  We study the nonlinear dynamics of trapped-ion models far away from the
Lamb-Dicke regime. This nonlinearity induces a sideband cooling blockade,
stopping the propagation of quantum information along the Hilbert space of the
Jaynes-Cummings and quantum Rabi models. We compare the linear and nonlinear
cases of these models in the ultrastrong and deep strong coupling regimes.
Moreover, we propose a scheme that simulates the nonlinear quantum Rabi model
in all coupling regimes. This can be done via off-resonant nonlinear red and
blue sideband interactions, yielding applications as a dynamical quantum
filter.
