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

76 Stationary states of a BEC in an optical lattice
As a consequence of the displacement property of the Wannier functions (6.27), we have
L/2
f
−L/2
3 L/2
j (z)fj(z ± d)dz = fj(z)f
−L/2
3 j (z ± d)dz , (6.28)
and we obtain the result
L/2
εj(k) = fj(z) −
−L/2
¯h2 ∂
2m
2
2 πz
+ sERsin +
∂z2 d
gnd
2 f 2
j (z) fj(z)dz
L/2
+2cos(kd) fj(z) −
−L/2
¯h2 ∂
2m
2
2 πz
+ sERsin +2gndf
∂z2 d
2
j (z) fj(z − d)dz
= ε0j − δj cos(kd) , (6.29)
where in the last step we have defined the quantities
ε0j =
L/2
fj(z) − ¯h2 ∂
2m
2
2 πz
+ sERsin +
∂z2 d
gnd
2 f 2
j (z) fj(z)dz , (6.30)
−L/2
L/2
δj = −2
−L/2
fj(z) − ¯h2 ∂
2m
2
2
+ sERsin
∂z2
πz
+2gndf
d
2
j (z) fj(z − d)dz .
(6.31)
Comparison of Eq.(6.29) with Eq.(4.29) reveals that in the tight binding regime, the energy
bands of a condensate have the same form as in the single particle case. The first term ε0j
is an off-set, while the second describes the formation of a band of height 2δj and a cos(kd)dependence
on the quasi-momentum. In particular, expression (6.31) generalizes the definition
of the tunneling parameter (4.30) to a condensate in presence of interactions.
In contrast to the single particle case, the off-set ε0j and the tunneling parameter δj do not
only depend on lattice depth, but also on density. This density-dependence shows up in two
ways: implicitly through the density-dependence of the Wannier function fj, which can often
be neglected, and explicitly through the interaction term.
We can apply the same considerations to the calculation of the chemical potential in the
tight binding regime. In this way, we obtain
d/2
µj(k) = ϕ
−d/2
∗
jk(z) − ¯h2 ∂
2m
2
2 πz
+ sERsin + gnd|ϕjk(z)|
∂z2 d
2
ϕjk(z)dz
= µ0j − δµ,j cos(kd) , (6.32)
where we have defined the quantities
µ0j =
L/2
−L/2
fj(z) − ¯h2 ∂
2m
2
2 πz
+ sERsin + gndf
∂z2 d
2
j (z) fj(z)dz , (6.33)
fj(z) − ¯h2 ∂
2m
2
2 πz
+ sERsin +4gndf
∂z2 d
2
j (z) fj(z − d)dz . (6.34)
L/2
δµ,j = −2
−L/2