The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate. (English) Zbl 1106.58009

The wave function of a Bose-Einstein condensate that is confined in the direction of the rotation axis by a harmonic trap decouples and can be analyzed in a two-dimensional setting. In this case, minimizing a modified Gross-Pitaevskii functional on a weighted Sobolev space under the unit mass constraint, the authors give an asymptotic estimate of the critical angular velocity for the nucleation of vortices in the interior of the region occupied by the condensate. In addition, near the critical velocity several estimates on the energy and the shape of the minimizer, and on the location of its vortex are proved.


58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
35Q55 NLS equations (nonlinear Schrödinger equations)
47J30 Variational methods involving nonlinear operators
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
Full Text: DOI


[1] Abo-Shaeer, J. R.; Raman, C.; Vogels, J. M.; Ketterle, W., Observation of vortex lattices in Bose-Einstein condensate, Science, 292 (2001)
[3] Aftalion, A.; Du, Q., Vortices in a rotating Bose-Einstein condensate: critical angular velocities and energy diagrams in the Thomas-Fermi regime, Phys. Rev. A, 64 (2001)
[4] Aftalion, A.; Jerrard, R. L., Shape of vortices for a rotating Bose-Einstein condensate, Phys. Rev. A, 66 (2002)
[5] Aftalion, A.; Rivière, T., Vortex energy and vortex bending for a rotating Bose-Einstein condensate, Phys. Rev. A, 64 (2001)
[6] Bethuel, F.; Brezis, H.; Hélein, F., Asymptotics for the minimization of a Ginzburg-Landau functional, Calculus Variations Partial Differential Equations, 1, 123-148 (1993) · Zbl 0834.35014
[7] Bethuel, F.; Brezis, H.; Hélein, F., Ginzburg-Landau Vortices (1993), Birkhäuser: Birkhäuser Basel · Zbl 0783.35014
[8] Brezis, H., Semilinear equations in \(R^N\) without conditions at infinity, Appl. Math. Optim., 12, 271-282 (1984) · Zbl 0562.35035
[9] Brezis, H.; Oswald, L., Remarks on sublinear elliptic equations, Nonlinear Anal., 10, 55-64 (1986) · Zbl 0593.35045
[10] Butts, D.; Rokhsar, D., Predicted signatures of rotating Bose-Einstein condensates, Nature, 397 (1999)
[11] Castin, Y.; Dum, R., Bose-Einstein condensates with vortices in rotating traps, European Phys. J. D, 7, 399-412 (1999)
[12] Farina, A., From Ginzburg-Landau to Gross-Pitaevskii, Monatsh. Math., 139, 265-269 (2003) · Zbl 1126.35063
[13] Ignat, R.; Millot, V., Vortices in a 2D rotating Bose-Einstein condensate, C. R. Acad. Sci. Paris Série I, 340, 571-576 (2005) · Zbl 1072.35566
[16] Lassoued, L.; Mironescu, P., Ginzburg-Landau type energy with discontinuous constraint, J. Anal. Math., 77, 1-26 (1999) · Zbl 0930.35073
[17] Lieb, E. H.; Seiringer, R.; Yngvason, J., A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys., 224, 17-31 (2001) · Zbl 0996.82010
[18] Madison, K.; Chevy, F.; Dalibard, J.; Wohlleben, W., Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett., 84 (2000)
[19] Madison, K.; Chevy, F.; Dalibard, J.; Wohlleben, W., Vortices in a stirred Bose-Einstein condensate, J. Mod. Opt., 47, 1-10 (2000)
[20] Sandier, E., Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal., 152, 119-145 (1998)
[21] Sandier, E.; Serfaty, S., Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 17, 119-145 (2000) · Zbl 0947.49004
[22] Sandier, E.; Serfaty, S., A rigorous derivation of a free boundary problem arising in superconductivity, Ann. Sci. École Norm. Sup. (4), 33, 561-592 (2000) · Zbl 1174.35552
[23] Sandier, E.; Serfaty, S., Ginzburg-Landau minimizers near the first critical field have bounded vorticity, Calculus Variations Partial Differential Equations, 17, 17-28 (2003) · Zbl 1037.49001
[25] Serfaty, S., On a model of rotating superfluids, ESAIM: Control Optim. Calculus Variations, 6, 201-238 (2001) · Zbl 0964.35142
[27] Struwe, M., An asymptotic estimate for the Ginzburg-Landau model, C. R. Acad. Sci. Paris Sér. I Math., 317, 677-680 (1993) · Zbl 0789.49005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.