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Moyal phase-space analysis of nonlinear optical Kerr media

Nonlinear optical media of Kerr type are described by a particular version of an anharmonic quantum harmonic oscillator. The dynamics of this system can be described using the Moyal equations of motion, which correspond to a quantum phase space representation of the Heisenberg equations of motion. For the Kerr system we derive exact solutions of the Moyal equations for a complete set of observables formed from the photon creation and annihilation operators. These Moyal solutions incorporate the asymptotics of the classical limit in a simple explicit form. An unusual feature of these solutions is that they exhibit periodic singularities in the time variable. These singularities are removed by the phase space averaging required to construct the expectation value for an arbitrary initial state. Nevertheless, for strongly number-squeezed initial states the effects of the singularity remain observable.