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

124 Macroscopic Dynamics
Let us consider the example of a cylindrically symmetric trap with ωx = ωy = ω⊥: For
ω⊥ ≫ ˜ωz the system behaves like an elongated cigar shaped condensate for which the lowest
frequency solutions are given by the center-of-mass motion (“dipole mode”)
ωD =
m
ωz
(9.51)
m∗ and by the axial breathing mode (“quadrupole mode”)
ωQ =
5 m
2 m∗ ωz . (9.52)
These frequencies do not depend on the chemical potential, and therefore on the coupling
constant. Still, the occurrence of the factor 5/2 in the quadrupole frequency ωQ is a nontrivial
consequence of the mean-field interaction [43]. In addition to the low frequency axial
motion the system exhibits radial oscillations at high frequency, of the order of ω⊥. Themost
important ones are the transverse breathing and quadrupole oscillations occuring at ω =2ω⊥
and ω = √ 2ω⊥ respectively. For elongated traps with ω⊥ ≫ ˜ωz we predict the frequencies
of these modes not to be affected by the optical lattice. Note that the result for the dipole
frequency (9.51) was obtained for the tight binding regime in [73].
A different scenario is obtained for ˜ωz ≫ ω⊥. In such a setting the system moves like a
disk shaped condensate: While the dipole mode, of course, still has the frequency (9.51), the
breathing mode oscillates at
ωQ = √
m
3 ωz
(9.53)
m∗ instead of (9.52). The low frequency modes involving the radial direction are, apart from the
dipole mode ω = ω⊥ given by ω = 10/3ω⊥ and ω = √ 2ω⊥ respectively.
Effective change of geometry
In chapter 5.4 we have found that the aspect ratio of a static harmonically trapped condensate
does not change when a lattice is superimposed, i.e. its shape of the condensate remains
the same. Yet, the above discussion shows that the dynamics of the system is governed
by an effective change of geometry: The condensate oscillates as if it was a condensate
without lattice in a trap with the frequencies ˜ωz = m/m ∗ ωz. Accordingly, an elongated
condensate (ωz > ω⊥) can behave as an effectively spherical ( m/m ∗ ωz = ω⊥) oreven
cigar-shaped system ( m/m ∗ ωz >> ω⊥). In the latter example, the effective change of
geometry manifests itself in a change of the axial breathing mode frequency from √ 3 m/m ∗ ωz
to 5/2 m/m ∗ ωz.
The results obtained in this section can easily be generalized to cubic two-dimensional
lattices. The frequencies of the low energy collective modes are then obtained from those in