Quantum Theory of a Laser Model

We derive the kinetic equations for the coupled single-particle density matrix ρ and the electromagnetic density matrix R to lowest order in the dimensionless coupling constant β²≡(ωLωD)². The laser frequency ωL is (4π)^-1/2(Nr₀λ²)1/2ω₀ where N is the number of two-level systems per unit volume, r₀ is the classical electron radius, λ is the wavelength of the radiation, and kω₀ is the two-level energy difference. The Doppler frequency ωD characterizes the center-of-mass motion. For gas lasers β² is much less than 1 and, consequently, we generalize and use the Bogoliubov derivation of kinetic equations for weak interactions. We find solutions when the average field vanishes and which include spontaneous emission correctly. The single-particle density matrix and the radiation density matrix are coupled through their second moments. When we substitute the solution of the second-moment equations into the density-matrix equations, we find that each density matrix satisfies an uncoupled linear equation with known time-dependent coefficients. We introduce and discuss dissipation from the density-matrix point of view. With the use of the density-matrix formalism we indicate that the correct expansion parameter for higher order kinetic equations is β².