Squeezed states: Operators for two types of one- and two-mode squeezing transformations

We find that the exponential operators U=exp[-i(${\lambda }_1$ ${\mathit{Q}}_1$ ${\mathit{P}}_2$+${\lambda }^2$ ${\mathit{Q}}_2$ ${\mathit{P}}_1$] and W=exp[-i(${\lambda }_1$ ${\mathit{Q}}_1$ ${\mathit{Q}}_2$-${\lambda }_2$p${\mathrm{P}}_1$ ${\mathit{P}}_2$)] (where ${\lambda }_1$ and ${\lambda }_2$ are real; $P_i$ and $Q_i$ are coordinate and momentum operators, respectively; and i=1,2) are two types of generalized squeezing operators responsible for generating two types of one- and two-mode combination squeezed states, respectively. The coordinate and/or momentum representations of U and W are presented, and their normal product forms are first derived in terms of the newly developed technique of integration within an ordered product, which provides us with a direct approach to constructing these new squeezed states. The fluctuations in quadrature phases for these states are analyzed.