Upper bound for the quantum coupling between free electrons and photons

The quantum interaction between free electrons and photons is fundamental to free-electron based light sources and free-electron quantum optics applications. A large coupling between free electrons and photons is generally desired. In this manuscript, I obtain the upper bound for the quantum coupling between free electrons and photons. The upper bound has a straightforward expression and can be applied to a broad range of optical materials, especially widely used low-loss photonic materials. The upper bound depends on the optical medium, the free-electron velocity, and the separation between the free electron and the optical medium. With simple structures, the numerically calculated coupling coefficient can reach ~99% of the upper bound. This study provides simple and practical guidance to reach the strong coupling between free electrons and photons.