Polariton spectrum of the Dicke-Ising model

The Dicke-Ising model describes cavity quantum electrodynamics setups in which dipoles couple not only with the photonic cavity field but also to each other through dipole-dipole interaction. In this work we diagonalise such a model in terms of bosonic polaritonic operators for arbitrarily large values of the light-matter coupling and for values of the dipole-dipole interaction until the onset of the ferromagnetic Ising phase transition. In order to accomplish this task we exploit higher order terms of the Holstein-Primakoff transformation, developing a general method which allows to solve the normal phase of the Ising model in term of bosonic excitations for large values of the dipole-dipole interaction. Our results shed light on the interplay between the dipole-dipole and the light-matter coupling strengths, and their effect on the virtual excitations which populate the ground-state when the interactions become comparable with the bare transition frequency.