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

7.3 Tight binding regime of the lowest Bogoliubov band 95
s with fixed gn, we can neglect the term of O(δ2 ) under the squareroot in Eq.(7.38)
and the spectrum takes the form
¯hω(q) =2 √ δκ−1
sin
qd
, (7.39)
2¯h
Of course for large gn, the proper density-dependence of δ and κ −1 has to be taken
into account in evaluating (7.39). Note that the Bogoliubov band becomes very flat
since δ decreases exponentially for large lattice depth s. Yet, its height decreases more
slowly than the one of the lowest Bloch band (6.29) whose width decreases linearly in
δ. To illustrate this characteristic difference in the behaviour of the lowest Bogoliubov
and Bloch bands, we compare in Fig.7.7 the numerically obtained band heights.
• for small enough gn, one can neglect the density dependence of δ and use the approximation
κ −1 =˜gn for the compressibility, where ˜g takes the form (6.45) in the tight
binding regime. This yields
¯hω(q) =
2 δ0 sin 2
qd
2 δ0 sin
2¯h
2
qd
+2˜gn , (7.40)
2¯h
which was first obtained in [122] (see also [115, 123, 124]). Eq.(7.40) has a form similar
to the well-known Bogoliubov spectrum of uniform gases, the energy 2δ0sin 2 (qd/2¯h)
replacing the free particle energy q 2 /2m.
¯hω/ER
1.5
1
0.5
0
5 10 15 20 25 30
Figure 7.7: Height of the lowest Bogoliubov band ¯hω(q) (solid line) and the lowest Bloch
energy band (6.4) (dashed line) for gn =1ER as a function of lattice depth s.
s