Quantum Theory of a Laser Model

We derive the kinetic equations for the coupled single-particle density matrix $\rho$ and the electromagnetic density matrix $R$ to lowest order in the dimensionless coupling constant ${\beta }^2\equiv {\left(\frac{{\omega }_L}{{\omega }_D}\right)}^2$. The laser frequency ${\omega }_L$ is ${\left(4\mathrm{}\pi \right)}^{-\frac{1}{2}}{\left(\mathfrak{N}{r}_0{\lambda }^2\right)}^{\frac{1}{2}}{\omega }_0$ where $\mathfrak{N}$ is the number of two-level systems per unit volume, $r_0$ is the classical electron radius, $\lambda$ is the wavelength of the radiation, and $k{\omega }_0$ is the two-level energy difference. The Doppler frequency ${\omega }_D$ characterizes the center-of-mass motion. For gas lasers ${\beta }^2$ 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 ${\beta }^2$.