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

92 Bogoliubov excitations of Bloch state condensates
d|bjql| 2 ,d|cjql| 2
6
4
2
0
6
4
2
0
6
4
2
0
j =1, ¯hq =0.1qB
−2 −1 0 1 2
j =1, ¯hq =0.5qB
−2 −1 0 1 2
j =1, ¯hq =0.9qB
−2 −1 0 1 2
6
4
2
0
6
4
2
0
6
4
2
0
j =2, ¯hq =0.1qB
−2 −1 0 1 2
j =2, ¯hq =0.5qB
−2 −1 0 1 2
j =2, ¯hq =0.9qB
−2 −1 0 1 2
l l
Figure 7.6: Square modulus of the Fourier coefficients bjql (white bars) and cjql (black bars)
of ũjq(z) and ˜vjq(z) respectively as defined in (7.17,7.18) for the lowest Bogoliubov band
(j =1, left column) and the first excited Bogoliubov band (j =2, right column) at q =
0.1qB, 0.5qB, 0.9qB as obtained from the numerical solution of Eqs.(7.14,7.15) with gn =
0.5ER and lattice depth s =10.
7.3 Tight binding regime of the lowest Bogoliubov band
Since the Bogoliubov amplitudes have Bloch form (see Eqs.(7.11,7.12)), we can find Wannier
functions fu,j(z) and f (v)
l,j (z) for the ujq and the vjq-amplitudes respectively. In their respective
Wannier basis, the Bogoliubov amplitudes read
ujq(z) = 1
√
Nw
vjq(z) = 1
√ Nw
l
l
f (u)
l,j (z − ld)eiqld , (7.20)
f (v)
l,j (z − ld)eiqld . (7.21)
The Wannier functions f (u) (v)
l,j and f l,j are in general different from each other and from the
Wannier functions fj (see Eq.(6.25)) of the condensate in a Bloch state. Yet, in the tight