Early-time behavior of quantum subharmonic generation

A few years ago Avetissian et al. [Phys. Rev. Lett. 113, 023904 (2014); Phys. Rev. A 92, 023820 (2015)] discovered that 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, not to the number density itself. In order to elucidate this surprising result obtained via a field-theoretical analysis, we point out that the basic physics involved is the same as that of resonant subharmonic transitions between two quantum oscillators. Using this model, we show that nonlinearities of the type discovered by Avetissian et al. are not unique to positronium and in fact will be encountered in a wide range of systems that can be modeled as nonlinearly coupled quantum oscillators.

Figure 1

The time dependence of the expectation values of the occupation number, $N_b\left(t\right)$, with $N_b\left(0\right)=0$, are shown for different values of $k$ with $l=2$ when the two different initial values (a) $N_a=10$ (small $N_a$) and (b) $N_a=69$ (large $N_a$) are given. The plots show the normalized scale, $N_b/N_a$, and compare the analytic approximations from Eq. (15) (star-marked green curves) with the numerical results (blue curves without star markers).