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 $S^z{S}^z$-type interaction where $S^z$ is the total spin. The geometric phase gate can be executed in times of the order of $2\pi \hslash /G$, 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 the Bose-Einstein condensates are analyzed.

• Received 7 June 2013
• Revised 16 April 2014

DOI:https://doi.org/10.1103/PhysRevA.89.053607