Quantum synchronization and correlations of two mechanical resonators in a dissipative optomechanical system

Quantum synchronization and its connection with other quantum correlations have attracted considerable attention. Here we present a theoretical scheme to simultaneously represent and significantly enhance the level of quantum synchronization and entanglement between two indirectly coupled mechanical membranes, which are coupled to a common optical field within a cavity. By applying a two-tone driving laser with weighted amplitudes and specific frequencies, both synchronization gauged by Mari's criterion and entanglement estimated by logarithmic negativity can be greatly enhanced. We then clarify the relationship between quantum synchronization and entanglement in detail. Numerical simulation results show that the influence of the coupling asymmetry ${G}_{2}/{G}_{1}$ on quantum complete synchronization behaves similarly to that on the purity while the influence of the coupling asymmetry ${G}_{2}/{G}_{1}$ on quantum phase synchronization is more similar to that on quantum entanglement. Besides, we demonstrate that although quantum synchronization and quantum entanglement are not directly related, both of them are sensitive to the squeezing parameter and the cooling effect. Furthermore, it is also shown that detuning the frequencies of two mechanical oscillators can actually help quantum synchronization and entanglement, which is somewhat similar to the case of quantum synchronization blockade.