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arXiv:2010.14948 · Science Advances

Optical Coherence Transfer Mediated by Free Electrons

Ofer Kfir, Valerio Di Giulio, F. Javier Garcı́a de Abajo, Claus Ropers

Published Jan 1, 2021• 94 citations
arXivHTMLPDF

Abstract

We theoretically investigate the quantum-coherence properties of the cathodoluminescence (CL) emission produced by a temporally modulated electron beam. Specifically, we consider the quantum-optical correlations of CL produced by electrons that are previously shaped by a laser field. Our main prediction is the presence of phase correlations between the emitted CL field and the electron-modulating laser, even though the emission intensity and spectral profile are independent of the electron state. In addition, the coherence of the CL field extends to harmonics of the laser frequency. Since electron beams can be focused to below 1 Å, their ability to transfer optical coherence could enable the ultra-precise excitation, manipulation, and spectrally resolved probing of nanoscale quantum systems.

DOI
10.1126/sciadv.abf6380

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We theoretically investigate the quantum-coherence properties of the cathodoluminescence (CL) emission produced by a temporally modulated electron beam. Specifically, we consider the quantum-optical correlations of CL produced by electrons that are previously shaped by a laser field. Our main prediction is the presence of phase correlations between the emitted CL field and the electron-modulating laser, even though the emission intensity and spectral profile are independent of the electron state. In addition, the coherence of the CL field extends to harmonics of the laser frequency. Since electron beams can be focused to below 1 Å, their ability to transfer optical coherence could enable the ultra-precise excitation, manipulation, and spectrally resolved probing of nanoscale quantum systems.
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.
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Since the introduction of time-resolved direct electron detectors to transmission electron microscopy (TEM), there has been a growing interest in studying the correlations between an electron and the cathodoluminescence (CL) photons it produces [1,2,3]. Recent studies using coincidence measurements in both continuous [4] and discrete variables [5] have shown correlations that exceed the classical limits for separable states, thus demonstrating quantum entanglement between free electrons and photons. We present our continuous-variable approach [4], showing how ghost imaging [6], a method adapted from photonic quantum optics, can be used to demonstrate entanglement and even EPR-like behaviour of the joint electron- photon state. For this purpose, we generate correlated electron-photon pairs via transition radiation from 200 keV electrons passing a 50 nm monocrystalline silicon membrane. We detect the CL photons in a time-resolved manner, using a custom parabolic mirror and a single-photon counter. Employing a Timepix3 based direct electron detector we are able to determine which photons arrive in time with a corresponding electron. These electron-photon pairs exhibit precise (anti-)correlations in momentum and position. Placing a known absorptive mask in the photon beam path while measuring the electron in imaging and diffraction mode allows us to extract the joint position and momentum uncertainties of the electron–photon pair. We use these measurements to evaluate entanglement criteria such as the MGTV inequality [7] and the Reid- EPR bound [8]. We hope that these results will help to make entanglement available as a resource for electron microscopy, enabling novel measurement schemes inspired by photonic quantum optics.