Understanding the interactions between relativistic electrons, matter, and optical fields is an exciting field of research. The development of microscopes capable of focusing electrons traveling at half the speed of light onto sub-nanometer areas has unlocked new possibilities in nano-optics, enabling the study of optical phenomena at unprecedented spatial scales. Our group has explored these interactions at deep spatial resolution, achieving results such as mapping phonon modes on nanoparticles or demonstrating strong plasmon-phonon coupling. These studies are described within a classical framework. Recently, injecting laser radiation into nanostructures has enabled new coupling regimes between electromagnetic fields, relativistic electrons, and matter. This has opened up a new research field where quantum optics experiments can be conducted with free electrons rather than bound ones. We have recently studied high-coherence cavities with fast electrons and shown that photonic bandgap materials with ultra-high finesse cavities couple efficiently to free electrons. This internship will focus on studying the quantum properties of these cavities under laser illumination, contributing to a deeper understanding of nonlinear effects and quantum interactions. The project will take place in the Solid State Physics Laboratory at the University of Paris-Saclay, within the STEM group, which offers cutting-edge electron spectromicroscopy facilities.