Two-photon sideband transition in a driven quantum Rabi model : Quantitative discussions with derived longitudinal drives and beyond the rotating wave approximation

In this work, we analytically and numerically study the sideband transition dynamics of the driven quantum Rabi model (QRM). We focus in particular on the conditions when the external transverse drive fields induce first-order sideband transitions. Inducing sideband transitions between two different systems is an essential technique for various physical models, including the QRM. However, despite its importance, a precise analytical study has not been reported yet that successfully explains the sideband transition rates in a driven QRM applicable for all system parameter configurations. In our study, we analytically derive the sideband transition rates based on second-order perturbation theory, not relying on the rotating wave approximation (RWA) \cite{RWA}. Our formula are valid for all ranges of drive frequencies and system's parameters. Our analytical derived formula agrees well with the numerical results in a regime of moderate drive amplitudes. Interestingly, we have found a non-trivial longitudinal drive effect derived from the transverse drive Hamiltonian. This accounts for significant corrections to the sideband transition rates that are expected without considering the derived longitudinal effect. Using this approach, one can precisely estimate the sideband transition rates in the QRM not confining themselves within specific parameter regimes. This provides important contributions for understanding experiments described by the driven QRM.