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arXiv:1606.08816 · Physical Review A

Two-site Bose-Hubbard model with nonlinear tunneling: Classical and quantum analysis

Diefferson Rubeni da Rosa de Lima, Jon Links, Phillip S. Isaac, Angela Foerster

Published Apr 6, 2017• 24 citations
arXivHTMLPDF

Abstract

The extended Bose-Hubbard model for a double-well potential with atom-pair tunneling is studied. Starting with a classical analysis we determine the existence of three quantum phases: self-trapping, phase-locking, and Josephson states. From this analysis we build the parameter space of quantum phase transitions between degenerate and nondegenerate ground states driven by the atom-pair tunneling. Considering only the repulsive case, we confirm the phase transition by the measure of the energy gap between the ground state and the first excited state. We study the structure of the solutions of the Bethe ansatz equations for a small number of atoms. An inspection of the roots for the ground state suggests a relationship to the physical properties of the system. By studying the energy gap we find that the profile of the roots of the Bethe ansatz equations is related to a quantum phase transition.

DOI
10.1103/physreva.95.043607

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The extended Bose-Hubbard model for a double-well potential with atom-pair tunneling is studied. Starting with a classical analysis we determine the existence of three quantum phases: self-trapping, phase-locking, and Josephson states. From this analysis we build the parameter space of quantum phase transitions between degenerate and nondegenerate ground states driven by the atom-pair tunneling. Considering only the repulsive case, we confirm the phase transition by the measure of the energy gap between the ground state and the first excited state. We study the structure of the solutions of the Bethe ansatz equations for a small number of atoms. An inspection of the roots for the ground state suggests a relationship to the physical properties of the system. By studying the energy gap we find that the profile of the roots of the Bethe ansatz equations is related to a quantum phase transition.
Exactly solvable models provide a unique method, via qualitative changes in the distribution of the ground-state roots of the Bethe ansatz equations, to identify quantum phase transitions. Here we expand on this approach, in a quantitative manner, for two models of Bose–Einstein condensates. The first model deals with the interconversion of bosonic atoms and molecules. The second is the two-site Bose–Hubbard model, widely used to describe tunneling phenomena in Bose–Einstein condensates. For these systems we calculate the ground-state root density. This facilitates the determination of analytic forms for the ground-state energy, and associated correlation functions through the Hellmann–Feynman theorem. These calculations provide a clear identification of the quantum phase transition in each model. For the first model we obtain an expression for the molecular fraction expectation value. For the two-site Bose–Hubbard model we find that there is a simple characterization of condensate fragmentation.
Abstract The extended Bose–Hubbard model for a double-well potential with pair tunneling is studied through both exact diagonalization and mean field theory (MFT). When pair tunneling is strong enough, the ground state wavefunction predicted by the MFT is complex and doubly degenerate while the quantum ground state wavefunction is always real and unique. The time reversal symmetry is spontaneously broken when the system transfers from the quantum ground state into one of the mean field ground states upon a small perturbation. As the gap between the lowest two levels decreases exponentially with particle number, the required perturbation inducing the spontaneous symmetry breaking (SSB) is infinitesimal for particle number of typical cold atom systems. The quantum ground state is further analyzed with the Penrose–Onsager criterion, and is found to be a fragmented condensate. The state also develops the pair correlation and has non-vanishing pair order parameter instead of the conventional single particle order parameter. When this model is generalized to optical lattice, a pair superfluid can be generated. The mean field ground state can be regarded as effective ground state in this simple model. The detailed computation for this model enables us to offer an in-depth discussion of the relation between SSB and effective ground state, giving a glimpse on how nonlinearity arises in the SSB of a quantum system.
The attractive two-site Bose-Hubbard model is studied within the framework of the analytical solution obtained by the application of the quantum inverse scattering method. The structure of the ground and excited states is analyzed in terms of solutions of Bethe equations, and an approximate solution for the Bethe roots is given. This yields approximate formulas for the ground-state energy and for the first excited-state energy. The obtained formulas work with remarkable precision for a wide range of parameters of the model, and are confirmed numerically. An expansion of the Bethe state vectors into a Fock space is also provided for evaluation of expectation values, although this does not have accuracy similar to that of the energies.
In this paper we theoretically investigate the nonlinear tunnelling of two weakly linked Bose–Einstein condensates in a double-well trap in a strong interaction regime with the two-body interaction extended to neighbouring lattice sites. As a consequence the boson Josephson-junction tunnelling depends on the atom–atom interaction and the total atom number as well. A crucial atom-pair tunnelling term, obtained directly from the extension, results in significant energy spectrum corrections and an abrupt change of the ground state viewed as quantum phase transition. 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 in an interaction region. The quantum phase transitions between degenerate and non-degenerate ground states driven by the atom-pair tunnelling are analysed. We show the scaling behaviours and the critical exponents of fidelity susceptibility, which can classify the universality.
The quantum effect of nonlinear co-tunnelling process, which is dependent on atom-pair tunneling and asymmetry of an double-well trap, is studied by using an asymmetrical extended Bose–Hubbard model. Due to the existence of atom-pair tunneling that describes quantum phenomena of ultracold atom-gas clouds in an asymmetrical double-well trap, the asymmetrical extended Bose–Hubbard model is better than the previous Bose–Hubbard model model by comparing with the experimental data cited from the literature. The dependence of dynamics and quantum phase transition on atom-pair tunneling and asymmetry are investigated. Importantly, it shows that the asymmetry of the extended Bose–Hubbard model, corresponding to the bias between double wells, leads to a number of resonance tunneling processes, which tunneling is renamed conditional resonance tunneling, and corrects the atom-number parity effect by controlling the bias between double wells.
We propose a Hamiltonian of ultracold spinless atom in optical lattices including 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 explains naturally 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. Finally a dynamics of oscillations induced by the atom-pair tunneling is found in the strong interaction regime, where the Bose-Hubbard model gives rise to the insulator state with fixed time-averaged value of atom-occupation number only. Quantum phase transitions between two quantum phases characterized by degenerate and nondegenerate ground states are shown to be coinciding with the Landau second-order phase-transition theory. 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.