The degenerate parametric amplifier in a cavity and the unitary transformation which generates the external field
This paper concerns with a degenerate parametric amplifier in a single-ended cavity, interacting with the external modes of an electromagnetic field. The quantum theory of damping and the Langevin approach are used to work out and solve the equations of motion for both the internal and the external modes. The use of two different boundary conditions for the external field allows to derive the unitary transformation which relates the external field operators in stationary conditions to the initial field operators. The unitary transformation belongs to the class of operators which generate or preserve superpositions of two-mode gaussian states. It is found that the unitary transformation for each pair of modes factorizes in terms of a squeeze and a rotation operator.