  We consider a simple cubic array of metallic nanoparticles supporting
extended collective plasmons that arise from the near-field dipolar interaction
between localized surface plasmons in each nanoparticle. We develop a fully
analytical quantum theory of the strong-coupling regime between these
collective plasmons and photons resulting in plasmon polaritons in the
nanoparticle array. Remarkably, we show that the polaritonic band gap and the
dielectric function of the metamaterial can be significantly modulated by the
polarization of light. We unveil how such an anisotropic behavior in the
plasmonic metamaterial is crucially mediated by the dipolar interactions
between the nanoparticles despite the symmetry of the underlying lattice. Our
results thus pave the way towards the realization of tunable quantum plasmonic
metamaterials presenting interaction-driven birefringence.
