We study a two-dimensional rotating Bose-Einstein condensate confined by an anharmonic trap in the framework of the Gross-Pitaevskii theory. We consider a rapid rotation regime close to the transition to a giant vortex state. It was proven in Correggi et al. [J. Math. Phys. 53, 095203 (2012)] that such a transition occurs when the angular velocity is of order ε−4, with ε−2 denoting the coefficient of the nonlinear term in the Gross-Pitaevskii functional and ε ≪ 1 (Thomas-Fermi regime). In this paper, we identify a finite value Ωc such that if Ω = Ω0/ε4 with Ω0 > Ωc, the condensate is in the giant vortex phase. Under the same condition, we prove a refined energy asymptotics and an estimate of the winding number of any Gross-Pitaevskii minimizer.
REFERENCES
1.
Abo-Shaeer
, J. R.
, Raman
, C.
, Vogels
, J. M.
, and Ketterle
, W.
, “Observation of vortex lattices in Bose-Einstein condensates
,” Science
292
, 476
–479
(2001
).2.
Abramovitz
, M.
and Stegun
, I. A.
, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables
(Dover
, New York
, 1965
).3.
Aftalion
, A.
, Blanc
, X.
, and Dalibard
, J.
, “Vortex patterns in a fast rotating Bose-Einstein condensate
,” Phys. Rev. A
71
, 023611
(2005
).4.
Aftalion
, A.
, Blanc
, X.
, and Nier
, F.
, “Vortex distributions in the lowest Landau level
,” Phys. Rev. A
73
, 011601
(2006
).5.
Aftalion
, A.
, Jerrard
, R. L.
, and Royo-Letelier
, J.
, “Non-existence of vortices in the small density region of a condensate
,” J. Funct. Anal.
260
, 2387
–2406
(2011
).6.
Antoine
, X.
and Duboscq
, R.
, “GPELab, a matlab toolbox to solve Gross-Pitaevskii equations. I: Computation of stationary solutions
,” Comput. Phys. Commun.
185
, 2969
–2991
(2014
).7.
Antoine
, X.
and Duboscq
, R.
, “GPELab, a matlab toolbox to solve Gross-Pitaevskii equations. II: Dynamics and stochastic simulations
,” Comput. Phys. Commun.
193
, 95
–117
(2015
).8.
Bretin
, V.
, Stock
, S.
, Seurin
, Y.
, and Dalibard
, J.
, “Fast rotation of a Bose-Einstein condensate
,” Phys. Rev. Lett.
92
, 050403
(2004
).9.
Bru
, J.-B.
, Correggi
, M.
, Pickl
, P.
, and Yngvason
, J.
, “The TF limit for rapidly rotating Bose gases in anharmonic traps
,” Commun. Math. Phys.
280
, 517
–544
(2008
).10.
Castin
, Y.
and Dum
, R.
, “Bose-Einstein condensates with vortices in rotating traps
,” Eur. Phys. J. D
7
, 399
–412
(1999
).11.
Coddington
, I.
, Haljan
, P. C.
, Engels
, P.
, Schweikhard
, V.
, Tung
, S.
, and Cornell
, E. A.
, “Experimental studies of equilibrium vortex properties in a Bose-condensed gas
,” Phys. Rev. A
70
, 063607
(2004
).12.
Cooper
, N. R.
, “Rapidly rotating atomic gases
,” Adv. Phys.
57
, 539
–616
(2008
).13.
Correggi
, M.
, Pinsker
, F.
, Rougerie
, N.
, and Yngvason
, J.
, “Critical rotational speeds in the Gross-Pitaevskii theory on a disc with Dirichlet boundary conditions
,” J. Stat. Phys.
143
, 261
–305
(2011
).14.
Correggi
, M.
, Pinsker
, F.
, Rougerie
, N.
, and Yngvason
, J.
, “Rotating superfluids in anharmonic traps: From vortex lattices to giant vortices
,” Phys. Rev. A
84
, 053614
(2011
).15.
Correggi
, M.
, Pinsker
, F.
, Rougerie
, N.
, and Yngvason
, J.
, “Critical rotational speeds for superfluids in homogeneous traps
,” J. Math. Phys.
53
, 095203
(2012
).16.
Correggi
, M.
, Pinsker
, F.
, Rougerie
, N.
, and Yngvason
, J.
, “Giant vortex phase transition in rapidly rotating trapped Bose-Einstein condensates
,” Eur. Phys. J.: Spec. Top.
217
, 183
–188
(2013
).17.
Correggi
, M.
, Pinsker
, F.
, Rougerie
, N.
, and Yngvason
, J.
, “Vortex phases of rotating superfluids
,” J. Phys.: Conf. Ser.
414
, 012034
(2013
).18.
Correggi
, M.
and Rougerie
, N.
, “Inhomogeneous vortex patterns in rotating Bose-Einstein condensates
,” Commun. Math. Phys.
321
, 817
–860
(2013
).19.
Correggi
,M.
and Rougerie
,N.
, “On the Ginzburg-Landau functional in the surface superconductivity regime
,” Commun. Math. Phys.
332
, 1297
–1343
(2014
);Erratum,
Correggi
, M.
and Rougerie
, N.
, Commun. Math. Phys.
338
, 1451
–1452
(2015
).20.
Correggi
, M.
and Rougerie
, N.
, “Boundary behavior of the Ginzburg-Landau order parameter in the surface superconductivity regime
,” Arch. Rational Mech. Anal.
