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

7.3 Tight binding regime of the lowest Bogoliubov band 93
binding regime, the situation is simpler in the case of the lowest band: Since the excitation
energies go to zero as s is increased, the Bogoliubov amplitudes are approximately proportional
to the condensate wavefunction of the groundstate. So we set
uq(z) = 1
√ Uqϕq(z) (7.22)
Nw
vq(z) = 1
√ Vqϕq(z) , (7.23)
Nw
where Uq and Vq are numbers that depend on the quasi-momentum q. The normalization
condition (7.8) implies that
|Uq| 2 −|Vq| 2 =1. (7.24)
Expanding the condensate wavefunction in its Wannier basis we can write
uq(z) = 1
√ Uq
Nw
l
fl(x)e iqld
(7.25)
vq(z) = 1
√ Vq fl(x)e
Nw l
iqld . (7.26)
Our discussion of elementary excitations in the tight binding regime of the lowest Bogoliubov
band will be based on these expressions for the Bogoliubov amplitudes. We presuppose the
lowest Bloch band to be within the tight binding regime so that only next-neighbour overlap
has to be considered.
Lowest Bogoliubov band
To solve for the lowest Bogoliubov band ¯hω(q), we first add and subtract the two equations
(7.14,7.15) yielding two coupled equations for uq + vq and uq − vq
L1(uq + vq) =¯hω(q)(uq − vq) , (7.27)
L3(uq − vq) =¯hω(q)(uq + vq) , (7.28)
where
L1 = − ¯h2 ∂
2m
2
∂z2 + sER sin 2
πz
+ dgn|ϕ(z)|
d
2 − µ, (7.29)
L3 = − ¯h2 ∂
2m
2
∂z2 + sER sin 2
πz
+3dgn|ϕ(z)|
d
2 − µ. (7.30)
Eliminating uq + vq or uq − vq from (7.27,7.28), we obtain
The matrix elements of L1 and L3 take the form
L3 L1(uq + vq) =¯h 2 ω(q) 2 (uq + vq) , (7.31)
L1 L3(uq − vq) =¯h 2 ω(q) 2 (uq − vq) . (7.32)
〈fl|L1|fl〉 = δ, 〈fl|L1|fl±1〉 = − δ
, (7.33)
2
〈fl|L3|fl〉 =3δ− 2δµ +2gnd dzf 4 (z) , 〈fl|L3|fl±1〉 = δ
2 − δµ . (7.34)