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

166 Condensate fraction
12.4 Tight binding regime
In the regime of deep optical lattices, one can neglect contributions to the depletion from
higher bands, because high energy excitations are particle-like (see discussion section 7.2) and
hence
vj=1,q,px,py (r)
≈ 0 . (12.50)
uj=1,q,px,py (r)
We are then allowed to restrict the sum in (12.1) to j =1.
In the tight binding limit, one can easily generalize expressions (7.25,7.26) for the Bogoliubov
amplitudes in the lowest band to account for transverse excitations
uq,px,py(r) = ei(pxx+pyy)/¯h
L
vq,px,py(r) = ei(pxx+pyy)/¯h
L
Uq,px,py
√ Nw
Vq,px,py
√ Nw
e iqld/¯h f(z−ld), (12.51)
l
l
e iqld/¯h f(z−ld), (12.52)
where f(z) is the condensate Wannier function (see Eq.(6.26) with j =1). Starting from this
ansatz and for simplicity neglecting contributions arising from n∂δ/∂n (see discussion section
7.2), Eq.(7.38) can be generalized to 3D in a straightforward way, yielding
where
¯hω(p⊥,q)=
ε(p⊥,q)(ε(p⊥,q)+2κ −1 ), (12.53)
ε(p⊥,q)= p2 ⊥
m +2δ sin2 (qd/2) (12.54)
and p 2 ⊥ = p2 x + p 2 y. For the amplitudes Uq,px,py and Vq,px,py in (12.51,12.52) we find the result
Uq,px,py =
Vq,px,py =
ε +¯hω
2 √ ,
¯hωε
(12.55)
ε − ¯hω
2 √ ,
¯hωε
(12.56)
ensuring the 3D normalization condition (12.10).
Omitting the sum over j>1 in (12.1) we are left with the expression
∆Ntot
Ntot
= 1
Ntot
= 1
Ntot
q,px,py
q,px,py
[ε(px,py,q) − ¯hω(px,py,q)] 2
4¯hω(px,py,q)ε(px,py,q)
1 ε(px,py,q)+κ
2
−1
− 1
¯hω(px,py,q)
(12.57)
with ε(px,py,q) and ω(px,py,q) given by (12.54) and (12.53) respectively. The spectrum of
values of q and px, py is given by (12.2) and (12.3 ) to satisfy the correct boundary conditions.