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arXiv:2610.35467

Electron-photon entanglement in real space from quantum electrodynamics

Ivan P. Christov

quant-phSubmitted Oct 26, 2026• Announced Oct 27, 2026• 0 citations
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Abstract

We study the entanglement between an electron and the photon it scatters, resolved in real space and time. Starting from nonrelativistic QED in Fermi's oscillator representation, we write Fock's coupled equations for components with fixed photon number in the photon configuration space, in the length gauge, where the photon is created around the atom through a nonlocal kernel. Truncated at one photon, the equations are solved numerically without the rotating-wave approximation for a one-dimensional model atom in a waveguide with a single transverse mode and enhanced coupling. The atom is driven by a short, off-resonant laser pulse, with the electron and photon represented on real-space grids. The global electron-photon entanglement is largely transient. It peaks during the pulse and decreases as the polarized electron cloud returns to the core, leaving a state close to a product of the electron state and a superposition of the field vacuum with a single scattered photon. Local entanglement, obtained from walkers sampled across both photon-number components, exhibits opposite spatial patterns for the two particles. The electron entropy is smallest at the nucleus and maps the local strength of the dipole coupling, whereas the photon entropy is largest near the atom, where photons are created and absorbed, and smaller in the outgoing wave packets.

AI Summary

How does entanglement between an electron and scattered radiation develop and propagate in real space and time? This paper studies that question using nonrelativistic QED in Fermi’s oscillator representation. It formulates Fock’s coupled equations for fixed-photon-number components in photon configuration space, using the length gauge, in which photon creation around the atom is described by a nonlocal kernel. The equations are truncated to the vacuum and one-photon sectors and solved numerically without the rotating-wave approximation. The model consists of a one-dimensional atom in a waveguide with one transverse mode and enhanced coupling, driven by a short, off-resonant laser pulse. Both electron and photon coordinates are represented on real-space grids. The reported global electron–photon entanglement is predominantly transient: it peaks during the drive and declines as the polarized electron cloud returns to the core. The remaining state is close to a product of an electron state and a field state containing a superposition of vacuum and one scattered photon. Local entanglement, evaluated using walkers sampled across both photon-number components, has contrasting spatial profiles. Electron entropy is lowest at the nucleus and follows the local dipole-coupling strength; photon entropy is highest near the atom and lower in the outgoing wave packets. The stated model restrictions are the one-photon truncation, one-dimensional atomic dynamics, single transverse waveguide mode, and enhanced coupling. The abstract provides no quantitative entanglement values, convergence estimates, or parameter ranges; the full text is unavailable here.

