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Nonlinear Dynamics

, Volume 98, Issue 3, pp 1905–1918 | Cite as

Modulational instability in addition to discrete breathers in 2D quantum ultracold atoms loaded in optical lattices

  • Z. I. DjoufackEmail author
  • F. Fotsa-Ngaffo
  • E. Tala-Tebue
  • E. Fendzi-Donfack
  • F. Kapche-Tagne
Original paper
  • 71 Downloads

Abstract

The modulational instability associated with discrete breathers in 2D quantum ultracold atoms is studied by using the Glauber’s coherent state combined with a semi-discrete approximation and multiple-scale methods. The linear stability analysis exhibits an intriguing threshold amplitude and instability regions associated with modulational growth rate. In addition, we demonstrate a coexistence of two bright intrinsic localized modes namely, the radial symmetric and bilateral symmetric modes, at the center and at the edges of the Brillouin zone, respectively, by alternating the on-site parameter interaction. Numerical investigations reveal a good agreement with the theoretical analysis.

Keywords

Discrete breathers Modulational instability Optical lattice Quantum ultracold atoms 

Notes

Acknowledgements

Z. I. Djoufack is grateful to African Institute for Mathematical Sciences (AIMS) South Africa for research facilities and computer services.

Compliance with ethical standards

Conflict of interest

All authors declare that they have no conflict of interest.

