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Abstract:
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If there is a general theme to this thesis , it is the effects of strong correlations in both bosons and fermions . The bosonic system considered here consists of ultracold alkali atoms trapped by interfering lasers , so called optical lattices . Strong interactions , realized by increasing the depth of the lattice potential , or through the phenomenon of Feshbach resonances induce strong correlations amongst the atoms , rendering attempts to describe the systems in terms of single particle type physics unsuccessful . Of course strong correlations are not the exclusive domain of bosons , and also are not caused only by strong interactions . Other factors such as reduced dimensionality , in one -dimensional electron gases , or strong magnetic fields , in two -dimensional electron gases are known to induce strong correlations . In this thesis , we explore the manifestations of strong correlations in ultracold atoms in optical lattices and interacting electron gases . Optical lattices provide a near -perfect realization of lattice models , such as the bosonic Hubbard model (BHM ) that have been formulated to study solid state systems . This follows from the absence of defects or impurities that usually plague real solid state systems . Another novel feature of optical lattices is the unprecedented control experimenters have in tuning the different lattice parameters , such as the lattice spacing and the intensity of the lasers . This control enables one to study the model Hamiltonians over a wide range of variables , such as the interaction strength between the atoms , thereby opening the door towards the observation of diverse and interesting phenomena . The BHM , and also its variants , predict various quantum phases , such as the strongly correlated Mott insulator (MI ) phase that appears as a function of the parameter t /U , the ratio of the nearest neighbor hopping amplitude to the on -site interaction , which one varies experimentally over a wide range of values simply by switching the intensity of the lasers . But as always , even in these designer -made "solid state" systems , practical considerations introduce complications that blur the theoretical interpretation of experimental results , such as inhomogeneities in the lattice structure . The first part of this thesis presents a quantum theory of ultracold bosonic atoms in optical lattices capable of describing the properties of the various phases and the transitions between them . Its usefulness , compared to other approaches , we believe rests in its broad applicability and in the relative ease it handles the complications while producing quantitatively accurate results . |