Macroscopic superposition states of a mechanical oscillator in an optomechanical system with quadratic coupling

We present a proposal to prepare the macroscopic superposition states of a mechanical oscillator in an optomechanical system with quadratic coupling. By coherently driving the optomechanical cavity with an external field, a Kerr-type interaction among phonons is induced. The macroscopic mechanical superposition can be generated by exploiting the Kerr nonlinearity in the mechanical oscillator, just as the generation of a nonclassical state of light in a Kerr medium. One merit of our proposal is that the Kerr nonlinearity can be strongly enhanced by adjusting the amplitude of the driving field. The influence of the system damping on the realization of mechanical non-Gaussian state is discussed by the negativity of the Wigner function. The results show that the macroscopic superposition states are robust against the mechanical thermal noise.