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

2.2 Role of Quadrupole deformations 15
between these two parameters by requiring that the quadrupole velocity field satisfies the
condition
∇ · [n(r)(vQ − Ω × r)] = 0 , (2.22)
where Ω =Ωˆz. As a consequence of the equation of continuity, this condition implies that in
the rotating frame the density of the gas is stationary except for the motion of the vortex core
which involves a change of the density at small length scales. Eq.(2.22) yields the relationship
[16]
α = −Ωδ (2.23)
which selects a natural class of paths that will be considered in the present investigation.
In the presence of the quadrupole velocity field (2.21), the energy of the condensate in the
rotating frame can then be expressed in terms of the deformation parameter δ only. Using the
formalism of [16] we find the expression
EQ(δ, ¯
2 1 − εδ −
Ω,ε,µ)=Nµ
7
¯ Ω2δ2 √
1 − ε2 √
3
+ . (2.24)
1 − δ2 7
Here, ¯ Ω=Ω/ω⊥ and we have neglected the change of the central density caused by the
velocity field (2.21). This assumption will be used throughout the paper5 . Keep in mind that
(2.24) is the energy in the rotating frame and hence already includes the angular momentum
term −m Ω · dr n(r)[r × vQ].
It is useful to expand Eq. (2.24) as a function of δ in the case of axisymmetric trapping
(ε =0). We find the result
EQ(δ, ¯
5 1
Ω,ε=0,µ) Nµ + δ2
7 7 (1 − 2¯ Ω 2
) + O(δ 3
) . (2.25)
which explicitly shows that for Ω >ω⊥/ √ 2 the symmetric configuration (δ =0) is energetically
unstable against the occurrence of quadrupole deformations.
A quadrupolar deformed vortex state
In order to explore the role of quadrupolar shape deformations for vortex nucleation we assume
the presence of a velocity field
v = vvortex + vQ , (2.26)
where vvortex is the velocity field (2.3) associated with a straight vortex line at distance d on
the x-axis while vQ is given by (2.21) (with α chosen according to (2.23)) and gives rise to a
quadrupolar shape deformation. The total energy of the condensate in the rotating frame in
the presence of the velocity field (2.26) reads
Etot(d/Rx,δ, ¯ Ω,ε,µ)=Ev(d/Rx,ε,µ)+EQ(δ, ¯ Ω,ε,µ)+E3(d/Rx,δ, ¯ Ω) , (2.27)
where Ev(d/Rx,ε,µ) is the energy (2.4) of the vortex in the lab frame and EQ is the energy
in the rotating frame (2.24) due to the presence of the irrotational velocity field vQ. Thethird
5 Imposing that the energy (2.24) be stationary with respect to δ yields the solutions derived in [16] apart
from small corrections due to the changes of the central density not accounted for in the present formalism.