Theory of Ideal Four-Wave Mixing in Bose-Einstein Condensates

Starting from a second-quantized Hamiltonian of many-particle systems, we derive the Gross-Pitaevskii (GP) equation in momentum space, which is suitable for studying the multi-wave mixing processes of coherent matter waves. The coupling equations are then applied to study ideal four-wave mixing (4WM), in which only four waves with definite wavevectors are involved. Some interesting problems of 4WM, such as the phase-matching condition, the collapse and revival behaviour, the effects of relative phase difference, and the conversion efficiency are discussed in detail. We also show that the main characters of recent 4WM experiment [Deng et al, Nature 398, 218 (1999)] can be undersood in the present simplified model.