Rapidly rotating Bose-Einstein condensatesâ Alexander Fetter ...

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Static behavior of a vortex line in axisymmetric trap V tr (r, z) = 1 2 M ( ω 2 ⊥r 2 + ω 2 zz 2) • If ω z ≫ ω ⊥ , strong axial confinement gives disk-shaped condensate • If ω ⊥ ≫ ω z , strong radial confinement gives cigar-shaped condensate • for vortex on axis, condensate wave function is Ψ(r,z) = e iφ |Ψ(r, z)| • velocity is v = (/Mr) ˆφ, like uniform condensate • centrifugal energy again forces wave function to vanish for r ξ • density is now toroidal; hole along symmetry axis • TF limit: separated length scales with ξ (vortex core) ≪ d 0 (mean oscillator length) d 0 (mean oscillator length) ≪ R 0 (mean condensate radius) • hence TF density is essentially unchanged by vortex apart from small hole along the vortex core 8
Energy of rotating TF condensate with one vortex • use density of vortex-free TF condensate; cut off the logarithmic divergence at core radius ξ • if condensate is in rotational equilibrium at angular velocity Ω, the appropriate energy functional is [8] E ′ [Ψ] = E[Ψ] − Ω · L[Ψ] where L is the angular momentum • let E ′ 0 be energy of rotating vortex-free condensate • let E ′ 1(r 0 , Ω) be energy of a rotating condensate with straight vortex that is displaced laterally by distance r 0 from symmetry axis • approximation of straight vortex works best for disk-shaped condensate (ω z ω ⊥ ) • Difference of these two energies is energy associated with formation of vortex ∆E ′ (r 0 , Ω) = E ′ 1(r 0 , Ω)−E ′ 0 • ∆E ′ (r 0 , Ω) depends on position r 0 of vortex and on Ω 9
Page 1 and 2: Rapidly rotating Bose-Einstein cond
Page 3 and 4: • assume an energy functional ⎡
Page 5 and 6: • general property: circulation a
Page 7: One vortex line in trapped BEC Firs
Page 11 and 12: curve (a) is ∆E ′ (r 0 , Ω) fo
Page 13 and 14: 2 Experimental creation/detection o
Page 15 and 16: École Normale Supérieure (ENS) in
Page 17 and 18: • MIT group has prepared consider
Page 19 and 20: As Ω increases, the mean vortex de
Page 21 and 22: For Ω along z, can complete square
Page 23 and 24: • TF formulas for condensate radi
Page 25 and 26: • variation with respect to u yie
Page 27 and 28: Tkachenko oscillations of the vorte
Page 29 and 30: • rough agreement with JILA exper
Page 31 and 32: General two-dimensional energy func
Page 33 and 34: • assume that the GP wave functio
Page 35 and 36: Take this LLL trial function seriou
Page 37 and 38: • if vortex lattice is exactly un
Page 39 and 40: • in either scenario, BEC and the
Page 41 and 42: What is physics of this phase trans
Page 43 and 44: References [1] F. Dalfovo, S. Giorg
Page 45: [27] I. Coddington, P. Engels, V. S

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