Rapidly rotating Bose-Einstein condensatesâ Alexander Fetter ...

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3 Vortex arrays in mean-field Thomas-Fermi regime Feynman’s mean vortex density in a <strong>rotating</strong> superfluid • solid-body rotation has v sb = Ω × r • v sb has constant vorticity ∇ × v sb = 2Ω • each quantized vortex at r j has localized vorticity ∇ × v = 2π M δ(2) (r − r j ) ẑ • assume N v vortices uniformly distributed in area A bounded by contour C • circulation around C is N v × 2π/M • but circulation in A is also 2ΩA • hence vortex density is n v = N v /A = MΩ/π • area per vortex 1/n v is π/MΩ ≡ πl 2 which defines radius l = √ /MΩ of circular cell • intervortex spacing ∼ 2l decreases like 1/ √ Ω • analogous to quantized flux lines (charged vortices) in type-II superconductors 18
As Ω increases, the mean vortex density n v = MΩ/π increases linearly following the Feynman relation • in addition, centrifugal forces expand the condensate radially, so that the area πR⊥ 2 also increases • hence the number of vortices N v = n v πR 2 ⊥ = MΩR2 ⊥ / increases faster than linearly with Ω • conservation of particles implies that the condensate also shrinks axially • TF approximation assumes that interaction energy 〈g|Ψ| 4 〉 and trap energy 〈V tr |Ψ| 2 〉 are large relative to kinetic energy for density variations ( 2 /M)〈(∇|Ψ|) 2 〉 • radial expansion of <strong>rotating</strong> condensate means that central density eventually becomes small 19
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: • MIT group has prepared consider
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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