Upper Bound for the Quantum Coupling between Free Electrons and Photons
Abstract
The fundamental limit of free-electron—light coupling is an important question relevant to frontier studies to reach the strong coupling regime, which is crucial for applications such as free-electron based photon sources, quantum sensing and quantum computing using free electrons [1], [2]. This study rigorously establishes the fundamental upper bound of free-electron—photon coupling [3]. The upper bound depends on the optical medium, the electron velocity, the separation distance, and the interaction length, which is applicable to arbitrary structures enclosed within a design region (Fig. 1(a)). When the photonic modes have a discrete spectrum as in the case of a lossless optical cavity, the upper bound for the free-electron—photon coupling coefficient ($g_{Q u, m}$ ) is $\left|g_{Q u, m}\right|^2< g_{\text{ub}}^2 \equiv \frac{q_e^2}{4 \pi \hbar c \varepsilon_0} \frac{|\chi|^2}{\varepsilon} \frac{k L}{2 \pi} g_{\text{geo}}^2$, where $g_{\text{geo}}^2=\int_{R_{\perp}} d^2 r_{\perp}\left[\frac{\alpha_e^4}{k^2} K_0^2\left(\alpha_e \rho\right)+\frac{k_e^2 \alpha_e^2}{k^2} K_1^2\left(\alpha_e \rho\right)\right],\varepsilon$ is the relative permittivity, $\chi=\varepsilon-1, L$ is the interaction length, $k=\omega/c=2\pi/\lambda,\omega$ is the optical frequency, $\lambda$ is the wavelength, $k_e=\omega / v, v$ is the free-electron velocity, $\alpha_{e}=\sqrt{k_{e}^{2}-k^{2}},\rho$ is the transverse distance towards the electron trajectory, $K_{0}, K_{1}$ are the modified Bessel…
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