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arXiv:2010.11396 · Physical Review Letters

Quantum Entanglement and Modulation Enhancement of Free-Electron–Bound-Electron Interaction

Zhexin Zhao, Xiao-Qi Sun, Shanhui Fan

Published Jun 11, 2021• 87 citations
arXivHTMLPDF

Abstract

The modulation and engineering of the free-electron wave function bring new ingredients to the electron-matter interaction. We consider the dynamics of a free-electron passing by a two-level system fully quantum mechanically and study the enhancement of interaction from the modulation of the free-electron wave function. In the presence of resonant modulation of the free-electron wave function, we show that the electron energy loss and gain spectrum is greatly enhanced for a coherent initial state of the two-level system. Thus, a modulated electron can function as a probe of the atomic coherence. We further find that distantly separated two-level atoms can be entangled through interacting with the same free electron. Effects of modulation-induced enhancement can also be observed using a dilute beam of modulated electrons.

DOI
10.1103/physrevlett.126.233402

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The modulation and engineering of the free-electron wave function bring new ingredients to the electron-matter interaction. We consider the dynamics of a free-electron passing by a two-level system fully quantum mechanically and study the enhancement of interaction from the modulation of the free-electron wave function. In the presence of resonant modulation of the free-electron wave function, we show that the electron energy loss and gain spectrum is greatly enhanced for a coherent initial state of the two-level system. Thus, a modulated electron can function as a probe of the atomic coherence. We further find that distantly separated two-level atoms can be entangled through interacting with the same free electron. Effects of modulation-induced enhancement can also be observed using a dilute beam of modulated electrons.
We present a comprehensive relativistic quantum-mechanical theory for interaction of a free electron with a bound electron in a model, where the free electron is represented as a finite-size quantum electron wave packet (QEW) and the bound electron is modeled by a quantum two-level system (TLS). The analysis reveals the wave-particle duality nature of the QEW, delineating the point-particle-like and wavelike interaction regimes and manifesting the physical reality of the wave function dimensions when interacting with matter. This QEW size dependence may be used for interrogation and coherent control of superposition states in a TLS and for enhancement of cathodoluminescence and electron energy-loss spectroscopy in electron microscopy.
We present a comprehensive theory for interrogation of the quantum state of a two-level system (TLS) based on a free-electron--bound-electron resonant interaction scheme. The scheme is based on free electrons, whose quantum electron wavefunction is preshaped or optically modulated by lasers in an electron microscope setup and then inelastically scattered by a quantum TLS target (e.g., atom, quantum dot, and crystal defect center) upon traversing in proximity to the target. Measurement of the postinteraction energy spectrum of the electrons probes and quantifies the full Bloch sphere parameters of a pre-excited TLS and enables coherent control of the qubit states. The exceptional advantage of this scheme over laser-based ones is atomic-scale spatial resolution of addressing individual TLS targets. Thus, this scheme opens horizons for electron microscopy in material interrogation and quantum information technology.
Electron beams are a formidable tool for probing nanophotonic systems [1]. Combining electron microscopy with external optical excitation in ultrafast transmission electron microscopes has enabled the mapping of optical near-fields [2] and the bunching of electrons in attosecond pulses [3, 4]. These developments are based on the stimulated inelastic scattering between electrons and light [5, 6], in which the electrons gain energy from or lose energy to the optical field. Recently, the underlying interaction strength of this process has been increased significantly by harnessing resonant enhancement of the optical field and velocity phase matching [7]. Such an increased interaction strength also benefits the spontaneous scattering of electrons with light, thus promising applications like generating complex quantum states of light [8]. In this presentation, we discuss a set of experiments on free electron quantum optics that investigate single-electron-single-photon interactions and probe the resulting entanglement. Specifically, we consider scenarios in which optical waveguide structures are placed inside a transmission electron microscope (TEM) facilitating the interaction. Electrons passing by, for example, an integrated photonic structure (cf. Fig. 1a) can inelastically scatter with the optical mode, leading to the emergence of electron-photon pair states [9]. The resulting correlation between energy-loss electrons and photons, illustrated in Fig. 1b, can be probed employing time- and energy-resolved detection of both particles, revealing a pronounced coincidence peak between their arrival times, schematically shown in Fig. 1c. Extending the photon detection setup by a second detector to