219
, 553
–606
(2016
).21.
Correggi
, M.
, Rindler-Daller
, T.
, and Yngvason
, J.
, “Rapidly rotating Bose-Einstein condensates in strongly anharmonic traps
,” J. Math. Phys.
48
, 042104
(2007
).22.
Correggi
, M.
, Rindler-Daller
, T.
, and Yngvason
, J.
, “Rapidly rotating Bose-Einstein condensates in homogeneous traps
,” J. Math. Phys.
48
, 102103
(2007
).23.
Correggi
, M.
, Rougerie
, N.
, and Yngvason
, J.
, “The transition to a giant vortex phase in a fast rotating Bose-Einstein condensate
,” Commun. Math. Phys.
303
, 451
–508
(2011
).24.
Correggi
, M.
and Yngvason
, J.
, “Energy and vorticity in fast rotating Bose-Einstein condensates
,” J. Phys. A: Math. Theor.
41
, 445002
(2008
).25.
Danaila
, I.
, “Three-dimensional vortex structure of a fast rotating Bose-Einstein condensate with harmonic-plus-quartic confinement
,” Phys. Rev. A
72
, 013605
(2005
).26.
Evans
, L. C.
, Partial Differential Equation
, Graduate Studies in Mathematics
Vol. 19
(AMS
, Providence
, 1998
).27.
Fetter
, A. L.
, “Rotating trapped Bose-Einstein condensates
,” Rev. Mod. Phys.
81
, 647
–691
(2009
).28.
Fetter
, A. L.
, “Rotating vortex lattice in a Bose-Einstein condensate trapped in combined quadratic and quartic radial potentials
,” Phy. Rev. A
64
, 063608
(2001
).29.
Fetter
, A. L.
, Jackson
, N.
, and Stringari
, S.
, “Rapid rotation of a Bose-Einstein condensate in a harmonic plus quartic trap
,” Phys. Rev. A
71
, 013605
(2005
).30.
Fischer
, U. R.
and Baym
, G.
, “Vortex states of rapidly rotating dilute Bose-Einstein condensates
,” Phys. Rev. Lett.
90
, 140402
(2003
).31.
Ignat
, R.
and Millot
, V.
, “The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate
,” J. Funct. Anal.
233
, 260
–306
(2006
).32.
Ignat
, R.
and Millot
, V.
, “Energy expansion and vortex location for a two dimensional rotating Bose-Einstein condensate
,” Rev. Math. Phys.
18
, 119
–162
(2006
).33.
Kasamatsu
, K.
, Tsubota
, M.
, and Ueda
, M.
, “Giant hole and circular superflow in a fast rotating Bose-Einstein condensate
,” Phys. Rev. A
66
, 050606
(2002
).34.
Kachmar
, A.
, “Energy of a rotating Bose-Einstein condensate in a harmonic trap
,” preprint arXiv:1306.3296 [math.AP].35.
Kavoulakis
, G. M.
and Baym
, G.
, “Rapidly rotating Bose-Einstein condensates in anharmonic potentials
,” New J. Phys.
5
, 51.1
–51.11
(2003
).36.
Kim
, J. K.
and Fetter
, A. L.
, “Dynamics of a rapidly rotating Bose-Einstein condensate in a harmonic plus quartic trap
,” Phys. Rev. A
72
, 023619
(2005
).37.
Lassoued
, L.
and Mironescu
, P.
, “Ginzburg-Landau type energy with discontinuous constraint
,” J. Anal. Math.
77
, 1
–26
(1999
).38.
Lieb
, E. H.
and Seiringer
, R.
, “Derivation of the Gross-Pitaevskii equation for rotating Bose gases
,” Commun. Math. Phys.
264
, 505
–537
(2006
).39.
Lieb
, E. H.
, Seiringer
, R.
, Solovej
, J. P.
, and Yngvason
, J.
, The Mathematics of the Bose Gas and its Condensation
, Oberwolfach Seminar Series
Vol. 34
(Birkhäuser
, Basel
, 2005
), expanded version available at e-print arXiv:cond-mat/0610117.40.
Lieb
, E. H.
, Seiringer
, R.
, and Yngvason
, J.
, “Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional
,” Phys. Rev. A
61
, 043602
(2000
).41.
Lieb
, E. H.
, Seiringer
, R.
, and Yngvason
, J.
, “The yrast line of a rapidly rotating Bose gas: The Gross-Pitaevskii regime
,” Phys. Rev. A
79
, 063626
(2009
).42.
Madison
, K. W.
, Chevy
, F.
, Wohlleben
, W.
, and Dalibard
, J.
, “Vortex formation in a stirred Bose-Einstein condensate
,” Phys. Rev. Lett.
84
, 806
–809
(2000
).43.
Raman
, C.
, Abo-Shaeer
, J. R.
, Vogels
, J. M.
, Xu
, K.
, and Ketterle
, W.
, “Vortex nucleation in a stirred Bose-Einstein condensate
,” Phys. Rev. Lett.
87
, 210402
(2001
).44.
Rougerie
, N.
, “The giant vortex state for a Bose-Einstein condensate in a rotating anharmonic trap: Extreme rotation regimes
,” J. Math. Pures Appl.
95
, 296
–347
(2011
).45.
Rougerie
, N.
, “Vortex rings in fast rotating Bose-Einstein condensates
,” Arch. Rational Mech. Anal.
203
, 69
–135
(2012
).© 2016 Author(s).
2016
Author(s)
You do not currently have access to this content.
