, which depends on the angle θ (tan θ = k x /k z ) of the external probe. Here, c is the speed of light in vacuum, n c is the refractive index inside the cavity, and ħk is the momentum of the photon. While the collective Hamiltonian (Eq. 17) is an extremely important physical object to describe experimental conditions, we will focus the majority of our discussion on the single-molecule, single-mode case (Eq. 13) and return to the collective case in Sec. 5 where we will describe extensions toward the realistic modeling of polaritonic states and their dynamics.