Exact Photocount Statistics: Lasers near Threshold

A previously constructed laser model with quantum (noncommuting) noise sources was shown to lead near threshold to a quantum rotating-wave Van der Pol oscillator. A full dynamical correspondence between quantum and classical random processes allows one to compute the average of any time-ordered, normal-ordered operator function by averaging the associated function of classical random variables. Numerical calculations for the associated classical Van der Pol oscillator of the steady-state distribution, the total intensity fluctuations, and the linewidth versus operating point were amply confirmed experimentally. Measurements (and calculations) of higher than two-time correlations were sparse and contradictory. Photocount distributions, at times short compared to the intensity correlation time, confirm only the steady state of the laser. Photocount distributions at intermediate and longer times are difficult to compute because they involve multitime correlations of high ($\infty$) order. By providing an exact solution for photocount distributions and their moments for all times, we expected to stimulate measurements near threshold which would provide an adequate test of the Van der Pol laser model. Comparison of the results reported here with recent photocount experiments of Meltzer, Davis, and Mandel and of Jakeman, Oliver, and Pike provides gratifying agreement and confirmation of our statistical understanding of laser fluctuations near threshold.