Reconciling quantum and classical spectral theories of ultrastrong coupling: Role of cavity bath coupling and gauge corrections

Focusing on the widely adopted Hopfield model with cavity dissipation, we show how the linear spectrum of an ultrastrongly coupled cavity and a dipole can be described either classically or quantum mechanically, but only when the quantum model includes (i) corrections to maintain gauge invariance, and (ii) a specific type of cavity bath coupling. We also show the impact of this bath model on the quantum Rabi model, which has no classical analogue in ultrastrong coupling.