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

10 Vortex nucleation
2.1 Single vortex line configuration
A quantized vortex is characterized by the appearance of a velocity field associated with a
non-vanishing, quantized circulation (see Eq.(1.14)). If we assume that the vortex with κ =1
is straight and oriented along the z-axis the quantization of circulation takes the simple form
∇ × vvortex = 2π¯h
m δ(2) (r − d)ˆz (2.2)
for a vortex located at distance d ≡|d| from the axis. The general solution of (2.2) can be
written in the form
vvortex = ∇ (ϕd + S) (2.3)
where ϕd is the azimuthal angle around the vortex line at position d and S is a single-valued
function which gives rise to an irrotational component of the velocity field. The irrotational
component may be important in the case of a vortex displaced from the symmetry axis and its
inclusion permits to optimize the energy cost associated with the vortex line 2 . Considering a
straight vortex line is a first important assumption that we introduce in our description 3 .
Vortex energy
The energy cost associated with a straight vortex line at the center of the trap is given by [31]
Ev(d =0,ε,µ)= 4πn0 ¯h
3
2
0.671R⊥
5 ¯hω⊥
Z log
= N¯hω⊥ 1 − ε
m ξ0
4 µ
2
log 1.342 µ
,
¯hω⊥
(2.4)
Here, Z is the TF-radius in z-direction, ξ0 is the healing length calculated with the central
density n0, µ is the TF-chemical potential and ε is the deformation of the trap introduced above
(see Eq.(1.6)). The factor 4n0Z/3 in (2.4) corresponds to the column density dz n(d =0,z)
evaluated within the TF-approximation at the trap center. Noting that the column density at
distance d from the center along the x-axis is given by
dz n(d, z) = 4n0Z
3
2 3/2
d
1 −
, (2.5)
Rx
we introduce the following simple description of the energy of a vortex located at distance d
from the center on the x-axis 4
2 3/2
d
Ev(d/Rx,ε,µ)=Ev(d =0,ε,µ) 1 −
, (2.6)
Rx
where Ev(d =0,ε,µ) is given by Eq.(2.4). This expression is expected to be correct within
logarithmic accuracy (see [27] and references therein). It could be improved by including an
2 For a uniform superfluid confined in a cylinder, this extra irrotational velocity field is crucial in order to
satisfy the proper boundary conditions and its effects can be exactly accounted for by the inclusion of an image
vortex located outside the cylinder. See for example [45].
3 The inclusion of curvature effects in the description of quantized vortices in trapped condensates has been
the subject of recent theoretical studies. See, for example, [46, 47, 48]. See also [27] and references therein.
4 Our results do not change if d/Rx is replaced by d/Ry. We expect that predicting the preferable direction
for a vortex to enter the condensate demands the calculation of the vortex energy beyond logarithmic accuracy.