Elastic quantum coupling between free electrons and photons

The quantum coupling between free electrons and photons enables the application of quantum optics techniques in electron microscopy. For example, the phase of the electron wave function can be manipulated by passing the electron through an intense laser field. We develop here a quantum-optical framework for elastic coupling, which reveals the hidden back action: the phase shift that a passing electron induces on the photonic mode. The so-called "phase kickback" can be quantified as a refractive index of a free electron or be formulated as a dispersive Hamiltonian. This elastic scattering paves the way toward electron number states by counting free electrons nondestructively, and thus, allowing for subshot-noise sensing and imaging at the angstrom-scale.