Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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