Matter waves with special boundary conditions

Abstract

This cumulative thesis is based on three publications [I, II, III] presenting a selection of the author’s research activities under the supervision of apl. Prof. Dr. Maxim A. Efremov at the Institut für Quantenphysik of Universität Ulm during the years 2016 to 2024. In all three articles, the dynamics of non-trivial quantum systems, lacking a closed-form analytical solution, is studied. Rather than investigating the systems with purely numerical methods, the partially approximate symmetries and constraints are explored in order to find deliberately simplified version of these systems with eigenstates adhering to the corresponding boundary conditions and symmetries. Subsequently, these states are used as a basis to examine the dynamics of the original, more complex, systems in an interpretable way, as the number of the participating relevant states is highly reduced. Publication I is devoted to the analysis of diffractive waveguides. Such structures repeatedly focus a propagating wave, while the partial focusing is accomplished solely by the truncation of the edges with a slit, i. e. a pure amplitude modulation in contrast to the phase modulation of a refractive lens. The transmission efficiency is analyzed in terms of the waveguides' periodic eigenmodes, whose existence is also successfully verified in the experiments. Publication II compares the classical and quantum-mechanical phase-space dynamics of electrons in a free-electron laser. Pseudo-periodic eigenstates are leveraged to study the transition from the quantum regime, featuring a reduction of the electron to an effective two-level system, to the classical dynamics. Two parameters governing this transition are identified and subsequently their the effect on the gain of the laser field is examined. Publication III explores the scattering of two cold atoms that are confined to spherical shells. The focus is on the occurrence and characterization of confinement-induced resonances resulting from the coupling of center-of-mass and relative degrees of freedom. The dependence of the resonances on the scattering length and the shell radius is investigated. For this purpose, the exact spectrum is calculated so that the resonances can be determined from avoided crossings. The involved states are identified with the help of approximate eigenstates that exhibit a hyperspherical symmetry and incorporate the two-particle interaction as a boundary condition