State-agnostic approach to certifying electron–photon entanglement in electron microscopy
Phila Rembold, Santiago Beltrán-Romero, Alexander Preimesberger, Sergei Bogdanov, Isobel Claire Bicket, Nicolai Friis, Elizabeth Agudelo, Dennis Rätzel, Philipp Haslinger
Abstract Transmission electron microscopes (TEMs) enable atomic-scale imaging and characterisation, driving advances across fields from materials science to biology. Quantum correlations, specifically entanglement, may provide a basis for novel hybrid sensing techniques to make TEMs compatible with sensitive samples prone to radiation damage. We present a protocol to certify entanglement between electrons and photons naturally arising from certain coherent cathodoluminescence processes. Using mutually unbiased bases in position and momentum, our method allows robust, state-agnostic entanglement verification and provides a lower bound on the entanglement of formation, enabling quantitative comparisons across platforms. Simulations under experiment-inspired conditions and preliminary experimental data highlight the feasibility of implementing this approach in modern TEM systems with optical specimen access. Our work integrates photonic quantum information techniques with electron microscopy. It establishes a foundation for entanglement-based imaging at the atomic scale, offering a potential pathway to reduce radiation exposure.
Abstract Transmission electron microscopes (TEMs) enable atomic-scale imaging and characterisation, driving advances across fields from materials science to biology. Quantum correlations, specifically entanglement, may provide a basis for novel hybrid sensing techniques to make TEMs compatible with sensitive samples prone to radiation damage. We present a protocol to certify entanglement between electrons and photons naturally arising from certain coherent cathodoluminescence processes. Using mutually unbiased bases in position and momentum, our method allows robust, state-agnostic entanglement verification and provides a lower bound on the entanglement of formation, enabling quantitative comparisons across platforms. Simulations under experiment-inspired conditions and preliminary experimental data highlight the feasibility of implementing this approach in modern TEM systems with optical specimen access. Our work integrates photonic quantum information techniques with electron microscopy. It establishes a foundation for entanglement-based imaging at the atomic scale, offering a potential pathway to reduce radiation exposure.
The integration of quantum correlations into transmission electron microscopy (TEM) offers new possibilities for advanced imaging and hybrid quantum sensing, with potential benefits such as enhanced contrast mechanisms and reduced radiation dose for sensitive samples [1-4]. Recent experiments have demonstrated electron photon entanglement for the first time [1, 2]. However, these demonstrations often rely on specific assumptions about the states. In this work, we present an approach for entanglement certification that does not depend on such assumptions. Specifically, we propose and analyze a robust, state-agnostic protocol for certifying electron photon entanglement using mutually unbiased bases in position and momentum [3, 5]. Our simulations, based on a Cherenkov radiation model, describe the entangling interaction between a fast electron and the electromagnetic field generated by a dielectric medium. Conservation of energy and momentum naturally correlates the electron and the emitted photon [6-8]. We quantify these correlations, establish a lower bound on the entanglement of formation, and demonstrate that certification is feasible under realistic experimental conditions including uncertainties in both position and momentum measurements [1, 3].
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<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.
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.
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.
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]. In this contribution, we demonstrate “ghost imaging” [4], a method adapted from photonic quantum optics that leverages position or momentum correlations within electron-photon pairs for imaging. In a TEM, working at 200 keV, we generate electron-photon pairs via the process of transition radiation using a thin silicon membrane (see Fig. 1). We collect the photons using a custom-made parabolic mirror and let the photons interact with an arbitrary absorptive mask placed in the image plane of the collection system (see grating mask in Fig. 1b), thus imprinting the shape of this mask in the measured photon distribution. Using a single photon counting module and time tagger, we can match each detected photon to its emitting electron and retrieve the mask shape in the distribution of the corresponding electrons (Fig. 1e). This is remarkable, as the electron did not interact with the object; the information about the mask shape is mediated purely through the correlations between the two particles. In a recent publication [5], we employed ghost imaging in order to demonstrate electron-photon entanglement. We will present ghost imaging with considerably better spatial resolution and explain how this might allow us to show the Einstein-Podolsky-Rosen paradox using electron-photon pairs.
In electron microscopy, the pursuit of atomic-scale resolution for imaging biological [1] samples and soft-matter systems [2] is perpetually constrained by beam-induced sample degredation. While classical approaches attempt to optimize the trade-off between dose efficiency and image quality, they remain fundamentally limited by the physical interactions between electrons and the sample. Quantum technologies, however, present a transformative opportunity by enhancing information extraction per electron without increasing exposure [3], thereby breaking through classical constraints. In this work, we integrate for the first time a trapped-ion qubit into a transmission electron microscope, leveraging the strong Coulomb interaction between swift electrons and Ca⁺ ions [4], along with the precise control and readout capabilities of trapped Ca⁺ ions [5]. By harnessing quantum coherence and controlled interactions, this hybrid approach seeks to advance the capabilities of low-dose imaging, improving the analysis of radiation-sensitive samples while minimizing damage. For this purpose, we customize a transmission electron microscope to provide the required ultra-high-vacuum integrity and adapt the Ca⁺ trap design, including laser cooling and state readout/manipulation, to the geometric constraints of the microscope’s pole-piece gap.