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 />
In one dimension, there is only the backscattering contribution k → −k<br />
1<br />
klB<br />
= V 2<br />
0<br />
2µ 2<br />
kσ<br />
(1 + k 2 ξ 2 ) 2C1(2kσ).<br />
(4.13)<br />
Due to the finite support <strong>of</strong> the speckle correlation function Cd(2k) (3.12a),<br />
there is no backscattering at all for kσ > 1, and the inverse Boltzmann mean<br />
free path goes to zero, at least within the scope <strong>of</strong> the Born approximation.<br />
In dimensions d ≥ 2, there are still contributions from scattering angles<br />
between 0 and π/2 [see insets <strong>of</strong> figure 4.3(b)], thus the inverse Boltzmann<br />
transport length does not go abruptly to zero at kσ = 1, figure 4.3(b).<br />
Conclusion<br />
How strongly are the <strong>Bogoliubov</strong> excitations affected by elastic scattering?<br />
In order to justify the description in terms <strong>of</strong> plane-wave states with a<br />
well-defined sound velocity, the Boltzmann length lB and the localization<br />
length lloc should be much larger than the wave length λ = 2π/k. This<br />
is easily fulfilled for the Boltzmann length (3.51). The curves shown in<br />
figure 4.3(b) are bounded and the scaling with V 2<br />
0 /µ 2 guarantees lB ≫ λ.<br />
The localization lengths are equal to or even exponentially larger than the<br />
Boltzmann length. The only thing to keep in mind is that for large values<br />
<strong>of</strong> kσ, forward scattering events, which produce incoherence in the phase,<br />
may occur frequently.<br />
4.1.3. Speed <strong>of</strong> sound<br />
We compute the relative correction <strong>of</strong> the speed <strong>of</strong> sound Λ = Re� Σ(k,ck)µ 2<br />
2c2k2V 2<br />
0<br />
from the self-energy (4.9)<br />
Λ = − 1<br />
2 P<br />
� d ′ d k σ<br />
(2π) d Cd(|k − k ′ |σ)<br />
(k · k ′ ) 2<br />
k 2 (k ′2 − k 2 )<br />
. (4.14)<br />
This formula has simplified significantly compared with the principal-value<br />
integral (3.60).<br />
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