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

SIGMA 3 (2007), 040, 14 pages      nlin.SI/0701030      https://doi.org/10.3842/SIGMA.2007.040
Contribution to the Vadim Kuznetsov Memorial Issue

q-Boson in Quantum Integrable Systems

Anjan Kundu
Saha Institute of Nuclear Physics, Theory Group & Centre for Applied Mathematics & Computational Science, 1/AF Bidhan Nagar, Calcutta 700 064, India

Received November 14, 2006, in final form January 15, 2007; Published online March 05, 2007

Abstract
q-bosonic realization of the underlying Yang-Baxter algebra is identified for a series of quantum integrable systems, including some new models like two-mode q-bosonic model leading to a coupled two-component derivative NLS model, wide range of q-deformed matter-radiation models, q-anyon model etc. Result on a new exactly solvable interacting anyon gas, linked to q-anyons on the lattice is reported.

Key words: quantum integrable systems; Yang-Baxter algebra; quantum group, q-bosonic integrable models; q-deformed matter-radiation models; q-anyon; derivative-δ-function anyon gas.

pdf (287 kb)   ps (184 kb)   tex (21 kb)

References

  1. Drinfel'd V.G., Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR 283 (1985), 1060-1064.
  2. Drinfel'd V.G., Quantum group, in Proc. Int. Cong. Mathematicians, Berkeley, 1986, Vol. 1, 798-820.
  3. Faddeev L.D., Algebraic aspects of Bethe ansatz, Internat. J. Modern Phys. A 10 (1995), 1845-1878, hep-th/9404013.
  4. MacFarlane A.J., On q-analogues of the quantum harmonic oscillator and the quantum group SU(2)q, J. Phys. A: Math. Gen. 22 (1989), 4581-4588.
  5. Biedenharn L.C., The quantum group SUq(2) and a q-analogue of the boson operators, J. Phys. A: Math. Gen. 22 (1989), L873-L878.
  6. Sun C.P., Fu H.C., The q-deformed boson realisation of the quantum group SU(n)q and its representations, J. Phys. A: Math. Gen. 22 (1989), L983-L986.
  7. Chaichian M., Ellinas D., Kulish P., Quantum algebra as the dynamical symmetry of the deformed Jaynes-Cummings model, Phys. Rev. Lett. 65 (1990), 980-983.
  8. Chang Z., Generalized Jaynes-Cummings model with an intensity-dependent coupling interacting with a quantum group-theoretic coherent state, Phys. Rev. A 47 (1993), 5017-5023.
  9. Basu-Mallick B., Kundu A., Single q-oscillator mode realisation of the quantum group through canonical bosonisation of SU(2) and q-oscillators, Modern Phys. Lett. A 6 (1991), 701-705.
  10. Bogoliubov N.M., Bullough R.K., A q-deformed completely integrable bose gas model, J. Phys. A: Math. Gen. 25 (1992), 4057-4071.
  11. Bonatsos D., Raychev P.P, Roussev R.P., Smirnov Yu.F., Description of rotational molecular spectra by the quantum algebra SUq(2), Chem. Phys. Lett. 175 (1990), 300-306.
  12. Raychev P.P., Roussev R.P., Smirnov Yu.F., The quantum algebra SUq(2) and rotational spectra of deformed nuclei, J. Phys. G 16 (1990), L137-L142.
  13. Chang Z., Yan H., Quantum group theoretic approach to vibrating and rotating diatomic molecules, Phys. Lett. A 158 (1991), 242-246.
  14. Kundu A., Ng J., Deformation of angular momentum and its application to the spectra of triatomic molecules, Phys. Lett. A 197 (1995), 221-226.
  15. Ghosh A., Mitra P., Kundu A., Multidimensional isotropic and anisotropic q-oscillator models, J. Phys. A: Math. Gen. 29 (1996), 115-124.
  16. Kundu A., Basu-Mallick B., Construction of integrable quantum lattice models through Sklyanin like algebras, Modern Phys. Lett. A 7 (1992), 61-69.
  17. Kundu A., Algebraic approach in unifying quantum integrable models, Phys. Rev. Lett. 82 (1999), 3936-3939, hep-th/9810220.
  18. Kundu A., Unifying approaches in integrable systems: quantum and statistical, ultralocal and non-ultralocal, in Classical & Quantum Integrable Systems, IOP, Bristol, 2003, 147-181, hep-th/0303263.
  19. Izergin A.G., Korepin V.E., Lattice versions of quantum field theory models in two dimensions, Nuclear Phys. B 205 (1982), 401-413.
