Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2. The Inhomogeneous Bogoliubov Hamiltonian
Im[γ (s) (θ)]
f
f
2
0
− f
2
−f
0
π
4
π
2
kξ = 0.2
0.5
1.0
2.0
5.0
Figure 2.8.: Elastic scattering amplitude (2.61) in units of f = γ0 V0/(8ξµ) for different
values of kξ, at fixed potential radius kr0 = 0.5. Symbols: Results from numerical
integration of the full GP equation. Solid curves: analytical prediction (2.61). With
increasing k, the node moves to the left, according to (2.60). The overall amplitude has
a maximum at kξ ≈ 1.
• The inverse group velocity (∂k/∂ɛk) behaves like the constant c −1 for
sound waves (kξ ≪ 1) and decreases as k −1 for particles (kξ ≫ 1).
The product of both contributions therefore has limiting behavior k and k −1 ,
respectively, with a maximum around the crossover kξ ≈ 1 from phonons
to particles.
2.4.5. One-dimensional setting
In a one-dimensional setting, only forward scattering and backscattering
are present. This allows easily computing the transmission of a Bogoliubov
excitation across an impurity. Now we consider the reflection of an incident
Bogoliubov excitation γ (0)
k ′ = δk ′ kγ0 and compute the elastically reflected
wave γ (r)
k ′ = δk ′ (−k)γr in the Born approximation. The elastically scattered
wave is again singled out by the imaginary part of the Green function, such
that (2.61) applies
Im γr
γ0
= L 1
2V−2k
4µξ
3π
4
θ
π
kξ
k2ξ2 . (2.62)
+ 1
For a point like scatterer, the right hand side is proportional to the weight
of the impurity L 1
2Vk ≈ � dxV (x) =: 8µξB. From the elastically reflected
wave, we obtain the reflection amplitude r = 2Imγr/γ0. The factor 2 is
because the reflected wave exists only on one side of the impurity. For
a narrow potential, the transmission intensity T = 1 − |r| 2 is shown in
figure 2.9. The impurity is perfectly transparent at long wave lengths. At the
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