Abstract
The quantum coupling between free-electrons and photons enables applying quantum optics techniques in electron microscopy. Here, we formulate the elastic electron-photon quantum coupling and its possible implications. Our analysis shows that when an electron traverses the field of an optical cavity, it induces a phase shift onto its confined photonic mode, which can be quantified as a refractive index of a free electron. This principle can be applied to counting electrons in a beam without changing its quantum states. The elastic scattering operator forms an electron-counting dispersive Hamiltonian for electron-photon systems within electron microscope, and it could enable quantum- and sub-shot-noise sensing and imaging at the {\AA}-scale.