Binary mixture of Bose-Einstein condensates in a double-well potential: Berry phase and two-mode entanglement

A binary mixture of Bose-Einstein condensate structures exhibits an incredible richness in terms of holding different kinds of phases. Depending on the ratio of the inter- and intra-atomic interactions, the transition from the mixed to separated phase, which is also known as the miscibility-immiscibility transition, has been reported in different setups. Here, we describe such a quantum phase transition (QPT) in an effective Hamiltonian approach by applying the Holstein-Primakoff transformation in the limit of a large number of particles. We demonstrate that a nontrivial geometric phase near the critical coupling is present, which confirms the connection between the Berry phase and QPT. We also show that, by using the spin form of the Hillery and Zubairy criterion, a two-mode entanglement accompanies this transition in the limit of a large, but not infinite, number of particles.