Rapidly Rotating Bose-Einstein Condensates in Strongly Anharmonic Traps. Michele Correggi. T. Rindler-Daller, J. Yngvason math-ph/

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1 Rapidly Rotating Bose-Einstein Condensates in Strongly Anharmonic Traps Michele Correggi Erwin Schrödinger Institute, Vienna T. Rindler-Daller, J. Yngvason math-ph/ in collaboration with preprint ESI Workshop Complex Quantum, Classical Systems and Effective Equations July 2006

2 The Model A 2d rotating Bose-Einstein Condensate (BEC) in a cylindrical trap is described by the Gross Pitaevskii (GP) Theory [Lieb, Seiringer 05] [GP Functional] E GP [Ψ] = B d r { Ψ 2 Ω(ε)Ψ LΨ + Ψ 4 ε 2 } [Ground State] E GP ε = inf Ψ =1 EGP [Ψ] = E GP [Ψ GP ε ] D GP = H 1 (B), L = i third component of the angular momentum, ϑ Ω(ε) angular velocity, ε small (semiclassical) parameter, B unit disc 2

3 Motivations [Anharmonic Trap] - The harmonic trap case is very special (upper bound for Ω(ε), Ψ GP ε in the lowest Landau level...) [Aftalion et al. 06] - In anharmonic traps very large angular velocities (Ω 1) can be reached - The flat trap is the most anharmonic trap [Very Large Angular Velocity] - Macroscopic effects of the rotation (holes, giant vortices...) [Baym, Fischer 03] 3

4 GP Theory vs. Ginzburg-Landau (GL) Theory The minimization of the GP functional is equivalent to the minimization of a GL-type functional E GP [Ψ] = B d r ( i Aε ) Ψ 2 Ω(ε) 2 r 2 Ψ ( 1 Ψ 2 ) 2 ε 2 A ε ( r) = Ω(ε) 2 ẑ r Ψ L 2 (B) = 1 [Analogies] - A ε GL magnetic field - Ψ GL order parameter [Differences] - L 2 normalization of Ψ (chemical potential) - Electric field Ω(ε) 2 4

5 Summary 1. Spontaneous Symmetry Breaking in the Ground State 2. Ω(ε) 1/ε 3. Ω(ε) 1/ε - No macroscopic effect of the rotation - Ψ GP 1 ε π (minimizer of E GP [Ψ] with Ω = 0) ε 0 - Ψ GP ε ε 0 ρ TF 1 π - Above a certain threshold ρ TF = 0 in a macroscopic region (hole) 4. Ω(ε) 1/ε - Ψ GP ε - Ψ GP ε lives near the boundary Ψ GP ε e inεϑ (giant vortex) [?] 5

6 Hole Giant Vortex [Kasamatsu et al. 02] 6

7 Spontaneous Symmetry Breaking Proposition 1 For ε sufficiently small, no minimizer of E GP [Ψ] is an eigenfunction of the angular momentum, if 6 log ε + 3 < Ω(ε) 1 ε 2 The symmetry breaking is due to the occurrence of vortices (i.e. isolated zeros of Ψ GP ε ) In the symmetry breaking regime the minimizer is not unique The first critical velocity is expected to be 2 log ε [Ignat, Millot 05] The result is expected to hold true even above 1/ε 2 In a generic trap (homogeneous potential), for any fixed Ω(ε) and for ε sufficiently small there is symmetry breaking [Seiringer 02] 7

8 Ω(ε) 1/ε Proposition 2 For any Ω(ε) such that lim ε 0 εω(ε) = 0 and for ε sufficiently small ε 2 E GP ε = 1 π O(ε2 Ω(ε) 2 ) Ψ GP ε 2 1/π L 1 (B) = O(εΩ(ε)) Proof: 1. [upper bound] trial function 1 π 2. [lower bound] E GP [Ψ] = B d r { ( i A ε )Ψ 2 Ω(ε)2 r 2 Ψ 2 4 } + Ψ 4 ε 2 Ψ 4 4 ε 2 Ω(ε)2 4 1 πε Ω(ε)

9 Ω(ε) 1/ε Fine Structure of Ψ GP ε Ω(ε) < 2 log ε - The minimizer is unique - The minimizer is a (positive) radial function Ω(ε) 2 log ε - Vortices of degree 1 start to appear in Ψ GP ε - The number of vortices is uniformly bounded - The average vortex size is ε log ε Ω(ε) 1/ε - The number of vortices is Ω(ε) - The average area of a vortex core is ε - The area covered by vortices is of order εω(ε) 0 (in accord with the convergence of the density profile) 9

