Advertisement

Journal of Statistical Physics

, Volume 143, Issue 2, pp 261–305 | Cite as

Critical Rotational Speeds in the Gross-Pitaevskii Theory on a Disc with Dirichlet Boundary Conditions

  • M. Correggi
  • F. Pinsker
  • N. Rougerie
  • J. Yngvason
Article

Abstract

We study the two-dimensional Gross-Pitaevskii theory of a rotating Bose gas in a disc-shaped trap with Dirichlet boundary conditions, generalizing and extending previous results that were obtained under Neumann boundary conditions. The focus is on the energy asymptotics, vorticity and qualitative properties of the minimizers in the parameter range |log ε|≪Ωε −2|log ε|−1 where Ω is the rotational velocity and the coupling parameter is written as ε −2 with ε≪1. Three critical speeds can be identified. At \(\varOmega=\varOmega_{\mathrm{c_{1}}}\sim |\log\varepsilon|\) vortices start to appear and for \(|\log\varepsilon|\ll\varOmega< \varOmega_{\mathrm{c_{2}}}\sim \varepsilon^{-1}\) the vorticity is uniformly distributed over the disc. For \(\varOmega\geq\varOmega _{\mathrm{c_{2}}}\) the centrifugal forces create a hole around the center with strongly depleted density. For Ωε −2|log ε|−1 vorticity is still uniformly distributed in an annulus containing the bulk of the density, but at \(\varOmega=\varOmega_{\mathrm {c_{3}}}\sim\varepsilon ^{-2}|\log\varepsilon |^{-1}\) there is a transition to a giant vortex state where the vorticity disappears from the bulk. The energy is then well approximated by a trial function that is an eigenfunction of angular momentum but one of our results is that the true minimizers break rotational symmetry in the whole parameter range, including the giant vortex phase.

Keywords

Bose-Einstein condensates Superfluidity Vortices Giant vortex 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aftalion, A.: Vortices in Bose-Einstein Condensates. Progress in Nonlinear Differential Equations and their Applications, vol. 67. Birkhäuser, Basel (2006) MATHGoogle Scholar
  2. 2.
    Aftalion, A., Jerrard, R.L., Royo-Letelier, J.: Non existence of vortices in the small density region of a condensate. J. Funct. Anal. 260, 2387–2406 (2011) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Béthuel, F., Brézis, H., Hélein, F.: Ginzburg-Landau Vortices. Progress in Nonlinear Differential Equations and Their Applications, vol. 13. Birkhäuser, Basel (1994) MATHCrossRefGoogle Scholar
  4. 4.
    Cooper, N.R.: Rapidly rotating atomic gases. Adv. Phys. 57, 539–616 (2008) CrossRefADSGoogle Scholar
  5. 5.
    Correggi, M., Rindler-Daller, T., Yngvason, J.: Rapidly rotating Bose-Einstein condensates in strongly anharmonic traps. J. Math. Phys. 48, 042104 (2007) CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Correggi, M., Rindler-Daller, T., Yngvason, J.: Rapidly rotating Bose-Einstein condensates in homogeneous traps. J. Math. Phys. 48, 102103 (2007) CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Correggi, M., Pinsker, F., Rougerie, N., Yngvason, J.: in preparation Google Scholar
  8. 8.
    Correggi, M., Rougerie, N., Yngvason, J.: The transition to a giant vortex phase in a fast rotating Bose-Einstein condensate. Commun. Math. Phys. 303, 451–508 (2011) CrossRefMathSciNetADSMATHGoogle Scholar
  9. 9.
    Correggi, M., Yngvason, J.: Energy and vorticity in fast rotating Bose-Einstein condensates. J. Phys. A, Math. Theor. 41, 445002 (2008) CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Evans, L.C.: Partial Differential Equation. Graduate Studies in Mathematics, vol. 19. AMS, Providence (1998) Google Scholar
  11. 11.
    Fetter, A.L.: Rotating trapped Bose-Einstein condensates. Rev. Mod. Phys. 81, 647–691 (2009) CrossRefADSGoogle Scholar
  12. 12.
    Fetter, A.L., Jackson, N., Stringari, S.: Rapid rotation of a Bose-Einstein condensate in a harmonic plus quartic trap. Phys. Rev. A 71, 013605 (2005) CrossRefADSGoogle Scholar
  13. 13.
    Fischer, U.R., Baym, G.: Vortex states of rapidly rotating dilute Bose-Einstein condensates. Phys. Rev. Lett. 90, 140402 (2003) CrossRefADSGoogle Scholar
  14. 14.
    Fournais, S., Helffer, B.: Spectral Methods in Surface Superconductivity. Progress in Nonlinear Differential Equations and Their Applications, vol. 77. Birkhäuser, Basel (2010) MATHGoogle Scholar
  15. 15.
    Ignat, R., Millot, V.: The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate. J. Funct. Anal. 233, 260–306 (2006) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Ignat, R., Millot, V.: Energy expansion and vortex location for a two dimensional rotating Bose-Einstein condensate. Rev. Math. Phys. 18, 119–162 (2006) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Jerrard, R.L.: Lower bounds for generalized Ginzburg-Landau functionals. SIAM J. Math. Anal. 30, 721–746 (1999) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Jerrard, R.L., Soner, H.M.: The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differ. Equ. 14, 524–561 (2002) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Kavoulakis, G.M., Baym, G.: Rapidly rotating Bose-Einstein condensates in anharmonic potentials. New J. Phys. 5, 51.1–51.11 (2003) CrossRefGoogle Scholar
  20. 20.
    Lassoued, L., Mironescu, P.: Ginzburg-Landau type energy with discontinuous constraint. J. Anal. Math. 77, 1–26 (1999) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Rougerie, N.: Vortex rings in fast rotating Bose-Einstein condensates (2010). arXiv:1009.1982 [math-ph]
  22. 22.
    Sandier, E.: Lower bounds for the energy of unit vector fields and applications. J. Funct. Anal. 152, 349–358 (1998) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Sandier, E., Serfaty, S.: On the energy of type-II superconductors in the mixed phase. Rev. Math. Phys. 12, 1219–1257 (2000) CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Sandier, E., Serfaty, S.: Vortices in the Magnetic Ginzburg-Landau Model. Progress in Nonlinear Differential Equations and Their Applications, vol. 70. Birkhäuser, Basel (2007). Erratum available at http://www.ann.jussieu.fr/serfaty/publis.html MATHGoogle Scholar
  25. 25.
    Seiringer, R.: Gross-Pitaevskii theory of the rotating Bose gas. Commun. Math. Phys. 229, 491–509 (2002) CrossRefMATHADSMathSciNetGoogle Scholar
  26. 26.
    Serfaty, S.: On a model of rotating superfluids. ESAIM Control Optim. Calc. Var. 6, 201–238 (2001) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • M. Correggi
    • 1
  • F. Pinsker
    • 2
  • N. Rougerie
    • 3
  • J. Yngvason
    • 4
    • 5
  1. 1.CIRMFondazione Bruno KesslerTrentoItaly
  2. 2.DAMTPUniversity of CambridgeCambridgeUK
  3. 3.Département de Mathématiques, CNRS-UMR 8088CNRS and Université de Cergy-PontoiseCergy-Pontoise CedexFrance
  4. 4.Fakultät für PhysikUniversität WienViennaAustria
  5. 5.Erwin Schrödinger Institute for Mathematical PhysicsViennaAustria

Personalised recommendations