Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 095, 8 pages      nlin.SI/0612063      https://doi.org/10.3842/SIGMA.2006.095
Contribution to the Vadim Kuznetsov Memorial Issue

Bethe Ansatz Solutions of the Bose-Hubbard Dimer

Jon Links and Katrina E. Hibberd
Centre for Mathematical Physics, School of Physical Sciences, The University of Queensland, 4072, Australia

Received October 26, 2006, in final form December 19, 2006; Published online December 29, 2006

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.

Key words: Bose-Hubbard dimer; Bethe ansatz.

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References

  1. Albiez M., Gati R., Fölling J., Hunsman S., Cristiani M., Oberthaler M., Direct observation of tunneling and non-linear self-trapping in a single bosonic Josephson junction, Phys. Rev. Lett., 2005, V.95, 010402, 4 pages, cond-mat/0411757.
  2. Cirac J.I., Lewenstein M., Molmer K., Zoller P., Quantum superposition states of Bose-Einstein condensates, Phys. Rev. A, 1998, V.57, 1208-1218, quant-ph/9706034.
  3. Dunning C., Hibberd K.E., Links J., On quantum phase crossovers in finite systems, J. Stat. Mech.: Theor. Exp., 2006, P11005, 11 pages, quant-ph/0602098.
  4. Enol'skii V.Z., Salerno M., Kostov N.A., Scott A.C., Alternate quantizations of the discrete self-trapping dimer, Phys. Scripta, 1991, V.43, 229-235.
  5. Enol'skii V.Z., Salerno M., Scott A.C., Eilbeck J.C., There's more than one way to skin Schrödinger's cat, Phys. D, 1992, V.59, 1-24.
  6. Enol'skii V.Z., Kuznetsov V.B., Salerno M., On the quantum inverse scattering method for the DST dimer, Phys. D, 1993, V.68, 138-152.
  7. Kohler S., Sols F., Oscillatory decay of a two-component Bose-Einstein condensate, Phys. Rev. Lett., 2002, V.89, 060403, 4 pages, cond-mat/0107568.
  8. Kuznetsov V.B., Tsiganov A.V., A special case of Neumann's system and the Kowalewski-Chaplygin-Goryachev top, J. Phys. A: Math. Gen., 1989, V.22, L73-L79.
  9. Leggett A.J., Bose-Einstein condensation in the alkali gases: some fundamental concepts, Rev. Modern Phys., 2001, V.73, 307-356.
  10. Links J., Zhou H.-Q., McKenzie R.H., Gould M.D., Algebraic Bethe ansatz method for the exact calculation of energy spectra and form factors: applications to models of Bose-Einstein condensates and metallic nanograins, J. Phys. A: Math. Gen., 2003, V.36, R63-R104, nlin.SI/0305049.
  11. Matthews M.R., Anderson B.P., Haljan P.C., Hall D.S., Holland M.J., Williams J.E., Wieman C.E., Cornell E.A., Watching a superfluid untwist itself: recurrence of Rabi oscillations in a Bose-Einstein condensate, Phys. Rev. Lett., 1999, V.83, 3358-3361, cond-mat/9906288.
  12. Milburn G.J., Corney J., Wright E.M., Walls D.F., Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential, Phys. Rev. A, 1997, V.55, 4318-4324.
  13. Ortiz G., Somma R., Dukelsky J., Rombouts S., Exactly-solvable models derived from a generalized Gaudin algebra, Nuclear Phys. B, 2005, V.707, 421-457, cond-mat/0407429.
  14. Pan F., Draayer J.P., Quantum critical behavior of two coupled Bose-Einstein condensates, Phys. Lett. A, 2005, V.339, 403-407, cond-mat/0410423.
  15. Tonel A.P., Links J., Foerster A., Quantum dynamics of a model for two Josephson-coupled Bose-Einstein condensates, J. Phys. A: Math. Gen., 2005, V.38, 1235-1245, quant-ph/0408161.
  16. Tonel A.P., Links J., Foerster A., Behaviour of the energy gap in a model of Josephson-coupled Bose-Einstein condensates, J. Phys. A: Math. Gen., 2005, V.38, 6879-6891, cond-mat/0412214.
  17. Ulyanov V.V., Zaslavskii O.B., New methods in the theory of quantum spin systems, Phys. Rep., 1992, V.216, 179-251.

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