Structural transitions in ion traps dispersively coupled to optical cavities

In this work we consider an optomechanical system composed of trapped ions dispersively coupled to a single mode of a pumped optical cavity. Firstly, we focus on a single ion, particularly on a parameter range that exhibits bistability of different configurations in the semiclassical description. This semiclassical description, however, is not valid in close proximity to the system transitions or when the mean photon number is low. Here, we provide a numerical analysis of the full quantum state in the few-photon regime, exploring the features of the asymptotic state across the transition and analyzing possible markers of semiclassical bistability. For larger mean photon numbers, we resort to an approach based on an approximated evolution in phase space. We compare the results of this technique with the ones from numerical diagonalization and find that although the treatment leads to a smoothing and a slight shift of the transitions in the system, it still provides a clear improvement over localized semiclassical approximations. Secondly, we provide a characterization of steady-state entanglement in a one-dimensional chain of three ions when the cavity wave vector is parallel to the ion chain axis, giving rise to dynamics similar to that of the Frenkel-Kontorova model, where a transition from a sliding to a pinned phase is observable. Within a semiclassical approximation in terms of Gaussian states, we describe the relation between entanglement and the ion chain vibrational modes, as well as the effect of defect formation upon entanglement, identifying the different situations leading to entangled steady-states of vibrational modes and cavity field fluctuations. Moreover, we observe the presence of genuine multipartite quantum correlations within the pinned phase.