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The excitation of photonic vacuum by the aloof propagation of free electrons near dielectric surfaces, as demonstrated in waveguides [1] and microresonators [2], has been considered to be directly linked to the electric-field component parallel to the electron propagation direction [3,4]. Our quantum theory reveals a more general interaction that also depends on the transverse electric field. Originating from the ponderomotive potential and the coupling between canonical momentum and the vector potential, free electrons can emit photon pairs simultaneously rather than one photon at a time. When the inelastic linear coupling parameter vanishes, $g_{\rm Qu}=0$, the scattering operator reduces pure squeeze operators, and the interaction is essentially governed by the SU(1,1) group [5,6]. The squeezing strength of photonic vacuum is characterized by a nonlinear coupling parameter, $ξ_{\rm Qu}$, which depends on the interaction time and the photonic mode volume.
To illustrate the SU(1,1) interaction, we present the simulations of photon-pair generation. Pure single-mode squeezing of photonic vacuum is demonstrated in an optical nanobeam that confines light to a small mode volume. Pure two-mode squeezing appears in the interaction in a waveguide microresonator, where it exhibits entanglement between two different frequency modes. Furthermore, for low-speed electrons, the interaction is restricted to two energy levels, and each electron can emit at most one photon pair. The emitted photons are therefore superbunched, which can be characterized by measuring the second-order correlation function. The SU(1,1) interaction presented here reveals a new mechanism of electron energy loss and photon emission, and may be used to characterize photonic modes while providing a new route toward quantum electron and photon sources.
[1] G. Arend et al., Electrons herald non-classical light, Nat. Phys. 21, 1855 (2025).
[2] A. Feist et al., Cavity-mediated electron-photon pairs, Science 377, 777 (2022).
[3] O. Kfir, Entanglements of Electrons and Cavity Photons in the Strong-Coupling Regime, Phys. Rev. Lett. 123, (2019).
[4] F. J. García De Abajo, Optical excitations in electron microscopy, Rev. Mod. Phys. 82, 209 (2010).
[5] B. Yurke, S. L. McCall, and J. R. Klauder, SU(2) and SU(1,1) interferometers, Phys. Rev. A 33, 4033 (1986).
[6] Z. Y. Ou and X. Li, Quantum SU(1,1) interferometers: Basic principles and applications, APL Photonics 5, 080902 (2020).