Stimulated multiphoton emission from exciton Bose condensate

The coherent recombination of several $(N)$ Bose-condensed excitons with simultaneous creation of $N$ photons is considered. Due to the momentum conservation law, the total momentum of created photons is zero because of the zero momentum of excitons in Bose condensate. This requirement, in conjunction with the fact that the photon wavenumbers are fixed and equal to approximately $E_g /c$ ($E_g$ is the semiconductor gap and $c$ is the speed of light), determines the mutual orientation of the wavevectors of emitted photons. This can be seen from the photon angular correlation in the experiments with several appropriately oriented detectors operating in the time-coincidence mode. It is shown that, if these processes are induced by $N-1$ external laser beams (each with wavevector $k_i$), then a unidirectional radiation with recoil wavevector $k = -\sum_{i} k_i$ should emerge from the exciton system. The intensities of coherent three-and four-exciton recombination are estimated for the exciton system in Cu$_2$O.