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

98 Bogoliubov excitations of Bloch state condensates
Sound in a condensate at rest
In section 7.2, we have presented the results for the Bogoliubov bands of a condensate in the
groundstate. By determining the slope of the lowest Bogoliubov band at q =0, we obtain the
sound velocity as a function of s and gn/ER. The results are presented in Fig.7.9 where we
plot the ratio c(s)/c(s =0)for different values of gn/ER. The presence of the lattice leads
to a slow-down of sound. For gn =0.5ER the decrease of the sound velocity amounts to
about 34% and 71% at s =10and s =20respectively. In fact, the hydrodynamic formalism
developed below in section 9 allows us to derive the relation
c = 1
√ m ∗ κ . (7.48)
Hence, the decrease of the sound velocity is a consequence of the exponential increase of the
effective mass m ∗ . which overcompensates the decrease of the compressibility κ. It is not the
enhanced rigidity which governs the sound velocity, but the fact that the atoms are slowed
down by the potential barriers. To underline this point we also display in Fig. 7.9 the function
m/m ∗ obtained for gn =0.5ER. Clearly, this quantity reproduces the characteristic features
of the ratio c(s)/c(s =0).
Fig.7.9 shows that the density-dependencies of m ∗ and κ −1 lead to a slight increase of
the ratio c(s)/c(s =0)with gn/ER for fixed s which can be understood in terms of the
screening effect of interactions. This effect is due to the decrease of m ∗ with increasing
density which overcompensates the decrease of 1/gnκ (see Figs. 5.5 and 6.8). For small but
nonzero interaction gn one obtains the law c(s)/c(s =0)= (m/m ∗ )(˜g/g) with the effective
coupling constant ˜g as defined in Eq.(5.15).
Note that in the tight binding regime the sound velocity (7.48) can also be obtained from
the low-q limit of the expression for the lowest Bogoliubov band (7.38).
The results for the sound velocity presented in this section concern sound waves of small
amplitude. Such sound waves exist as long as the system is superfluid and hence at all lattice
depths for which GP-theory can be applied. Yet, it remains a question whether sound waves
of finite amplitude can propagate at a certain s. This problem will be discussed in chapter 11.
Sound in a moving condensate
The energy of a sound wave excitation in a moving uniform condensate observed from the lab
frame is related to its energy in the rest frame by the Galilei transformation
¯hω(q) =c¯h|q|± ¯h|k|
¯h|q| . (7.49)
m
Here, c is the sound velocity in the rest frame, ¯hk is the momentum associated with the relative
motion of the two frames and q is the wave number of the sound wave in the rest frame. The
plus- and minus-sign hold when the sound wave propagates in the same and in the opposite
direction as the condensate respectively. Hence, the sound velocities observed in the lab frame
are obtained by simply adding or subtracting c to/from the velocity ¯h|k|/m of the moving