The collective interaction of the charge polarization modes with the vacuum excitations, photons leads to the perturbation of both the linear dispersion relation of photons and constant dispersion of charge waves by the avoided crossing between the two dispersion lines of polaritons.
[1] Similar to the acoustic and the optical phonons and far from the resonance one branch is photon-like while the other charge is wave-like.
Expressing it in terms of the creation and annihilation operators for the harmonic oscillators we get Assuming oscillators to be on some kind of the regular solid lattice and applying the polaritonic Fourier transform and defining projections of oscillator charge waves onto the electromagnetic field polarization directions after dropping the longitudinal contributions not interacting with the electromagnetic field one may obtain the Hopfield Hamiltonian Because the interaction is not mixing polarizations this can be transformed to the normal form with the eigen-frequencies of two polaritonic branches: with the eigenvalue equation where with (vacuum photon dispersion) and is the dimensionless coupling constant proportional to the density
Mathematically the Hopfield dielectric for the one mode of excitation is equivalent to the trojan wave packet in the harmonic approximation.
The Hopfield model of the dielectric predicts the existence of eternal trapped frozen photons similar to the Hawking radiation inside the matter with the density proportional to the strength of the matter-field coupling.
One may notice that unlike in the vacuum of the electromagnetic field without matter the expectation value of the average photon number
similarly to the Hawking radiation in the neighbourhood of the black hole because of the Unruh–Davies effect.
which suggests that Hopfield dielectric will undergo the superradiant phase transition.