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arXiv:1511.04450 · New Journal of Physics

Single-electron entanglement and nonlocality

David Dasenbrook, Joseph Bowles, Jonatan Bohr Brask, Patrick P. Potts, Christian Flindt, Nicolas Brunner

Published Apr 26, 2016• 46 citations
arXivHTMLPDF

Abstract

Motivated by recent progress in electron quantum optics, we revisit the question of single-electron entanglement, specifically whether the state of a single electron in a superposition of two separate spatial modes should be considered entangled. We first discuss a gedanken experiment with single-electron sources and detectors, and demonstrate deterministic (i. e. without post-selection) Bell inequality violation. This implies that the single-electron state is indeed entangled and, furthermore, nonlocal. We then present an experimental scheme where single-electron entanglement can be observed via measurements of the average currents and zero-frequency current cross-correlators in an electronic Hanbury Brown–Twiss interferometer driven by Lorentzian voltage pulses. We show that single-electron entanglement is detectable under realistic operating conditions. Our work settles the question of single-electron entanglement and opens promising perspectives for future experiments.

DOI
10.1088/1367-2630/18/4/043036

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Motivated by recent progress in electron quantum optics, we revisit the question of single-electron entanglement, specifically whether the state of a single electron in a superposition of two separate spatial modes should be considered entangled. We first discuss a gedanken experiment with single-electron sources and detectors, and demonstrate deterministic (i. e. without post-selection) Bell inequality violation. This implies that the single-electron state is indeed entangled and, furthermore, nonlocal. We then present an experimental scheme where single-electron entanglement can be observed via measurements of the average currents and zero-frequency current cross-correlators in an electronic Hanbury Brown–Twiss interferometer driven by Lorentzian voltage pulses. We show that single-electron entanglement is detectable under realistic operating conditions. Our work settles the question of single-electron entanglement and opens promising perspectives for future experiments.
Engineering and studying few-electron states in ballistic conductors is a key step towards understanding entanglement in quantum electronic systems. In this Rapid Communication, we introduce the intrinsic two-electron coherence of an electronic source in quantum Hall edge channels and relate it to two-electron wave functions and to current noise in a Hanbury Brown--Twiss interferometer. Inspired by the analogy with photon quantum optics, we propose to measure the intrinsic two-electron coherence of a source using low-frequency current correlation measurements at the output of a Franson interferometer. To illustrate this protocol, we discuss how it can distinguish between a time-bin-entangled pure state and a statistical mixture of time-shifted electron pairs.
Experiments violating Bell inequalities have formed our belief that the world at its smallest is genuinely nonlocal. While many nonlocality experiments use the first quantized picture, the physics of fields of indistinguishable particles is captured most conveniently by second quantization. This implies the possibility of nonlocal correlations, such as entanglement, between modes of the field. In this paper, we propose an experimental scheme that tests the theoretically predicted entanglement between modes in space occupied by massive bosons. A successful test of this scheme would not only indicate that mode entanglement is as genuine as particle entanglement, despite the particle number superselection rule, but would also prove that this superselection rule can be overcome.
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).
Abstract Aside from other puzzling features of entanglement, it has been debated whether a physically meaningful notion of entanglement requires two (or more) particles as carriers of the correlated degrees-of-freedom, or if a single particle could be considered to be entangled as well. While the usefulness of single-boson entanglement has been demonstrated some time ago, the restrictions of superselection rules have previously thwarted attempts at similar arguments for single fermions. In Dasenbrook et al (2016 New J. Phys. 18 043036 ) this obstacle is overcome. The authors propose a scheme for a Bell test on two copies of single-electron states whose entanglement is individually not accessible. The discussed scheme, which makes use of recent progress in electronic quantum optics, provides an experimentally viable and theoretically unambiguous way to assert that certain single-electron states can be considered to be entangled.
We analyze a Hanbury Brown-Twiss geometry in which particles are injected from two independent sources into a mesoscopic conductor in the quantum Hall regime. All partial waves end in different reservoirs without generating any single-particle interference; in particular, there is no single-particle Aharonov-Bohm effect. However, exchange effects lead to two-particle Aharonov-Bohm oscillations in the zero-frequency current cross correlations. We demonstrate that this is related to two-particle orbital entanglement, detected via violation of a Bell inequality. The transport is along edge states and only adiabatic quantum point contacts and normal reservoirs are employed.
We consider a scheme for on-demand teleportation of a dual-rail electron qubit state, based on single-electron sources and detectors. The scheme has a maximal efficiency of 25%, which is limited both by the shared entangled state as well as the Bell-state measurement. We consider two experimental implementations, realizable with current technology. The first relies on surface acoustic waves, where all the ingredients are readily available. The second is based on Lorentzian voltage pulses in quantum Hall edge channels. As single-electron detection is not yet experimentally established in these systems, we consider a tomographic detection of teleportation using current correlators up to (and including) third order. For both implementations, we take into account environmental effects.