Theory of the decay luminescence spectrum of a Bose-condensed interacting exciton gas

Recent experimental studies by Wolfe, Snoke, and Lin have given strong evidence that paraexcitons in optically excited ${\mathrm{Cu}}_2$O crystals form a Bose condensate. In these studies, the observed recombination decay luminescence line shape I(ω) has been analyzed in terms of an ideal Bose-condensed gas. We derive a general expression for I(ω) in terms of the exact single-particle spectral density A(k,ω) of a Bose fluid. This expression is then used to calculate I(ω) for a dilute, weakly interacting Bose gas. Our analytic results, based on the approximate theory due to Popov, are valid in the region just below the Bose-Einstein transition temperature. When formally extended down to T=0, they reproduce the well-known Bogoliubov approximation.