Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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4. Disorder—Results and Limiting Cases<br />
20<br />
15<br />
10<br />
5<br />
0<br />
20<br />
15<br />
10<br />
5<br />
0<br />
v = −0.03<br />
v = −0.15<br />
v = +0.03<br />
v = +0.15<br />
-0.6 -0.4 -0.2 0 0.2 -0.6 -0.4 -0.2 0 0.2<br />
ΛN<br />
ΛN<br />
Figure 4.9.: Typical histograms <strong>of</strong> the correction to the speed <strong>of</strong> sound ΛN = ∆cµ 2 /(cV 2<br />
0 )<br />
at kξ = 0.05, kσ = 1, σ/ξ = 20 over 50 realizations <strong>of</strong> disorder. The mean value <strong>of</strong><br />
the respective distribution is marked with a solid line, and the Born prediction (4.15)<br />
ΛN = 1 1 ln 2 − 4 2 is shown as a dashed line. For V0/µ � 0.1, the width <strong>of</strong> the distribution<br />
is clearly narrower than the mean value, i.e. the speed <strong>of</strong> sound shows self-averaging<br />
behavior. For V0 = +0.15µ, the distribution becomes very broad, extending even to<br />
positive values. The repulsive potential is so strong that the condensate density gets<br />
depressed nearly to zero below the highest peaks. Then, it is impossible to imprint a<br />
plane-wave density modulation (4.34). In two out <strong>of</strong> 50 realizations, no result could be<br />
obtained.<br />
evolution is computed using again the fourth-order Runge-Kutta algorithm.<br />
The excitation propagates with a modified speed <strong>of</strong> sound and is slightly<br />
scattered at the same time. In order to extract the relevant information,<br />
the deviations <strong>of</strong> the wave function from the ground state are <strong>Bogoliubov</strong><br />
transformed. Then the phase velocity is extracted from the complex phase<br />
<strong>of</strong> γk ∝ e −iɛkt/� . This is done for many realizations <strong>of</strong> disorder and averaged<br />
over. By comparison with the phase velocity in the clean system, the change<br />
in the speed <strong>of</strong> sound is obtained. Furthermore, one can monitor the lifetime<br />
<strong>of</strong> the excitations by observing the elastically scattered amplitude γ−k.<br />
4.3.2. Disorder average and range <strong>of</strong> validity <strong>of</strong> the Born<br />
prediction<br />
The numerical results for the speed <strong>of</strong> sound and other quantities (and experimental<br />
results as well) depend on the particular realization <strong>of</strong> disorder.<br />
For an infinite system, the results would be perfectly self-averaging. For<br />
practical reasons the size <strong>of</strong> the system was chosen to be 200 disorder corre-<br />
98