Vortex patterns and sheets in segregated two component Bose–Einstein condensates

  • Amandine AftalionEmail author
  • Etienne Sandier


We study minimizers of a Gross–Pitaevskii energy describing a two-component Bose–Einstein condensate set into rotation. We consider the case of segregation of the components in the Thomas–Fermi regime, where a small parameter \(\varepsilon \) conveys a singular perturbation. We estimate the energy as a term due to a perimeter minimization and a term due to rotation. In particular, we prove a new estimate concerning the error of a Modica Mortola type energy away from the interface. For large rotations, we show that the interface between the components gets long, which is a first indication towards vortex sheets.

Mathematics Subject Classification

49J10 35B25 35J20 49N60 



  1. 1.
    Aftalion, A.: Vortices in Bose–Einstein Condensates. Birkhauser, Basel (2006)CrossRefGoogle Scholar
  2. 2.
    Aftalion, A., Mason, P.: Classification of the ground states and topological defects in a rotating two-component Bose–Einstein condensate. Phys. Rev. A 84, 033611 (2011)CrossRefGoogle Scholar
  3. 3.
    Aftalion, A., Noris, B., Sourdis, C.: Thomas–Fermi approximation for coexisting two component Bose–Einstein condensates and nonexistence of vortices for small rotation. Com. Math. Phys 336(2), 509–579 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Aftalion, A., Royo-Letelier, J.: A minimal interface problem arising from a two component Bose–Einstein condensate via Gamma-convergence. Calc. Var. PDE’s. 52, 165–197 (2015)CrossRefGoogle Scholar
  5. 5.
    Aftalion, A., Sourdis, C.: Interface layer of a two-component Bose–Einstein condensate. Commun. Contemp. Math. 19, 1650052–1650097 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Alama, S., Bronsard, L., Contreras, A., Pelinovsky, D.E.: Domain walls in the coupled Gross–Pitaevskii equations. Arch. Ration. Mech. Anal. 215(2), 579–610 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Alikakos, N.D., Faliagas, A.C.: Stability criteria for multiphase partitioning problems with volume constraints. Discrete Contin. Dyn. Syst. A 37(2), 663–683 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, vol. 24. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  9. 9.
    Ambrosio, L., Tortorelli, V.M.: On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B 6(1), 105–123 (1992)MathSciNetzbMATHGoogle Scholar
  10. 10.
    André, N., Shafrir, I.: Minimization of a Ginzburg–Landau type functional with nonvanishing Dirichlet boundary condition. Calc. Var. Partial. Differ. Equ. 7(3), 191–217 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Berestycki, H., Lin, T.-C., Wei, J., Zhao, C.: On phase-separation model: asymptotics and qualitative properties. Arch. Ration. Mech. Anal. 208, 163–200 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Berestycki, H., Terracini, S., Wang, K., Wei, J.: On entire solutions of an elliptic system modeling phase separations. Adv. Math. 243, 102–126 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Brezis, H.: Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris (1983)Google Scholar
  14. 14.
    Caffarelli, L.A.: The obstacle problem revisited. J. Fourier Anal. Appl. 4, 383–402 (1998)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Caffarelli, L.A., Lin, F.-H.: An optimal partition problem for eigenvalues. J. Sci. Comput. 31, 5–18 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Caffarelli, L.A., Lin, F.-H.: Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries. J. Am. Math. Soc. 21, 847–862 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Correggi, M., Pinsker, F., Rougerie, N., Yngvason, J.: Critical Rotational Speeds in the Gross–Pitaevskii Theory on a Disc with Dirichlet Boundary Conditions. J. Stat. Phys. 143, 261–305 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Cozzi, M., Figalli, A.: Regularity theory for local and nonlocal minimal surfaces: an overview. Nonlocal and nonlinear diffusions and interactions: new methods and directions, 117–158, Lecture Notes in Math., 2186, Fond. CIME/CIME Found. Subser., Springer, Cham, (2017)Google Scholar
  19. 19.
    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhuser, Basel (1984)CrossRefGoogle Scholar
  20. 20.
