Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

1. Interacting Bose Gases—Complex
Dynamics in Lattices and Disorder
This work is dedicated to the intriguing interplay of interaction and inhomogeneous
potentials. Often these elements tend to produce opposite physical
effects. Each of them separately is in general well understood, but together,
they lead to complicated physical problems. A good starting point is solving
the problem of one competitor alone, and then adding the other one.
In part I, we start with the homogeneous interacting Bose gas and then
add a disorder potential as perturbation. In part II, we proceed the other
way around: We start with the non-interacting Bloch oscillation in a tilted
lattice potential, and then switch on the particle-particle interaction.
The main ingredients disorder and interaction are ubiquitous in nature.
Both of them have dramatic effects on transport properties. Disorder can
induce Anderson localization of waves [1], which suppresses diffusion and
conduction. In lattice systems, described by the Bose-Hubbard model, repulsive
interaction drives the transition from superfluid to the Mott insulator
[2–4].
The physical system of choice is a Bose-Einstein condensate (BEC)
formed of an ultracold atomic gas. Bose-Einstein condensation is a quantumstatistical
effect that occurs at high phase-space density: at sufficiently low
temperature and high particle density, macroscopically many particles condense
into the single-particle ground state. With some efforts, this exotic
state of matter is achieved in the laboratory. The wave function of the condensate
is a macroscopic quantum object and features macroscopic phase
coherence. Thus, Bose-Einstein condensates can interfere coherently [5, 6],
just like the matter wave of a single particle or coherent light in Young’s
double-slit experiment.
Ultracold-atom experiments are not only a very interesting field of physics
by themselves, but can also serve as model systems for problems from other
fields of physics. For example, the Bose-Hubbard model [4], and the phenomenon
of Bloch oscillation [7, 8] are realized experimentally. There are
analogies with completely different fields of physics. For example, dilute
Bose-Einstein condensates are well described by the Gross-Pitaevskii equation,
a prototypical nonlinear wave equation, also known as the nonlinear
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