Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
1.2. Disorder<br />
1.2. Disorder<br />
Idealized models, like infinitely extended perfect crystals, can explain a great<br />
deal <strong>of</strong> physical phenomena, but some important features are missed. Disorder<br />
is nearly always present in nature and can have dramatic effects on<br />
transport properties in all kinds <strong>of</strong> media. In solid-state physics, disorder<br />
appears in the guise <strong>of</strong> impurities and displacements in crystals, which influence<br />
the dynamics <strong>of</strong> electrons and phonons.<br />
Anderson localization and weak localization<br />
Disorder can lead to coherent localization <strong>of</strong> waves, which means the suppression<br />
<strong>of</strong> diffusion [1, 17]. In the case <strong>of</strong> electrons, this implies the suppression<br />
<strong>of</strong> electrical conductivity. Localization relies on the interference <strong>of</strong><br />
waves and occurs also for other types <strong>of</strong> waves, like microwaves [18], light<br />
[19, 20], ultrasound [21], water waves [17, Sec. 3.5], and atomic matter waves<br />
[22–26].<br />
Localization phenomena depend crucially on dimension [17, 27]. Scaling<br />
theory [28] allows general statements on localization. In one dimension, all<br />
states are exponentially localized, no matter how weak the disorder. Also<br />
in two dimensions, all states are in principle localized, but the localization<br />
lengths are exponentially large and <strong>of</strong>ten exceed the relevant length scales <strong>of</strong><br />
experimental setups. In three dimensions, localized and delocalized states<br />
coexist. Phonons are delocalized at low energies, separated by the so-called<br />
mobility edge from high-energy localized states [29, 30]. Electron states are<br />
localized at both upper and lower band edge, with delocalized states in the<br />
center <strong>of</strong> the band [11, 29].<br />
Originally, Anderson localization is a linear phenomenon, without interactions<br />
between the particles playing a role. For electrons, things are complicated<br />
by the Coulomb interaction [11]. Pure localization without interaction<br />
effects can, for example, be observed for light propagating through a cold<br />
gas, with the disorder realized by the random positions <strong>of</strong> the atoms [19, 20].<br />
In cold-atom experiments, one can reverse the roles <strong>of</strong> light and atoms: in<br />
the speckle field <strong>of</strong> a laser, the atoms are subject to a random potential proportional<br />
to the laser intensity [22, 23]. In both settings, the constructive<br />
interference <strong>of</strong> paths along closed scattering paths survives the disorder average<br />
and leads to enhanced backscattering. This regime <strong>of</strong> enhanced return<br />
probability is termed weak localization.<br />
3