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

In this paper, we analytically and numerically study the sideband interaction 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 interactions. Inducing sideband interactions 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 interaction rates in a driven QRM applicable for all system parameter configurations. In our paper, we analytically derive the sideband interaction rates based on second-order perturbation theory, not relying on the rotating wave approximation (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 nontrivial longitudinal drive effect derived from the transverse drive Hamiltonian. This accounts for significant corrections to the sideband interaction rates that are expected without considering the derived longitudinal effect. Using this approach, one can precisely estimate the sideband interaction rates in the QRM not confining themselves within specific parameter regimes for moderate drive amplitudes. This provides important contributions for quantitatively understanding experiments described by the driven QRM.