The role of few-body physics in ultracold gases has become increasingly important. For instance, knowledge of two-body scattering has allowed tuning of the two-body interaction strength through a Fano-Feshbach resonance in both Bose and Fermi gases. A detailed understanding of three-body scattering provides information on the trap loss processes which can determine the lifetime of a gas. It can also be used to explore basic few-body behavior. The description of a gas of bosonic dimers relies on understanding the four-body dimer-dimer scattering process. This dissertation concerns two related concepts, the behavior of N-fermion systems with tunable two-body interactions and controllable few-body systems and their impact on ultracold gases. Both fall under the overarching area of controlled few-body physics in many-body systems.
The N-fermion system is described in using the basic methods of hyperspherical coordinates. This approach provides an intuitive picture of the qualitative behavior of a degenerate Fermi gas by finding an effective one-dimensional potential that predicts the ground state energy and the RMS size of the gas in fair agreement with other theoretical techniques. This study also provides an important initial connection between the techniques of few-body physics and many-body systems. By extending the method to multi-component gases, I also predict a dynamic instability in three- and four-component Fermi gases that is now within the range of experimental exploration.
In the second part of this dissertation, I explore three- and four-body problems with s-wave interactions. By applying a new method based on the Lippmann-Schwinger equation to the three-body problem, I develop scattering potentials for an arbitrary three-body system with zero-range pseudo-potential interactions. By extending this approach to multi-channel interactions, a new class of three-body quasi-bound states is predicted, creating true three-body Fano-Feshbach resonances. I also approximately find the four-fermion hyperradial scattering potential. These potentials are then used to find the zero-energy s-wave dimer-dimer scattering length to and accuracy larger than previous calculations. By exploring the dimer-dimer scattering potential, the full universal, energy-dependent, dimer-dimer scattering length is found, including the inelastic processes of dimer dissociation and relaxation.
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}, year = {2009}, publisher = {University of Colorado Boulder}, address = {Boulder}, }