Stable and Unstable Vortex Knots in a Trapped Bose Condensate

Order, Disorder, and Phase Transition in Condensed System
  • 3 Downloads

Abstract

The dynamics of a quantum vortex toric knot TP,Q and other analogous knots in an atomic Bose condensate at zero temperature in the Thomas–Fermi regime is considered in the hydrodynamic approximation. The condensate has a spatially inhomogeneous equilibrium density profile ρ(z, r) due to the action of an external axisymmetric potential. It is assumed that z*= 0, r*= 1 is the point of maximum of function rρ(z, r), so that δ(rρ) ≈ –(α–)z2/2–(α + )(δr)2/2 for small z and δr. The geometrical configuration of a knot in the cylindrical coordinates is determined by a complex 2πP-periodic function A(ϕ, t) = Z(ϕ, t) + i[R(ϕ, t))–1]. When |A| ≪ 1, the system can be described by relatively simple approximate equations for P rescaled functions \({W_n}(\varphi ) \propto A(2\pi n + \varphi ):i{W_{n,t}} = - ({W_{n,\varphi \varphi }} + \alpha {W_n} - \in W_n^*)/2 - \sum\nolimits_{j \ne n} {1/(W_n^* - W_j^*)} \). For = 0, examples of stable solutions of type W n = θ n (ϕ–γt)exp(–iωt) with a nontrivial topology are found numerically for P = 3. In addition, the dynamics of various unsteady knots with P = 3 is modeled, and the tendency to the formation of a singularity over a finite time interval is observed in some cases. For P = 2 and small ≠ 0, configurations of type W0W1B0exp(iζ) + C(B0, α)exp(–iζ) + D(B0, α)exp(3iζ), where B0 > 0 is an arbitrary constant, ζ = k0ϕ–Ω0t + ζ0, k0 = Q/2, and Ω0 = (–α)/2–2/B02, which rotate about the z axis, are investigated. Wide stability regions for such solutions are detected in the space of parameters (α, B0). In unstable zones, a vortex knot may return to a weakly excited state.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. L. Fetter, Rev. Mod. Phys. 81, 647 (2009).ADSCrossRefGoogle Scholar
  2. 2.
    A. A. Svidzinsky and A. L. Fetter, Phys. Rev. A 62, 063617 (2000).ADSCrossRefGoogle Scholar
  3. 3.
    A. L. Fetter and A. A. Svidzinsky, J. Phys.: Condens. Matter 13, R135 (2001).ADSGoogle Scholar
  4. 4.
    V. P. Ruban, Phys. Rev. E 64, 036305 (2001).ADSCrossRefGoogle Scholar
  5. 5.
    A. Aftalion and T. Riviere, Phys. Rev. A 64, 043611 (2001). eADSCrossRefGoogle Scholar
  6. 6.
    J. Garcia-Ripoll and V. Perez-Garcia, Phys. Rev. A 64, 053611 (2001).ADSCrossRefGoogle Scholar
  7. 7.
    J. R. Anglin, Phys. Rev. A 65, 063611 (2002).ADSCrossRefGoogle Scholar
  8. 8.
    A. Aftalion and R. L. Jerrard, Phys. Rev. A 66, 023611 (2002).ADSCrossRefGoogle Scholar
  9. 9.
    P. Rosenbusch, V. Bretin, and J. Dalibard, Phys. Rev. Lett. 89, 200403 (2002).ADSCrossRefGoogle Scholar
  10. 10.
    A. Aftalion and I. Danaila, Phys. Rev. A 68, 023603 (2003).ADSCrossRefGoogle Scholar
  11. 11.
    A. Aftalion and I. Danaila, Phys. Rev. A 69, 033608 (2004).ADSCrossRefGoogle Scholar
  12. 12.
    D. E. Sheehy and L. Radzihovsky, Phys. Rev. A 70, 063620 (2004).ADSCrossRefGoogle Scholar
  13. 13.
