Useful models of four-wave mixing in Bose–Einstein condensates

Abstract

A recent experiment demonstrated four-wave mixing of wavepackets in a sodiumBose–Einstein condensate (Deng et al 1999 Nature 398 218). This was followed by atheoretical and numerical treatment of the experiment (Trippenbach et al 2000 Phys. Rev.A 62 02368). In the experiment, a short period of free expansion of the condensate, afterrelease from the magnetic trap, was followed by a set of two Bragg pulses which createdmoving wavepackets. These wavepackets, due to nonlinear interaction and underphase-matching conditions, created a new momentum component in a four-wave mixingprocess. We propose simple mathematical models for this process. Next we suggest that,instead of exactly matching the frequencies as in the abovementioned experiments,we introduce a small mismatch in the energies, and therefore the frequenciesΔω.We show that such a small mismatch can compensate for the initial phases that are builton the condensate during free expansion. A physical explanation is offered. Thiscompensation can improve the efficiency of four-wave mixing; in some cases even increasingit by a factor of 2. We also deal with the situation where two strong wavepackets areaccompanied by a weak input beam applied as a seed both with and without a mismatch.Here the influence of the mismatch is less obviously beneficial. We also comment onrecent work by Ketterle’s group (Vogels et al 2002 Phys. Rev. Lett. 89 020401).