Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

ǫk
0.25 0.5 0.75 1 1.25 1.5 1.75
kσ
(a)
Λ
�1�6
�1�4
�1�2
4.1. Hydrodynamic limit I: ξ = 0
kσ
0 1 2
(b)
d = 1
d = 3
Figure 4.4.: (a) Schematic representation of the disorder-averaged dispersion relation (4.15)
in one dimension. According to (4.19) the relative correction to c k (gray) is − 1
2
for small kσ (dashed cyan) and − 3
8
�3�8
�4�8
�5�8
2 V0 /µ 2
2 V0 /µ 2 for large kσ (dotted blue). The full correction
(4.15) interpolates between the limits (solid black). This plot corresponds to the linear
regime of the Bogoliubov dispersion relation in figure 2.3. (b) Relative correction (4.14)
to the speed of sound in a d-dimensional speckle potential. The limits (4.19) are met at
the edges.
which leads to Λ = − 2+d
8
. Summarized, the limits are
Λ = − 1
2 ×
� −1
d , kσ ≪ 1 , (4.19a)
1
4 (2 + d), kσ ≫ 1 . (4.19b)
In figure 4.4(b), these limits are shown together with the full curves (4.14)
for speckle disorder. Note, however, that the limiting values are independent
of the particular type of disorder.
2D speckle disorder
In the case of a two-dimensional speckle disorder, the angular part of equation
(4.14) is solved analytically as a closed-path integral in the complex
plane z = eiβ , β = ∡(k, k − k ′ ), using the residue theorem
�
q
� �
2
A2(q) = 1 − + 1 −
2k
q2
2k2 �
k2 q � q2 − (2k) 2, q = |k − k′ |. (4.20)
The last term is imaginary for q < 2k and real for q > 2k. If kσ > 1,
there is no overlap of the correlator Cd(q) with the real part, such that the
correction of the speed of sound (4.14) takes the form
Λ2 = − 1
4π Re
�
dqσC2(qσ)A2(q) = − 1
�
1 −
2
1
8k2σ2 �
, for kσ > 1.
(4.21)
89