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arXiv:1706.03804 · Physical review. E

Continuous-variable approach to the spectral properties and quantum states of the two-component Bose-Hubbard dimer

Fabio Lingua, Vittorio Penna

Published Jun 30, 2017• 13 citations
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Abstract

A bosonic gas formed by two interacting species trapped in a double-well potential features macroscopic localization effects when the interspecies interaction becomes sufficiently strong. A repulsive interaction spatially separates the species into different wells while an attractive interaction confines both species in the same well. We perform a fully analytic study of the transitions from the weak- to the strong-interaction regime by exploiting the semiclassical method in which boson populations are represented in terms of continuous variables. We find an explicit description of low-energy eigenstates and spectrum in terms of the model parameters which includes the neighborhood of the transition point. To test the effectiveness of the continuous-variable method we compare its predictions with the exact results found numerically. Numerical calculations confirm the spectral collapse evidenced by this method when the space localization takes place.

DOI
10.1103/physreve.95.062142

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A bosonic gas formed by two interacting species trapped in a double-well potential features macroscopic localization effects when the interspecies interaction becomes sufficiently strong. A repulsive interaction spatially separates the species into different wells while an attractive interaction confines both species in the same well. We perform a fully analytic study of the transitions from the weak- to the strong-interaction regime by exploiting the semiclassical method in which boson populations are represented in terms of continuous variables. We find an explicit description of low-energy eigenstates and spectrum in terms of the model parameters which includes the neighborhood of the transition point. To test the effectiveness of the continuous-variable method we compare its predictions with the exact results found numerically. Numerical calculations confirm the spectral collapse evidenced by this method when the space localization takes place.
We investigate the ground-state properties of a two-species condensate of interacting bosons in a double-well potential. Each atomic species is described by a two-space-mode Bose-Hubbard model. The coupling of the two species is controlled by the interspecies interaction W. To analyze the ground state when W is varied in both the repulsive ( W > 0 ) and the attractive ( W < 0 ) regime, we apply two different approaches. First we solve the problem numerically (i) to obtain an exact description of the ground-state structure and (ii) to characterize its correlation properties by studying (the appropriate extensions to the present case of) the quantum Fisher information, the coherence visibility and the entanglement entropy as functions of W. Then we approach analytically the description of the low-energy scenario by means of the Bogoliubov scheme. In this framework the ground-state transition from delocalized to localized species (with space separation for W > 0 , and mixing for W < 0 ) is well reproduced. These predictions are qualitatively corroborated by our numerical results. We show that such a transition features a spectral collapse reflecting the dramatic change of the dynamical algebra of the four-mode model Hamiltonian.
Abstract We investigate the ground-state properties of a two-species condensate of interacting bosons in a double-well potential. Each atomic species is described by a two-space-mode Bose–Hubbard model. The coupling of the two species is controlled by the interspecies interaction W . To analyze the ground state when W is varied in both the repulsive ( W > 0 ) and the attractive ( W < 0 ) regime, we apply two different approaches. First we solve the problem numerically (i) to obtain an exact description of the ground-state structure and (ii) to characterize its correlation properties by studying (the appropriate extensions to the present case of) the quantum Fisher information, the coherence visibility and the entanglement entropy as functions of W . Then we approach analytically the description of the low-energy scenario by means of the Bogoliubov scheme. In this framework the ground-state transition from delocalized to localized species (with space separation for W > 0 , and mixing for W < 0 ) is well reproduced. These predictions are qualitatively corroborated by our numerical results. We show that such a transition features a spectral collapse reflecting the dramatic change of the dynamical algebra of the four-mode model Hamiltonian.
We determine the low-energy properties of a trapped Bose gas split in two by a potential barrier over the whole range of barrier heights and asymmetry between the wells. For either weak or strong coupling between the wells, our two-mode theory yields a two-site Bose-Hubbard Hamiltonian with the tunneling, interaction, and bias parameters calculated simply using an explicit form of two mode functions. When the potential barrier is relatively low, most of the particles occupy the condensate mode and our theory reduces to a two-mode version of the Bogoliubov theory, which gives a satisfactory estimate of the spatial shape and energy of the lowest collective excitation. When the barrier is high, our theory generalizes the standard two-site Bose-Hubbard model into the case of asymmetric modes, and correctly predicts a full separation of the modes in the limit of strong separation of the wells. We provide explicit analytic forms for the number squeezing and coherence as a function of particle number and temperature. We compare our theory to other two-mode theories for bosons in a double well and discuss their validity in different parameter regimes.
We investigate the ground state of one-dimensional few-atom Bose–Bose mixtures under harmonic confinement throughout the crossover from weak to strong inter-species attraction . The calculations are based on the numerically exact multi-configurational time-dependent Hartree method. For repulsive components, we detail the condition for the formation of a molecular Tonks–Girardeau gas in the regime of intermediate inter-species interactions, and the formation of a molecular condensate for stronger coupling. Beyond a critical inter-species attraction, the system collapses to an overall bound state. Different pathways emerge for unequal particle numbers and intra-species interactions. In particular, for mixtures with one attractive component, this species can be viewed as an effective potential dimple in the trap center for the other, repulsive component.
We study the dynamical properties of a few bosons confined in an one-dimensional split hard wall trap with the interaction strength varying from the weakly to strongly repulsive regime. The system is initially prepared in one side of the double well by setting the barrier strength of the split trap to be infinity and then the barrier strength is suddenly changed to a finite value. Both exact diagonalization method and Bose-Hubbard model (BHM) approximation are used to study the dynamical evolution of the initial system. The exact results based on exact diagonaliztion verify the enhancement of correlated tunneling in the strongly interacting regime. Comparing results obtained by two different methods, we conclude that one-band BHM approximation can well describe the dynamics in the weakly interacting regime, but is not efficient to give quantitatively consistent results in the strongly interacting regime. Despite of the quantitative discrepancy, we validate that the form of correlated tunneling gives an important contribution to tunneling in the large interaction regime. To get a quantitative description for the dynamics of bosons in the strongly interacting regime, we find that a multi-band BHM approximation is necessary.
We investigate the weak excitations of a system made up of two condensates trapped in a Bose-Hubbard ring and coupled by an interspecies repulsive interaction. Our approach, based on the Bogoliubov approximation scheme, shows that one can reduce the problem Hamiltonian to the sum of sub-Hamiltonians ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}$, each one associated to momentum modes $\ifmmode\pm\else\textpm\fi{}k$. Each ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}$ is then recognized to be an element of a dynamical algebra. This uncommon and remarkable property allows us to present a straightforward diagonalization scheme, to find constants of motion, to highlight the significant microscopic processes, and to compute their time evolution. The proposed solution scheme is applied to a simple but nontrivial closed circuit, the trimer. The dynamics of low-energy excitations, corresponding to weakly populated vortices, is investigated considering different choices of the initial conditions and the angular-momentum transfer between the two condensates is evidenced. Finally, the condition for which the spectral collapse and dynamical instability are observed is derived analytically.