Explicitly Correlated Gaussian Approach: Method Development and Application to Ultracold Few-Fermion Systems
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Few-body physics plays an important role in quantum gases. For example, the lifetime of ultracold atomic gases is limited by atom losses. While atom losses are, in general, undesirable, they can be measured experimentally and analyzed in terms of three- and higher-body recombination processes. This dissertation takes a theoretical approach to discuss trapped and free-space few-fermion systems with emphasis on the development and implementation of numerical techniques.To solve the Schrodinger equation for trapped few-body systems, we implement the explicitly correlated Gaussian technique, which is a basis set expansion-type approach. The technique is used to investigate equal-mass spin-balanced and spin-imbalanced four-body systems confined in a spherically symmetric trap. A large portion of the zero-range energy spectra for infinitely large interspecies s-wave scattering length are tabulated and analyzed. The results serve as a benchmark for other numerical techniques and are relevant for understanding few-body energy spectra, which can be measured experimentally using radio- frequency spectroscopy. The energies were used to calculate the fourth-order virial coefficient and to investigate inter-system degeneracies. In addition, we discuss unequal-mass four-body systems consisting of three identical fermions and one distinguishable impurity.Experiments on ultracold Fermi gases cannot only realize three-dimensional few-body systems but also effectively low-dimensional few-body systems. The properties of quasi one-dimensional systems can be described by effective one-dimensional model Hamiltonian. The explicitly correlated Gaussian technique is used to investigate the validity regime of different effective one-dimensional model Hamiltonian through direct comparison with three- dimensional results.This thesis also develops a hyperspherical explicitly correlated Gaussian approach, which solves- within the hyperspherical framework- the hyperangular Schrodinger equation for few- body systems with finite angular momentum. One major advantage of this technique is that the hyperspherical framework provides access to bound states as well as to scattering states. As a first application, we solved and analyzed the energetically lowest-lying hyperangular eigenvalues of equal- and unequal- mass four-particle systems with infinitely large interspecies s-wave scattering length. The analysis can be extended to study interesting few-body phenomena like four-body resonances and Efimov physics.