A model describing amplification and quantum statistics of the exciton polaritons with k = 0 in a non-resonantly excited semiconductor quantum well embedded in a microcavity is presented. Exploiting the bottleneck effect for exciton polaritons, it is assumed that the polaritons with k ≠ 0 act as a reservoir. The time evolution of the k = 0 polaritons is described by a master equation, in which two-and one-polariton transitions between the mode with k = 0 and the reservoir are accounted for. The k = 0 mode exhibits a threshold depending on the material parameters and on the injected exciton density. Below threshold the quantum statistics of the emission is characteristic of an incoherent process, while above threshold it approaches that of a laser.71.36.+c, 73.21.Fg
I INTRODUCTIONIn the last few years, non-linear scattering between exciton polaritons has been observed in semiconductor quantum wells embedded in microcavities under resonant as well as non-resonant excitation 1, 2 . Resonant non-linear effects have been investigated in experiment 3-5 and in theory [6][7][8] . It has been shown that for resonant excitation the experiments may be understood in terms of parametric processes (parametric amplification and oscillation) both in a pump and probe configuration and in a configuration in which only the resonant pump is present. It has been also shown that a spontaneous transition from an incoherent emission regime to a coherent one occurs under resonant excitation 9,10 The emission under non-resonant excitation has also been investigated in experiments. As early as 1998 it has been shown that an amplification regime is present in a CdTe quantum well when the non-resonant pump energy is sufficiently large 11 . The amplified emission is observed in correspondence to the polariton mode with k=0. It has also been shown 12, 13 that the amplification is obtained in the strong coupling regime. In more recent years other characteristics of this amplification effect have been investigated experimentally. In particular, the onset of a transition from incoherent to coherent emission has been demonstrated 14 and the quantum statistics of the emitted radiation has been measured 15,16 . This transition has analogies with the transition characterizing the onset of laser action. Furthermore, some exciting results on the spatial coherence of the emission have been recently published 17 . However, in spite of the large number of existing experimental results, the theoretical investigation of the non-resonant amplified emission is still in progress. The theoretical description of the amplification process has to reproduce the threshold effect observed in the experiments and has to allow calculating the photon statistics of the emitted radiation i.e. the quantum polariton statistics. It should include both the exciton-phonon interaction that is responsible for the energy losses of the excitons generated by non-resonant pumping as well as the exciton-photon and the excitonexciton interaction leading to the formation of a macroscop...