Journal of Low Temperature Physics

, Volume 196, Issue 1–2, pp 140–146 | Cite as

Decay of Phase-Imprinted Dark Soliton in Bose–Einstein Condensate at Nonzero Temperature

  • Hiroki Ohya
  • Shohei WatabeEmail author
  • Tetsuro Nikuni


We study relaxation dynamics of dark soliton, created by a phase-imprinted method, in a two-dimensional trapped Bose–Einstein condensate at nonzero temperatures by using the projected Gross–Pitaevskii equation. At absolute zero temperature, a dark soliton is known to decay with a snake instability. At nonzero temperature, as expected, we find that this snake instability cannot be seen as clearly as in the absolute zero temperature case because of the presence of thermal fluctuations. We find that the energy dependence of the decay rate, defined by the half-life of the fidelity with respect to the phase-imprinted initial state, shows a power law decay and approaches a nonzero value in the large energy limit.


Bose–Einstein condensate Soliton Decay Nonzero temperature 



Authors thank T. Sato for discussing implementation of the PGPE. S.W. was supported by JSPS KAKENHI Grant No. JP16K17774. T.N. was supported by JSPS KAKENHI Grant No. JP16K05504.


  1. 1.
    S. Burger et al., Dark solitons in Bose–Einstein condensates. Phys. Rev. Lett. 83, 5198 (1999)CrossRefGoogle Scholar
  2. 2.
    J. Denschlag et al., Generating solitons by phase engineering of a Bose–Einstein condensate. Science 287, 97 (2000)CrossRefGoogle Scholar
  3. 3.
    C. Becker et al., Oscillations and interactions of dark and dark-bright solitons in Bose–Einstein condensates. Nat. Phys. 4, 496 (2008)CrossRefGoogle Scholar
  4. 4.
    J. Brand, W.P. Reinhardt, Solitonic vortices and the fundamental modes of the snake instability: possibility of observation in the gaseous Bose–Einstein condensate. Phys. Rev. A 65, 043612 (2002)CrossRefGoogle Scholar
  5. 5.
    A.V. Mamaev, M. Saffman, A.A. Zozulya, Propagation of dark stripe beams in nonlinear media: snake instability and creation of optical vortices. Phys. Rev. Lett. 76, 2262 (1996)CrossRefGoogle Scholar
  6. 6.
    D.L. Feder et al., Dark-soliton states of Bose–Einstein condensates in anisotropic traps. Phys. Rev. A 62, 053606 (2000)CrossRefGoogle Scholar
  7. 7.
    Z. Dutton et al., Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose–Einstein condensate. Science 293, 663 (2001)CrossRefGoogle Scholar
  8. 8.
    L. Pitaevskii, S. Stringer, Bose–Einstein Condensation (Clarendon Press, Oxford, 2003)zbMATHGoogle Scholar
  9. 9.
    B. Jackson, N.P. Proukakis, C.F. Barenghi, Dark-soliton dynamics in Bose–Einstein condensates at finite temperature. Phys. Rev. A 75, 051601 (2007)CrossRefGoogle Scholar
  10. 10.
    M.J. Davis, S.A. Morgan, K. Burnett, Simulations of Bose fields at finite temperature. Phys. Rev. Lett. 87, 160402 (2001)CrossRefGoogle Scholar
  11. 11.
    M.J. Davis, S.A. Morgan, K. Burnett, Simulations of thermal Bose fields in the classical limit. Phys. Rev. A 66, 053618 (2002)CrossRefGoogle Scholar
  12. 12.
    M.J. Davis, S.A. Morgan, Microcanonical temperature for a classical field: application to Bose–Einstein condensation. Phys. Rev. A 68, 053615 (2003)CrossRefGoogle Scholar
  13. 13.
    P.B. Blakie, M.J. Davis, Projected Gross–Pitaevskii equation for harmonically confined Bose gases at finite temperature. Phys. Rev. A 72, 063608 (2005)CrossRefGoogle Scholar
  14. 14.
    P.B. Blakie, M.J. Davis, Classical region of a trapped Bose gas. J. Phys. B At. Mol. Opt. Phys. 40, 2043 (2007)CrossRefGoogle Scholar
  15. 15.
    P.B. Blakie et al., Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques. Adv. Phys. 57, 363 (2008)CrossRefGoogle Scholar
  16. 16.
    P.B. Blakie, Numerical method for evolving the projected Gross–Pitaevskii equation. Phys. Rev. E 78, 026704 (2008)CrossRefGoogle Scholar
  17. 17.
    T. Sato, T. Suzuki, N. Kawashima, Finite-temperature transition in a quasi-2D Bose gas trapped in the harmonic potential. J. Phys. Conf. Ser. 150, 032094 (2009)CrossRefGoogle Scholar
  18. 18.
    H.H. Rugh, Microthermodynamic formalism. Phys. Rev. E 64, 055101(R) (2001)MathSciNetCrossRefGoogle Scholar
  19. 19.
    T. Sato et al., Validity of projected Gross–Pitaevskii simulation: comparison with quantum Monte Carlo. Phys. Rev. E 85, 050105(R) (2012)CrossRefGoogle Scholar
  20. 20.
    J. Sato et al., Exact relaxation dynamics of a localized many-body state in the 1D Bose gas. Phys. Rev. Lett. 108, 110401 (2012)CrossRefGoogle Scholar
  21. 21.
    Z. Hadzibabic et al., Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas. Nature 441, 1118 (2006)CrossRefGoogle Scholar
  22. 22.
    C.L. Hung et al., Observation of scale invariance and universality in two-dimensional Bose gases. Nature 470, 236 (2011)CrossRefGoogle Scholar
  23. 23.
    R.J. Fletcher et al., Connecting Berezinskii–Kosterlitz–Thouless and BEC phase transitions by tuning interactions in a trapped gas. Phys. Rev. Lett. 114, 255302 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Tokyo University of ScienceTokyoJapan

Personalised recommendations