We introduce the coordinate-dependent one- and two-mode squeezing transformations and discuss the properties of the corresponding one-and two-mode squeezed states. We show that the coordinate-dependent one-and two-mode squeezing transformations can be constructed by the combination of two transformations, a coordinate-dependent displacement followed by the standard squeezed transformation. Such a decomposition turns a nonlinear problem into a linear one because all the calculations involving the nonlinear one- and two-mode squeezed transformation have been shown to be able to reduce to those only concerning the standard one- and two-mode squeezed states.
We discuss the unitary operator corresponding to the general two-mode coordinate-momentum mixed transformation , where and are arbitrary real numbers. Suitably selecting the parameters and , we obtain a new two-mode bosonic realization of the Lie algebra. We also study the squeezing effects of the squeezed vacuum associated with the new two-mode bosonic realization of the Lie algebra. The results show that the new squeezed vacuum does not possess second-order squeezing, but exhibits higher-order squeezing.