Three-Mode Nonlinear Bogoliubov Transformations

We introduce the three-mode nonlinear Bogoliubov transformations based on the work of Siena et al. (Phys. Rev. A 64:063803, 2001) and Ying Wu (Phys. Rev. A 66:025801, 2002) about nonlinear Bogoliubov transformations. We show that three-mode nonlinear Bogoliubov transformations can be constructed by the combination of two unitary transformations, a coordinate-dependent displacement followed by the standard squeezed transformation. Such decomposition turns all the nonlinear canonic coordinate-dependent Bogoliubov transformations into essentially linear problems as we shall prove and hence greatly facilitate calculations of the properties and the quantities related to the nonlinear transformations.