  We theoretically propose a scheme for the generation of a non-classical
single-mode motional state of a mechanical resonator (MR) in the three-mode
optomechanical systems, in which two optical modes of the cavities are linearly
coupled to each other and one mechanical mode of the MR is optomechanically
coupled to the two optical modes with the same coupling strength
simultaneously. One cavity is driven by a coherent laser light. By properly
tuning the frequency of the weak driving field, we obtain engineered
Liouvillian superoperator via engineering the selective interaction Hamiltonian
confined to the Fock subspaces. In this case, the motional state of the MR can
be prepared into a non-Gaussian state, which possesses the sub-Poisson
statistics although its Wigner function is positive.
