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

Classical coupled harmonic oscillator models have been proven capable to describe successfully the spectra of many nanophotonic systems where an optical mode couples to a molecular or matter excitation. Although models with distinct coupling terms have been proposed, they are used interchangeably due to their similar results in the weak and strong coupling regimes. However, in the ultrastrong coupling regime, each oscillator model leads to very different predictions. Further, it is important to determine which physical magnitude is associated to each harmonic oscillator of these models in order to reproduce appropriately experimentally measurable quantities in each system. To clarify which classical model must be used for a given experiment, we establish a connection with the quantum description of these systems based on cavity quantum electrodynamics. We show that the proper choice of the classical coupling term depends on the presence or absence of the diamagnetic term in the quantum models and on whether the electromagnetic modes involved in the coupling are transverse or longitudinal. The comparison with quantum models further enables to make the correspondence between quantum operators and classical variables in the oscillator models, in order to extract measurable information of the hybrid modes of the system.