In our previous work (Meng et al 2020 Phys. Rev. A 101 023838), we discover the phenomenon that the quantum entanglement on the driving threshold line remains a constant in a three-mode optomechanical phonon laser system. In this paper, to find the conditions under which the constant boundary entanglement shows up, we explicitly study how this boundary entanglement depends on various parameters through numerical integrations. The results show that the necessary and sufficient condition is a resonant frequency match between the optical frequency difference and the mechanical vibrational frequency, and this constant value is proportional to the multiplication of the square of the optomechanical coupling strength and the resonant driving threshold power.

在我们之前的工作 (Meng et al 2020 Phys. Rev. A 101 023838) 中，我们发现了驱动阈值线上的量子纠缠在三模光机械声子激光系统中保持恒定的现象。在本文中，为了找到恒定边界纠缠出现的条件，我们通过数值积分明确研究了这种边界纠缠如何依赖于各种参数。结果表明，充分必要条件是光频差与机械振动频率的谐振频率匹配，该常数值与光机耦合强度的平方和谐振驱动阈值功率的乘积成正比。