allow for Hanbury Brown and Twiss (HBT) measurements, the photon statistics heralded by such energy-loss electrons can be studied [10]. The observed anti-bunching behavior indicates a nonclassical optical state with single-photon properties. The inelastic electron-light scattering underlying the electron-photon pair generation is further expected to induce entanglement in the particle energies [11], however, direct observation of quantum entanglement would be exceedingly difficult given the large number of independent quantum states available in these degrees of freedom. Therefore, we propose experiments based on the quantum eraser scheme to demonstrate this free-electron-photon entanglement in different degrees of freedom [12,13]. Specifically, we consider a double-slit geometry coherently illuminated by an electron beam such that an interference pattern can be observed on a distant screen (cf. Fig. 2a). If electrons passing through the left or right (denoted |Land |R⁠, respectively) slit generate distinct marker photons with vertical (|V)or horizontal (|H) polarization, this interference pattern would disappear due to the entanglement of electron path and photon polarisation (cf. Fig. 2b). However, if this which-path information carried by the marker photon is erased, e.g. by mixing the polarisation states, the interference pattern can be recovered when detecting electrons and photons in coincidence. We describe an extension of this idea to an entanglement test, before discussing possible realistic implementations and presenting preliminary results on several experimental key aspects. In summary, we discuss the inelastic electron-light scattering at integrated photonic structures with applications ranging from coherent electron beam modulation using continuous-wave optical fields to the electron-driven generation of quantum optical states of light. Delving deeper into single electron-single photon interactions, we outline feasible experiments to generate and verify electron-photon entanglement with a potential extension to electron pairs based on entanglement swapping. All this adds to the toolbox of free electron quantum optics, potentially enabling novel sources of quantum light as well as quantum-enhanced sensing in electron microscopy. Generation of correlated electron-photon pairs. (a) Electrons pass over a photonic integrated circuit placed in the sample plane of a TEM, generating photons in the optical waveguides via inelastic scattering. Energy-loss electrons are detected and time-tagged behind an imaging spectrometer in coincidence with photons exiting the waveguide. (b) Inelastic scattering links the electron energy loss (green, right ladder) to the generation of photons (red, left ladder), creating correlations in energy and time. (c) A strong temporal correlation between the loss electrons and detected photons, indicated by the red dot, is observed relative to the uncorrelated zero-loss background electrons (blue line). Free-electron-photon quantum eraser. (a) Coherent electron illumination of a double-slit structure results in interference between the two pathways |Land |R.(b) If the electrons generate distinguishable marker photons, e.g. of orthogonal polarizations |Vand |H⁠, no interference pattern can be observed due to the electron-photon entanglement. (c) Collecting these marker photons and erasing the which-path information, e.g. using a half-wave plate (⁠λ/2⁠), the electron interference can be recovered using electron-photon coincidence detection with single-photon detectors (SPDs) placed behind a polarizing beam splitter (PBS).
We present a comprehensive theory for interrogation of the quantum state of a two-level system (TLS) based on a free-electron - bound-electron resonant interaction scheme. The scheme is based on free electrons, whose quantum electron wavefunction is pre-shaped or optically modulated by lasers in an electron microscope setup and then inelastically scattered by a quantum TLS target (e.g., atom, quantum dot, crystal defect center, etc.) upon traversing in proximity to the target. Measurement of the post-interaction energy spectrum of the electrons, probes and quantifies the full Bloch sphere parameters of a pre-excited TLS and enables coherent control of the qubit states. The exceptional advantage of this scheme over laser-based ones is the atomic-scale spatial resolution of addressing individual TLS targets. Thus, this scheme opens new horizons for electron microscopy in material interrogation and quantum information technology.
Recent advances in time-resolved cathodoluminescence have enabled ultrafast studies of single emitters in quantum materials with femtosecond temporal resolution. Here, we develop a quantum theory modeling the dynamics of free electrons interacting with quantum emitters in arbitrary initial states. Our analysis reveals that a free electron can induce transient coherent oscillations in the populations when the system is initially prepared in a coherent superposition of its states. Moreover, the electron energy spectrum exhibits a clear signature of the quantum coherence and sensitivity to the transition frequency of the emitter. These coherence effects manifest themselves as oscillations in the zero-loss peak of the spectral energy-loss probability. Our findings pave the way for characterization of quantum-coherent dynamics of individual quantum emitters by electron-probes.