Theory of Four Wave Mixing of Matter Waves from a Bose-Einstein Condensate

A recent experiment [Deng et al., Nature 398, 218(1999)] demonstrated fourwave mixing of matter wavepackets created from a Bose-Einstein condensate. The experiment utilized light pulses to create two high-momentum wavepackets via Bragg diffraction from a stationary Bose-Einstein condensate. The high-momentum components and the initial low momentum condensate interact to form a new momentum component due to the nonlinear self-interaction of the bosonic atoms. We develop a three-dimensional quantum mechanical description, based on the slowly-varying-envelope approximation, for fourwave mixing in Bose-Einstein condensates using the time-dependent GrossPitaevskii equation. We apply this description to describe the experimental observations and to make predictions. We examine the role of phasemodulation, momentum and energy conservation (i.e., phase-matching), and particle number conservation in four-wave mixing of matter waves, and develop simple models for understanding our numerical results. Typeset using REVTEX