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 />
The disordered <strong>Bogoliubov</strong> problem contains three different length scales:<br />
the excitation wave length λ = 2π/k, the healing length ξ, and the disorder<br />
correlation length σ. The physics is not determined by the absolute values<br />
<strong>of</strong> these length scales, but rather by their values relative to each other. The<br />
three relevant dimensionless parameters are the following:<br />
• The ratio σ/ξ indicates, whether the disordered condensate is in the<br />
Thomas-Fermi regime (σ ≫ ξ) or in the smoothing regime (σ ≪ ξ),<br />
see subsection 2.2.3.<br />
• The parameter kξ indicates, whether the excitations are sound-wave<br />
like (kξ ≪ 1) or particle like (kξ ≫ 1), see subsection 2.3.1.<br />
• The parameter kσ discriminates the point-scatterer regime (kσ ≪ 1)<br />
from a very smooth scattering potential kσ ≫ 1.<br />
These three relevant parameters are not independent, two <strong>of</strong> them fix the<br />
third one, e.g. σ/ξ = (kσ)/(kξ). The dimension <strong>of</strong> the parameter space<br />
is reduced from three to two dimensions. This is analog to RGB color<br />
space at fixed lightness. Each <strong>of</strong> the three dimensionless parameters can be<br />
mapped to one <strong>of</strong> the three RGB channels. The parameter kξ, for example,<br />
determines the red channel, where kξ → 0 is mapped to cyan and kξ →<br />
∞ to red. Similarly, kσ and σ/ξ define the green and the blue channel,<br />
respectively, which completes the color. The identity (σ/ξ)(kξ)/(kσ) = 1<br />
σ<br />
ξ<br />
= 0<br />
kξ = ∞<br />
σ = 0<br />
ξ = ∞<br />
kσ = 0<br />
k = ∞<br />
k = 0<br />
kσ = ∞<br />
σ = ∞<br />
ξ = 0<br />
kξ = 0<br />
σ<br />
ξ<br />
= ∞<br />
Figure 4.1: Illustration <strong>of</strong> the parameter<br />
space <strong>of</strong> the full <strong>Bogoliubov</strong> problem.<br />
The relevant dimensionless parameters<br />
σ/ξ, kξ, kσ are identified with the color<br />
space at fixed lightness. On the edges,<br />
the limiting cases <strong>of</strong> the following sections<br />
are found.<br />
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