Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3. Disorder
The difference resides in the order of switching on the disorder and applying
the transformation (2.32). Let us investigate which option better fulfills the
orthogonality conditions (2.75) and (2.76).
3.2.1. Bogoliubov basis in terms of free particle states
Let us start with the simpler looking option (3.13b). By comparison with
d − equation (2.67), we directly identify the functions ũk(r) = L 2uke ik·r and
d − ˜vk(r) = L 2vke ik·r . These are exactly the same functions as in the homogeneous
case, and obviously all Bogoliubov modes k �= 0 fulfill the biorthogonality
(2.75).
In presence of an external potential, however, the condensate state Φ is
deformed and is not orthogonal to the plane waves uk(r) and vk(r) any
more. Testing the condition (2.76), we indeed find
�
d d r Φ(r) � uk(r) − vk(r) � = (uk − vk)Φ−k �= 0.
This overlap with the ground state mixes the modes k �= 0 with the
zero-frequency mode that cannot be treated as a proper Bogoliubov mode
(subsection 2.5.3). If one tries nevertheless to work with operators ˆ bk =
� d d r � ũ ∗ ν(r)δ ˆ Ψ(r) + ˜v ∗ ν(r)δ ˆ Ψ † (r) � , the inelastic coupling matrices W k ′ k are
found to diverge for k → 0. Due to this diverging coupling to low-energy
modes, perturbation theory will break down, even for small values of the
external potential.
3.2.2. Bogoliubov basis in terms of density and phase
Let us try the other option (3.13a), where the disorder is switched on before
the transformation (2.32) is applied. Φ(r) is the disorder-deformed groundstate
and the field operator is expressed in terms of ˆγk and ˆγ †
k
64
δ ˆ Ψ(r) = Φ0 δˆn(r)
+
Φ(r) 2Φ0
Φ(r)
i Φ0δ ˆϕ(r)
Φ0
�
e ik·r
��
Φ(r)
+ Φ0ak
� �
Φ(r)
ˆγk −
Φ(r)
(2.32) 1
=
2L d
2
k
Φ0ak
Φ0ak
− Φ0ak
�
ˆγ
Φ(r)
†
�
−k ,
(3.14)