Dynamical theory of polariton amplifiers.

We present the theory of the dynamics of the polariton amplifier in the region of small polariton densities. We give an analytical solution for the polariton condensate density matrix and show that the formation of a coherent quantum state is possible. Once the condensate is formed, the coherence becomes macroscopically long living. Polariton amplifier represents, therefore, an optical memory element, where the input weak coherent signal can be amplified and kept.