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

68 Stationary states of a BEC in an optical lattice
potential (6.5) due to interactions. In order to have a look at the change of the k-dependence
brought about by interactions, we plot in Fig. 6.5 b) the same data as in Fig. 6.5 a), but
from each data set we subtract the respective groundstate value. We observe that interactions
most affect the k-dependence of the lowest bands, especially in a deep lattice where higher
bands are almost identical to the single particle case.
ε/ER, µ/ER
12
11
10
9
8
7
6
5
4
3
a) b)
2
−1 −0.5 0 0.5 1
¯hk/qB
16
14
12
10
8
6
4
2
−1 −0.5 0 0.5 1
¯hk/qB
Figure 6.4: Bloch band spectra εj(k) (6.4) (solid lines) and µj(k) (6.5) (dashed lines) for
gn =0.5ER at lattice depth a) s =5and b) s =10.
Gap in the band spectra
The opening up of gaps between the bands is a feature that physically characterizes a system
in presence of a periodic potential. Fig.6.6 displays this gap between first and second band
as a function of lattice depth for different values of gn/ER for both the energy per particle
(6.4) and the chemical potential (6.5). At fixed s, the gap becomes smaller when gn/ER is
increased which again can be understood as a screening effect. Yet, quantitatively the effect
is not very large. For large s the gap very slowly approaches the value 2 √ sER given by the
harmonic approximation of the potential well. Note that in Fig. 6.6 we do not include the
range of small potential depths where swallow tails exist (see comments at the end of this
section).
The gap in the energy spectrum has been studied experimentally in [67, 68, 69]