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 />
ΛN<br />
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kσ<br />
0<br />
0<br />
0.5 1 1.5 2<br />
kξ = 0.05<br />
10 20 30 40<br />
σ/ξ<br />
Figure 4.2.: Relative correction <strong>of</strong> the speed <strong>of</strong> sound ΛN = ∆c µ 2 /(c V 2<br />
0 ) for 1D speckle<br />
disorder. Full formula (3.60) for kξ = 0.05 (black line). Limiting formulae ΛN(kσ)|ξ=0<br />
[section 4.1, (4.15)] (dotted blue) and ΛN(σ/ξ)|k=0 [section 4.2, (4.31)] (dashed green).<br />
Numerical results <strong>of</strong> a Gross-Pitaevskii integration, as discussed in section 4.3, are shown<br />
for V0/µ = +0.03 (blue straight marks) and V0/µ = −0.03 (red triangular marks).<br />
fixes the value <strong>of</strong> the lightness, leaving the two-dimensional space <strong>of</strong> hue and<br />
saturation, as shown in Fig. 4.1.<br />
Limiting cases, where one <strong>of</strong> the length scales is zero or infinity, i.e. much<br />
shorter or much longer than the other length scales, are found on the edges.<br />
We focus mainly on the low-energy excitations kξ ≪ 1. Only in the last<br />
section, section 4.4, we consider the transition to particle-like excitations<br />
kξ ≫ 1.<br />
Low-energy excitations<br />
We are interested in the low-energy features, i.e. the<br />
sound-wave regime kξ ≪ 1. For practical purposes, we<br />
choose a small but finite value kξ = 0.05. In the parameter<br />
space, the curve defined by kξ = 0.05 appears as a<br />
smooth curve close to the ξ = 0 edge and the k = 0 edge,<br />
k = 0<br />
as shown in the illustration on the right. Let us consider<br />
the speed <strong>of</strong> sound, which is a significant physical quantity. In figure 4.2,<br />
the correction (3.60) <strong>of</strong> the speed <strong>of</strong> sound due to a one-dimensional speckle<br />
disorder potential is shown. The curve is rather complicated with three<br />
different regimes: a linear increase at very short correlation lengths, a nonmonotonic<br />
intermediate range and saturation at long correlation lengths. In<br />
82<br />
σ = 0<br />
kξ = 0.05<br />
ξ = 0<br />
σ = ∞