  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 the rotating
and counter-rotating terms produces multi-photon interactions whose resonance
frequencies depend, due to the Stark term, 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.
