Generation of maximally entangled states of a Bose-Einstein condensate and Heisenberg-limited phase resolution

We outline a procedure for Heisenberg-limited phase resolution between two Bose-Einstien condensates (BECs) defined as different hyperfine levels. The method involves first establishing a maximally entangled state using the ideas of nonlinear interferometry previously discussed in the optical domain [C. C. Gerry , Phys. Rev. A 66, 013804 (2002)]. In the case of the condensates, the nonlinear interactions are realized by the interatomic interactions within each condensate. Quarter cycle Raman pulses between hyperfine levels act as beam splitters. Parity measurements of one of the components of the BEC resolve the phase at the Heisenberg limit. We point out that parity measurements can be made by coupling the mode of interest with a third condensate where both components evolve under nonlinear interatomic interactions. After another Raman pulse, the components are populated according to parity. One need only determine which component is populated to determine the parity.