  Semiconductor microcavities with artificial single-photon emitters have
become one of the backbones of semiconductor quantum optics. In many cases
however, technical and physical issues limit the study of optical fields to
incoherently excited systems. We analyze the model of a two-level system in a
single-mode cavity, where the former is incoherently driven. The specific
structure of the applied master equation yields a recurrence relation for the
steady-state values of correlations of the intracavity field and the emitter.
We provide boundary conditions, that permit a systematical, easy to implement
solution, which is numerically less demanding than standard methods. Different
cavity systems from previous experiments are analyzed. The derived boundary
conditions also allow us direct analytical statements about the overall quantum
state and its higher order moments. With this we can give very good
approximations for the full quantum state of the field and show, that for every
physically reasonable set of system parameters, the state of the intracavity
field is nonclassical.