10 Ω(ε) < 2 log ε [Ignat, Millot 05] ε 2 Eε GP = 1 π + o(ε2 ) Ψ GP ε 1 π in H 1 (B) ε 0 The minimizer is unique and it is an eigenfunction of the angular momentum = no symmetry breaking Proof: construction of vortex balls (clusters of vortices) to localize the phase singularities of Ψ GP ε [Sandier, Serfaty 00] The balls are finite and disjoint All the vortices are contained inside the balls A suitable lower bound for the kinetic energy is satisfied 10

11 Ω(ε) > 2 log ε [Ignat. Millot 05] ε 2 E GP ε = 1 π + O(ε2 log ε ) Ψ GP 1 ε ε 0 π in L 2 (B) Critical velocities Ω d (ε) = 2 log ε + 2(d 1) log log ε : Ω d (ε) Ω(ε) Ω d+1 (ε) = Ψ GP ε of degree 1 has exaclty d vortices If d > 1 the minimizer is not unique (symmetry breaking) 11

12 Ω(ε) > 2 log ε [Ignat, Millot 05] Vortices are located at r ε i, i = 1,..., d Vortices are close to the origin, r ε i another, r ε i rε j C log ε C log ε and close one Setting x i = Ω(ε) r i ε, the configuration ( x 1,..., x d ) minimizes the renormalized energy W( x 1,..., x d ) = i j ln x i x j d x 2 i i=1 12

13 Ω(ε) = Ω 0 /ε 1 E TF [ρ] = 2π dr 0 The Thomas-Fermi (TF) Functional { ρ 2 (r) Ω2 0 r2 ρ(r) 4 } D TF = { ρ L 2 ([0, 1]) ρ 0 } E GP [Ψ] = ( d r iaε )Ψ 2 + ETF [ Ψ 2 ] B ε 2 The profile Ψ GP ε minimizes the TF functional The energy contribution of the TF part ( 1/ε 2 ) is much larger than the gap between the Landau levels ( 1/ε) = the minimizer is not in the lowest Landau level The kinetic contribution due to the bending of the profile Ψ GP ε is smaller The number of vortices is 1/ε and each vortex gives a kinetic contribution log ε = the total correction due to vortices is of lower order ( log ε /ε) 13

14 Ω(ε) = Ω 0 /ε The TF Functional E TF = min ρ=1 E TF [ρ] = E TF [ρ TF ] E TF = ρ TF (r) = 1 π Ω2 0 8 πω4 0 ( 768 Ω π Ω 0 ) if Ω 0 4 π if Ω 0 > 4 π 1 π Ω (1 2r2 ) if Ω 0 4 [ π Ω (r2 1) + Ω ] 0 2 if Ω 0 > 4 π π [Hole] if Ω 0 > 4 π = ρ TF = 0 for r R πω0 14

15 Ω(ε) = Ω 0 /ε Energy Asymptotics Theorem 1 For any Ω 0 > 0 and for ε sufficiently small, ε 2 E GP ε = E TF + O(ε log ε ) Proof: 1. [lower bound] E GP [Ψ] = B d r ETF [ Ψ 2 ] ε 2 ( ia ε )Ψ 2 + ETF [ Ψ 2 ] ETF ε 2 ε 2 2. [upper bound] trial function Ψ ε ( r) s.t. - Ψ ε 2 ε 0 ρ TF in L p (B) - Ψ ε has 1/ε vortices 15

16 Ω(ε) = Ω 0 /ε Energy Upper Bound (Trial Function) Ψ ε can not have the form Ψ ε ( r) = f(r)e inεθ, for n ε 1/ε: { 1 [ E GP [f(r)e inεθ ] = 2π dr f 2 nε + 0 r Ω ] } 2 0r f 2 + ETF [f 2 ] 2ε ε 2 Ψ ε can not be in the lowest Landau level (LLL): the energy correction would be too small, since for any function in the LLL ( d r iaε )Ψ 2 1 B ε while we expect it to be at least of order log ε /ε 16

17 Ω(ε) = Ω 0 /ε Energy Upper Bound [Trial Function] Ψ ε ( r) = f ε (r) χ ε ( r) g ε ( r) f ε (r) = ρ TF if Ω 0 4 π j ε ρ TF if Ω 0 > 4 π g ε (z) = z z i z z i l i l is a (square) regular lattice of spacing l ε = ε δ j ε is a cut-off function supported near R 0 χ ε ( r) = 1 if r r i ε α r r i ε α if r r i ε α α > 1/2 17

18 Ω(ε) = Ω 0 /ε Energy Upper Bound Proposition 3 For α > 1, Ω 0 > 0 and ε sufficiently small, E TF [ Ψ ε 2 ] E TF + C Ω0 ε Theorem 2 There exists a constant C Ω0,δ independent of ε, such that for ε sufficiently small d r ( ) iaε 2 π gε Λ 2ε 2 ( Ω0 ) 2 + C Ω 0,δ log ε 2 π δ 2 ε where Λ = B\ i l B i ε and B i ε is a ball of radius ε α, α > 5/2, centered at r i 18

