  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.
