Frontiers of Physics

, Volume 10, Issue 4, pp 1–7 | Cite as

Quadrupolar matter-wave soliton in two-dimensional free space

  • Jia-Sheng Huang
  • Xun-Da Jiang
  • Huai-Yu Chen
  • Zhi-Wei Fan
  • Wei Pang
  • Yong-Yao LiEmail author
Research Article


We study two-dimensional (2D) matter-wave solitons in the mean-field models formed by electric quadrupole particles with long-range quadrupole–quadrupole interaction (QQI) in 2D free space. The existence of 2D matter-wave solitons in the free space was predicted using the 2D Gross–Pitaevskii Equation (GPE). We find that the QQI solitons have a higher mass (smaller size and higher intensity) and stronger anisotropy than the dipole–dipole interaction (DDI) solitons under the same environmental parameters. Anisotropic soliton–soliton interaction between two identical QQI solitons in 2D free space is studied. Moreover, stable anisotropic dipole solitons are observed, to our knowledge, for the first time in 2D free space under anisotropic nonlocal cubic nonlinearity.


2D matter-wave solitons quadrupole–quadrupole interaction anisotropy soliton–soliton interaction dipole solitons 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. P. Pitaevskii and S. Sandro, Bose–Einstein Condensation, No. 116, Oxford University Press, 2003zbMATHGoogle Scholar
  2. 2.
    S. Burger, K. Bongs, S. Dettmer, W. Ertmer, and K. Sengstock, Dark solitons in Bose-Einstein condensates, Phys. Rev. Lett. 83(25), 5198 (1999)CrossRefADSGoogle Scholar
  3. 3.
    S. Song, L. Wen, C. Liu, S. Gou, and W. Liu, Ground states, solitons and spin textures in spin-1 Bose–Einstein condensates, Front. Phys. 8(3), 302 (2013)CrossRefGoogle Scholar
  4. 4.
    B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, Watching dark solitons decay into vortex rings in a Bose–Einstein condensate, Phys. Rev. Lett. 86(14), 2926 (2001)CrossRefADSGoogle Scholar
  5. 5.
    J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, Observation of vortex lattices in Bose–Einstein condensates, Science 292(5516), 476 (2001)CrossRefADSGoogle Scholar
  6. 6.
    F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, Josephson junction arrays with Bose-Einstein condensates, Science 293(5531), 843 (2001)CrossRefADSGoogle Scholar
  7. 7.
    C. Lee, W. Hai, L. Shi, X. Zhu, and K. Gao, Chaotic and frequency-locked atomic population oscillations between two coupled Bose-Einstein condensates, Phys. Rev. A 64(5), 053604 (2001)CrossRefADSGoogle Scholar
  8. 8.
    C. Lee, J. Huang, H. Deng, H. Dai, and J. Xu, Nonlinear quantum interferometry with Bose condensed atoms, Front. Phys. 7(1), 109 (2012)CrossRefGoogle Scholar
  9. 9.
    C. Lee, Universality and anomalous mean-field breakdown of symmetry-breaking transitions in a coupled two-component Bose–Einstein condensate, Phys. Rev. Lett. 102(7), 070401 (2009)CrossRefADSGoogle Scholar
  10. 10.
    H. Zheng and Q. Gu, Dynamics of Bose–Einstein condensates in a one-dimensional optical lattice with double-well potential, Front. Phys. 8(4), 375 (2013)CrossRefGoogle Scholar
  11. 11.
    Y. Li, J. Liu, W. Pang, and B. A. Malomed, Symmetry breaking in dipolar matter-wave solitons in dual-core couplers, Phys. Rev. A 87(1), 013604 (2013)CrossRefADSGoogle Scholar
  12. 12.
    Y. Li, W. Pang, and B. A. Malomed, Nonlinear modes and symmetry breaking in rotating double-well potentials, Phys. Rev. A 86(2), 023832 (2012)CrossRefADSGoogle Scholar
  13. 13.
    J. Denschlag, Generating solitons by phase engineering of a Bose–Einstein condensate, Science 287, 97 (2000)CrossRefADSGoogle Scholar
  14. 14.
    Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, Formation of a matter-wave bright soliton, Science 296, 1290 (2002)CrossRefADSGoogle Scholar
  15. 15.
