Using two equivalent methods, the Heisenberg-Langevin and the master equation, we investigate the properties of a single-emitter nanolaser: an incoherently pumped single two-level system interacting with a single cavity mode of finite finesse. We are interested in a stationary generation regime when the mean number of photons inside the laser cavity is much larger than unity. In this case we linearize the Heisenberg-Langevin equations around the stationary classical solution and obtain analytical results for the linewidth, the quadrature fluctuation spectra, and the intracavity Mandel
Q parameter. As the second method we use the master equation for the density operator of the laser field. Using the coherent-state representation of the density matrix we obtain the stationary solution for the Glauber
P function of the laser mode and investigate the quantum statistics of the laser field and its threshold behavior. We compare the results of both methods with numerical simulations and find very good agreement. In the end we use our theoretical results for analyzing experiment with single-ion nanolasers.
We generalize the method of deriving a kinetic equation for a gas laser developed in the first paper of this series to include radiation-matter correlations. As a consequence, we are able to show that the first Born approximation with asymptotic conditions which contain radiation-matter correlations is sufficient to explain the nonthermal photon distribution observed in photon-counting experiments. Our derivation includes the spatial and velocity effects of the motion of the gas atoms. After adiabatically eliminating the matter variables, we obtain a master equation for the radiation density matrix
R with nonlinear coefficients. Our radiation master equation for stationary atoms without spatial effects is the same as the generalized Fokker-Planck equation of the Langevin-noise-equation treatment of lasers. We show that the generalized Fokker-Planck equation is the result of the first-Born-approximation treatment of the j\ifmmode\cdot\else\textperiodcentered\fi{}A interaction and is not dependent upon the microscopic structure of the reservoirs. In the first Born approximation, the diagonal and off-diagonal matrix elements of
R do not interact with each other. Consequently, an initially diagonal radiation density matrix remains diagonal. However, we show that even though the stationary state is diagonal it is necessary to know the propagator of the off-diagonal part of
R to answer time-dependent questions about quantities, such as the line shape, that depend on phase.