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Optica

Quantum correlations in electron microscopy

Chen Mechel, Yaniv Kurman, Aviv Karnieli, Nicholas Rivera, Ady Arie, Ido E. Kaminer

Published Nov 23, 2020• 29 citations
Publisher

Abstract

Electron microscopes provide a powerful platform for exploring physical phenomena with nanoscale resolution, based on the interaction of free electrons with the excitations of a sample such as phonons, excitons, bulk plasmons, and surface plasmons. The interaction usually results in the absorption or emission of such excitations, which can be detected directly through cathodoluminescence or indirectly through electron energy loss spectroscopy (EELS). However, as we show here, the underlying interaction of a free electron and an arbitrary optical excitation goes beyond what was predicted or measured so far, due to the interplay of entanglement and decoherence of the electron-excitation system. The entanglement of electrons and optical excitations can provide new analytical tools in electron microscopy. For example, it can enable measurements of optical coherence, plasmonic lifetimes, and electronic length scales in matter (such as the Bohr radius of an exciton). We show how these can be achieved using common configurations in electron diffraction and EELS, revealing significant changes in the electron’s coherence, as well as in other quantum information theoretic measures such as purity. Specifically, we find that the purity after interaction with nanoparticles can only take discrete values, versus a continuum of values for interactions with surface plasmons. We quantify the post-interaction density matrix of the combined electron-excitation system by developing a framework based on macroscopic quantum electrodynamics. The framework enables a quantitative account of decoherence due to excitations in any general polarizable material (optical environment). This framework is thus applicable beyond electron microscopy. Particularly in electron microscopy, our work enriches analytical capabilities and informs the design of quantum information experiments with free electrons, allowing control over their quantum states and their decoherence by the optical environment.

