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

14 Vortex nucleation
excitation of the l =2quadrupolar surface mode. For a weakly deformed trap this instability
sets in at an angular velocity (see Eq.(1.20) with n =0,l =2)
ωcr(n =0,l =2)= ω0,l=2
,
2
(2.17)
where ω0,l=2 is the frequency of the surface oscillation with angular momentum l =2¯h. In
the Thomas-Fermi limit the critical angular velocity takes the simple form (see Eq.(1.21) with
l =2)
ωcr = ω⊥
√2 ≈ 0.707ω⊥ . (2.18)
The experimental evidence that Eq. (2.17) provides a good estimate for the critical angular
velocity for the nucleation of quantized vortices indicates that the quadrupolar shape deformation
of the condensate plays a crucial role, in agreement with the theoretical considerations
developed in [50, 51, 17, 52, 53, 54]. This is further supported by the observation of strong ellipsoidal
deformations observed temporarily during the process of nucleation in the experiments
[10, 9, 11, 13] and in simulations (see for example [54, 55, 56]). These observations indicate
that the evolution of the shape deformation serves as a mechanism for vortex nucleation.
A quadruplar deformation of the density distribution can be described by the parameter
δ = 〈y2 − x2 〉
〈y2 + x2 . (2.19)
〉
The expression (2.6) for the vortex energy regards a condensate whose deformation mirrors
the one of the trap (1.3). In this case δ = ε and hence δ is fixed by the external conditions.
When the system becomes unstable towards the creation of quadrupolar surface oscillations
it is appropriate to release this constraint and to let the condensate deformation δ be a free
parameter. In this way, we allow the system to take deformations different from the trap
deformation
δ = ε. (2.20)
In this section we will derive expressions for the energy and the angular momentum valid for
arbitrary condensate deformation δ. This allows us to show that at sufficiently large angular
velocities Ω the system can bypass the energy barrier by strongly deforming itself.
Quadrupolar velocity field
In order to describe properly the effects of the quadrupole deformation we introduce, in addition
to the vortical field (2.2), an irrotational quadrupolar velocity field given by
vQ = α∇(xy) , (2.21)
where α is a parameter. Note that vQ is the velocity field in the laboratory frame expressed in
terms of the coordinates of the rotating frame. The form (2.21) is suggested by the quadrupolar
class of irrotational solutions exhibited by the time-dependent Gross-Pitaevskii equation in
the rotating frame [16]. In Ref.[16] it has been shown that these solutions are associated
with a quadrupole deformation of the density described by the deformation parameter δ (see
Eq.(2.19)). The parameters δ and α characterize the quadrupole degrees of freedom that we
are including in our picture. In order to provide a simplified description, we fix a relationship