References

  1. 1.
    Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A.: Observation of Bose–Einstein condensation in a dilute atomic vapor. Science 269, 198 (1995)Google Scholar
  2. 2.
    DeMarco, B., Jin, D.S.: Onset of Fermi degeneracy in a trapped atomic gas. Science 285, 1703 (1999)Google Scholar
  3. 3.
    Bloch, I., Dalibard, J., Zwerger, W.: Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885 (2008)Google Scholar
  4. 4.
    Giuliano, D., Rossini, D., Sodano, P., Trombettoni, A.: XXZ spin-1/2 representation of a finite-U Bose–Hubbard chain at half-integer filling. Phys. Rev. B 87, 035104 (2013)Google Scholar
  5. 5.
    Mandel, O., Esslinger, T., Hansch, T.E., Greiner, M., Bloch, I.: Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39 (2002)Google Scholar
  6. 6.
    Fisher, M.P.A., Weichman, P.B., Grinstein, G., Fisher, D.S.: Boson localization and the superfluid–insulator transition. Phys. Rev. B 40, 546 (1989)Google Scholar
  7. 7.
    Bloch, I.: Ultracold quantum gases in optical lattices. Nat. Phys. 1, 23 (2005)Google Scholar
  8. 8.
    Jaksch, D., Zoller, P.: The cold atom Hubbard toolbox. Ann. Phys. 315, 52 (2005)zbMATHGoogle Scholar
  9. 9.
    Bloch, I.: Ultracold quantum gases in optical lattices. Nat. Phys. 1, 23 (2005)Google Scholar
  10. 10.
    Romanenko, V.I., Udovitskaya, YeG, Romanenko, A.V., Yatsenko, L.P.: Cooling and trapping of atoms and molecules by counterpropagating pulse trains. Phys. Rev. A 90, 053421 (2014)Google Scholar
  11. 11.
    Jaksch, D., Bruder, C., Cirac, J.I., Gardiner, C.W., Zoller, P.: Cold bosonic atoms in optical lattices. Phys. Rev. Lett. 81, 3108 (1998)Google Scholar
  12. 12.
    Morsch, O., Oberthaler, M.: Dynamics of Bose–Einstein condensates in optical lattices. Rev. Mod. Phys. 78, 179 (2006)Google Scholar
  13. 13.
    Brennen, G.K., Caves, C.M., Jessen, P.S., Deutsch, I.H.: Quantum logic gates in optical lattices. Phys. Rev. Lett. 82, 1060 (1999)Google Scholar
  14. 14.
    Scott, A.C., Eilbeck, J.C., Gilhøj, H.: Quantum lattice solitons. Physica D 78, 194 (1994)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Flach, S., Willis, C.R.: Discretes breathers. Phys. Rep. 295, 181 (1998)MathSciNetGoogle Scholar
  16. 16.
    Flach, S., Gorbach, A.V.: Discrete breathers-advances in theory and applications. Phys. Rep. 467, 1 (2008)zbMATHGoogle Scholar
  17. 17.
    MacKay, R.S., Aubry, S.: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7, 1623 (1994)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Flach, S., Kladko, K., MacKay, R.S.: Energy thresholds for discrete breathers in one-, two-, and three-dimensional lattices. Phys. Rev. Lett. 78(7), 1207 (1997)Google Scholar
  19. 19.
    Hennig, H., Dorignac, J., Campbell, D.K.: Transfer of Bose–Einstein condensates through discrete breathers in an optical lattice. Phys. Rev. A 82, 053604 (2010)Google Scholar
  20. 20.
    Hennig, H., Fleischmann, R.: Nature of self-localization of Bose–Einstein condensates in optical lattices. Phys. Rev. A 87, 033605 (2013)Google Scholar
  21. 21.
    Tang, B.: Quantum two-breathers formed by ultracold bosonic atoms in optical lattices. Int. J. Theor. Phys. (2016).  https://doi.org/10.1007/s10773-015-2903-9 CrossRefzbMATHGoogle Scholar
  22. 22.
    Djoufack, Z.I., Kenfack-Jiotsa, A., Nguenang, J.-P.: Quantum signature of breathers in 1D ultracold bosons in optical lattices involving next-nearest neighbor interactions. Int. J. Mod. Phys. B 31, 1750140 (2017)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Sarkar, R., Dey, B.: Energy localization and transport in two-dimensional Fermi–Pasta–Ulam lattice. Phys. Rev. E 76, 016605 (2007)MathSciNetGoogle Scholar
  24. 24.
    Yang, C., Zhou, Q., Triki, H., Mirzazadeh, M., Ekici, M., Liu, W.-J., Biswas, A., Belic, M.: Bright soliton interactions in a (2+1)-dimensional fourth-order variable-coefficient nonlinear Schrödinger equation for the Heisenberg ferromagnetic spin chain. Nonlinear Dyn. (2018).  https://doi.org/10.1007/s11071-018-4609-z CrossRefGoogle Scholar