DOI
10.1364/optica.402693

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•Jan 8, 2015•0 citations
Electron microscopes provide a powerful platform for exploring physical phenomena with nanoscale resolution, based on the interaction of free electrons with the excitations of a sample such as phonons, excitons, bulk plasmons, and surface plasmons. The interaction usually results in the absorption or emission of such excitations, which can be detected directly through cathodoluminescence or indirectly through electron energy loss spectroscopy (EELS). However, as we show here, the underlying interaction of a free electron and an arbitrary optical excitation goes beyond what was predicted or measured so far, due to the interplay of entanglement and decoherence of the electron-excitation system. The entanglement of electrons and optical excitations can provide new analytical tools in electron microscopy. For example, it can enable measurements of optical coherence, plasmonic lifetimes, and electronic length scales in matter (such as the Bohr radius of an exciton). We show how these can be achieved using common configurations in electron diffraction and EELS, revealing significant changes in the electron’s coherence, as well as in other quantum information theoretic measures such as purity. Specifically, we find that the purity after interaction with nanoparticles can only take discrete values, versus a continuum of values for interactions with surface plasmons. We quantify the post-interaction density matrix of the combined electron-excitation system by developing a framework based on macroscopic quantum electrodynamics. The framework enables a quantitative account of decoherence due to excitations in any general polarizable material (optical environment). This framework is thus applicable beyond electron microscopy. Particularly in electron microscopy, our work enriches analytical capabilities and informs the design of quantum information experiments with free electrons, allowing control over their quantum states and their decoherence by the optical environment.
Entanglement, a key resource of emerging quantum technologies, describes correlations between particles that defy classical physics. It has been studied extensively on various platforms, but has remained elusive in electron microscopy. Transmission electron microscopes are well-established tools for materials characterisation with unparalleled spatial resolution. They provide control over the preparation and detection of high energy electrons, with largely unexploited potential in the study of many-body quantum correlations. Here, we demonstrate entanglement in electron-photon pairs generated via cathodoluminescence in a transmission electron microscope. Employing coincidence imaging techniques adapted from photonic quantum optics, we reconstruct both near- and far-field ``ghost'' images of periodic transmission masks. By measuring spatial and momentum correlations, we show a violation of the classical uncertainty bound: Δx−2Δk+2=0.502±0.047<1Δx_-^2 Δk_+^2 = 0.502 \pm 0.047<1Δx−2​Δk+2​=0.502±0.047<1. Hence, we demonstrate entanglement in position and momentum -- the continuous variables at the base of most imaging methods, bridging the fields of electron microscopy and quantum optics. Our work paves the way for exploring quantum correlations in free-electron systems and their application to quantum-enhanced imaging techniques on the nanoscale.
Free electrons provide a powerful tool for probing material properties at atomic resolution. Recent advances in ultrafast electron microscopy enable the manipulation of free-electron wave functions using laser pulses. It would be of great importance if one could combine the spatial resolution of electron microscopes with the ability of laser pulses to probe coherent phenomena in quantum systems. To this end, we propose a novel concept that leverages free electrons that are coherently shaped by laser pulses to measure quantum coherence in materials. We develop the quantum theory of interactions between shaped electrons and arbitrary qubit states in materials, and show how the postinteraction electron energy spectrum enables measuring the qubit state (on the Bloch sphere) and the decoherence or relaxation times (T_{2}/T_{1}). Finally, we describe how such electrons can detect and quantify superradiance from multiple qubits. Our scheme can be implemented in ultrafast transmission electron microscopes (UTEM), opening the way toward the full characterization of the state of quantum systems at atomic resolution.
Electron--photon quantum entanglement in an electron microscope paves the way for a new quantum platform, enabling the integration of quantum functionalities into electron microscopy and opening opportunities for quantum imaging and quantum sensing at the nanoscale. To realize such a platform, it is crucial to understand the degree and nature of electron--photon entanglement in cathodoluminescence (CL). However, its dependence on electron-beam properties, particularly transverse coherence, remains unclear. Here, we present a theoretical framework describing the quantum state of an electron--photon pair generated in coherent CL from a system with translational invariance in the plane perpendicular to the electron beam, as can be realized, for example, in transition radiation at a planar interface. By expressing the scattered state directly in terms of the luminescence spectrum, we evaluate the entanglement using both subsystem purity and an Einstein--Podolsky--Rosen-type criterion. These two measures enable a clear distinction among wave-like, particle-like, and classical regimes in terms of spatial and momentum entanglement in the electron--photon system. Our analysis identifies the roles of the electron's transverse and longitudinal coherence, as well as the photon's spectral width, and reveals the conditions under which strong spatial entanglement emerges. This unified perspective clarifies the nature of electron--photon quantum correlations in coherent CL and advances the development of quantum-enabled functionalities in electron microscopy.
We provide a theoretical framework to describe the dynamics of a free-electron beam interacting with quantized bound systems in arbitrary electromagnetic environments. This expands the quantum optics toolbox to incorporate free-electron beams for applications in highly tunable quantum control, imaging, and spectroscopy at the nanoscale. The framework recovers previously studied results and shows that electromagnetic environments can amplify the intrinsically weak coupling between a free-electron and a bound electron to reach previously inaccessible interaction regimes. We leverage this enhanced coupling for experimentally feasible protocols in coherent qubit control and towards the nondestructive readout and projective control of the electron beam's quantum-number statistics. Our framework is broadly applicable to microwave-frequency qubits, optical nanophotonics, cavity quantum electrodynamics, and emerging platforms at the interface of electron microscopy and quantum information.
A recent pedagogical paper by Strauch concretely demonstrated how interaction-mediated entanglement can suppress fringe visibility in a one-dimensional model of the electron double-slit experiment. Here we extend that framework to model actual experimental data from electron biprism interferometry. Kerker et al. showed that the macroscopic QED model of Scheel and Buhmann successfully describes their measured results. We show that this Scheel-Buhmann model can be recovered from Strauch's simplified framework by employing a Markov approximation and including thermal effects. The resulting decoherence rate is expressed in terms of mode-resolved scattering probabilities familiar from electron energy-loss spectroscopy (EELS), directly relating EELS to decoherence. The thermal dependence is significant in its own right, as recent theoretical work suggests that visibility reduction could serve as a non-invasive thermal probe for nanoscale systems. This progression from a toy model, to a quantitative account of real data, to a measurement application offers a case study in how simplified models can be made experimentally relevant.
Surface plasma waves arise from the collective oscillations of billions of electrons at the surface of a metal in unison. The simplest way to quantize these waves is by direct analogy to electromagnetic fields in free space, with the surface plasmon, the quantum of the surface plasma wave, playing the same role as the photon. It follows that surface plasmons should exhibit all of the same quantum phenomena that photons do, including quantum interference and entanglement. Unlike photons, however, surface plasmons suffer strong losses that arise from the scattering of free electrons from other electrons, phonons, and surfaces. Under some circumstances, these interactions might also cause “pure dephasing,” which entails a loss of coherence without absorption. Quantum descriptions of plasmons usually do not account for these effects explicitly, and sometimes ignore them altogether. In light of this extra microscopic complexity, it is necessary for experiments to test quantum models of surface plasmons. In this thesis, I describe two such tests that my collaborators and I performed. The first was a plasmonic version of the Hong-Ou-Mandel experiment, in which we observed two-particle quantum interference between plasmons with a visibility of 93 ± 1%. This measurement confirms that surface plasmons faithfully reproduce this effect with the same visibility and mutual coherence time, to within measurement error, as in the photonic case. The second experiment demonstrated path entanglement between surface plasmons with a visibility of 95 ± 2%, confirming that a path-entangled state can indeed survive without measurable decoherence. This measurement suggests that elastic scattering mechanisms of the type that might cause pure dephasing must have been weak enough not to significantly perturb the state of the metal under the experimental conditions we investigated. These two experiments add quantum interference and path entanglement to a growing list of quantum phenomena that surface plasmons appear to exhibit just as clearly as photons, confirming the predictions of the simplest quantum models.