19 Ω(ε) = Ω 0 /ε Energy Upper Bound Proposition 3 = E TF [ Ψ ε ] ε 0 E TF choosing δ = 2π Ω 0, Theorem 2 = d r ( ) iaε 2 C Ω0,δ log ε Ψ ε B ε Remark The proof should work for any regular lattice, provided that the lattice spacing is of order ε. Indeed the estimate in Theorem 2 involves only the volume of the foundamental cell. A different choice of the lattice might improve the coefficient of the remainder. 19

20 Ω(ε) = Ω 0 /ε Convergence of the Profile Corollary 1 For any Ω 0 > 0 and for ε sufficiently small Ψ GP ε 2 ρ TF L 1 (B) = O( ε log ε ) Proposition 4 Let T ε be T ε = { r B r R 0 ε3 1 } Then for any Ω 0 > 4 π and ε sufficiently small, there exist two constants C Ω0 and C Ω 0, such that, for r T ε, Ψ GP ε ( r) 2 C Ω0 ε 1 6 log ε exp C Ω 0 dist( r, T ε ) 2 ε

21 Ω(ε) = Ω 0 /ε 1+α α > 0 The TF Profile E TF ε [ρ] = ε 2α B d r { ρ 2 Ω2 0 r2 ρ 4ε 2α } E TF ε = min Eε TF ρ=1 [ρ] = E TF ε [ρ TF ] = Ω2 0 4 ( 1 8εα 3 πω 0 ) ρ TF ε = Ω2 0 8ε 2α [ r 2 R 2 ε ] + R ε = 1 4εα πω0 The minimizer lives near the boundary: ρ TF ε δ(1 r) ε 0 21

22 Ω(ε) = Ω 0 /ε 1+α α > 0 Energy Asymptotics Theorem 3 For any Ω 0 > 0, α > 0 and ε sufficiently small ε 2+2α E GP ε = E TF ε + O(ε 2α ) + O(ε 2 log ε ) Proof: 1. [lower bound] E GP [Ψ] ETF ε [ Ψ 2 ] ε 2 ETF ε ε 2+2α 2. [upper bound] trial function Ψ ε ( r) = j ε (r) { [ ] } ρ TF Ω0 ε (r) exp i ϑ 2ε 1+α 22

23 Ω(ε) = Ω 0 /ε 1+α α > 0 Convergence of the Profile Corollary 2 Let D ε = { r r R ε }. If Ω 0 > 0, 0 < α < 2 and ε is sufficiently small Ψ GP ε Proposition 5 Let T ε = 2 L 2 (B\D ε ) = O(εα ) + O(ε 2 α log ε ) { r B r 1 ε α } 4 where α = min [α, 2 α]. Then for any Ω 0 > 0, 0 < α < 2 and ε sufficiently small, there exist constants C Ω0 and C Ω 0, such that, for r T ε, Ψ GP ε ( r) 2 C Ω0 ε α 3 log ε exp C Ω dist( r, T 0 ε ) 2 ε 1+α 2 23

24 Perspectives Ω(ε) = Ω 0 ε - The number of vortices is Ω(ε)/2 1/ε - Analysis of the fine structure of the minimizer, i.e. the vortex lattice (triangular? distorted?...) Ω(ε) 1 ε - There exists N ε Ω(ε)/2 s.t. e inεϑ Ψ GP ε is almost a radial function, i.e. L ( e inεϑ Ψ GP ) ε Cε β - The number of vortices in D ε is essentially zero, i.e. almost the whole vorticity of Ψ GP ε is concentrated where it is exponentially small Extension to generic anharmonic and anisotropic traps 24

25 References [1] A. Aftalion, X. Blanc, SIAM Math. Anal. in press (2006) [2] A. Aftalion, X. Blanc, F. Nier, Phys. Rev. A 73 (2006) [3] G. Baym, U.R. Fischer, Phys. Rev. Lett. 80 (2003) [4] R. Ignat, V. Millot, J. Funct. Anal. 233 (2006) [5] R. Ignat, V. Millot, Rev. Math. Phys. 18 (2006) [5] K. Kasamatsu, M. Tsuboda, M. Ueda, Phys. Rev. A 66 (2002) [7] E.H. Lieb, R. Seiringer, Commun. Math. Phys. 264 (2006) [8] E. Sandier, J. Funct. Anal. 152 (1998) [9] E. Sandier, S. Serfaty, Ann. Sci. Ecole Norm. Sup. 33 (2000) [10] R. Seiringer, Commun. Math. Phys. 229 (2002) [11] R. Seiringer, J. Phys. A: Math. Gen. 36 (2003) 25