Efficient two-mode interferometers with spinor Bose-Einstein condensates

We consider general three-mode interferometers using a spin-1 atomic Bose-Einstein condensate with macroscopic magnetization. We show that these interferometers, combined with the measurement of the number of particles in each output port, provide an ultrahigh phase sensitivity. We construct effective two-mode interferometers which involve two Zeeman modes showing that they also provide an ultrahigh phase sensitivity but of a bit reduced factor in the corresponding Fisher information. A special case of zero magnetization is shown to persist the efficiency of the two-mode interferometry.