Analytical results from the quantum theory of a single-emitter nanolaser. II
Using two equivalent approaches, Heisenberg-Langevin and density operator, we investigate the properties of nanolaser: an incoherently pumped single two-level system interacting with a single-cavity mode of finite finesse. We show that in the case of good-cavity regime the Heisenberg-Langevin approach provides the analytical results for linewidth, amplitude fluctuation spectrum, and intracavity Mandel Q-parameter. In the bad-cavity regime we estimate the frequency at which the peak of relaxation oscillations appears. With the help of the master equation for density operator written in terms of coherent states we obtain approximated expression for Glauber-Sudarshan P function. This solution can be used in the case of good-cavity regime and allows us to investigate in more detail the thresholdless behaviour of the nanolaser. All two approaches are in a very good agreement with each other and numerical simulations.