  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 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 crystals we show that the dipole and
quadrupole states mix in many high symmetry directions of the Brilouin 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 for metal fill fractions above
$0.8$. In addition to the plasmon-polariton energies we analyse 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.
