where c is the speed of the light. When k ∥ = 0, hω k (0) = hck ⊥ ≡ hω c is the cavity frequency at normal incidence. The photonic Hamiltonian Ĥph is expressed as Ĥph = k ∥ hω k (â † k âk + 1 2 ), and â † k and âk are the photonic raising and lowering operators for mode k, respectively. We consider k ∥ with discrete (but still quasi-continuous) values k α = 2π N L α, where the mode indexes α ∈ [-M-1 2 , ...0, ... M-1 2 ], and M = 283 is the total number of cavity modes needed to capture the relevant energies for the hybrid system.