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

54 Groundstate of a BEC in an optical lattice
1/κgn
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0 5 10 15 20 25 30
Figure 5.5: Comparison of κ −1 /gn as obtained from Eq.(5.11) for gn =0.1ER (dashed line)
and gn =0.5ER (dash-dotted line) with the approximate formula (5.12) (solid line) evaluated
using (5.15) as a function of s.
5.3 Momentum distribution
Since the groundstate solution of the GPE (5.4) is periodic with period d, it can be expanded
in the Fourier series
ϕ(z) =
al e il2πz/d , (5.16)
l
showing that the contributing momenta are multiples of 2qB justasforasingleparticlein
a periodic potential (see discussion section 4.1). The values of the coefficients al involve
interaction effects: The contribution of momenta with l = 0is slightly reduced in the presence
of repulsive interactions, due to the screening effect on the lattice.
The fact that the momentum distribution in the direction of the lattice is characterized by
several momentum components is an interesting difference with respect to the uniform system
where the presence of a BEC is associated with a single peak in the momentum distribution.
This feature governs the expansion of atoms released from an optical lattice: During the time
of flight the different momentum components are separated spatially resulting in a density
distribution featuring several peaks (see Fig. 5.6). Due to the large kinetic energy contained
in (5.8) the role of interactions can be neglected after the potential has been switched off, in
contrast to a situation without lattice where the expansion is governed by mean-field effects.
This issue has been discussed in [70] with regard to both theory and experiment.
s