Entanglement : quantification via uncertainties and search among ultracold bosons in optical lattices

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.