Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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1<br />
0.5<br />
0<br />
-0.5<br />
0 20 40 60 80 100<br />
x/σ<br />
(a)<br />
4.3. Numerical study <strong>of</strong> the speed <strong>of</strong> sound<br />
1<br />
0.5<br />
0<br />
-0.5<br />
0 20 40 60 80 100<br />
x/σ<br />
(b)<br />
Figure 4.8.: Speckle potential V (x)/µ (red) and ground state density pr<strong>of</strong>ile n0(x)/n∞<br />
(green) for (a) a repulsive speckle potential V0 = +0.1µ, (b) an attractive speckle potential<br />
V0 = −0.1µ. The asymmetric distribution (3.10) <strong>of</strong> the speckle potential can lead<br />
to occasional deep density depressions (a) or density peaks (b). At σ = ξ, the density<br />
pr<strong>of</strong>iles are slightly smoothed, compared with the Thomas-Fermi pr<strong>of</strong>ile (2.20).<br />
healing length, disorder correlation length and wave length. Due to the finite<br />
system size L, Fourier space becomes discrete, with ∆k = 2π/L. The<br />
first step is to generate the speckle disorder potential. For all k ≤ σ−1 , the<br />
independent complex field amplitudes Ek from (3.7) are populated using a<br />
Gaussian random number generator from the gsl library. In real space, the<br />
amplitudes are squared and the mean value is compensated, which gives the<br />
speckle disorder potential (3.9). Then the condensate ground-state is computed<br />
by imaginary time evolution, using the fourth-order Runge-Kutta algorithm<br />
[104]. Starting point is the homogeneous condensate, which adapts<br />
to the disorder potential during the imaginary time evolution, figure 4.8.<br />
The imaginary time evolution is not unitary, thus it violates the normalization<br />
<strong>of</strong> the wave function. This is compensated by re-normalizing the wave<br />
function after each imaginary time step, which corresponds to the shift <strong>of</strong><br />
the chemical potential discussed in subsection 3.4.5.<br />
Next, the disordered ground state is superposed with a plane-wave excitation<br />
γk <strong>of</strong> the homogeneous system. According to the <strong>Bogoliubov</strong> transformation<br />
(2.33) the imprints in density and phase read<br />
δn(x) = 2 √ n∞<br />
�<br />
ɛ 0 k<br />
ɛk<br />
Γ cos(kx) δϕ(x) = 1<br />
�<br />
ɛk<br />
√<br />
n∞ ɛ0 Γ sin(kx). (4.34)<br />
k<br />
The amplitude <strong>of</strong> the phase modulation is much larger than the amplitude in<br />
the density, because <strong>of</strong> ɛk ≫ ɛ0 k in the sound-wave regime. Thus, we choose<br />
a small value for the amplitude � ɛk/ɛ0 k Γ = 0.3√n∞V0/µ. Then, the time<br />
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