Protected ultrastrong coupling regime of the two-photon quantum Rabi model with trapped ions

We propose a robust realization of the two-photon quantum Rabi model in a trapped-ion setting based on a continuous dynamical decoupling scheme. In this manner the magnetic dephasing noise, which is identified as the main obstacle to achieve long-time coherent dynamics in ion-trap simulators, can be safely eliminated. More specifically, we investigate the ultrastrong coupling regime of the two-photon quantum Rabi model whose realization in trapped ions involves second-order sideband processes. Hence, the resulting dynamics becomes unavoidably slow and more exposed to magnetic noise requiring an appropriate scheme for its elimination. Furthermore, we discuss how dynamical decoupling methods take a dual role in our protocol, namely they remove the main source of decoherence from the dynamics while they actively define the parameter regime of the simulated model.