Squeezing of output field in a two-mechanical-mode optomechanical system with a nonlinear medium

We consider the squeezing of output field in a two-mechanical-mode optomechanical cavity, which contains a Kerr-down conversion nonlinear crystal. It is found that the squeezing magnitude of the output field is enhanced by injection of the optical parametric amplifier (OPA) and the Kerr medium, and the squeezing region is shifted with the Kerr parameter. When the frequencies of the two movable mirrors are set by different values, the sharp peak with high contrast due to excess noise from thermally occupied mechanical mode will be split into two parts, which are located at the frequencies of the movable mirrors. This can be applied to high precision measurement of the frequencies of the mechanical oscillators. Finally, we consider the influence of the environment temperature on the squeezing spectrum. This is helpful in generating controllable squeezing in the optomechanical system by using nonlinear media.