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

18 Vortex nucleation
0.00
⊕
+0.02 ⊘
d/Rx
0
-0.38
⊗
0.1 0.2
0.3
-0.03
▽
⊙
-0.01
δ
0.4
Ω=0.7 ω⊥
Figure 2.3: Above the critical angular velocity of vortex nucleation: the plot shows the dependence
of the total energy (2.27) minus the energy of the initial non-deformed vortexfree
state E(d/Rx =1,δ =0)on the quadrupolar shape deformation δ and on the vortex
displacement d/Rx from the center. Energy is given in units of N¯hω⊥. The dashed
line corresponds to Etot − E(d/Rx = 1,δ = 0) = 0, while the solid curve refers to
Etot − E(d/Rx = 1,δ = 0) = 0.015N¯hω⊥. In this plot ε = 0.04, µ = 10¯hω⊥, as in
Fig. 2.2, and Ω=0.7ω⊥. Important states are indicated as in Fig. 2.2. At the chosen Ω,
the saddle ⊙ lies lower than the initial state ⊕ allowing the system to bypass the barrier ⊘
by taking a quadrupolar deformation δ and reach the preferable vortex state ⊗. Notealsothe
existence of a favorable deformed and vortex-free state labeled by ▽ [16].
parameters ε and µ/¯hω⊥. Fig. 2.4 shows the dependence of ¯ Ωc on ε for different choices of
µ/¯hω⊥. To lowest order in ε and ¯hω⊥/µ the dependence is given by
¯Ωc(ε)− 1
√ ≈
2 1
(Aη) 1/2 ε
√ −
2 4 (Aη) 1/4
, (2.31)
where
η = log 1.342 µ
5/2 7/2 ¯hω⊥
,
¯hω⊥ µ
and A =2−5/421 √ 3. This formula shows that the relevant parameters of the expansion are
η1/2 and ε/η1/4 . Fig. 2.4 demonstrates that it is applicable also at rather small values of the
chemical potential.
At ε = 0 the vortex nucleation, according to the present scenario, is possible only at
angular velocities slightly higher than the value ω⊥/ √ 2. For non-vanishing ε the preferable
configuration will be always deformed even for small values of ¯ Ω where δ depends linearly on
ε. At higher ¯ Ω the condensate gains energy by increasing its deformation in a nonlinear way
0.5
0.6
0.7