Abstract

An important new development in the field of ultracold atomic gases is
the study of the properties of these gases in a so-called optical
lattice. An optical lattice is a periodic trapping potential for the
atoms that is formed by the interference pattern of a few laser
beams. A reason for the interest in
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these systems is that the effects
of the interatomic interactions can be strongly enhanced. More
specifically, it has been shown in a beautiful experiment by Greiner
et al. in 2002 that by loading a Bose-Einstein condensate into an
optical lattice it is possible for the system to undergo a quantum
phase transition to a new quantum phase of matter, the so-called Mott
insulator phase. Within this Mott insulator phase each lattice site is
occupied by exactly one atom. This makes the Mott insulator phase
especially well suited for applications in the field of quantum
computation and quantum information processing.
We have theoretically investigated the above mentioned quantum phase
transition
and our formalism allows for a description of the Mott insulator phase at
nonzero temperatures.
Another important experimental
development in the field of ultracold atomic gases is the use of
Feshbach resonances to control the interatomic interactions. Such a
resonance occurs whenever two colliding atoms form a long-lived
molecule for some time. The crucial point of a Feshbach resonance is
that the above mentioned molecule has a magnetic moment that is not
equal to twice the magnetic moment of the atom. As a consequence the
energy difference between the two atoms and the molecule and hence the
interactions between the atoms can be controlled by using an external
magnetic field. By combining these two techniques, i.e, by trapping
ultracold atomic gases in an optical lattice and by tuning a magnetic
field near a Feshbach resonance there can be a new quantum phase
transition between two superfluid phases.
We gave derived the the theory for the description of these Feshbach
resonances
in optical lattices and applied it to various systems.
To be a bit more precise, if
we tune the external magnetic field such that the energy difference
between a molecule and two atoms is sufficiently negative, then the
gas consists of a Bose-Einstein condensate of molecules. In contrast,
if the energy difference is large enough and positive we have a gas
that consists primarily of a Bose-Einstein condensate of atoms. It
turns out that these two limits are separated by an Ising-like quantum
phase transition. By using atomic Bose gases near a Feshbach resonance
detailed experimental studies which test the theoretical predictions
of the statistical and dynamical properties of these quantum phase
transitions can be made.
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