Effective equilibrium theory of quantum light-matter interaction in cavities for extended systems and the long wavelength approximation

When light and matter interact strongly, the resulting hybrid system inherits properties from both constituents, allowing one to modify the material behavior by engineering the surrounding electromagnetic environment. This concept underlies the emerging paradigm of cavity materials engineering, which aims at the control of material properties via tailored vacuum fluctuations of dark photonic environments. The theoretical description of such systems is challenging due to the combined complexity of extended electronic states and quantum electromagnetic fields. Here, we derive an effective, non-perturbative theory for low-dimensional crystals embedded in a Fabry-Perot resonator. Starting from the full Pauli-Fierz Hamiltonian, we reduce the cavity field to an effective single-mode description within the long-wavelength limit, while retrieving the correct scaling of the light-matter interaction when the system size is scaled up to the extended system limit. This scaling is akin to the case where all the full continuum of cavity states are included in the light-matter interaction Hamiltonian. By explicitly accounting for the finite reflectivity of cavity mirrors, our theory also avoids double counting the contribution from free-space light-matter coupling. Our method provides a fully ab-initio framework to simplify the description of cavity-matter interactions for extended systems. The hybrid behavior of strongly interacting light and matter in cavities can be engineered by tailoring the cavity parameters, but simulating such systems is hard due to the complexity of the matter and quantum light. In this work, the authors derive an effective ab-initio theory reducing the light description to a single degree of freedom while ensuring finite light-matter coupling even in macroscopic systems.