Coupling-type-dependent selection rule in the deep strong-coupling regime

The quantum Rabi model describes the fundamental interaction between a two-level system and an electromagnetic mode in a resonator. One of the characteristic features of the quantum Rabi model is that it breaks the gauge symmetry, and such a breakdown becomes important in the deep-strong coupling regime, where the coupling strength is comparable to or even greater than the transition energies of the two-level system and resonator. We find that, apart from the well-known parity selection rule, there exists another type of selection rule. The selection rule is coupling-type dependent in the sense that whether it applies or not depends on the phase of quadrature through which the resonator is coupled to the waveguide, which originates from the breakdown of gauge symmetry. We demonstrate this selection rule by calculating the transmission coefficients in the one-tone and two-tone spectroscopy experiments.