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arXiv:1309.2956 · Journal of Physics A Mathematical and Theoretical

A bosonic multi-state two-well model

G Santos, Angela Foerster, I. Roditi

Published Jun 12, 2013• 3 citations
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Abstract

Inspired by the increasing possibility of experimental control in ultracold atomic physics, we introduce a new Lax operator and use it to construct and solve models with two wells and two on-well states together with its generalization for n on-well states.The models are solved by the algebraic Bethe ansatz method and can be viewed as describing two Bose-Einstein condensates allowing for an exchange interaction through Josephson tunnelling.

DOI
10.1088/1751-8113/46/26/265206

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Abstract In this work we present a general construction of integrable models describing boson tunneling in multi-well systems. We show how the models may be derived through the quantum inverse scattering method and solved by algebraic Bethe ansatz means. From the transfer matrix we find only two conserved operators. However, we construct additional conserved operators through an alternative approach. As a consequence the models admit multiple pseudovacua, each associated with a set of Bethe ansatz equations (BAEs). We show that all sets of BAEs are needed to obtain a complete set of eigenstates.
We present the exact solution of a family of two-spins models. The models are solved by the algebraic Bethe ansatz method using the gl(2)gl(2)gl(2)-invariant RRR-matrix and a multi-spins Lax operator. The interactions are by the Heisenberg spins exchange. We are also considering magnetic BBB-fields and a term for Haldane single spin anisotropy.
We present three families of integrable models for Bose–Einstein condensates in a double-well potential which are based on the combination, two by two, of three algebraic realizations of the Lax operator: the bosonic realization and the su (2) and su (1, 1) Holstein–Primakoff realizations. The families of Hamiltonians describe Josephson tunneling between two condensates in a double-well potential with boson distribution dependence in the tunneling term. Two families describe models with interacting bosons and the other does not. The integrability is shown by using the quantum inverse scattering method and algebraic Bethe ansatz. The energy eigenvalues and the Bethe ansatz equations are obtained for each model proposed. In the end, we use the energy gap as a testing function to compare the proposed integrable models with the well-known integrable two-site Bose–Hubbard model to show the effect of the boson distribution in the tunneling.
The two-site version of the Bose-Hubbard model can be used as a simplified model of N bosons confined to two quantum wells. We study the behavior of low-temperature Bose-Einstein condensates as a function of the strength of the repulsive interactions between the bosons and of the size of the quantum-mechanical tunneling between the quantum wells. The ground-state of weakly interacting bosons tends to be one in which almost all particles are in the same quantum-mechanical state (‘Bose-Einstein Condensate’, or ‘BEC’). It is found that even very small amounts of interwell tunneling lead to a BEC, but that zero tunneling leads to two separate BECs, one in each well (‘fragmentation’). We study systematically the transition between a single BEC comprised of bosons in both wells, and a fragmented condensate consisting of two BECs, one for each well. Our analysis will use matrix computational methods to solve the Schrödinger equation of the two-well boson system, and obtain the energies and wavefunctions as eigenvalues and eigenvectors of the Hamiltonian matrix. We will compare the results obtained with previous analyses, and against analytical approximations developed for the limiting case of a large number of bosons (large N).