Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3. Disorder
3.3. Effective medium and diagrammatic
perturbation theory
Finally, all preparations for tackling the disordered Bogoliubov problem
have been made. In section 2.3, the Bogoliubov Hamiltonian (2.42) has
been derived and in the previous sections, we have discussed the statistical
properties of the disorder potential. Now it is time to characterize the dynamics
of Bogoliubov excitations in presence of disorder. The key idea is to
average over the disorder and to understand the disordered Bose-Einstein
condensate as an effective medium for the propagation of Bogoliubov excitations.
The effective medium is characterized by quantities like the index
of refraction [subsection 3.4.5] describing the propagation speed, the mean
free path [subsection 3.4.2] or the diffusion constant [Boltzmann transport
length, subsection 3.4.2]. The main information is contained in the single-
(quasi)particle Green-function, also called propagator or resolvent, of the
system.
3.3.1. Green functions
The Bogoliubov excitations are bosonic quasiparticles with the commutator
relations
�
ˆγk, ˆγ †
k ′
� � � � †
= δkk ′,
ˆγk, ˆγ ′
k = ˆγ k , ˆγ† k ′
�
= 0. (3.18)
We define the single-(quasi)particle Green-function [16, Chapter 8]
Gkk ′(t) := 1 ��
ˆγk(t), ˆγ
i�
†
k ′(0) �� Θ(t) . (3.19)
The average 〈·〉 denotes the thermal average. Here, T = 0 is considered, so
it is simply the expectation value in the ground state. The Green function
contains information, how a quasiparticle created at time 0 in some state k ′
propagates to some state k where it is destroyed at time t.
Before starting with the disordered problem, let us determine the unperturbed
Green function, which is the starting point for the perturbation
theory. In absence of the perturbation terms in (2.43), the time evolution
of the Bogoliubov operators
ˆγk(t) = exp(−iɛkt/�)ˆγk(0) (3.20)
is trivial and the free Green function (G0(t)) kk ′ = G0(k, t)δ kk ′ is found as
66
G0(k, t) = 1
i� exp(−iɛkt/�)Θ(t). (3.21)