Universal and non-universal properties of ultracold few-atom systems
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Ultracold atomic systems have been of great theoretical and experimental interest in the past few decades. They provide ideal paradigms to study quantum phenomena, simulate condensed matter systems, and could potentially become a platform for quantum computations. This thesis focuses on the theoretical investigation of universal and non-universal properties of ultracold few-atom systems. The goal of this thesis is to answer or provide insights into several interesting open questions. First, what is the role of the non-universal three-body parameter in weakly-interacting harmonically trapped Bose gases and at which order does it kick in? Second, how large is the ground state energy of small two-component Fermi gases with infinitely strong interactions compared to their non-interacting counterparts? Lastly, what is the interplay between the spin-orbit coupling and the $s$-wave two-body interaction and how does it affect the energy spectrum of the two-atom system?In order to answer these questions, we develop and employ different analytical and numerical techniques. A perturbative effective field theory treatment to study the weakly-interacting trapped Bose gas is employed. The treatment is extended to up to fourth-order perturbation theory and techniques to regularize the diverging sums are developed. A logarithmic divergence in the three-body sector is identified and the need for a non-universal three-body parameter is shown.The explicitly correlated Gaussian method is used to calculate the ground state energy of small trapped two-component Fermi gases with up to ten particles. A code is developed from scratch and parallelized. An improved optimization scheme is developed. A novel technique to remove the finite-range dependence is proposed and tested. Our calculations serve as benchmarks for other numerical techniques. The explicitly correlated Gaussian method is extended to treat systems with periodic boundary conditions for the first time.Standard perturbation theory is used to treat the spin-orbit coupling term in trapped two-atom systems. Our treatment is based on the exact two-body zero-range solution in the trap. The interplay between the spin-orbit coupling and the two-body $s$-wave interaction is investigated and shown to have a great impact on the energy spectrum for certain parameter combinations.