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

100 Bogoliubov excitations of Bloch state condensates
where c is the sound velocity (7.48) in the condensate at rest and m∗ µ is the effective mass
obtained from the chemical potential band spectrum (see Eq.(6.23)). Hence, in the lab frame
the sound velocity is given by
ck = c ± ¯h|k|
m∗ . (7.53)
µ
The plus- and minus-sign hold when the sound wave propagates in the same and in the opposite
direction as the condensate respectively. Since the quantity ¯hk/m∗ µ is generally different from
the group velocity ¯v = ¯hk/m∗ , the sound velocity is not generally obtained by adding ¯v
and c in the case of copropagating sound waves, or by subtracting ¯v from c in the case of
counterpropagation. This is the case only for s =0where m∗ = m∗ µ = m. It is interesting to
note that the result (7.53) gives a physical meaning to the quantity m∗ µ.
In the tight binding regime, we can write ck = c ± ¯v +(δµ − δ)d2k/¯h , where δµ − δ =
n∂δ/∂n (see Eqs.(6.38,6.36)), in order to show that the density-derivative of δ plays an
important role. We remind that, in contrast, contributions arising from this quantity are
negligible as far as excitations of groundstate condensate are concerned (see discussion in
section 7.3).
Expression (7.53) has been derived assuming a small value of the condensate quasimomentum
k. The hydrodynamic formalism can also be employed to calculate the sound
velocity for any value of k. The result reads [107, 125] (see also section 9 below)
n
ck =
m∗ ∂µ(k)
(k) ∂n ±
∂µ(k)
∂k , (7.54)
where m ∗ (k) is the generalized effective mass (6.22) with j =1. Expanding this expression
up to O(k) at k =0, we recover result (7.53).