Upper Bound for the Quantum Coupling between Free Electrons and Photons

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 (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$g_{Q u, m}$</tex> ) is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\left|g_{Q u, m}\right|^2&lt; g_{\text{ub}}^2 \equiv \frac{q_e^2}{4 \pi \hbar c \varepsilon_0} \frac{|\chi|^2}{\varepsilon} \frac{k L}{2 \pi} g_{\text{geo}}^2$</tex>, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$g_{\text{geo}}^2=\int_{R_{\perp}} d^2 r_{\perp}\left[\frac{\alpha_e^4}{k^2} K_0^2\left(\alpha_e \rho\right)+\frac{k_e^2 \alpha_e^2}{k^2} K_1^2\left(\alpha_e \rho\right)\right],\varepsilon$</tex> is the relative permittivity, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\chi=\varepsilon-1, L$</tex> is the interaction length, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k=\omega/c=2\pi/\lambda,\omega$</tex> is the optical frequency, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\lambda$</tex> is the wavelength, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k_e=\omega / v, v$</tex> is the free-electron velocity, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha_{e}=\sqrt{k_{e}^{2}-k^{2}},\rho$</tex> is the transverse distance towards the electron trajectory, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K_{0}, K_{1}$</tex> are the modified Bessel functions of the second kind with order 0 and 1 respectively, and the design region <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R$</tex> is expanded by a transverse cross section <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R_{\perp}$</tex> and a longitudinal length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L$</tex>.