Quantisation of limit cycles in a P representation of a dissipative driven anharmonic oscillator

The non-linear polarisability model of dispersive optical bistability with periodic driving field is shown to have non-trivial limit cycle solutions in phase space. The model is quantised rigorously in the positive P representation. For a particular (but typical) limit cycle, the associated steady state probability distribution is shown to be compatible with a bivariate Gaussian. The inverse widths of the Gaussian are calculated as functions of position in both phase space and time.