Rapidly rotating Bose-Einstein condensatesâ Alexander Fetter ...

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4 Vortex arrays in mean-field quantum-Hall regime Lowest-Landau-Level (quantum-Hall) behavior When the vortex cores overlap, kinetic energy associated with density variation around each vortex core becomes important • hence the TF approximation breaks down (it ignores this kinetic energy from density variations) • return to full GP energy E ′ [Ψ] in the rotating frame. • in this limit of rapid rotations (Ω ω ⊥ ), Ho [28] incorporated kinetic energy exactly • condensate expands and is effectively two dimensional • for simplicity, treat a two-dimensional condensate that is uniform in the z direction over a length Z • condensate wave function Ψ(r,z) can be written as √ N/Z ψ(r), where ψ(r) is a two-dimensional wave function with unit normalization ∫ d 2 r |ψ| 2 = 1 30
General two-dimensional energy functional in rotating frame becomes ⎛ ⎞ ∫ E ′ [ψ] = d 2 r ψ ∗ ⎜ p 2 ⎝2M + 1 2 Mω2 ⊥r 2 − ΩL z + 1 } {{ } 2 g 2D|ψ| 2 ⎟ ⎠ } {{ } ψ, one−body oscillator H 0 interaction where p = −i∇, L z = ẑ · r × p, and g 2D = Ng/Z One-body oscillator hamiltonian in rotating frame H 0 is exactly soluble and has eigenvalues [29] ɛ nm = [ω ⊥ + n (ω ⊥ + Ω) + m (ω ⊥ − Ω)] where n and m are non-negative integers • in limit Ω → ω ⊥ , these eigenvalues are essentially independent of m (massive degeneracy) • n becomes the Landau level index • lowest Landau level with n = 0 is separated from higher states by gap ∼ 2ω ⊥ 31
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 and 8: One vortex line in trapped BEC Firs
Page 9 and 10: Energy of rotating TF condensate wi
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: • rough agreement with JILA exper
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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