Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

ΛN
�0.1
�0.2
�0.3
�0.4
4.3. Numerical study of the speed of sound
V0/µ
�0.2 �0.1 0.0 0.1 0.2
0.0
Figure 4.10.: Correction to the speed of sound at kξ = 0.05, kσ = 1, σ/ξ = 20 as function
of the disorder strength. The perturbative prediction appears as a horizontal line. The
values were obtained by averaging over 50 realizations of disorder, the error bars show
the estimated error of this mean value, which is given by the width of the distribution
divided by the square root of the number of realizations. The histograms corresponding
to the points at v = ±0.03, ±0.15 are shown in figure 4.9. At large values of v, the
potential is likely to fragment the condensate. At v = 0.15, this happened in two out of
50 realizations; At v = 0.2 it happened already in five out of 30 realizations, making the
disorder average questionable.
lation lengths. In figure 4.9 the distribution of results is shown for different
disorder strengths V0 at kξ = 0.05 and kσ = 1. As long as the disorder is not
too strong, the distributions are clearly single-peaked with well-defined averages.
For larger values of |V0| � 0.1µ, the highest potential peaks or wells
reach the value of the chemical potential µ, which violates the assumptions
of perturbation theory of the Born approximation from chapter 3. In the
case of an attractive (red-detuned) potential with deep wells, V0 < 0, the effect
is not very dramatic. In the opposite case of a repulsive (blue-detuned)
speckle potential, however, the rare high peaks of the speckle potential may
fragment the condensate. In the corresponding panel of V0 = +0.15µ, the
distribution is strongly broadened, including even some points with opposite
sign.
Also the mean values are shifted as function of the disorder strength V0.
This is investigated in figure 4.10, where the mean values of the distributions
in figure 4.9 are shown together with their estimated error. The correction
is shown in units of v 2 = V 2
0 /µ 2 , such that the Born approximation from
subsection 3.4.5 appears as the horizontal line as function of the disorder
strength V0. At small values of |V0/µ|, the agreement is very good. Then
there is a clear negative linear trend, which is due to the third moment
of the speckle distribution function (3.10). At larger values v � 0.15, the
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