    Ph. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen, and B. J. Verhaar, Observation of a Feshbach resonance in cold atom scattering, Phys. Rev. Lett. 81, 69 (1998)CrossRefADSGoogle Scholar
  16. 16.
    M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, Tuning the scattering length with an optically induced Feshbach resonance, Phys. Rev. Lett. 93(1), 123001 (2004)CrossRefADSGoogle Scholar
  17. 17.
    X. Zhang, X. Hu, D. Wang, X. Liu, and W. Liu, Dynamics of Bose-Einstein condensates near Feshbach resonance in external potential, Front. Phys. China 6(1), 46 (2011)ADSGoogle Scholar
  18. 18.
    S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn and W. Ketterle, Observation of Feshbach resonances in a Bose–Einstein condensate, Nature 293(6672), 151 (1998)ADSGoogle Scholar
  19. 19.
    P. Courteille, R. S. Freeland, D. J. He inzen, F. A. van Abeelen and B. J. Verhaar, Observation of Feshbach resonances in cold atom scattering, Phys. Rev. Lett. 81, 69 (1998)CrossRefADSGoogle Scholar
  20. 20.
    J. Huang, H. Li, X. Zhang, and Y. Li, Transmission, reflection, scattering, and trapping of traveling discrete solitons by Ƈ and Ʋ point defects, Front. Phys. 10(2), 104201 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Z. Chen, J. Huang, J. Chai, X. Zhang, Y. Li, and B. A. Malomed, Discrete solitons in self-defocusing systems with PT -symmetric defects, Phys. Rev. A 91(5), 053821 (2015)CrossRefADSGoogle Scholar
  22. 22.
    M. Saha, A. K. Sarma, Modulation instability in nonlinear metamaterials induced by cubic–quintic nonlinearities and higher order dispersive effects, Opt. Commun. 291, 321 (2013)CrossRefADSGoogle Scholar
  23. 23.
    J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature 422(6928), 147 (2003)CrossRefADSGoogle Scholar
  24. 24.
    G. Chen, Z. Hong, and Z. Mai, Two-dimensional discrete Anderson location in waveguide matrix, J. Nonlinear Optic. Phys. Mat. 23(03), 1450033 (2014)CrossRefADSGoogle Scholar
  25. 25.
    X. Zhang, J. Chai, D. Ou, and Y. Li, Antisymmetry breaking of discrete dipole gap solitons induced by a phase-slip defect, Mod. Phys. Lett. B 28(12), 1450097 (2014)CrossRefADSGoogle Scholar
  26. 26.
    N. K. Efremidis and D. N. Christodoulides, Lattice solitons in Bose–Einstein condensates, Phys. Rev. A 67(6), 063608 (2003).CrossRefADSGoogle Scholar
  27. 27.
    N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Two-dimensional optical lattice solitons, Phys. Rev. Lett. 91(21), 213906 (2003)CrossRefADSGoogle Scholar
  28. 28.
    W. Pang, J. Wu, Z. Yuan, Y. Liu, and G. Chen, Lattice solitons in optical lattice controlled by electromagnetically induced transparency, J. Phys. Soc. Jpn. 80(11), 113401 (2011)CrossRefADSGoogle Scholar
  29. 29.
    J. T. Cole and Z. H. Musslimani, Band gaps and lattice solitons for the higher-order nonlinear Schrödinger equation with a periodic potential, Phys. Rev. A 90(1), 013815 (2014)CrossRefADSGoogle Scholar
  30. 30.
    X. Gan, P. Zhang, S. Liu, F. Xiao, and J. Zhao, Beam steering and topological transformations driven by interactions between a discrete vortex soliton and a discrete fundamental soliton, Phys. Rev. A 89(1), 013844 (2014)CrossRefADSGoogle Scholar
  31. 31.
    G. Chen, H. Huang, and M. Wu, Solitary vortices in twodimensional waveguide matrix, J. Nonlinear Optic. Phys. Mat. 24(01), 1550012 (2015)CrossRefADSGoogle Scholar
  32. 32.
    G. Chen, H. Huang, and M. Wu, Discrete vortices on anisotropic lattices, Front. Phys. 10, 104206 (2015)Google Scholar
  33. 33.
    R. Heidemann, U. Raitzsch, V. Bendkowsky, B. Butscher, R. Low, and T. Pfau, Rydberg excitation of Bose–Einstein condensates, Phys. Rev. Lett. 100(3), 033601 (2008)CrossRefADSGoogle Scholar
  34. 34.
