A general framework for interactions between electron beams and quantum optical systems

TL;DR

The paper tackles the challenge of weak coupling between free-electron beams and bound electrons by proposing a universal framework that couples a bound qubit, an electromagnetic environment, and an electron beam. It derives effective $\hat{H}_{\text{int}}$ and scattering operators for free-space and cavity-mediated interactions, introducing coupling strengths $\phi_0$ and $\phi_{\text{cav}}$ and showing environment-induced enhancement enabling unitary dynamics and FEBERI-like entanglement with electron-number. It then presents non-destructive readout and state-engineering protocols that map the electron-number distribution $p(n)$ onto qubit observables via $N(\phi)=\sum_n p(n) e^{-i2\phi n}$, and outlines inverse-Fourier reconstruction and projection schemes. The work broadens the quantum optics toolbox for nanoscale quantum control and proposes experimentally feasible routes for quantum-controlled electron beams in microwave to optical platforms.

Abstract

We provide a theoretical framework to describe the dynamics of a free-electron beam interacting with quantized bound systems in arbitrary electromagnetic environments. This expands the quantum optics toolbox to incorporate free-electron beams for applications in highly tunable quantum control, imaging, and spectroscopy at the nanoscale. The framework recovers previously studied results and shows that electromagnetic environments can amplify the intrinsically weak coupling between a free-electron and a bound electron to reach previously inaccessible interaction regimes. We leverage this enhanced coupling for experimentally feasible protocols in coherent qubit control and towards the nondestructive readout and projective control of the electron beam's quantum-number statistics. Our framework is broadly applicable to microwave-frequency qubits, optical nanophotonics, cavity quantum electrodynamics, and emerging platforms at the interface of electron microscopy and quantum information.

Figures (3)

• Figure 1: General electron-beam spin qubit interaction a. System Schematic: an electron beam with an electron number distribution $p(n)$ and engineered wavefunction interacts with a qubit in an arbitrary electromagnetic environment. b. Effects of the interaction on both the spin qubit (for resonant and non-resonant cases, top) and the electron beam (bottom). The interaction entangles qubit precession and the electron beam's quantized number distribution. Projective readout of the qubit thus modifies the electron beam distribution (bottom right).
• Figure 2: Electron Statistics Discrimination and Determination a. Three stage experiment: (i): Spin initialization in $\ket{\downarrow}$ state; (ii): Electron beam-spin interaction, entangling spin evolution with electron beam number distribution; (iii): Spin readout of $\hat{\sigma}_z$, with an optional rotation prior to readout to select a measurement axis. b. $\langle \hat{\sigma}_y\rangle$ readout for different effective interaction strengths $\phi_0$, for free space (left) and an optimized cavity with localized fields (right). c. $\langle \hat{\sigma}_z\rangle$ for various distributions, characterized by the mean electron number $\mu$, and the Fano number $F \equiv \sigma_N^2/\mu$ with different interaction strengths. d. Kullback-Leibler (KL) divergence between the true electron number distribution $p(n)$ and the recovered distribution $\hat{p}(n)$ using the inverse Fourier Transform, as a function of maximum attainable interaction strength $\phi_{\text{max}}$ for the experiment.
• Figure 3: Electron Number Projection a. Protocol for non-destructive electron number projection. The initial and target distribution set the experimental parameters $\phi_i, \theta_i$ for a series of electron beam-qubit interactions. At each interaction, the following are performed: i) Qubit initialized in $\ket{\downarrow}$ and rotated by angle $-\theta_i$. ii) Electron-beam interaction induces number-dependent spin precession. The targeted electron number rotates the spin back to $\ket{\downarrow}$. iii) Projectively measure the qubit in the $\ket{\downarrow}, \ket{\uparrow}$ basis. If all spins are measured in $\ket{\downarrow}$, the electron beam is successfully projected in the target state. b. Projection protocol with uniform interaction parameters $\phi_i$ and $\theta_i = -\phi_i n^*$. The electron beam is initially Poissonian with mean $\mu = 50$ and target state is $\ket{n^*} = \ket{50}$. c. Optimal protocol to reach $\ket{n^*}$ with minimal number of scattering interactions. The electron beam is initially Poissonian with mean $\mu = 50$. The interaction and preparation parameters vary as $\phi_i = \pi/2^i$ and $\theta = n^*\phi_i$.