  25. 25.
    Tang, G., Wang, S., Wang, G.: Solitons and complexitons solutions of an integrable model of (2+1)-dimensional Heisenberg ferromagnetic spin chain. Nonlinear Dyn. (2017).  https://doi.org/10.1007/s11071-017-3379-3 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wu, H.-Y., Jiang, L.-H.: Spatiotemporal bright and dark vector multipole and vortex solitons for coupled nonlinear Schrödinger equation with spatially modulated quintic nonlinearity. Nonlinear Dyn. (2017).  https://doi.org/10.1007/s11071-017-3735-3 CrossRefGoogle Scholar
  27. 27.
    Sun, H.-Q., Chen, A.-H.: Interactional solutions of a lump and a solitary wave for two higher-dimensional equations. Nonlinear Dyn. (2018).  https://doi.org/10.1007/s11071-018-4454-0 CrossRefGoogle Scholar
  28. 28.
    Darvishi, M.T., Kavitha, L., Najafi, M., Senthil Kumar, V.: Elastic collision of mobile solitons of a (3+1)-dimensional soliton equation. Nonlinear Dyn. (2016).  https://doi.org/10.1007/s11071-016-2920-0 CrossRefGoogle Scholar
  29. 29.
    Gadzhimuradov, T.A.: Envelope solitons in a nonlinear string with mirror nonlocality. Nonlinear Dyn. (2019).  https://doi.org/10.1007/s11071-019-04896-9 CrossRefGoogle Scholar
  30. 30.
    Zhang, Y., Liu, Y.: Breather and lump solutions for nonlocal Davey–Stewartson II equation. Nonlinear Dyn. (2019).  https://doi.org/10.1007/s11071-019-04777-1 CrossRefGoogle Scholar
  31. 31.
    Tan, W., Dai, Z.-D., Yin, Z.-Y.: Dynamics of multi-breathers, N-solitons and M-lump solutions in the (2+1)-dimensional KdV equation. Nonlinear Dyn. (2019).  https://doi.org/10.1007/s11071-019-04873-2 CrossRefGoogle Scholar
  32. 32.
    Lai, R., Sieves, A.J.: Modulational instability of nonlinear spin waves in easy-axis antiferromagnetic chains. Phys. Rev. B 57, 3433 (1998)Google Scholar
  33. 33.
    Stockhofe, J., Schmelcher, P.: Modulational instability and localized breather modes in the discrete nonlinear Schrödinger equation with helicoidal hopping. Physica D 328, 9 (2016)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Tabi, C.B., Mohamadou, A., Kofane, T.C.: Modulational instability in the anharmonic Peyrard–Bishop model of DNA. Eur. Phys. J. B 74, 151 (2010)Google Scholar
  35. 35.
    Baronio, F., Chen, S., Grelu, P., Wabnitz, S., Conforti, M.: Baseband modulation instability as the origin of rogue waves. Phys. Rev. A 91, 033804 (2015)Google Scholar
  36. 36.
    Gor, G., Macrì, T., Trombettoni, A.: Modulational instabilities in lattices with powerlaw hoppings and interactions. Phys. Rev. E 87, 032905 (2013)Google Scholar
  37. 37.
    Wang, L., Zhu, Y.-J., Qi, F.-H., Li, M., Guo, R.: Modulational instability, higher-order localized wave structures, and nonlinear wave interactions for a nonautonomous Lenells–Fokas equation in inhomogeneous fibers. Chaos 25, 063111 (2015)MathSciNetGoogle Scholar
  38. 38.
    Wang, L., Zhang, J.-H., Wang, Z.-Q., Liu, C., Li, M., Qi, F.-H., Guo, R.: Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation. Phys. Rev. E 93, 012214 (2016)MathSciNetGoogle Scholar
  39. 39.
    Wang, L., Zhang, J.-H., Liu, C., Li, M., Qi, F.-H.: Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects. Phys. Rev. E 93, 062217 (2016)MathSciNetGoogle Scholar
  40. 40.
    Kivshar, Y.S., Peyrard, M.: Modulational instabilities in discrete lattices. Phys. Rev. A 46, 3198 (1992)Google Scholar
  41. 41.
    Kivshar, Y.S.: Localized modes in a chain with nonlinear on-site potential. Phys. Lett. A 173, 172 (1993)Google Scholar
  42. 42.
    Guo, D., Tian, S.-F., Zhang, T.-T., Li, J.: Modulation instability analysis and soliton solutions of an integrable coupled nonlinear Schrödinger system. Nonlinear Dyn. (2018).  https://doi.org/10.1007/s11071-018-4522-5 CrossRefGoogle Scholar
  43. 43.
