Geometric phase gate for entangling two Bose-Einstein condensates

We propose a method of entangling two spinor Bose-Einstein condensates using a geometric phase gate. The scheme relies upon only the ac Stark shift and a common controllable optical mode coupled to the spins. Our scheme allows for the creation of an SzSz type interaction where Sz is the total spin. The geometric phase gate can be executed in times of the order of 2{\pi} /G, where G is the magnitude of the Stark shift. In contrast to related schemes which relied on a fourth order interaction to produce entanglement this is a second order interaction in the number of atomic transitions. Closed expressions for the entangling phase are derived and the effects of decoherence due to cavity decay, spontaneous emission and incomplete de-entangling of the light to BEC are analyzed.