Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

6. Bloch Oscillations and
Time-Dependent Interactions
The second part of this work is somewhat separate from the Bogoliubov
part, as now a lattice system is considered (figure 6.1), in contrast to the
continuous system in part I. Instead of the disorder leading to scattering
and localization, now the tilted lattice localizes the wave packet by the phenomenon
of Bloch oscillation. The perturbation comes as a time-dependent
interaction, which in general destroys the wave packet.
There are also strong links to part I. After the smooth-envelope approximation
(6.7), the discrete Gross-Pitaevskii equation takes the form of
the continuous Gross-Pitaevskii equation (which is also known as nonlinear
Schrödinger equation), but now with a time-dependent effective mass.
Additionally to the time-dependent mass, also the interaction can be made
time-dependent by means of a Feshbach resonance. Together they provide a
source of energy for the growth of excitations (dynamical instability). These
excitations are of the same type as the Bogoliubov excitations in part I. The
growth of these excitations is the main mechanism for the destruction of the
coherent wave packet.
The work of this part was done in collaboration with the group of Francisco
Domínguez-Adame at Complutense University in Madrid. The essentials
have been published in [129, 130].
6.1. Introduction
Historically, the interest in electrons and phonons in crystals has lead to
the investigation of lattice systems. Understanding quantum particles as
waves is the key to understanding the dramatic effects a lattice can have
Figure 6.1: Schematic representation of the
setting for Bloch oscillations. The wave
packet (blue) is located in the wells of a
deep tilted lattice potential (gray)
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