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We study the entanglement between an electron and the photon it scatters, resolved in real space and time. Starting from nonrelativistic QED in Fermi's oscillator representation, we write Fock's coupled equations for components with fixed photon number in the photon configuration space, in the length gauge, where the photon is created around the atom through a nonlocal kernel. Truncated at one photon, the equations are solved numerically without the rotating-wave approximation for a one-dimensional model atom in a waveguide with a single transverse mode and enhanced coupling. The atom is driven by a short, off-resonant laser pulse, with the electron and photon represented on real-space grids. The global electron-photon entanglement is largely transient. It peaks during the pulse and decreases as the polarized electron cloud returns to the core, leaving a state close to a product of the electron state and a superposition of the field vacuum with a single scattered photon. Local entanglement, obtained from walkers sampled across both photon-number components, exhibits opposite spatial patterns for the two particles. The electron entropy is smallest at the nucleus and maps the local strength of the dipole coupling, whereas the photon entropy is largest near the atom, where photons are created and absorbed, and smaller in the outgoing wave packets.
How does entanglement between an electron and scattered radiation develop and propagate in real space and time? This paper studies that question using nonrelativistic QED in Fermi’s oscillator representation. It formulates Fock’s coupled equations for fixed-photon-number components in photon configuration space, using the length gauge, in which photon creation around the atom is described by a nonlocal kernel.
The equations are truncated to the vacuum and one-photon sectors and solved numerically without the rotating-wave approximation. The model consists of a one-dimensional atom in a waveguide with one transverse mode and enhanced coupling, driven by a short, off-resonant laser pulse. Both electron and photon coordinates are represented on real-space grids.
The reported global electron–photon entanglement is predominantly transient: it peaks during the drive and declines as the polarized electron cloud returns to the core. The remaining state is close to a product of an electron state and a field state containing a superposition of vacuum and one scattered photon. Local entanglement, evaluated using walkers sampled across both photon-number components, has contrasting spatial profiles. Electron entropy is lowest at the nucleus and follows the local dipole-coupling strength; photon entropy is highest near the atom and lower in the outgoing wave packets.
The stated model restrictions are the one-photon truncation, one-dimensional atomic dynamics, single transverse waveguide mode, and enhanced coupling. The abstract provides no quantitative entanglement values, convergence estimates, or parameter ranges; the full text is unavailable here.
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 A local dynamical evolution equation for a two-spinor photon wave function is established. The vectorial spinor components are those of the two helicity components A T , ± ( + ) of the analytical part of the gauge invariant transverse vector potential. The theory’s Hamilton operator is spatially local and without singularities. Thereby previously established field-based formulations leading to various nonlocal integro-differential equations for the dynamical evolution of the photon wave function are simplified. The dynamical evolution equations for the spinor components can be written in the compact form F ± ( + ) ( r , t ) = 0 , where F ± ( + ) ( r , t ) are the analytical Riemann–Silberstein–Oppenheimer–Bialynicki vectors. The theory after extension to the QED level, allows one to describe the dynamical time evolution of single-photon wave packets. The bridge between the wave mechanical and the QED formulations is an operationally suitable mean position state | R ⟩ ( r , t ) for a transverse photon in Hilbert space. The inner product ⟨ R ( r , t ) ⟩ R ( r , t ) relates to our inability to localize a photon in space. The normalization condition for the two-spinor wave packet is established. A dynamical evolution equation describing the photon-source entanglement is established starting from the Heisenberg equation of motion for the annihilation operators associated to the two helicity species. We show how each released photon is labelled by the transverse current density operator dynamics of the charged source particles. The basic theory is applied to a study of single-photon emission from a rectlinear magnetic dipole tube (string) carrying a rotationally symmetric surface current. Numerical results are obtained for photon wave train emission. The transition from the photon wave mechanical formulation to the magnetostatic Aharonov–Bohm vector potential field is established. The generality of the A T , ± ( + ) -thoery is underlined linking it to the Minimal Coupling Principle, magnetic monopole electrodynamics, spatial photon localization, the perfe
Based on the idea of reverse engineering, we design an optimal laser pulse to control strong-field multiphoton atomic transitions. Starting from the time-dependent Schr\"odinger equation of the full system, we adiabatically eliminate the nonessential states and apply the rotating-wave approximation to arrive at an effective two-state representation that involves dynamic Stark shifts and multiphoton coupling. Solving this equation inversely for the field, we obtain an analytical laser pulse shape that is expected to induce the full system's evolution according to user-defined quantum pathways. In our procedure, the amplitude and phase of the laser pulse are engineered such that the dynamically shifted electronic states are resonantly coupled during the action of the pulse at each moment of time. As a result, the driven system evolves from an arbitrary initial population distribution to any desired final quantum state superposition at a predefined rate. The proposed scheme is demonstrated using the example of the $3s\ensuremath{\rightarrow}4s$ two-photon transition of atomic sodium. By solving the time-dependent Schr\"odinger equation of the single-active electron with two different methods, either propagating time-dependent coefficients of many field-free states or directly propagating the three-dimensional electronic wave packet on a grid, we demonstrate the robustness as well as the limitations of the presented reverse engineering scheme.
Starting from the postulate that the electromagnetic field appearing in the transverse set of microscopic Maxwell-Lorentz equations governing field-matter interactions, properly normalized, can be looked upon as describing one-photon-emission and -absorption processes in space and time, a first-quantized initiation of photon emission by a single atom is presented. The wave function for the emerging photon is introduced as a six-vector object constructed from the complex analytical signals of the Riemann-Silberstein vectors belonging to opposite photon helicities. When the atom is no longer electrodynamically active, the emitted photon is described in first quantization by the so-called energy wave function well known for photons in free space. From the momentum representation of the emerging photon wave function a condition on the analytical part of the transverse atomic current density is established which ensures that precisely one photon is emitted. A propagator description of the emerged photon dynamics in the coordinate representation is established. The photon propagator is introduced as a two-component spinor, where upper and lower tensor components are constructed, respectively, from positive and negative helicity combinations of the propagators describing the time-space evolution of the transverse electric and magnetic fields. It is shown that the emission region for the photon coincides with the region in space where the transverse atomic current density is nonvanishing. For a photon emitted in an electric dipole transition the emission region essentially is the near-field zone of the atom, and this zone therefore determines the initial (and best) spatial confinement of the photon. The photon emerging from an atom active for a finite time necessarily is of the polychromatic sort and the associated wave packet essentially is confined between spherical shells moving outwards with the vacuum speed of light. To illustrate the main principles of the fundamental theory in a heuristic fashion we apply it to a study of the emission of a one-photon sinusoidal wavetrain from a pointlike atom. It is found that the atomic current density needed to create just one photon is independent of the oscillation period in the train and thus depends only on the number of periods in the wave train. An explicit expression for the one-photon energy is derived, and it is shown that only for extremely short pulse trains pronounced deviations from the textbook result, $E=\ensuremath{\Elzxh}{\ensuremath{\omega}}_{0},$ occur. The radial energy flow in the coupled atom-photon system in the near-field zone of the atom is investigated, and the cycle-averaged outwards energy transport carried by the emerging photon in a given distance from the atom is determined.