TIME-DEPENDENT AND STEADY-STATE STATISTICS OF PHOTONS AT NONZERO CHEMICAL-POTENTIAL

The paper analyses some time-dependent and steady-state properties of coupled photons and electrons in two-level systems, by taking into consideration the radiative transitions and photon dissipation. The main assumption is the slaving principle for adiabatic elimination of the electrons. A master equation is derived for the photon number. The stability of the photon population is studied and the laser threshold concentration of the upper-level electrons determined. A first-order phase transition that occurs at threshold is established explicitly, and it is shown that the transition requires the neglect of the spontaneous emission and of the ambient influence. The steady-state photon distribution is not Poissonian and is characterized by large fluctuations. Fluctuations become negligible when the stimulated emission is zero. The steady-state distribution can obey the Bose-Einstein law with a non-zero chemical potential. The equation of state of the photon population is derived. When dissipation is weak and dilution increases, the equation of state is similar to that of a degenerate Bose gas in equilibrium.