  We propose the quantum simulation of the quantum Rabi model in all parameter
regimes by means of detuned bichromatic sideband excitations of a single
trapped ion. We show that current setups can reproduce, in particular, the
ultrastrong and deep strong coupling regimes of such a paradigmatic
light-matter interaction. Furthermore, associated with these extreme dipolar
regimes, we study the controlled generation and detection of their entangled
ground states by means of adiabatic methods. Ion traps have arguably performed
the first quantum simulation of the Jaynes-Cummings model, a restricted regime
of the quantum Rabi model where the rotating-wave approximation holds. We show
that one can go beyond and experimentally investigate the quantum simulation of
coupling regimes of the quantum Rabi model that are difficult to achieve with
natural dipolar interactions.
