bulletin of the American Physical Society

Modeling degenerate pulsed electron matter waves with partial quantum coherence

Sam Keramati, Eric Jones, Herman Batelaan

Published May 31, 2018• 0 citations
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The use of electron beams is ubiquitous; electron microscopy, scanning tunneling microscopy, electron lithography, and electron diffractometry all use well-collimated and focused beams. On the other hand, quantum degenerate electron beams do currently not exist. The realization of such beams may impact all electron beam technologies and are interesting to pursue. Past attempts to reach degeneracy were hampered by the low degeneracy of continuously emitting electron sources. With the recent advent of ultra-short electron pulses, high degeneracy is expected. Coulomb repulsion and low quantum coherence are hurdles that need to be overcome. A quantum analysis of the electron degeneracy for partially coherent pulsed electron sources is presented. Two-particle coincidence spectra are obtained for source parameters that are currently available. The conclusive demonstration of the fermionic Hanbury-Brown Twiss (HBT) effect for free electrons is shown to be within reach, and our results support the claim that femto-second nanotip electron sources, both polarized and unpolarized, can manifest partial to complete quantum degeneracy with appreciable signal-to-noise-ratios for free electron pulses notwithstanding their small particle numbers.
We employ ptychography, a phase-retrieval imaging technique, to show experimentally that a partially coherent high-energy matter (electron) wave emanating from an extended source can be decomposed into a set of mutually independent modes of minimal rank. Partial coherence significantly determines the optical transfer properties of an electron microscope and so there has been much work on this subject. However, previous studies have employed forms of interferometry to determine spatial coherence between discrete points in the wave field. Here we use the density matrix to derive a formal quantum mechanical description of electron ptychography and use it to measure a full description of the spatial coherence of a propagating matter wave field, at least within the fundamental uncertainties of the measurements we can obtain.
Intensities in high-resolution phase-contrast images from electron microscopes build up discretely in time by detecting single electrons. A wave description of pulse-like coherent-inelastic interaction of an electron with matter implies a time-dependent coexistence of coherent partial waves. Their superposition forms a wave package by phase decoherence of 0.5 - 1 radian with Heisenbergs energy uncertainty ΔEH = ħ/2 Δt-1 matching the energy loss ΔE of a coherent-inelastic interaction and sets the interaction time Δt. In these circumstances, the product of Planck's constant and the speed of light hc is given by the product of the expression for temporal coherence λ2/Δλ and the energy loss ΔE. Experimentally, the self-coherence length was measured by detecting the energy-dependent localization of scattered, plane matter waves in surface proximity exploiting the Goos-Hänchen shift. Chromatic-aberration Cc-corrected electron microscopy on boron nitride (BN) proves that the coherent crystal illumination and phase contrast are lost if the self-coherence length shrinks below the size of the crystal unit cell at ΔE > 200 eV. In perspective, the interaction time of any matter wave compares with the lifetime of a virtual particle of any elemental interaction, suggesting the present concept of coherent-inelastic interactions of matter waves might be generalizable.