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

2.3 Critical angular velocity for vortex nucleation 19
Ωc/ω⊥
0.8
0.6
0.4
0.2
0.0
0.00
0.05
1√ 2
µ =9¯hω⊥ analytic
µ =9¯hω⊥
µ =18¯hω⊥
Figure 2.4: Critical angular velocity of vortex nucleation in units of ω⊥ as a function of the
trap deformation ε. The long dashed and short dashed curves correspond to the numerical
calculation satisfying condition (2.30) for µ =9¯hω⊥ and µ =18¯hω⊥ respectively, while the
solid line is the analytic prediction (2.31) evaluated with µ =9¯hω⊥. The arrow indicates the
angular velocity at which the quadrupole surface mode becomes unstable in the case ε =0.
The value µ =9¯hω⊥ is close to the experimental setting of [6].
(see [16]). Eq. (2.31) and Fig. 2.4 show that for non-vanishing ε the saddle point on the
energy ridge can be surpassed at angular velocities smaller than ω⊥/ √ 2. The critical angular
velocity can be lowered further by increasing the value of µ/¯hω⊥.
The above exemplified scenario is in reasonable agreement with the experiments. For
example, for ε =0.045 ,µ =8.71¯hω⊥ 6 a critical angular velocity Ωc =0.64ω⊥ is obtained
from the measurement of the angular momentum (see data reported in Fig. 2 of [7]). For this
value of ε and µ =9¯hω⊥ we find Ωc =0.68ω⊥ (see Fig.2.4). Further on, increasing the value
of ε is found to lower the critical angular velocity: In [9] a decrease of ≈ 6% of Ωc is observed
when increasing ε from 0.01 to 0.019. For this setting we find a decrease of 2%. Several
papers note that the nucleation range extends to lower Ω for larger ε or longer stirring times
[12, 10, 11, 9]. The experiment [11] demonstrates particularly clearly that a strong stirrer with
l =2-symmetry corresponding to a large ε shifts Ωc to angular velocities significantly (≈ 30%)
below ω⊥/ √ 2.
The data reported in the experimental papers is not sufficient to verify our prediction for the
dependence of Ωc on µ/¯hω⊥. For the small values of ε used in the experiments [6, 7, 8, 10, 9]
the calculated dependence on µ/¯hω⊥ is in fact very weak. This seems in agreement with
6
Note that in this experiment the static magnetic trap has an anisotropy of ε ≈ 0.01 which is not taken into
account in our comparison.
ε
0.10
0.15