Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

30 Introduction
We develop simple theoretic frameworks for the description of statical and dynamical properties
and discuss quantities which are crucial for their characterization. We also evaluate the validity
of GP-theory in order to demonstrate the applicability of our results.
In particular, after a short review of some standard results concerning a single particle in a
one-dimensional periodic potential, we will discuss
• the groundstate with and without harmonic trap (chapter 5)
• stationary states of Bloch form (chapter 6)
• the Bogoliubov excitations of the groundstate (chapter 7)
• the linear response to a density perturbation (chapter 8)
• the macroscopic dynamics with and without harmonic trap (chapter 9)
• the description of the system as an array of Josephson junctions (chapter 10)
• the propagation of sound signals (chapter 11)
• the quantum depletion (chapter 12)
Here is a more detailed overview of the thesis:
Chapter 5 discusses the groundstate of a condensate confined in a one-dimensional optical
lattice. We first consider a system without harmonic trapping which is uniform in the direction
transverse to the lattice. We study the chemical potential, the energy, the density profile and
the compressibility as a function of lattice depth s and the interaction parameter gn, where
g is the two-body coupling constant (3.11) and n is the average density. In a second step,
we allow for the additional presence of radial and axial harmonic trapping. We use the results
obtained for the purely periodic potential as an input to calculate the groundstate properties
in the combined trap.
In chapter 6, we extend the discussion of stationary condensates in a one-dimensional
lattice to non-groundstate solutions of the GP-equation. We focus on solutions which take
the form of Bloch states and investigate the associated band spectra for the energy and the
chemical potential in dependence on lattice depth and interaction strength carrying out a
detailed comparison both with the non-interacting and the uniform cases. The energy Bloch
band spectrum determines the current and therefore the group velocity and the effective mass.
In the tight binding regime, the energy and chemical potential bands take a simple analytic
form. From the expression for the Bloch energy bands we find equations for the current and
the group velocity. Exploiting the tight binding formalism, we also derive simple expressions
for the compressibility of the groundstate.
Chapter 7 deals with small perturbations of a stationary Bloch state condensate in the
periodic potential. We study in detail the Bogoliubv band spectrum of the groundstate both
numerically for all lattice depths and analytically in the tight binding regime. Special attention
is paid to the behavior of the sound velocity both for the groundstate and for a condensate
moving with non-zero group velocity.
The Bogoliubov band structure can be probed by studying the linear response of the condensate
to a weak external perturbation. In chapter 8 we consider the particular case in which