Quantum noise VII: The rate equations and amplitude noise in lasers

In this paper we start from our Quantum Noise IV model of a homogenously broadened maser and specialize to the case in which the number of photons and the population difference are the slowest changing variables. In spite of the high coherence of the system, we find that by eliminating all other variables adiabatically we obtain the usual rate equations plus a set of noise sources with precisely those moments appropriate to shot noise. Thus we justify McCumber's model for amplitude noise in a laser, and after making a quasi-linear approximation obtain his amplitude spectrum. Comparison with exact calculations of Hempstead and Lax for the rotating wave van der Pol oscillator suggests that quasi-linear methods are accurate except within a factor 10 (in photon number) of threshold.