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

6 Introduction
Using the same experimental scheme a strong anharmonic rotating deformation of the atomic
cloud has been used by [12, 11]. Experiments with a potential of the form (1.6) have also
been reported in [13, 14]. In these cases the rotation of the trap is achieved by rotating the
magnetic trap itself.
This chapter focuses on Bose-Einstein condensates loaded into a trap of the form (1.6).
The discussion can be generalized to describe systems in set-ups of the kind used in [12, 11].
Many properties of low temperature dilute-gas Bose-Einstein condensates can be understood
assuming zero temperature and working within the framework of Gross-Pitaevskii (GP) theory
[1, 15]. Within this framework all atoms are condensed into a single mode ϕ(r,t) often denoted
as the condensate wavefunction. The quantity Ψ(r,t)= √ Ntotϕ(r,t) constitutes an order
parameter. Its modulus and phase S are closely related to the density distribution and the
velocity field respectively
n(r,t)=|Ψ(r,t)| 2 , (1.7)
v(r,t)= ¯h
∇S(r,t) . (1.8)
m
The order parameter’s temporal evolution in the external potential Vext(r,t) obeys the Gross-
Pitaevskii equation (GPE)
i¯h ∂Ψ(r,t)
= −
∂t
¯h2
2m ∇2 + Vext(r,t)+g |Ψ(r,t)| 2
Ψ(r,t) . (1.9)
Two-body interaction between atoms is accounted for by the nonlinear term which is governed
by the coupling constant
g = 4π¯h2 a
, (1.10)
m
where a is the s-wave scattering length.
To describe a condensate in the rotating trap (1.3) we set Vext = Vlab in (1.9). The GPE
for the order parameter ΨR = exp(iΩˆ Lzt/¯h)Ψ in the reference frame rotating with angular
velocity Ω around the z-axis takes the form
i¯h ∂ΨR(r,t)
∂t
=
− ¯h2
2m ∇2 + Vrot(r)+g |ΨR(r,t)| 2 − Ω ˆ Lz
ΨR(r,t) , (1.11)
where ˆ Lz is the z-component of the angular momentum operator and Vrot(r) is the timeindependent
potential (1.6). Stationary solutions of (1.11) satisfy the equation i¯h∂ΨR/∂t =
µΨR.
A first class of stationary solutions is associated with an irrotational velocity field
∇×v =0. (1.12)
A condensate in such a state can carry angular momentum. Yet, the circulation of the velocity
field is zero everywhere, i.e.
dl · v =0, (1.13)
for any closed contour. Such states have been studied theoretically in [16, 17, 18] and investigated
experimentally in [10, 14].