Strong correlation effect in the second harmonic generation of a Bose-Einstein condensate

The effect of Bose-Einstein condensation, achieved recently for alkaline gases in magnetic traps, should lead to a strong enhancement of non-linear harmonic generation. A nearly resonant light imposed on the gas of two-level atoms excites the intra-atomic transitions and causes generation of various harmonics. Normally, in the thermal state of the gas, a contribution of the inter-atomic interaction to the non-linear harmonic generation is negligible. However, if a Bose-Einstein condensate (BEC) is present, the inter-particle interaction cannot be ignored. Considering second harmonic generation, we show that, in the presence of the BEC, the intensity of the second harmonic is proportional to (aN n)², where N and n are the total number and the effective density of the condensed atoms, respectively, and a stands for the scattering length describing the process of scattering of two excited atoms by each other so that, after the collision, one excited atom enters the condensate. We consider the interaction between light, represented by the electric field E, and the atomic Bose field Ψ as well as the two body interaction treated within the scattering length approximation