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

11.4 Experimental observability 157
11.4 Experimental observability
In order to discuss the experimental observability of the discovered effects, a few things need
to be considered: first of all, an experimental set-up involves a trapping potential in the
longitudinal and radial directions. In our model, we have neglected completely the radial
degrees of freedom, which might play an important role in sound propagation. First of all,
even in the uniform case, a radial TF-density distribution can change the sound velocity to
c = gn/2m [152, 153, 154, 139]. Such effects can be simply taken into account by correctly
changing the density dependence of the interaction term in our equations [116]. The second
important thing to consider is the external harmonic potential along the direction of sound
propagation. One should require that the time needed by the sound packet to travel along an
observable distance L>>dis shorter than the oscillation period in the trap. For example
for 87 Rb atoms, for s =10÷ 20, gn =0.5ER and L =50d, trapping frequencies smaller
than 2π × 25 ÷ 80 Hz are required. Last but not least, one should of course require that the
signal is measurable. In order to mimic the experimental resolution of the detection system,
one has to perform a convolution over a few lattice sites of the GPE solution. This gives
a maximum observable amplitude for the density variation which is in good agreement with
our DNLS prediction ∆nmax =0.2π δ/U. For gn =0.5ER, the density variations which
can be observed for a few different values of the optical potential depth are: for s =10,
∆nmax =0.13; s =15, ∆nmax =0.08; s =20, ∆nmax =0.05. Finally, one should also
keep in mind, that the saturation effect is clearly evident only if the shock waves move with
a velocity much lower than the sound velocity, which requires δ/U ≪ 1/3. Obtaining a clear
saturation effect (small δ/U) associated with a large signal amplitude ∆n (large δ/U) atfixed
lattice depth requires a compromise in the choice of lattice depth s and interaction strength
gn.