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arXiv:nlin/0612063 · Symmetry Integrability and Geometry Methods and Applications

Bethe Ansatz Solutions of the Bose-Hubbard Dimer

Jon Links

Published Dec 29, 2006• 12 citations
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Abstract

The Bose-Hubbard dimer Hamiltonian is a simple yet effective model for describing tunneling phenomena of Bose-Einstein condensates.One of the significant mathematical properties of the model is that it can be exactly solved by Bethe ansatz methods.Here we review the known exact solutions, highlighting the contributions of V.B. Kuznetsov to this field.Two of the exact solutions arise in the context of the Quantum Inverse Scattering Method, while the third solution uses a differential operator realisation of the su(2) Lie algebra.

DOI
10.3842/sigma.2006.095

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The Bose-Hubbard dimer Hamiltonian is a simple yet effective model for describing tunneling phenomena of Bose-Einstein condensates.One of the significant mathematical properties of the model is that it can be exactly solved by Bethe ansatz methods.Here we review the known exact solutions, highlighting the contributions of V.B. Kuznetsov to this field.Two of the exact solutions arise in the context of the Quantum Inverse Scattering Method, while the third solution uses a differential operator realisation of the su(2) Lie algebra.
I introduce a new parametrization of a bosonic Lax operator for the algebraic Bethe ansatz method with the gl(2)gl(2)gl(2)-invariant RRR-matrix and use it to present the exact solution of a generalized two-sites Bose-Hubbard model with asymmetric tunnelling. In the no interaction limit I show that the Bethe ansatz equations can be written as a SN−1S^{N-1}SN−1 sphere, where NNN is the total number of atoms in the condensate.
Abstract It is shown that the extended one-dimensional dimer Bose–Hubbard model with multi-body interactions can be solved exactly by using the algebraic Bethe ansatz mainly due to the site-permutation S 2 symmetry. The solution for the model with up to three-particle hopping and three-body on-site interaction is explicitly shown. As an example of the application, lower part of the excitation energy levels and the ground-state entanglement measure of the standard Bose–Hubbard Hamiltonian with the attractive two-body on-site interaction plus the three-body on-site interaction for 100 bosons with variation of the control parameter are calculated by using the exact solution. It is shown that the attractive three-body on-site interaction reinforces the critical point entanglement of the system, which may be helpful for design of an optical lattice for ultracold atoms or a tuneable superconducting quantum interference device with maximal entanglement.