The numerical scheme

σ= 0

kξ= 0.05

In the previous two sections, we have worked out the low-energy behavior of the Bogoliubov excitations in the limits ξ = 0 and k = 0, which reproduce nicely the results of the full theory in the respective regimes. In this section, we confront the previous results with a direct numerical inte-gration of the time-dependent Gross-Pitaevskii equation

(2.16). The numerical procedure is similar to the simulation of the single scattering process in subsection 2.4.4. Essentially, the impurity is replaced by a disorder potential extending over the whole system and the setup is reduced from two dimensions to one dimension. Again, the simulation relies neither on the linearization in the excitations, nor on perturbation theory in the disorder potential. Thus, it is possible to go beyond leading-order perturbation theory for weak disorder. Also, we investigate the statistical distribution of the disorder average and the self-averaging properties of the speed of sound. As expected, the predictions from the Born approximation are confirmed very well for sufficiently weak disorder.

In the numerical integration of the original Gross-Pitaevskii equation (2.16) it is impossible to perform the strict limit k = 0 or ξ = 0 as it was done in the previous sections. Instead, the parameter kξ is fixed at a small but finite value. Then the ratio σ/ξ is varied, which allows reaching both regimes ξ σ, λ from section 4.1 and λ σ, ξ from section 4.2.

4.3.1. The numerical scheme

The integration is done in a one-dimensional system of length L with pe-riodic boundary conditions. The discretization ∆x is chosen smaller than

-0.5

Figure 4.8.: Speckle potential V(x)/µ (red) and ground state density profile n0(x)/n_∞ (green) for (a) a repulsive speckle potentialV0 = +0.1µ, (b) an attractive speckle poten-tial V0 = −0.1µ. The asymmetric distribution (3.10) of the speckle potential can lead to occasional deep density depressions (a) or density peaks (b). At σ = ξ, the density profiles are slightly smoothed, compared with the Thomas-Fermi profile (2.20).

healing length, disorder correlation length and wave length. Due to the fi-nite system size L, Fourier space becomes discrete, with ∆k = 2π/L. The first step is to generate the speckle disorder potential. For all k ≤ σ⁻¹, the independent complex field amplitudes Ek from (3.7) are populated using a Gaussian random number generator from the gsl library. In real space, the amplitudes are squared and the mean value is compensated, which gives the speckle disorder potential (3.9). Then the condensate ground-state is com-puted by imaginary time evolution, using the fourth-order Runge-Kutta al-gorithm [104]. Starting point is the homogeneous condensate, which adapts to the disorder potential during the imaginary time evolution, figure 4.8.

The imaginary time evolution is not unitary, thus it violates the normaliza-tion of the wave funcnormaliza-tion. This is compensated by re-normalizing the wave function after each imaginary time step, which corresponds to the shift of the chemical potential discussed in subsection 3.4.5.

Next, the disordered ground state is superposed with a plane-wave exci-tation γk of the homogeneous system. According to the Bogoliubov trans-formation (2.33) the imprints in density and phase read

δn(x) = 2√ The amplitude of the phase modulation is much larger than the amplitude in the density, because of k ⁰_k in the sound-wave regime. Thus, we choose a small value for the amplitude p

k/⁰_kΓ = 0.3√

n_∞V0/µ. Then, the time

0 5 10 15 20

0 5 10 15 20

-0.6 -0.4 -0.2 0 0.2 -0.6 -0.4 -0.2 0 0.2

ΛN

ΛN

v= +0.03 v=−0.03

v= +0.15 v=−0.15

Figure 4.9.: Typical histograms of the correction to the speed of sound ΛN = ∆cµ²/(cV₀²) at kξ = 0.05, kσ = 1, σ/ξ = 20 over 50 realizations of disorder. The mean value of the respective distribution is marked with a solid line, and the Born prediction (4.15) Λ_N = ¹₄ln 2−¹₂ is shown as a dashed line. ForV₀/µ.0.1, the width of the distribution is clearly narrower than the mean value, i.e. the speed of sound shows self-averaging behavior. For V0 = +0.15µ, the distribution becomes very broad, extending even to positive values. The repulsive potential is so strong that the condensate density gets depressed nearly to zero below the highest peaks. Then, it is impossible to imprint a plane-wave density modulation (4.34). In two out of 50 realizations, no result could be obtained.

evolution is computed using again the fourth-order Runge-Kutta algorithm.

The excitation propagates with a modified speed of sound and is slightly scattered at the same time. In order to extract the relevant information, the deviations of the wave function from the ground state are Bogoliubov transformed. Then the phase velocity is extracted from the complex phase of γk ∝e^−ikt/~. This is done for many realizations of disorder and averaged over. By comparison with the phase velocity in the clean system, the change in the speed of sound is obtained. Furthermore, one can monitor the life-time of the excitations by observing the elastically scattered amplitude γ₋k.

4.3.2. Disorder average and range of validity of the Born prediction

The numerical results for the speed of sound and other quantities (and ex-perimental results as well) depend on the particular realization of disorder.

For an infinite system, the results would be perfectly self-averaging. For practical reasons the size of the system was chosen to be 200 disorder

corre--0.4 -0.3 -0.2 -0.1

0.0-0.2 -0.1 0.0 0.1 0.2

V₀/µ

ΛN

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; Atv = 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 strengthsV0 atkξ = 0.05 and kσ = 1. As long as the disorder is not too strong, the distributions are clearly single-peaked with well-defined av-erages. 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 ef-fect 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 infigure 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 v² = V₀²/µ², 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

condensate gets already fragmented, due to rare high potential peaks from the exponential tail in (3.10), see also figure 4.8. This is reflected in the positive deviation of the mean values and also in the increased width of the distribution. To sum things up: Beyond-Born effects are clearly visible, but the Born result remains useful in a rather wide range. The correction stays negative, even for nearly fragmented condensates.