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A Fokker-Planck equation for the Wigner function evolution in a noisy Kerrmedium ($\chi^{(3)}$ non-linearity) is presented. We numerically solved thisequation taking a coherent state as an initial condition. The dissipationeffects are discussed. We provide examples of quantum interference, sub-Planckphase space structures, and Gaussian versus non-Gaussian dynamical evolution ofthe state. The results also apply to the description of a nanomechanicalresonator with an intrinsic Duffing nonlinearity.Comment: 10 pages, 11 figure

Wigner function evolution of quantum states in the presence of self-Kerr interaction