Indirect Entanglement Transfer and Steering Between Two Atomic Ensembles in a Multi-Mode Hybrid Optomechanical System

We give a theoretical study of entanglement transfer and steering between two atomic ensembles in a multi-mode optomechanical system. The indirect coupling between two atomic ensembles is transferred by interactions of cavity-atomic ensemble and cavity-MR, and sets up an indirect correlation in the optomechanical system. Using numerical solutions, we show that the steady-state entanglement of two atomic ensembles depends on the coupling strengths of cavity-atomic ensemble and cavity-MR. Furthermore, we also find that (both one-way and two-way) Gaussian steerable correlations of two atomic ensembles can be built up with indirect coupling. Finally, we show that the two atomic ensembles can get entangled and steered after finite time and display similar behaviors with the time evolution.