    Hao, H.-Q., Guo, R., Zhang, J.-W.: Modulation instability, conservation laws and soliton solutions for an inhomogeneous discrete nonlinear Schrödinger equation. Nonlinear Dyn. (2017).  https://doi.org/10.1007/s11071-017-3333-4 CrossRefzbMATHGoogle Scholar
  44. 44.
    Asaoka, R., Tsuchiura, H., Yamashita, M., Toga, Y.: Density modulations associated with the dynamical instability in the Bose–Hubbard model. J. Phys. Soc. Jpn. 83, 124001 (2014) Google Scholar
  45. 45.
    Asaoka, R., Tsuchiura, H., Yamashita, M., Toga, Y.: Dynamical instability in the Bose–Hubbard model. Phys. Rev. A 93, 013628 (2016)Google Scholar
  46. 46.
    Hennig, H., Dorignac, J., Campbell, D.K.: Transfer of Bose–Einstein condensates through discrete breathers in an optical lattice. Phys. Rev. A 82, 053604 (2010)Google Scholar
  47. 47.
    Zhou, Q., Liu, S.: Dark optical solitons in quadratic nonlinear media with spatio-temporal dispersion. Nonlinear Dyn. (2015).  https://doi.org/10.1007/s11071-015-2023-3 MathSciNetCrossRefGoogle Scholar
  48. 48.
    Zhang, T., Li, J.: Exact solitons, periodic peakons and compactons in an optical soliton model. Nonlinear Dyn. (2017).  https://doi.org/10.1007/s11071-017-3950-y CrossRefzbMATHGoogle Scholar
  49. 49.
    Djoufack, Z.I., Tala-Tebue, E., Fotsa-Ngaffo, F., Djimeli Tsajio, A.B., Kapche-Tagne, F.: Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving next-nearest neighbor interactions. Optik 164, 575 (2018)Google Scholar
  50. 50.
    Toda, M.: Waves in nonlinear lattice. Prog. Theor. Phys. Suppl. 45, 174–200 (1970)Google Scholar
  51. 51.
    Burlakov, V.M., Kiselev, S.A., Pyrkov, V.N.: Computer simulation of intrinsic localized modes in one-dimensional and two-dimensional anharmonic lattices. Phys. Rev. B 42, 8 (1990)Google Scholar
  52. 52.
    Pouget, J., Remoissenet, M., Tamga, J.M.: Energy self-localization and gap local pulses in a two-dimensional nonlinear lattice. Phys. Rev. B 47, 22 (1993)zbMATHGoogle Scholar
  53. 53.
    Flach, S., Kaldko, K., Willis, C.R.: localized excitations in 2D Hamiltonian lattices. Phys. Rev. B 50, 3 (1994)Google Scholar
  54. 54.
    Flach, S., Kaldko, K., Takeno, S.: Acoustic breathers in two-dimensional lattices. Phys. Rev. Lett. 79, 24 (1994)Google Scholar
  55. 55.
    Marin, J.L., Eilbeck, J.C., Russell, E.M.: Localized moving breathers in a 2D hexagonal lattice. Phys. Lett. A 248, 225–229 (1998)Google Scholar
  56. 56.
    Marin, J.L., Russell, E.M., Eilbeck, J.C.: Breathers in cuprate-like lattices. Phys. Lett. A 281, 21 (2001)zbMATHGoogle Scholar
  57. 57.
    Butt, I.A., Wattis, J.A.D.: Discrete breathers in a two-dimensional hexagonal Fermi–Pasta–Ulam lattice. J. Phys. A Math. Theor. 40, 1239 (2007)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Wattis, J.A.D., James, L.M.: Discrete breathers in honeycomb Fermi–Pasta–Ulam lattices. J. Phys. A Math. Theor. 47, 345101 (2014)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Wang, W., Liu, L.: Solitary waves in two-dimensional nonlinear lattices. Acta Mech. 228, 3155–3171 (2017)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Zaera, R., Vila, J., Fernandez-Saez, J., Ruzzene, M.: Propagation of solitons in a two-dimensional nonlinear square lattice. Int. J. Non Linear Mech. (2018).  https://doi.org/10.1016/j.ijnonlinmec.2018.08.002 CrossRefGoogle Scholar
  61. 61.
    Alba, V., Haque, M., Andreas, M.: Lauchli: entanglement spectrum of the two-dimensional Bose–Hubbard model. PRL 110, 260403 (2013)Google Scholar
  62. 62.
    Schonmeier-Kromer, J., Pollet, L.: Ground-state phase diagram of the two-dimensional Bose–Hubbard model with anisotropic hopping. Phys. Rev. A 89, 023605 (2014)Google Scholar
  63. 63.
    Glauber, R.J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131, 2766 (1963)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Smith, H.: Introduction to Quantum Mechanics, pp. 108–109. World Scientific, Singapore (1991)Google Scholar
  65. 65.