    M. Viteau, M. G. Bason, J. Radogostowicz, N. Malossi, D. Ciampini, O. Morsch, and E. Arimondo, Rydberg excitations in Bose–Einstein condensates in quasi-one-dimensional potentials and optical lattices, Phys. Rev. Lett. 107(6), 060402 (2011)CrossRefADSGoogle Scholar
  35. 35.
    S. Giovanazzi, A. Gorlitz, and T. Pfau, Tuning the dipolar interaction in quantum gases, Phys. Rev. Lett. 89(13), 130401 (2002)CrossRefADSGoogle Scholar
  36. 36.
    P. Pedri and L. Santos, Two-dimensional bright solitons in dipolar Bose–Einstein condensates, Phys. Rev. Lett. 95(20), 200404 (2005)CrossRefADSGoogle Scholar
  37. 37.
    T. Koch, T. Lahaye, J. Metz, B. Frohlich, A. Griesmaier, and T. Pfau, Stabilization of a purely dipolar quantum gas against collapse, Nat. Phys. 4(3), 218 (2008)CrossRefGoogle Scholar
  38. 38.
    I. Tikhonenkov, B. A. Malomed, and A. Vardi, Anisotropic solitons in dipolar Bose–Einstein condensates, Phys. Rev. Lett. 100(9), 090406 (2008)CrossRefADSGoogle Scholar
  39. 39.
    Y. Li, J. Liu, W. Pang, and B. A. Malomed, Matter-wave solitons supported by field-induced dipole–dipole repulsion with spatially modulated strength, Phys. Rev. A 88(5), 053630 (2013).CrossRefADSGoogle Scholar
  40. 40.
    Z. Luo, Y. Li, W. Pang, and Y. Liu, Dipolar matter-wave soliton in one-dimensional optical lattice with tunable local and nonlocal nonlinearities, J. Phys. Soc. Jpn. 82(9), 094401 (2013)CrossRefADSGoogle Scholar
  41. 41.
    S. E. Pollack, D. Dries, M. Junker, Y. P. Chen, T. A. Corcovilos, and R. G. Hulet, Extreme tunability of interactions in a Li7 Bose–Einstein condensate, Phys. Rev. Lett. 102(9), 090402 (2009)CrossRefADSGoogle Scholar
  42. 42.
    L. P. Pitaevskii and A. Stringari, Bose–Einstein condensation, Clarendon Press, Oxford, 2003zbMATHGoogle Scholar
  43. 43.
    Y. Li, J. Liu, W. Pang, and B. A. Malomed, Lattice solitons with quadrupolar intersite interactions, Phys. Rev. A 88(6), 063635 (2013)CrossRefADSGoogle Scholar
  44. 44.
    L. D. Landau and E. M. Lifshitz, The Field Theory, Nauka-Publishers, Moscow, 1988Google Scholar
  45. 45.
    H. Sakaguchi and B. A. Malomed, Solitons in combined linear and nonlinear lattice potentials, Phys. Rev. A 81(1), 013624 (2010)CrossRefADSGoogle Scholar
  46. 46.
    X. Zhang, J. Chai, J. Huang, Z. Chen, Y. Li, and B. A. Malomed, Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear ƤƬ -symmetric defect, Opt. Exp. 22(11), 13927 (2014)CrossRefADSGoogle Scholar
  47. 47.
    M. L. Chiofalo, S. Succi, and M. P. Tosi, Ground state of trapped interacting Bose–Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E 62(5), 7438 (2000)CrossRefADSGoogle Scholar
  48. 48.
    J. Yang and T. I. Lakoba, Accelerated imaginary-time evolution methods for the computation of solitary waves, Stud. Appl. Math. 120(3), 265 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    J. Yang and T. I. Lakoba, Universally-convergent squaredoperator iteration methods for solitary waves in general nonlinear wave equations, Stud. Appl. Math. 118(2), 153 (2007)MathSciNetCrossRefGoogle Scholar
  50. 50.
    G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, 2007Google Scholar
  51. 51.
    Z. Chen, J. Liu, S. Fu, Y. Li, and B. A. Malomed, Discrete solitons and vortices on two-dimensional lattices of ƤƬ -symmetric couplers, Opt. Exp. 22(24), 029679 (2014)CrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Jia-Sheng Huang
    • 1
  • Xun-Da Jiang
    • 1
  • Huai-Yu Chen
    • 1
  • Zhi-Wei Fan
    • 1
  • Wei Pang
    • 2
  • Yong-Yao Li
    • 1
    Email author
  1. 1.Department of Applied Physics, College of Electronic EngineeringSouth China Agricultural UniversityGuangzhouChina
  2. 2.Department of Experiment TeachingGuangdong University of TechnologyGuangzhouChina

Personalised recommendations