Restoring Bloch's Theorem for Exciton Polaron-Polaritons

We introduce a novel representation for hybrid photon-exciton-phonon quantum electrodynamics (QED) Hamiltonians that restores Bloch's theorem. Due to the dynamic interchange of momentum between the fermionic and bosonic degrees of freedom, periodic excitonic systems lose their translational symmetry when strongly coupled to photons. The Hamiltonian, no longer block diagonal, is much more expensive to simulate, especially in calculations such as energy transport dynamics and the dielectric function. We derive a unitary transformation that restores the translational symmetry of the minimal coupling Hamiltonian with Fr\"ohlich electron--phonon coupling without making any additional assumptions such as the long-wavelength approximation. 

We can thus compute full polaron-polariton dispersion relations. We additionally obtain the zero-temperature dielectric function of the hybrid system, which only requires diagonalizing the Hamiltonian at the Γ-point.