Early-time behavior of quantum subharmonic generation

A few years ago Avetissian {\it et al.} \cite{Avetissian2014,Avetissian2015} discovered that the exponential growth rate of stimulated annihilation photons from a singlet positronium Bose-Einstein condensate should be proportional to square root number density, not density itself. In order elucidate this surprising result obtained via field-theoretical analysis, we point out basic physics involved is same as resonant subharmonic transitions between two quantum oscillators. Using model, show nonlinearities type by are unique and in fact will encountered wide range systems can modeled nonlinearly coupled