On the early-time behaviour of quantum subharmonic generation

A few years ago, the exponential growth rate of the stimulated annihilation photons from a singlet positronium Bose-Einstein condensate should be proportional to the square root of the positronium number density [Avetissian et al., Phys. Rev. Lett. 113, 23904 (2014)], not to the number density itself as had previously been assumed by many authors. Here we consider a quantum analysis of stimulated emission by modeling the conversion of k-ordered initial quanta of in a first oscillator to l-ordered final quanta of in a second oscillator. Our result shows that nonlinearities of this type are not unique to positronium and in fact are encountered in many systems that can be modeled as nonlinearly coupled quantum oscillators.