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This dissertation presents my research covering the field of ultracold atoms loaded in optical lattices. The static and dynamical properties of atoms in combined periodic and parabolic potentials are studied, with a focus on the strongly interacting regimes. Because parabolic magnetic and optical potentials are routinely used to confine atoms, the results of this research are directly relevant to ongoing experimental endeavours in atomic physics.

After a review of the basic theory of atoms in homogeneous periodic potentials, the equilibrium and non-equilibrium properties of non-interacting and interacting atoms in periodic plus parabolic potentials are studied. The problem of the localization of the many-body wavefunction for systems with arbitrary peak onsite density is presented in Chapters 3 and 4. The physics pertaining to the experimental realization of Mott insulator states with one or more atoms per sites in inhomogeneous lattices is elucidated by introducing an intuitive model for strongly interacting bosons in one dimension. This model is then utilized to study the decay of the dipole oscillations of atomic ensembles subject to a small displacement of the parabolic potential. Good agreement is found with results of recent experiments.

Chapters 5 and 6 are dedicated to the characterization of the Mott insulator state with unit filling, which plays a central role in proposed schemes for neutral atom quantum computation. The usefulness of Bragg spectroscopy to probe the excitation spectrum of the Mott state in homogeneous lattices is analyzed in Chapter 5, where the limits of validity of linear response theory in this strongly correlated regime are delimited. In Chapter 6 the effects of finite temperature on the confined Mott insulator state are studied, and a scheme is devised for possibly estimating the system's temperature, at energies of the order of the inter-particle interaction energy.

Finally, in Chapter 7, a proposal is introduced to utilize the Mott state as a robust register for neutral atom quantum computation. Unwanted residual quantum coherences inherent to the Mott insulator ground state are eliminated by a judicious choice of the trapping potentials and a selective measurement on a molecular photo-associative transition.