EVOLUTION UNDER POLYNOMIAL HAMILTONIANS IN QUANTUM AND OPTICAL-PHASE SPACES

We analyze the difference between classical and quantum nonlinear dyna mics by computing the time evolution of the Wigner functions for the s implest polynomial Hamiltonians of fourth degree in coordinate and mom entum. This class of Hamiltonians contains examples which are importan t in wave and quantum optics. The Hamiltonians under study describe th e third-order aberrations to the paraxial approximation and the nonlin ear Kerr medium. Special attention is given to the quantum analog of t he conservation of the volume element in classical phase space.