A new method for detection of exciton Bose condensation using stimulated two-photon emission

Stimulated two-photon emission by Bose-condensed excitons accompanied by a coherent two-exciton recombination, i.e., by simultaneous recombination of two excitons with opposite momenta leaving unchanged the occupation numbers of excitonic states with nonzero momenta, is investigated. Raman light scattering accompanied by a similar two-exciton recombination (or generation of two excitons) is also analyzed. The processes under consideration can occur only if a system contains Bose condensate, therefore, their detection can be used as a new method to reveal Bose condensation of excitons. The recoil momentum, which corresponds to a change in the momentum of the electromagnetic field in the processes, is transferred to phonons or impurities. If the recoil momentum is transmitted to optical phonons with frequency $\omega_0^s$, the stimulated two-photon emission with the coherent two-exciton recombination leads to the appearance of a line at $2\Omega'-\omega$, where $\Omega'=\Omega-\omega_0^s$ and $\Omega$ is the light frequency corresponding to the recombination of an exciton with zero momentum. Formulas for the cross sections at finite temperatures are obtained for the processes under consideration. Our estimates indicate that a spectral line, corresponding to the stimulated two-photon emission accompanied by the coherent optical phonon-assisted two-exciton recombination can be experimentally detected in Cu$_2$O.