More nonlocality with less entanglement in a tripartite atom-optomechanical system

We study quantum effects in hybrid atomic optomechanics in a system comprising a cloud of atoms and a mobile mirror mediated by a single-mode cavity. Tripartite nonlocality is observed in the atom-light-mirror system, as demonstrated by the violation of the Mermin-Klyshko (MK) inequality. It has been shown [C. Genes, et al., PRA 77, 050307 (R) (2008)] that tripartite entanglement is optimized when the cavity is resonant with the anti-Stokes sideband of the driving laser and the atomic frequency matches the Stokes one. However, we show that this is not the case for the nonlocality. The MK function achieves {\it minima} when the atoms are resonant with both the Stokes and anti-Stokes sidebands, and unexpectedly, we find violation of the MK inequality only in a parameter region where entanglement is far from being maximum. A negative relation exists between nonlocality and entanglement with consideration of the possibility of bipartite nonlocality in the violation of the MK inequality. We also study the non-classicality of the mirror by post-selected measurements, e.g. Geiger-like detection, on the cavity and/or the atoms. We show that with feasible parameters Geiger-like detection on the atoms can effectively induce mechanical non-classicality.