  We develop a quantum model for non-equilibrium Bose-Einstein condensation of
photons and polaritons in planar microcavity devices. The model builds upon
laser theory and includes the spatial dynamics of the cavity field, a
saturation mechanism and some frequency-dependence of the gain: quantum
Langevin equations are written for a cavity field coupled to a continuous
distribution of externally pumped two-level emitters with a well-defined
frequency. As a an example of application, the method is used to study the
linearised quantum fluctuations around a steady-state condensed state. In the
good-cavity regime, an effective equation for the cavity field only is proposed
in terms of a stochastic Gross-Pitaevskii equation. Perspectives in view of a
full quantum simulation of the non-equilibrium condensation process are finally
sketched.
