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

Abstract

We find that the exponential operators U=exp[-i(${\ensuremath{\lambda}}_{1}$${\mathit{Q}}_{1}$${\mathit{P}}_{2}$+${\ensuremath{\lambda}}^{2}$${\mathit{Q}}_{2}$${\mathit{P}}_{1}$] and W=exp[-i(${\ensuremath{\lambda}}_{1}$${\mathit{Q}}_{1}$${\mathit{Q}}_{2}$-${\ensuremath{\lambda}}_{2}$p${\mathrm{P}}_{1}$${\mathit{P}}_{2}$)] (where ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ 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.