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

4.2 Tight binding regime 45
parameter (4.30) (see for example [101]). The result reads
1
δj = ER
2 (−1)j
2
π
1/2 s
4
(j−1)/2+3/4 24(j−1)+5 e
(j − 1)!
−2√s , (4.35)
where j =1, 2, ... is the index of the band for which the tight binding regime is considered. In
order for expression (4.35) to be valid, the considered energy band must be a slowly varying
function of the quasi-momentum. Hence, the potential depth s must be sufficiently large.
Fig. 4.8 depicts (4.35) as a function of potential depth for the bands j = 1, 2. The
tunneling parameter δj=1 of the lowest band is much smaller than δj=2. This reflects the fact
that the Wannier functions of the lowest band concentrate at one site more easily than those
of higher bands.
In Fig.4.9 we also plot the ratio between the approximative δ (4.35) for j =1and the exact
δ as obtained from Eq.(4.33) using the numerical data for m∗ .Wefindthatfors>30 (4.35)
differs from the numeric result by less than 10%.
δ/ER
10 1
10 0
10 −1
10 −2
10
10 15 20 25 30
−3
Figure 4.8: Tunneling parameter (4.35) of the 1D Mathieu problem as a function of potential
depth for the bands j =1(solid line) and j =2(dashed line).
s