Ankara : The Department of Physics and the Institute of Engineering and Science of Bilkent University, 2009.Thesis (Ph. D.) -- Bilkent University, 2009.Includes bibliographical references leaves 63-69.In the first part of the Thesis, the known measures of entanglement for finite dimensional
systems are reviewed. Both the simplest case of pure states that belong to bipartite systems
and more general case of mixed states are discussed. The multipartite extensions are also
mentioned. In addition to the already existing ones, we propose a new measure of entanglement
for pure states of bipartite systems. It is based on the dynamical symmetry group
approach to quantum systems. The new measure is given in terms of the total uncertainty
of basic observables for the corresponding state. Unlike conventional measures concurrence
and 3-tangle, which measure the amount of entanglement of different groups of correlated
parties, our measure gives the total amount of multipartite entanglement in a specific state.
In the second part of the Thesis, the trapping of bosonic atoms in optical lattices is reviewed.
The band structure together with Bloch functions and Wannier basis are discussed
for this system. In relation with that, the corresponding Bose-Hubbard model and by the use
of this model, the resulting superfluid to Mott-insulator quantum phase transition is summarized.
In this regard, the Bose-Hubbard Hamiltonian of a specific system, namely ultracold
spin-1 atoms with coupled ground states in an optical lattice is considered. For this system
we examine particle entanglement, that is characterized by pseudo-spin squeezing both for
the superfluid and Mott-insulator phases in the case of ferromagnetic and antiferromagnetic
interactions. The role of a small but nonzero angle between the polarization vectors of counterpropagating
lasers forming the optical lattice on quantum correlations is investigated as
well.Öztop, BarışPh.D