Quantum simulation is the study of one quantum mechanical system via analog with another. In this thesis we explore a simple model of interacting bosons on a lattice, known as the Bose-Hubbard model, by experimental investigation of ultracold rubidium atoms in an optical lattice made from laser light. We describe the construction and stabilization of an optical superlattice with threefold symmetry, and its use in studying the Bose-Hubbard model on triangular and trimerized kagome lattices. We study the short range phase coherence of a Mott insulator on the triangular lattice, and develop a scheme to mitigate out-of-equilibrium effects arising from the state preparation. We show the first experimental realization of an optical trimerized kagome lattice for cold atoms, and discuss experiments characterizing this lattice. Finally, we provide evidence for a Mott insulating state with fractional average particle number per site with measurements of the nearest-neighbor phase coherence of strongly interacting atoms in the trimerized kagome lattice.
We produce a trimerized kagome lattice for ultracold atoms using an optical superlattice formed by overlaying triangular lattices generated with two colors of light at a 2:1 wavelength ratio. Adjusting the depth of each lattice tunes the strong intratrimer (
J) and weak intertrimer (${J}^{\ensuremath{'}}$) tunneling energies, and also the on-site interaction energy
U. Two different trimerization patterns are distinguished using matter-wave diffraction. We characterize the coherence of a strongly interacting Bose gas in this lattice, observing persistent nearest-neighbor spatial coherence in the large $U/{J}^{\ensuremath{'}}$ limit, and that such coherence displays asymmetry between the strongly and the weakly coupled bonds.