  20. Kundu A., Basu-Mallick B., Classical and quantum integrability of a novel derivative NLS model related to quantum group structures, J. Math. Phys. 34 (1993), 1052-1062.
  21. Cheng Z., Chen S.X., Thermodynamics of a deformed Bose gas, J. Phys. A: Math. Gen. 35 (2002), 9731-9749, cond-mat/0205208.
  22. Basu-Mallick B., Kundu A., Hidden quantum group structure in a relativistic quantum integrable model, Phys. Lett. B 287 (1992), 149-153.
  23. Kulish P., Sklyanin E.K., Quantum spectral transform method. Recent development, in Integrable Quantum Field Theories, Lect. Notes in Phys., Vol. 151, Editor J. Hietarinta, Springer, Berlin, 1982, 61-119.
  24. Ablowitz M., Lectures on the inverse scattering transform, Studies in Appl. Math. 58 (1978) 17-94.
  25. Bullough R.K., Bogoliubov N.M., Pang G.D., Timonen J., Quantum repulsive Schrödinger equations and their superfluidity, in Special Issue "Solitons in Science & Engineering: Theory & Applications", Editor M. Lakshmanan, Chaos Solitons Fractals 5 (1995), 2639-2656.
  26. Jaynes E.T., Cummings F.W., Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE 51 (1963), 89-109.
  27. Buck B., Sukumar C.V., Exactly soluble model of atom-phonon coupling showing periodic decay and revival, Phys. Lett. A 81 (1981), 132-135.
  28. Blockley C.A., Walls D.F., Risken H., Quantum collapes and revivals in a quantized, Eur. Phys. Lett. 17 (1992), 509-514.
  29. Buzek V., The Jaynes-Cummings model with a q analogue of a coherent state, J. Modern Opt. 39 (1992), 949-959.
  30. Chang Z., Dynamics of generalized photonic fields interacting with atoms, Phys. Rev. A 46 (1992), 5865-5873.
  31. Bogoliubov N.M., Rybin A.V., Timonen J., An algebraic q-deformed model for bosons interacting with spin impurities, J. Phys. A: Math. Gen. 27 (1994), L363-L367.
  32. Bogoliubov N.M., Rybin A.V., Bullough R.K., Timonen J., The Maxwell-Bloch system on a lattice, Phys. Rev. A 52 (1995), 1487-1493.
  33. Vogel W., de Mitos Filho R., Nonlinear Jaynes-Cummings dynamics of a trapped ion, Phys. Rev. A 52 (1995), 4214-4217.
  34. Zheng H.S., Kuang L.M., Gao K.L., Jaynes-Cummings model dynamics in two trapped ions, quant-ph/0106020.
  35. Chang Z., Minimum-uncertainty states, a two-photon system, and quantum-group symmetry, Phys. Rev. A 46 (1992), 5860-5864.
  36. Chang Z., Quantum group and quantum symmetry, Phys. Rep. 262 (1995), 137-225.
  37. Bogoliubov N.M., Izergin A.G., Kitane N.A., Correlation function for a strongly correlated boson system, Nuclear Phys. B 516 (1998), 501-528, solv-int/9710002.
  38. Kundu A., Quantum integrability and Bethe ansatz solution for interacting matter-radiation systems, J. Phys. A: Math. Gen. 37 (2004), L281-L287, quant-ph/0307102.
  39. Kundu A., Exact solution of double d function Bose gas through an interacting anyon gas, Phys. Rev. Lett. 83 (1999), 1275-1278, hep-th/9811247.
  40. Batchelor M.T., Guan X.W., Oelkers N., 1D interacting anyon gas: low energy properties and Haldane exclusion statistics, Phys. Rev. Lett. 96 (2006), 210402, 4 pages, cond-mat/0603643.
  41. Kundu A., Generation of a quantum integrable class of discrete-time or relativistic periodic Toda chain, Phys. Lett. A 190 (1994), 79-84, hep-th/9403001.
  42. Kuznetsov V.B., Tsiganov A.V., Separation of variables for the quantum relativistic Toda lattice, hep-th/9402111.
  43. Faddeev L.D., Quantum completely integrable models in field theory, Sov. Sci. Rev. Math. Phys. C 1 (1980), 107-155.
  44. Takhtajan L., Introduction to algebraic Bethe ansatz, in Exactly Solvable Problems in Condensed Matter and Relativistic Field Theory, Editors B.S. Shatry, S.S. Jha and V. Singh, Springer Verlag, 1985, 175-219.
  45. Shnirman A.G., Malomed B.A., Ben-Jacob E., Nonpertubative studies of a quantum higher order nonlinear Schrödinger model, Phys. Rev. A 50 (1994), 3453-3463.

Previous article   Next article   Contents of Volume 3 (2007)