The Squeezing Effect of Three-Mode Operator as an Extension from Two-Mode Squeezing Operator

Abstract

We theoretically study the squeezing effect in a 3-wave mixing process, generated by the operator \(S_{3}\equiv \exp[\mu(a_{1}a_{2}-a_{1}^{\dagger}a_{2}^{\dagger})+\nu(a_{1}a_{3}-a_{1}^{\dagger}a_{3}^{\dagger})]\). The corresponding 3-mode squeezed vacuum state in Fock space and its uncertainty relation are presented. It turns out that S3 may exhibit enhanced squeezing. By virtue of integration within an ordered product (IWOP) of operators, we also give the S3’s normally ordered expansion. Finally, we calculate the Wigner function of 3-mode squeezed vacuum state by using the Weyl ordering invariance under similar transformations.