Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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4.1. Hydrodynamic limit I: ξ = 0<br />
the following sections, we will illuminate the different regimes and elaborate<br />
limiting results.<br />
In addition to excitation wave length and healing length, the disorder correlation<br />
length provides a third length scale σ with no a priori constraints.<br />
Thus, the sound limit kξ → 0 can be understood in two different ways:<br />
1. Interaction dominates, and the healing length is the shortest <strong>of</strong> all<br />
length scales and drops out, ξ → 0. This so-called hydrodynamic<br />
limit leaves kσ as free variable. It is found at the lower right edge in<br />
the parameter space figure 4.1 and is discussed in detail in section 4.1.<br />
2. Conversely, the excitation wave length can be taken as the longest <strong>of</strong><br />
all length scales, i.e. k → 0, leaving σ/ξ as free variable. This limit is<br />
found at the lowermost edge <strong>of</strong> the parameter space and discussed in<br />
section 4.2.<br />
Analytical results from both limits reproduce the exact correction (3.60) for<br />
large σ/ξ and small kσ, respectively, cf. figure 4.2. Finally, in section 4.3, we<br />
verify the perturbative predictions in all regimes with a numerical integration<br />
<strong>of</strong> the time-dependent Gross-Pitaevskii equation in a one-dimensional<br />
disordered system.<br />
4.1. Hydrodynamic limit I: ξ = 0<br />
Here, the so-called hydrodynamic regime is discussed,<br />
where the interaction gn = µ dominates over the<br />
kσ = ∞<br />
quantum pressure, i.e. kinetic energy. For the ground<br />
state this means that the Thomas-Fermi approximation<br />
(2.20) holds, and the excitations are in the sound-wave<br />
kσ = 0<br />
regime kξ ≪ 1. The fact that the interaction µ is the<br />
largest energy scale implies that the healing length ξ = �/ √ 2mµ is the<br />
shortest length scale. Thus, the physical results from section 3.4 can be<br />
taken in the limit ξ → 0. All results will then depend only on the ratio σ/λ<br />
<strong>of</strong> the remaining length scales, respectively on kσ = 2πσ/λ.<br />
Physically, it is more instructive to perform the hydrodynamic limit in<br />
the very beginning and to derive the results from the much simpler hydrodynamic<br />
equations <strong>of</strong> motion [115].<br />
ξ = 0<br />
83