Cavity QED Beyond the Long Wavelength Approximation for Periodic Systems in the Ultrastrong Coupling Regime and Beyond

In this work, we develop a novel representation of a recently proposed Hamiltonian for describing cavity quantum electrodynamics (QED), optimized such that it efficiently converges for arbitrarily strong coupling strengths and is naturally applicable to periodic systems. Typically used Hamiltonians make the long wavelength approximation on the EM field inside the cavity. We show that many implementations of this approximation fundamentally contradict the conservation of momentum in the total light-matter system. Building on previous work on developing a light-matter Hamiltonian representation that is numerically advantageous for arbitrary coupling strengths, we then take this model beyond the long wavelength approximation, which is particularly salient for extended periodic systems coupled to the cavity. By doing this, we restore the conservation of momentum, allowing us to gain new intuitive insights into how photons and charged particles exchange momentum and how these exchange processes affect absorption and emmission phenomena in the ultrastrong coupling regime and beyond.