  The dynamics of the quantum Rabi model in the deep strong coupling regime is
theoretically analyzed in a trapped-ion setup. Recognizably, the main hallmark
of this regime is the emergence of collapses and revivals, whose faithful
observation is hindered under realistic magnetic dephasing noise. Here we
discuss how to attain a faithful implementation of the quantum Rabi model in
the deep strong coupling regime which is robust against magnetic field
fluctuations and at the same time provides a large tunability of the simulated
parameters. This is achieved by combining standing wave laser configuration
with continuous dynamical decoupling. In addition, we study the role that
amplitude fluctuations play to correctly attain the quantum Rabi model using
the proposed method. In this manner the present work further supports the
suitability of continuous dynamical decoupling techniques in trapped-ion
settings to faithfully realize different interacting dynamics.