    Goldman, M., Merlet, B.: Phase segregation for binary mixtures of Bose–Einstein Condensates. SIAM J. Math. Anal. 49(3), 1947–1981 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Goldman, M., Royo-Letelier, J.: Sharp interface limit for two components Bose–Einstein condensates. ESAIM Control Optim. Calc. Var. 21(3), 603–624 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ignat, R., Millot, V.: The critical velocity for vortex existence in a two-dimensional rotating Bose–Einstein condensate. J. Func. Anal. 233, 260–306 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ignat, R., Millot, V.: Energy expansion and vortex location for a two-dimensional rotating Bose–Einstein condensate. Rev. Math. Phys. 18(2), 119–162 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Jerrard, R.L., Soner, H.M.: The Jacobian and the Ginzburg–Landau functional. Calc. Var. 14(2), 151–191 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Jerrard, R.L., Soner, H.M.: Limiting behavior of the Ginzburg–Landau energy. J. Funct. Anal. 192(2), 524–561 (2002)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kachmar, A.: Magnetic vortices for a Ginzburg–Landau type energy with discontinuous constraint. ESAIM Control Optim. Calc. Var. 16(3), 545–580 (2010)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kasamatsu, K., Tsubota, M.: Vortex sheet in rotating two-component Bose–Einstein condensates. Phys. Rev. A 79, 023606 (2009)CrossRefGoogle Scholar
  28. 28.
    Lassoued, L., Mironescu, P.: Ginzburg–Landau type energy with discontinuous constraint. J. dAnalyse Mathé. 77(1), 1–26 (1999)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Leoni, G., Murray, R.: Second-order \(\Gamma \)-limit for the Cahn–Hilliard functional. Arch. Ration. Mech. Anal. 219(3), 1383–1451 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Luo, S., Ren, X., Wei, J.: Non hexagonal lattices from a two species interacting system. Preprint (2018)Google Scholar
  31. 31.
    Modica, L., Mortola, S.: Il limite nella G-convergenza di una famiglia di funzionali ellittici A (5). Boll. Un. Mat. Ital. 14(3), 526–529 (1977)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Parts, U., Ruutu, V.M.H., Koivuniemi, J.H., Krusius, M., Thuneberg, E.V., Volovik, G.E.: Measurements on the vortex sheet in rotating superfluid 3He-A. Phys. B Condens. Matter 210(34), 311–333 (1995)CrossRefGoogle Scholar
  33. 33.
    Ros, A.: The isoperimetric problem. Global Theory Min. Surf. 2, 175–209 (2001)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Sandier, E., Serfaty, S.: Global minimizers for the Ginzburg–Landau functional below the first critical magnetic field. Ann. IHP Anal. non linéaire 17(1), 119–145 (2000)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sandier, E., Serfaty, S.: On the energy of Type-II superconductors in the mixed phase. Reviews in Math. Phys. 12(9), 1219–1257 (2000)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Sandier, E., Serfaty, S.: A rigorous derivation of a free-boundary problem arising in superconductivity. Ann. Sci. Ecole Normale Sup. 4(33), 561–592 (2000)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Sandier, E., Serfaty, S.: Vortices in the magnetic Ginzburg–Landau model, vol. 70. Springer, Berlin (2008)zbMATHGoogle Scholar
  38. 38.
    Sandier, E., Serfaty, S.: From the Ginzburg–Landau model to vortex lattice problems. Commun. Math. Phys. 313, 635–743 (2012)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Serfaty, S.: On a model of rotating superfluids. ESAIM: Control Optim. Calc. Var. 6, 201–238 (2001)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Soave, N., Zilio, A.: On phase separation in systems of coupled elliptic equations: asymptotic analysis and geometric aspects. Annales de l’Institut Henri Poincaré (C) Analyse Non Linéaire (2016)Google Scholar
  41. 41.
    Sourdis, C.: Weak separation limit of a two-component Bose-Einstein condensate. Electron. J. Differ. Equat. 2018(40), 1–12 (2018)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Sternberg, P.: The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101(3), 209–260 (1988)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Sternberg, P., Zumbrun, K.: Connectivity of phase boundaries in strictly convex domains. Arch. Ration. Mech. Anal. 141(4), 375–400 (1998)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CNRS UMR 8557, Centre d’Analyse et de Mathématique SocialesEcole des Hautes Etudes en Sciences SocialesParisFrance
  2. 2.LAMA, Univ Paris Est CreteilUniv Gustave Eiffel, UPEM, CNRSCréteilFrance

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