Tunneling Theory for Few-Body Systems in One-Dimensional Traps

Abstract

In recent experiments, carried out at the University of Heidelberg, tunneling rates of ultracold distinguishable fermions out of an optomagnetical trap have been measured. The fermions interact by a tunable short-ranged interaction, and the trap is asymmetric making the trapped quantum system effectively one-dimensional. In this thesis, a method for calculating the energy levels and tunneling rates of one and two interacting particles out of a very general onedimensional potential well is devised. The method is based on expanding the Schr odinger equation of the system in a complex-momentum basis. This is done utilizing the so-called Berggren completeness relation. Ultimately, the basis expansion leads to a complex symmetric non-Hermitian eigenvalue problem for a large, dense matrix. The general method is applied to a system of trapped, ultracold fermionic atoms in a setup that closely resembles the Heidelberg experiments. The short-ranged interaction is modeled as a point-interaction, and the trap potential is regularized at large distances from the interesting region. The obtained energies and decay rates are contrasted to results obtained using the Wentzel-Kramers-Brillouin (WKB) approximation. Notable differences can be observed, and these may be due to insufficiency of the WKB approximation to accurately describe the system in question.