Nearly invariant boundary entanglement in optomechanical systems

In order to understand our previous numerical finding that steady-state entanglement along the instability boundary remains unchanged in a three-mode optomechanical system[Phys.Rev.A 101023838(2020)],we investigate in detail the boundary entanglement in a simpler two-mode optomechanical system.Studies show that both the mechanism to generate entanglement and the parameter dependence of boundary entanglement are quite similar in these two models.Therefore,the two-mode system has captured the main features in the three-mode system.With the help of analytical calculations and discussing in a much bigger parameter interval,we find that the unchanging behavior previously discovered is actually an extremely slow changing behavior of the boundary entanglement function,and most importantly,this nearly invariant boundary entanglement is a general phenomenon via parametric down conversion process in the weak dissipation regime.This is by itself interesting as threshold quantum signatures in optomechanical phonon lasers,or may have potential value in related applications based on boundary quantum properties.