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

Chapter 11
Sound propagation in presence of a
one-dimensional optical lattice
The propagation of sound in a harmonically trapped condensate without lattice, has been
observed in the experiment [148]. We study the effect of a one-dimensional optical lattice on
sound propagation in a set-up analogous to this experiment, yet without harmonic trap.
The spectrum of elementary excitations shows a linear behavior ¯hω = c¯h|q| in the lowest
Bogoliubov band at small quasi-momenta ¯hq, indicating the existence of phonons (see chapter
7.2 above). The sound velocity c decreases for increasing lattice depth, due to reduced
tunneling or equivalently to increased effective mass (see chapter 7.4 above). In the regime of
validity of the linearized GP-equation, sound propagation is expected to be possible. We confirm
numerically that for sufficiently small sound signal amplitude the sound velocity decreases
by increasing the lattice depth, as predicted by Bogoliubov theory.
However, it is not obvious a priori whether a sound signal of finite amplitude is able to
propagate also in deep lattices, where the tunneling rate is very small. For deep lattices,
nonlinear effects are very different from the uniform case: first of all, shock waves propagate
slower than sound waves (see section 11.3). This is due to the negative curvature of the
Bogoliubov dispersion relation in the lowest Bogoliubov band. The most striking effect is that
non-linearities can play a role also at very small density variations and induce a saturation
of the sound signal, which goes along with dephased currents in the back of the signal (see
section 11.3). This effect has no analogon in the uniform case.
In conclusion we find that sound signals propagate, but that the maximal attainable signal
amplitude (density variation) decreases very strongly with the optical lattice depth, making
it in practise observable only up to a certain lattice depth which depends on the interaction
strength. We show that there exists a range of optical potential depths where the signal is still
large enough and where the change in sound velocity induced by the lattice can be measured
(see section 11.4).
A paper containing the results presented in this chapter is in preparation.
143