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.

Figure 1

The displacement amounts (a) ${\beta }_a$ and (b) ${\beta }_b$ as a function of coupling strength $g_{ab}$ for different values of $g_b$. Here, we use equal tunneling amplitudes, $t_{\alpha }/j_{\alpha }=0.1g_a$, and in the insets, we consider the infinite number of particles case and take $t_{\alpha }/j_{\alpha }=0$.

Figure 2

The scaled Berry phase of the system for (a) a large and (b) small number of particles and as a function of coupling strength $g_{ab}$. Here, we use equal tunneling amplitudes for (a), $t_{\alpha }/j_{\alpha }=0.1g_a$, and an equal number of particles, $N_a=N_b={10}^4$. In the inset, we take $t_{\alpha }/j_{\alpha }=0$ (or, equivalently, $j_{\alpha }=\infty )$. For a finite number of particles (b), we use $t_{\alpha }=100g_a$ and obtain the results by solving the eigenstates of Eq. (3) numerically.

Figure 3

Hillery-Zubairy entanglement criterion [Eq. (36)] as a function of coupling strength $g_{ab}$. $E_{{}_{\mathrm{HZ}}}<0$ witnesses the presence of entanglement between the two collective spins. Here, we use $t_{\alpha }=100g_a$ and $g_b=g_a$. In the inset, we take $N_a=N_b=N$ and observe that entanglement decays faster as $N$ increases.