Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3. Disorder
In the previous chapter, we have set up the general formalism for describing
Bogoliubov excitations in a weak external potential, and, as a first application,
we have applied it to the scattering of Bogoliubov waves at a single
impurity. Now it is time to enter the disordered world by specifying the
potential V (r) as a disordered potential with certain statistical properties.
The disorder potential causes an imprint in the condensate density, which
forms an inhomogeneous background for the excitations (figure 3.1).
In section 3.1, the experimentally relevant optical speckle potential is
presented, and its statistical properties are derived. The speckle disorder will
be used throughout this work. In the final results, the two-point correlation
function of the speckle disorder can be replaced with the correlation function
of any other kind of disorder.
The goal of this chapter is to give a useful characterization of the dynamics
of Bogoliubov excitations in presence of disorder. Of course, the results
should not depend on the particular realization of disorder, so suitable disorder
averages are necessary. Before doing so, some considerations about
the choice of the basis for the excitations have to be made. Until now, the
free Bogoliubov basis in terms of density and phase fluctuations has been
used rather intuitively. In section 3.2, it is discussed in detail, why this
basis actually is the only reasonable choice for the disordered Bogoliubov
problem.
All considerations made so far meet in section 3.3, the essential of this
chapter. The propagation of Bogoliubov excitations in the disorder averaged
effective medium is described by the average Green function, which
reveals the disorder-broadened dispersion relation, with a finite lifetime and
a renormalized propagation speed of the excitations. The optical analog of
this is the index of refraction and the absorption coefficient in a medium
like glass or water.
Figure 3.1: Schematic representation of the
disordered Bogoliubov setting. The disorder
potential (red) leaves an imprint in
the condensate (blue). On top of that,
wave excitations (green) are considered.
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