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

7.4 Velocity of sound 99
c(s)/c(s=0)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 5 10 15 20 25 30
Figure 7.9: Ratio between the sound velocity c(s) at lattice depth s and the sound velocity
c(s =0)for gn =0.02ER (solid line), gn =0.1ER (dashed line) and gn =0.5ER as a
function of lattice depth s. The dotted line depicts the quantity m/m ∗ for gn =0.5ER.
condensate
ck = c ± ¯h|k|
. (7.50)
m
To generalize this result to account for the presence of a lattice, we first consider a condensate
in a Bloch state of the lowest band moving with a small group velocity ¯v with respect to the
lab frame (see discussion section 6.1)
¯v = ¯hk
, (7.51)
m∗ where ¯hk is the quasi-momentum of the condensate and m∗ is its effective mass in the lowest
band at k =0for a given lattice depth s. As mentioned above, the sound velocity in such a
condensate can be obtained by solving the Bogoliubov equations (7.6,7.7) with j =1and small
k and by extracting the slopes of the resulting lowest Bogoliubov band ¯hω(q) for q →±0. As
in the case s =0(see Eq.(7.50)), the Bogoliubov bands are asymmetric with respect to q =0
if k = 0and the sound velocity depends on the sign of q.
An alternative to the numeric solution of (7.6,7.7) is offered by the hydrodynamic formalism
developed below in chapter 9. Its application allows us to derive the analytic result
s
¯hω(q) =c¯h|q|± ¯h|k|
m∗ ¯h|q| , (7.52)
µ