Crystals of plasmonic metal nanoparticles have intriguing optical properties. They reach the regimes of ultrastrong and deep strong light-matter coupling, where the photonic states need to be included in the simulation of material properties. We propose a quantum description of the plasmon polaritons in supercrystals that starts from the dipole and quadrupole excitations of the nanoparticle building blocks and their coupling to photons. Our model excellently reproduces the results of finite difference time domain simulations. It provides detailed insight into the emergence of the polariton states. Using the example of a face centered cubic crystal we show that the dipole and quadrupole states mix in many high-symmetry directions of the Brillouin zone. A proper description of the plasmon and plasmon-polariton band structure is only possible when including the quadrupole-derived states. Our model leads to an expression of the reduced coupling strength in nanoparticle supercrystals that we show to enter the deep strong-coupling regime if the metal nanoparticles take up >60% of the total volume. In addition to the plasmon-polariton energies, we analyze the relative contributions of the dipole, quadrupole, and photonic states to their eigenfunctions and are able to demonstrate the decoupling of light in the deep strong-coupling regime. Our results pave the way for a better understanding of the quantum properties of metallic nanoparticle supercrystals in the ultrastrong- and deep strong-coupling regime.
