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

6.2 Tight binding regime 79
2m¯v/qB
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0.2
0.1
0
−0.1
a)
b)
−0.2
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
¯hk/qB
Figure 6.11: Comparison of the tight binding expression (6.39) for the group velocity in the
lowest band with the respective numerical solution for gn =0.5ER at a) s =5,b)s =10.
Compressibility and effective coupling
The compressibility κ in the tight binding regime can be calculated by inserting the tight
binding expression for the chemical potential (6.32) with j =1,k =0into the definition
κ −1 = n∂µ/∂n. Taking into account result (6.38) and (6.36), we find
κ −1 L/2
= gnd
−L/2
f 4 L/2
(z)+8gnd
−L/2
f 3 (z)f(z − d)dz , (6.43)
where µ0 was defined in Eq.(6.33). The first term is the leading order on-site contribution,
while the second contribution is due to the small overlap of neighbouring Wannier functions
and can be often neglected. In deriving (6.43) we also discarded terms involving the derivating
∂f(z; n)/∂n. This presupposes the lattice to be deep enough to ensure that the effect of
interactions on the wavefunction is negligible. Expression (6.43) then takes the form
κ −1 L/2
= gnd
−L/2
f 4 gn=0(z) (6.44)
where fgn=0 is the single particle Wannier function for the lowest band. This expression is
expected to give a good account of the compressibility in a sufficiently deep lattice. It follows