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

7.2 Bogoliubov bands and Bogoliubov Bloch amplitudes 87
¯hω/ER
10
9
8
7
6
5
4
3
2
1
0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
¯hq/qB
Figure 7.1: Bogoliubov bands ¯hωj(q) obtained from the solution of (7.14,7.15) in the first
Brillouin zone for s =1, gn =0(solid line) and gn =0.5ER (dash-dotted line). Note that
for such a small potential, the gap between second and third band is still very small.
while high bands differ from the non-interacting ones mainly by an energy shift gn. Thisisa
general feature: For a given gn and s, sufficiently high bands are not affected by the lattice,
but are governed by the behavior of the Bogoliubov spectrum of the uniform gas at momenta
much larger than the inverse healing length ¯h/ξ
¯hωuni(q) =
q 2 /2m(q 2 /2m +2gn) ≈ ¯hq 2 /2m + gn , (7.16)
where momenta lying in the j-th Brillouin zone are mapped to the j-th band.
In Fig.7.2 we compare the lowest Bogoliubov and Bloch bands with the single particle
energy. Clearly, the lowest Bloch band is less affected by the presence of interactions than the
Bogoliubov band. Similarly, in the uniform case the Bogoliubov dispersion is strongly affected
by interactions while the Bloch dispersion, obtained simply from a Galilei transformation, does
not involve interaction effects at all. The enhanced effect of interaction on the Bogoliubov
bands can be understood by recalling that the Bogoliubov band gives the energy of a small
perturbation propagating in a large background condensate while the Bloch band gives the
energy per particle related to the motion of the condensate as a whole.