    Abdullaev, FKh, Bouketir, A., Messikh, A., Umarova, B.A.: Modulational instability and discrete breathers in the discrete cubic quintic nonlinear Schrödinger equation. Physica D 232, 54 (2007)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Daumont, I., Dauxois, T., Peyrard, M.: Modulational instability: first step towards energy localization in nonlinear lattices. Nonlinearity 10, 617 (1997)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Dang Koko, A., Tabi, C.B., Ekobena Fouda, H.P., Mohamadou, A., Kofane, T.C.: Nonlinear charge transport in the helicoidal DNA molecule. Chaos 22, 043110 (2012)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Mohamadou, A., Kofané, T.C.: Modulational instability and pattern formation in discrete dissipative systems. Phys. Rev. E 73, 046607 (2006)Google Scholar
  69. 69.
    Tabi, C.B., Mohamadou, A., Kofané, T.C.: Formation of localized structures in the Peyrard–Bishop–Dauxois model. J. Phys. Condens. Matter 20, 415104 (2008)Google Scholar
  70. 70.
    Tabi, C.B., Mohamadou, A., Kofané, T.C.: Modulation instability and pattern formation in damped molecular systems. J. Comput. Theor. Nanosci. 6, 583 (2009)Google Scholar
  71. 71.
    Remoissenet, M.: Low-amplitude breather and envelope solitons in quasi-one-dimensional physical models. Phys. Rev. B 33, 2386 (1986)Google Scholar
  72. 72.
    Schneider, G., Wayne, C.E.: Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi–Pasta–Ulam model. In: International Conference on Differential Equations, World Scientific, River Edge, NJ, pp. 390–404 (2000)Google Scholar
  73. 73.
    Remoissenet, M.: Waves Called Solitons, Concepts and Experiments. Springer, Berlin (1994)zbMATHGoogle Scholar
  74. 74.
    Butt, A.I., Wattis, J.A.D.: Discrete breathers in a two-dimensional Fermi–Pasta–Ulam lattice. J. Phys. A Math. Gen. 39, 4955 (2006)MathSciNetzbMATHGoogle Scholar
  75. 75.
    Fibich, G., Papanicolaou, G.: Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension SIAM. J. Appl. Math. 60, 183 (1999)zbMATHGoogle Scholar
  76. 76.
    Kuznetsov, E.A., Rubenchik, A.M., Zakharov, V.E.: Soliton stabilty in plamas and hydrodynamics. Phys. Rep. 142(3), 103 (1986)MathSciNetGoogle Scholar
  77. 77.
    Davydova, T.A., Yakimenko, A.I., Zaliznyak, YuA: Two-dimensional solitons and vortices in normal and anomalous dispersive media. Phys. Rev. E 67, 026402 (2003)MathSciNetGoogle Scholar
  78. 78.
    Berge, L., Rasmussen, J.J.: Multisplitting and collapse of self-focusing anisotropic beams in normal/anomalous dispersive media. Phys. Plasmas 3(3), 824 (1996)Google Scholar
  79. 79.
    Stenzel, R.L.: Filamentation instability of a large amplitude whistler wave. Phys. Fluids 19(6), 865 (1976)Google Scholar
  80. 80.
    Balmashno, A.A.: On the self-focusing of whistler waves in a radial. Phys. Lett. A 79(5–6), 402 (1980)Google Scholar
  81. 81.
    Kivshar, YuS, Pelinovsky, D.E.: Self-focusing and transverse instabilities of solitary waves. Phys. Rep. 331, 117 (2000)MathSciNetGoogle Scholar
  82. 82.
    Tang, B., Li, D.-J., Tang, Y.: Spin discrete breathers in two-dimensional square anisotropic ferromagnets. Phys. Scr. 89, 095208 (2014)Google Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  • Z. I. Djoufack
    • 1
    • 2
    • 3
    Email author
  • F. Fotsa-Ngaffo
    • 4
    • 5
  • E. Tala-Tebue
    • 1
  • E. Fendzi-Donfack
    • 6
  • F. Kapche-Tagne
    • 1
  1. 1.Laboratoire d’Automatique et Informatique Apliquée (LAIA), Department of Telecommunication and Network EngineeringFotso Victor University Institute of TechnologyBandjounCameroon
  2. 2.Université de DschangDschangCameroon
  3. 3.African Institute for Mathematical SciencesCape TownSouth Africa
  4. 4.Institute of Wood TechnologiesUniversity of Yaounde IMbalmayoCameroon
  5. 5.Department of Physics, Faculty of ScienceUniversity of BueaBueaCameroon
  6. 6.Pure Physics Laboratory: Group of Nonlinear Physics and Complex Systems, Department of PhysicsUniversity of DoualaDoualaCameroon

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