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

Chapter 1
Introduction
Superfluids differ from normal fluids in their behavior under rotation [1] (chapter 6 and 14).
Prominent examples are the reduction of the momentum of inertia and the occurrence of
quantized vortices. Since the achievement of Bose-Einstein condensation in trapped ultra-cold
dilute atomic gases in 1995 [2, 3, 4], exploring the superfluid properties of these systems has
been the primary motivation for setting up rotating traps.
One of the experimental methods [5, 6, 7, 8, 9, 10, 11] consists in shining a laser beam
along the axis of a cylindrically symmetric magnetic trap
Vmag = m
2
ω 2 r
x 2 + y 2
+ ω 2 zz 2
. (1.1)
The laser beam axis is moved back and forth very rapidly between two positions symmetric
with respect to the z-axis. The atoms feel a time averaged dipole potential which is anisotropic
in the xy plane
δV (r) = 1
2 mω2
r ɛxx 2 + ɛyy 2
, (1.2)
where ɛx and ɛy depend on the intensity, the beam waist and on the spacing between the
extreme positions of the beam with respect to the z-axis. In addition to the fast movement
of the beam the xy axes can be rotated at an angular velocity Ω. In the lab frame the total
potential V = Vmag + δV is given by
Vlab = m
ω
2
2
⊥ x 2 + y 2
+ ω 2 zz 2
+ m
2 εω2
⊥ x 2 − y 2
cos (2Ωt)+2xy sin (2Ωt) . (1.3)
Here, ε describes the ellipsoidal deformation of the trapping potential in the rotating xy plane
ε = ω2 x − ω2 y
ω2 x + ω2 , (1.4)
y
where ω2 x,y = ω2 r(1 ± ɛx,y), andω⊥is an average transverse oscillator frequency
ω 2 ⊥ = ω2 x + ω 2 y
2
In a corotating coordinate system the potential takes the form
Vrot(r) = m
2
. (1.5)
(1 + ε) ω 2 ⊥x 2 +(1− ε) ω 2 ⊥y 2 + ω 2 zz 2
. (1.6)
5