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

78 Stationary states of a BEC in an optical lattice
mainly due to the explicit dependence of the tunneling parameter on gn in (6.31) and secondly
to the dependence of the Wannier function f on the density. In Fig. 6.9 one can see that the
deviation of m∗ (gn) from the single particle effective mass is approximately proportional to
gn. The departure from this linear law in gn is due to the density dependence of the Wannier
function f. For the considered interaction gn =0.5ER it has a small effect of about 5% on
the effective mass, to be compared with the 30% shift due to the explicit dependence on gn.
The generalized k-dependent effective mass (6.22) takes the form
1
m ∗ j (k) = d2 δj
¯h
2 cos(kd) . (6.42)
In Figs. 6.10 and 6.11 we compare the tight binding expressions for the lowest energy band
and the associated group velocity with the respective numerical solution. To evaluate (6.29)
and (6.39), the tunneling parameter (6.31) is obtained by inserting the numerical results for
m ∗ in Eq.(6.41). We find that the tight binding results provide a good description already at
s =10for gn =0.5ER. To obtain the same degree of agreement at a higher value of gn/ER
one has to go to larger s.
ε/ER
1
0.8
0.6
0.4
0.2
0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0.4
0.3
0.2
0.1
0.05
0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0.2
0.15
0.1
a)
b)
c)
0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
¯hk/qB
Figure 6.10: Comparison of the tight binding expression (6.29) for the lowest energy band with
the respective numerical solution for gn =0.5ER at a) s =1,b)s =5and c) s =10.