Quantum Critical Behavior of Two Coupled Bose-Einstein Condensates

The quantum critical behavior of the Bose-Hubbard model for a description of two coupled Bose-Einstein condensates is studied within the framework of an algebraic theory. Energy levels, wavefunction overlaps with those of the Rabi and Fock regimes, and the entanglement are calculated exactly as functions of the phase parameter and the number of bosons. The results show that the system goes though a phase transition and that the critical behavior is enhanced in the thermodynamic limit. PACS numbers: 73.43.Nq, 05.30.Jp, 03.75.Fi, 03.65.Ud As is well-known, the two-site Bose-Hubbard model can be used to describe pair tunneling between two superconductors through an insulating junction, trapped ultra-cold bosonic gases, etc. Furthermore, it can also be used to prepare macroscopically entangled states. The two-site Bose-Hubbard model has been investigated widely by many authors using various methods, such as the Gross-Pitaevskii approximation, mean-field theory, the quantum phase model, and the Bethe ansatz method. In [8], the temporal evolution of the expectation value for the relative number of particles between the two condensates for different choices of the coupling parameter and distinct initial states is analyzed. Also, quantum phase transitions that occur at zero temperature as a function of a coupling constant have become important in connection with various quantum many-body systems, such as the quantum Ising and rotor models, Fermi liquids,, and atomic nuclei. There are distinct features in these systems at the critical points. The main purpose of the present paper is to study the critical behavior of the two-site Bose-Hubbard model as a function of the coupling parameter and the total number of bosons. Specifically, we consider the two-site Bose-Hubbard Hamiltonian with Ĥ = −EJ(cd+ dc) + Ec(cccc+ dddd), (1) where c (c) and d (d) are boson creation (annihilation) operators in two traps or different hyperfine states. EJ is related to the Josephson coupling exchanging bosons between the two states, and Ec is related to the charging energy. In the present work we focus on the case where the effective interaction energy for the internal Josephson dynamics is negative, Ec < 0. In order to study the transitional patterns of the system, the Hamiltonian (1) is re-parameterized as Ĥ = Ĥ/E0 = −(1− x)(cd+ dc)− 4x n+ 1 (cccc+ dddd), (2) where E0 is a constant in arbitrary unit, n is the total number of bosons, and 0 ≤ x ≤ 1 is the phase parameter. The division of the second term in (2) by n + 1 serves to ensure that the boson rank of the two terms in Ĥ is the same, thereby making comparisons of results as a function of boson number more meaningful. For large values of n, this system is in the Rabi regime when x ∼ 0, the Fock regime when x ∼ 1, and the Josephson regime when 0 < x < 1. When x = 0, the system is in the Rabi regime with eigenstates given by |x = 0;n1, n2〉 = 1 2n/2 (c + d)1(c − d)2 |0〉, (3) where n = n1 + n2, and |0〉 is the boson vacuum state which is never degenerate. The corresponding eigenenergy of (2) is En1,n2(x = 0) = n2 − n1. (4)