Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

2.2 Role of Quadrupole deformations 13
Ev(d/R⊥, Ω,µ)/Ev(d =0,µ)
0.2
0
−0.2
−0.4
0
Ω=Ωv(µ)
0.2
Ω= 3
2 Ωv(µ)
0.4 0.6
d/R⊥
Figure 2.1: Vortex excitation energy in the rotating frame (2.10) for an axisymmetric configuration
as a function of the reduced vortex displacement d/R⊥ from the center. The curves
refer to two different choices for the angular velocity of the trap: Ω=Ωv(µ) (solid line), and
Ω=3Ωv(µ)/2 (dotted line), where Ωv(µ) =Ev(d =0,µ)/N ¯h. The initial vortex-free state
corresponds to d/R⊥ =1.ForΩ > Ωv(µ), the state with a vortex at the center (d/R⊥ =0)
is preferable. However, in this configuration the nucleation of the vortex is inhibited by a
barrier separating the vortex-free state (d/R⊥ =1) from the energetically favored vortex state
(d/R⊥ =0).
Equation (2.16) emphasizes the fact that the nucleation of the vortex is associated with
an increase of angular momentum from zero (no vortex) to Ntot¯h (one centered vortex),
accompanied by an initial energy increase (barrier) and a subsequent monotonous energy
decrease. In this form the TF-result (2.10) can be compared with alternative approaches
based on microscopic calculations of the vortex energy.
2.2 Role of Quadrupole deformations
In the previous section we have shown that in order to enter a vortex-state the system has
to overcome a barrier associated with a macroscopic energy cost. This barrier exists at any
angular velocity Ω > Ωv, withΩv given by (2.11), if the position d of the vortex line is the only
degree of freedom of the system. We need to go beyond this description in order to explain
vortex nucleation.
Hints as to what are the crucial degrees of freedom involved are close at hand: The critical
angular velocity (2.1) observed in the experiments [6, 7, 8, 10, 9] for small ε turns out to be
close to the value associated with the energetic instability of the vortex-free state towards the
0.8
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