Description of ultrastrong light-matter interaction through coupled harmonic oscillator models and their connection with cavity-QED Hamiltonians.

Classical coupled harmonic oscillator models are capable of describing the optical and infrared response of nanophotonic systems where a cavity photon couples to dipolar matter excitations. The distinct forms of coupling adopted in these classical models lead to different results in the ultrastrong coupling regime. To clarify the specific classical model required to address particular configurations, we establish a connection between each oscillator model and the equivalent cavity Quantum Electrodynamics description. We show that the proper choice of coupled harmonic oscillator model depends on the presence or absence of the diamagnetic term in the quantum models, linked to whether transverse or longitudinal electromagnetic fields mediate the coupling. This analysis also shows how to exploit the classical oscillator models to extract measurable information of the optical response, as demonstrated in three canonical photonic systems, and to describe the opening of the Reststrahlen band in the bulk dispersion of phononic materials.