Nonequilibrium kinetics of coupled photons and electrons in two-level systems of the laser type

Rate equations for photons and for electrons in systems of two groups of energy levels are set up allowing for absorption, stimulated and spontaneous emission, nonradiative transitions, photon loss or dissipation as well as photon gain from the ambient, and an externally applied pumping rate. The rate equations are discussed qualitatively, and the stability of the solutions is obtained by the use of phase portraits. The quantitative discussion includes steady-state photon distributions. Earlier work showing that these distributions can obey the Bose-Einstein law with a nonzero chemical potential kTF is taken into the laser regime, which was not covered by recent relevant work. The second-order phase transition that occurs at threshold is established explicitly, and it is shown that the transition requires the neglect of certain generation terms in the rate equations. Some time-dependent solutions of the rate equations are also discussed, using the slaving principle of adiabatic elimination for the electrons. The application of this work to semiconductors is discussed.