Nonlinear phase mismatch and optimal input combination in atomic four-wave mixing in Bose-Einstein condensates

This work treats four-wave mixing (4WM) in Bose-Einstein condensates (BEC), focusing on the nonlinear phase mismatch, maximum output, and optimal input combination. We show that the nonlinear phase mismatch decreases the 4WM efficiency. It was found that the 4WM efficiency depends on both the coupling coefficient (i.e., the product of the total number of atoms, the scattering length, and the overlap integral) and the ratios among the three initial input beams. The 4WM efficiency increases with the increase of the coupling coefficient when it is small, then saturates, and finally decreases at high coupling coefficient due to both pump depletion and phase-modulation effects. A maximum output efficiency of about 50% in our case is predicted. In order to get the maximum output, the two pump beams should have equal amplitude and the probe beam should be as small as possible. In addition, a large coupling coefficient (>pi/2), which is determined by the ratio of the probe beam to the total input, is required. On the other hand, when the coupling coefficient is fixed, a maximum output for this case can be obtained by optimizing the input ratios among the three input beams. Other ratio combinations will decrease the 4WM efficiency.