Theory of Quantum Light-Matter Interaction in Cavities: Extended Systems and the Long Wavelength Approximation

When light and matter interact strongly, the coupled system inherits properties from both constituents. It is consequently possible to alter the properties of either by engineering the other. This intriguing possibility has lead to the emergence of the cavity-materials-engineering paradigm which seeks to tailor material properties by engineering the fluctuations of a dark electromagnetic environment. The theoretical description of hybrid light-matter systems is complicated by the combined complexity of a realistic description of the extended electronic and quantum electromagnetic fields. Here we derive an effective, non-perturbative theory for low dimensional crystals embedded in a paradigmatic Fabry-P\'erot resonator in the long-wavelength limit. The theory encodes the multi-mode nature of the electromagnetic field into an effective single-mode scheme and it naturally follows from requiring a negligible momentum transfer from the photonic system to the matter. Crucially, in the effective theory the single light mode is characterized by a finite effective mode volume even in the limit of bulk cavity-matter systems and can be directly determined by realistic cavity parameters. As a consequence, the coupling of the effective mode to matter remains finite for bulk materials. By leveraging on the realistic description of the cavity system we make our effective theory free from the double counting of the coupling of matter to the electromagnetic vacuum fluctuations of free space. Our results provide a substantial step towards the realistic description of interacting cavity-matter systems at the level of the fundamental Hamiltonian, by effectively including the electromagnetic environment and going beyond the perfect mirrors approximation.