In this paper, we investigate the steady-state of quantum correlation measurement of hybrid optomechanical systems. The first system consists of a single optomechanical system simultaneously coupled to a mechanical oscillator. While the second system is a hybrid optomechanical system consisting of an atomic ensemble placed in between the optical cavity and mirror. For both optomechanical systems, we formulate the Hamiltonian and the explicit expression of the covariance matrix leading to the dynamic of the system. Under the linearization approximation, we investigate the steady-state quantum correlations which are quantified through the correlation function of non-Hermitian operators, while the logarithmic negativity is used to quantify the amount of quantum entanglement between the subsystems. Furthermore, our proposed quantum correlation function can be used to quantify the entangled bipartite states that are correlative and transfer information. It is found that the transfer of quantum correlations between the subsystem is related to the detuning and coupling strength. Our results provide a realistic route toward remote quantum entanglement detection and a framework of future realistic fiber-optic quantum network operating applications.
Determinant representation for a quantum correlation function of the lattice sine - Gordon model
