MULTI-BAND BOSE HUBBARD MODELS AND EFFECTIVE THREE-BODY INTERACTIONS

Experiments with ultracold atoms in optical lattice have become a versatile testing ground to study diverse quantum many-body Hamiltonians. A single-band Bose-Hubbard (BH) Hamiltonian was first proposed to describe these systems in 1998 and its associated quantum phase-transition was subsequently observed in 2002. Over the years, there has been a rapid progress in experimental realizations of more complex lattice geometries, leading to more exotic BH Hamiltonians with contributions from excited bands, and modified tunneling and interaction energies. There has also been interesting theoretical insights and experimental studies on “un- conventional” Bose-Einstein condensates in optical lattices and predictions of rich orbital physics in higher bands. In this thesis, I present our results on several multi- band BH models and emergent quantum phenomena. In particular, I study optical lattices with two local minima per unit cell and show that the low energy states of a multi-band BH Hamiltonian with only pairwise interactions is equivalent to an effec- tive single-band Hamiltonian with strong three-body interactions. I also propose a second method to create three-body interactions in ultracold gases of bosonic atoms in a optical lattice. In this case, this is achieved by a careful cancellation of two contributions in the pair-wise interaction between the atoms, one proportional to the zero-energy scattering length and a second proportional to the effective range. I subsequently study the physics of Bose-Einstein condensation in the second band of a double-well 2D lattice and show that the collision aided decay rate of the con- densate to the ground band is smaller than the tunneling rate between neighboring unit cells. Finally, I propose a numerical method using the discrete variable repre- sentation for constructing real-valued Wannier functions localized in a unit cell for optical lattices. The developed numerical method is general and can be applied to a wide array of optical lattice geometries in one, two or three dimensions.

African-American Composers for the Concert Saxophone: A Look at Three Prolific Composers

African-American composers within the field of classical music have made very profound contributions to the literature. In the field of chamber music, Scott Joplin, William Grant Still, Adolphus Hailstork and other composers illustrious composers have created an established and well-documented body of repertoire for many orchestral wind instruments. The saxophone repertoire, however, has not been developed as fully due to its limited tradition as an orchestral instrument and its prominence in the tradition of jazz and popular music. African-American composers in particular appear to be significantly under-represented within the standard concert saxophone literature. My personal experiences with saxophone repertoire in academic settings, solo recitals, conferences and in surveys of standard repertoire from nationally-recognized saxophone teachers support this assertion. There are many African-American composers who have made substantial contributions to the body of repertoire for the concert saxophone. This dissertation examines the works of three prolific African-American composers for the concert saxophone; Dr. Yusef A. Lateef, Andrew N. White III, and Dr. David N. Baker. All have composed more than five separate works featuring the concert saxophone. This project comprises three recitals, each dedicated to one of the three composers selected for this dissertation. Each recital presented will present their compositions featuring the saxophone as a soloist with various types of accompaniment. The project also includes newly-created piano reductions of Dr. David Baker's works for saxophone and orchestra made collaboratively with Baker and arranger John Leszczynski.