Evolution under polynomial Hamiltonians in quantum and optical phase spaces

Abstract

We analyze the difference between classical and quantum nonlinear dynamics by computing the time evolution of the Wigner functions for the simplest polynomial Hamiltonians of fourth degree in coordinate and momentum. This class of Hamiltonians contains examples which are important in wave and quantum optics. The Hamiltonians under study describe the third-order aberrations to the paraxial approximation and the nonlinear Kerr medium. Special attention is given to the quantum analog of the conservation of the volume element in classical phase space.