Time dependent quadratic Hamiltonians SU(2) and SU(2,1)

The properties of SU(1,1), SU(2), and SU(3) have often been used in quantum optics. In this Letter we make use of the symmetries of SU(2) and SU(2,1). The group properties of SU(2), and SU(2,1) are used to find the transition probabilities of time dependent quadratic Hamiltonians. We consider Hamiltonians representing the frequency converter and Raman scattering. These Hamiltonians are used to describe optical coupling in nonlinear crystals.