Slow-fast dynamics caused by exceptional points in non-hermitian quantum mechanics

Abstract

In this thesis, we analyze a Schrödinger equation with a non-Hermitian Hamiltonian involving an exceptional point. In the exceptional point at the origin the eigenvalues and eigenstates of the Hamiltonian coalesce. The governing non-Hermitian Hamiltonian depends on a complex parameter, which is assumed to vary slowly along a circle. Depending on whether the circle encloses the exceptional point, or not, different adiabatic or non-adiabatic phenomena occur. If the exceptional point is enclosed by the circle a so called “delayed bifurcation” or “stability loss delay” occurs, i.e. solutions follow a seemingly unstable quantum state for a considerable time followed by a rapid non-adiabatic transition to a new stable state.We analyze this problem by methods from dynamical systems theory, in particular “geometric singular perturbation theory” (GSPT) and the “blow-up method”. We show that, if the circle lies to the right of the exceptional point, unique stable and unstable periodic orbits exist, which correspond to purely adiabatic solution. We show that the phenomenon of delayed stability loss occurs, if the circle encloses the exceptional point. This leads to a delayed rapid non-adiabatic transition. Our approach is based on extending the relevant solutions into the complex time plane along a suitable contour, along which GSPT and the blow-up method are applicable.