Effective Hamiltonians in quantum optics:: a systematic approach

A general and systematic method for obtaining effective Hamiltonians that describe different nonlinear optical processes is discussed. The method exploits the existence of a nonlinear deformation of the usual su( 2) algebra that arises as the dynamical symmetry of the original model. When some physical parameter, dictated by the process under consideration, becomes small, a diagonal effective Hamiltonian is obtained immediately, that correctly represents the dynamics for arbitrary states and long times. The technique is extended to su(3) and su(N), finding the corresponding effective Hamiltonians when some resonance conditions are fulfilled.