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

9.4 Sound Waves 121
9.4 Sound Waves
For Vext =0, the small-current hydrodynamic equations (9.17,9.18) have sound wave solutions.
These correspond to small amplitude plane wave perturbations ∆n(r,t), ∆v(r,t) of a Bloch
state associated with constant macroscopic density ¯nM and macroscopic velocity ¯vM. Here,
we restrict ourselves to sound waves moving in the lattice direction in a Bloch state condensate
with quasi-momentum ¯hk and velocity ¯vM =¯hk/m oriented in the same direction. In this
case, the perturbation takes the form
∆n(z,t) ∝ ∆n(z) e i(qz−ω(q)t)
(9.26)
∆v(z,t) ∝ ∆v(z) e i(qz−ω(q)t)
(9.27)
To find the dispersion ω(q), we linearize Eqs.(9.17,9.18) in ∆n(z,t), ∆v(z,t) with Vext =0.
The resulting equations read
∂
m
∆n(z,t)+
∂t m∗ k
¯n∂z∆v(z,t)+
(¯n) m∗ µ(¯n) ∂z∆n(z,t) =0, (9.28)
∂
1 ∂µopt
∆v(z,t)+ ∂z∆n(z,t)+
∂t m ∂n ¯n
k
m∗ µ(¯n) ∂z∆v(z,t) =0, (9.29)
where m∗ and m∗ µ are the effective masses obtained from the lowest energy and chemical
potential Bloch band respectively (see Eqs.(6.19,6.23)) and we have neglected terms of order
higher than O(k). Inserting (9.26,9.27) into (9.28,9.29), we find
ω(q) = ¯h|k|
m∗ 1
|q|± √ |q| , (9.30)
µ(¯nM) κm∗ where κ is the groundstate compressibility (see Eq.(5.11)). According to this result, sound
waves in a groundstate condensate (k =0)travelatvelocity
c = 1
√ . (9.31)
κm∗ This expression and in particular the implied dependence of the sound velocity on average
density and on lattice depth has been discussed above in section 7.4. On the other hand, if
the condensate carrying the sound wave has non-zero quasi-momentum ¯hk, Eq.(9.30) yields
the sound velocity
ck = c ± ¯h|k|
m ∗ µ
, (9.32)
where the plus- and minus-sign hold for k parallel and anti-parallel q respectively. This expression
has been discussed above in chapter 7.4 (see Eq.(7.53)).
The result (9.30) describes the excitation energy spectrum associated with small perturbations
of a Bloch state condensate at any lattice depth in the limit of small q and k. Using the
set of hydrodynamic equations (9.24,9.25) valid also for large currents, the spectrum for small
q and any k has been obtained in [107]. The result reads [107, 125]
n
ck =
m∗ ∂µ(k)
(k) ∂n ±
∂µ(k)
∂k , (9.33)
where m ∗ (k) is the generalized effective mass (6.22) with j =1.