Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3.2. A suitable basis for the disordered problem
with ak = � ɛ0 k /ɛk. By comparison to equation (2.67), we identify the functions
uk(r) = 1
�
Φ(r)
+
2 Φ0ak
Φ0ak
� ik·r e
Φ(r) Ld/2 , vk(r) = 1
�
Φ(r)
−
2 Φ0ak
Φ0ak
� ik·r e
(3.15)
Φ(r) Ld/2 as disorder-deformed plane-waves. Now we can check their bi-orthogonality
(2.75), which is indeed fulfilled:
�
d d r [u ∗ k(r)u k ′(r) − v ∗ k(r)v k ′(r)] =
� d d r
2Ld �
ak
ak ′
+ ak ′
ak
�
e −i(k−k′ )·r = δkk ′ .
(3.16)
Also the orthogonality with respect to the ground state (2.76) is fulfilled,
because Φ(r) [uk(r) − vk(r)] is a plane wave with zero average for all k �= 0.
In the sense of the paragraph “Mean-field total particle number” in subsection
2.5.5, this compliance was expected, because the Bogoliubov modes
(2.32) are constructed from plane-wave density modulations with zero spatial
average, which cannot affect the mean-field particle number.
Conclusion
Only the Bogoliubov quasiparticles (2.32) in terms of density and phase
ˆγk = 1
ak
δˆnk
2Φ0
+ iakΦ0δ ˆϕk
(3.17)
fulfill the requirements for the study of the disordered Bogoliubov problem.
They are labeled by their momentum k, which is independent of the disorder
realization V (r). Only the ground state Φ(r), from where the fluctuations
are measured, is shifted. The Bogoliubov quasiparticles (2.32) fulfill the
bi-orthogonality relation (2.75), which is necessary for using them as basis,
but most importantly, they decouple from the zero-frequency mode.
The equations of motion (2.43) for the ˆγk for k �= 0 are coupled, which
will be subject of the perturbation theory in the next section.
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