Collapses and revivals of entanglement in phase space in an optomechanical cavity

We analyse the dynamics of an optomechanical cavity considering two modes of the electromagnetic field and a mechanical resonator. As a result of this interaction, the initial non-entangled state evolves into a final entangled state. In this work, we use the Q ( α ) function to observe the behavior of the system in the phase space, finding that such entanglement is a function of time showing collapses and revivals. To carry out this analysis, we use the generalization of the single-party Q ( α ) function towards three-party systems Q ( α , μ , ν ) and we ascertain that the entanglement perfectly maps into the Q ( α , μ , ν ) function because it becomes non-separable too. To prove that the Q function accurately certifies entanglement, we compare its behaviour with the plots of the van Loock inequalities. Additionally, as the optomechanical systems were proposed to carry out quantum information tasks, we investigate the kind of phase gate that is produced when we keep the Kerr term that arises in the evolution operator.