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

7.4 Velocity of sound 97
d|bjql| 2
6
5
4
3
2
1
0
1.5
1
0.5
0
1
0.8
0.6
0.4
0.2
0
¯hq =0.1qB
−2 −1 0 1 2
¯hq =0.5qB
−2 −1 0 1 2
¯hq =0.9qB
−2 −1 0 1 2
5
4
3
2
1
0
0.8
0.6
d|cjql| 2
0.4
0.2
0
0.4
0.3
0.2
0.1
0
¯hq =0.1qB
−2 −1 0 1 2
¯hq =0.5qB
−2 −1 0 1 2
¯hq =0.9qB
−2 −1 0 1 2
l l
Figure 7.8: Fourier coefficients bql, cql of the Bogoliubov Bloch waves ũq and ˜vq respectively:
Comparison of the tight binding approximation (7.43,7.44) (black bars) with the restults obtained
from the numerical solutions of the Bogoliubov equations (7.14,7.15) (white bars) at
lattice depth s =10for gn =0.5ER. Left column: |bql| 2 . Right column: |cql| 2 .
7.4 Velocity of sound
The low energy excitations of a stable stationary Bloch state ϕjk are sound waves. The
corresponding dispersion law is linear in the quasi-momentum ¯hq of the excitation. In general,
the spectrum ¯hω(q) is not symmetric with respect to q =0giving rise to two sound velocities
c+ and c−
¯hω(q) → c+¯hq , for q → 0 + , (7.46)
¯hω(q) → c−¯hq , for q → 0 − . (7.47)
For a carrier condensate with quasi-momentum ¯hk > 0, the velocities c+ and c− refer to sound
waves propagating in the same and in the opposite direction as stationary current respectively.
Their values depend on the quantum numbers j, k of the stationary condensate, on lattice
depth s and interaction strength gn. They can be determined from the slope of the lowest
Bogoliubov band at q =0. We first address in detail the case k =0and then discuss k = 0.