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