    I. Danaila, Phys. Rev. A 72, 013605 (2005).ADSCrossRefGoogle Scholar
  14. 14.
    A. Fetter, Phys. Rev. A 69, 043617 (2004).ADSCrossRefGoogle Scholar
  15. 15.
    T.-L. Horng, S.-C. Gou, and T.-C. Lin, Phys. Rev. A 74, 041603 (2006).ADSCrossRefGoogle Scholar
  16. 16.
    S. Serafini, M. Barbiero, M. Debortoli, S. Donadello, F. Larcher, F. Dalfovo, G. Lamporesi, and G. Ferrari, Phys. Rev. Lett. 115, 170402 (2015).ADSCrossRefGoogle Scholar
  17. 17.
    S. Serafini, L. Galantucci, E. Iseni, T. Bienaime, R. N. Bisset, C. F. Barenghi, F. Dalfovo, G. Lamporesi, and G. Ferrari, Phys. Rev. X 7, 021031 (2017).Google Scholar
  18. 18.
    R. N. Bisset, S. Serafini, E. Iseni, M. Barbiero, T. Bienaime, G. Lamporesi, G. Ferrari, and F. Dalfovo, Phys. Rev. A 96, 053605 (2017).ADSCrossRefGoogle Scholar
  19. 19.
    R. L. Ricca, D. C. Samuels, and C. F. Barenghi, J. Fluid Mech. 391, 29 (1999).ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    F. Maggioni, S. Alamri, C. F. Barenghi, and R. L. Ricca, Phys. Rev. E 82, 026309 (2010).ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    D. Proment, M. Onorato, and C. F. Barenghi, Phys. Rev. E 85, 036306 (2012).ADSCrossRefGoogle Scholar
  22. 22.
    D. Kleckner and W. T. M. Irvine, Nat. Phys. 9, 253 (2013).CrossRefGoogle Scholar
  23. 23.
    D. Proment, M. Onorato, and C. F. Barenghi, J. Phys.: Conf. Ser. 544, 012022 (2014).Google Scholar
  24. 24.
    P. Clark di Leoni, P. D. Mininni, and M. E. Brachet, Phys. Rev. A 94, 043605 (2016).ADSCrossRefGoogle Scholar
  25. 25.
    D. Kleckner, L. H. Kauffman, and W. T. M. Irvine, Nat. Phys. 12, 650 (2016).CrossRefGoogle Scholar
  26. 26.
    M. D. Bustamante and S. Nazarenko, Phys. Rev. E 92, 053019 (2015).ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    V. P. Ruban, JETP Lett. 105, 458 (2017).ADSCrossRefGoogle Scholar
  28. 28.
    V. P. Ruban, J. Exp. Theor. Phys. 124, 932 (2017).ADSCrossRefGoogle Scholar
  29. 29.
    H. Hasimoto, J. Fluid Mech. 51, 477 (1972).ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    V. P. Ruban, JETP Lett. 103, 780 (2016).ADSCrossRefGoogle Scholar
  31. 31.
    V. P. Ruban, JETP Lett. 104, 868 (2016).ADSCrossRefGoogle Scholar
  32. 32.
    V. P. Ruban, JETP Lett. 106, 223 (2017).ADSCrossRefGoogle Scholar
  33. 33.
    V. E. Zakharov, Sov. Phys. Usp. 31, 672 (1988).ADSCrossRefGoogle Scholar
  34. 34.
    R. Klein, A. J. Majda, and K. Damodaran, J. Fluid Mech. 288, 201 (1995).ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    C. E. Kenig, G. Ponce, and L. Vega, Commun. Math. Phys. 243, 471 (2003).ADSCrossRefGoogle Scholar
  36. 36.
    N. Hietala, R. Hanninen, H. Salman, and C. F. Barenghi, Phys. Rev. Fluids 1, 084501 (2016).ADSCrossRefGoogle Scholar
  37. 37.
    H. Aref, J. Fluid Mech. 290, 167 (1995).ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2018

Authors and Affiliations

  1. 1.Landau Institute of Theoretical PhysicsRussian Academy of SciencesChernogolovka, Moscow oblastRussia

Personalised recommendations