Non-linear Quantum Tunneling Of Bose-Einstein Condensates

With great development of laser cooling and optical trapping technology, two weakly linked Bose-Einstein condensates in a double-well trap can be realized and the interaction strength between particles can be manipulated according to Feshbach resonance technique. In strong interaction region we derive a Hamiltonian of ultracold spinless atoms in optical lattices including an atom-pair tunneling term and a single-particle tunneling term, due to the two-body interaction of nearest neighbors, which reduces to the Bose-Hubbard model in weak-interaction limit. An atom-pair hopping term appearing in the Hamiltonian is comfirmed by the recent experimental observation of correlated tunneling in a double-well trap with strong atom-atom interactions and moreover leads to a dynamic process of atom-pair tunneling where strongly interacting atoms can tunnel back and forth as a fragmented pair. With the two-body interaction of nearest neighbors taken into account, the Bose-Hubbard Model is extended as an extended Bose-Hubbard Model.First of all, for the extended Bose-Hubbard Model, the dynamics of oscillations induced by the atom-pair tunneling is found in the strong interaction regime, where the extended Bose-Hubbard Model gives rise to the insulator state with fixed time-averaged value of atom-occupation number only. With the Landau second-order phase-transition theory, quantum phase transitions characterized by degenerate and nondegenerate ground states are shown to be coinciding. For the system of small atom number, an atom-number effect has to be taken into account. Using the method of instantons, we calculate the energy-level split due to quantum tunneling. In the system of finite atom number, the degeneracy of ground states can be removed by quantum tunneling for the even number of atoms but not for the odd number.Secondly, within mean-field approximation, the extended Bose-Hubbard Model in double-well trap can be reduced to the extended boson Josephson-junction Model. As non-linear tunneling terms appearing in the Hamiltonian, the boson Josephson-junction tunnelling depends on the atom-atom interaction and the total atom number as well. A crucial atom-pair tunnelling term results in significant energy spectrum corrections and an abrupt change of the ground state viewed as quantum phase transition. In a strong interaction region, an atom-number oscillation state with a tunable relative phase between two Bose-Einstein condensates predicted for the first time is seen to be a degenerate ground state. We analyse the quantum phase transition between degenerate and non-degenerate ground states driven by the atom-pair tunneling, and obtain the scaling behaviours and the critical exponents of fidelity susceptibility, which can classify the universality.Finally, it is theoretically investigated the dynamic-effect induced by nonlinear tunnelling of two weakly linked Bose-Einstein condensates in an extended boson Josephson-Junction Model in strong interaction regime. Compared with the usual Josephson-junction model, the extended boson Josephson-Junction Model includes an atom-pair tunneling term and a single particle tunneling term, due to the two-body interaction of nearest neighbors. The atom-pair tunneling term leads to the macroscopic quantum-states of coherent relative-phase between two condensates controlled by system parameters. The new dynamic states are seen to be effective Schrodinger cat states and can be used to realize the qubit in quantum computation.