Selective Interactions in the Quantum Rabi Model

We demonstrate the emergence of selective $k$-photon interactions in the strong and ultrastrong coupling regimes of the quantum Rabi model with a Stark coupling term. In particular, we show that the interplay between rotating and counter-rotating terms produces $k$-photon interactions whose resonance frequencies depend on the state of the bosonic mode. We develop an analytical framework to explain these $k$-photon interactions by using time-dependent perturbation theory. Finally, we propose a method to achieve the quantum simulation of the quantum Rabi model with a Stark term by using the internal and vibrational degrees of freedom of a trapped ion, and demonstrate its performance with numerical simulations considering realistic physical parameters.