Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

Abstract
In this thesis, different aspects of interacting ultracold bosons in presence
of inhomogeneous external potentials are studied. The first part deals with
repulsively interacting Bose-Einstein condensates in speckle disorder potentials.
In the Bogoliubov approach, the many-body problem is split into the
Gross-Pitaevskii condensate (mean-field) and the Bogoliubov excitations,
which are bosonic quasiparticles. The disorder potential causes an imprint
in the condensate, which makes the Hamiltonian for the Bogoliubov excitations
inhomogeneous. The inhomogeneous Bogoliubov Hamiltonian is
the starting point for a diagrammatic perturbation theory that leads to the
renormalized Bogoliubov dispersion relation. From this effective dispersion
relation, physical quantities are derived, e.g. the mean free path and disorder
corrections to the speed of sound and the density of states. The analytical
results are supported by a numerical integration of the Gross-Pitaevskii
equation and by an exact diagonalization of the disordered Bogoliubov problem.
In the second part, Bloch oscillations of Bose-Einstein condensates in
presence of time-dependent interactions are considered. In general, the interaction
leads to dephasing and destroys the Bloch oscillation. Feshbach
resonances allow the atom-atom interaction to be manipulated as function
of time. In particular, modulations around zero are considered. Different
modulations lead to very different behavior: either the wave packet evolves
periodically with time or it decays rapidly. The former is explained by a periodic
time-reversal argument. The decay in the other cases can be described
by a dynamical instability with respect to small perturbations, which are
similar to the Bogoliubov excitations in the first part.