Ph.D Thesis


Ph.D StudentRaam Uzdin
SubjectThe Effects of Non-Hermitian Degeneracies on Light and
Matter Interactions
DepartmentDepartment of Physics
Supervisor Professor Emeritus Moiseyev Nimrod
Full Thesis textFull thesis text - English Version


Abstract

Non-Hermitian (NH) quantum mechanics (QM) is an extremely powerful tool in the study of meta-stable quantum systems. For example, the Hermitian description of ionization requires the use of an infinite number of states (continuum), while in NH formalism it may be described by a single state. In our view, NHQM is not a generalization of QM, but simply an efficient tool that helps study and discover new effects associated with meta-stable/resonance states. Findings based on NHQM should be reproducible in Hermitian simulations.


Perhaps the most exciting unique feature in NHQM is the NH degeneracy (also known as “Branch point” or “Exceptional point”). The main goal of this thesis is to find new effects associated with NH degeneracies and discuss their physical realization. While in Hermitian degeneracy only the eigenvalues coalesce, in NH degeneracies, the eigenvectors coalesce as well. This gives rise to interesting geometrical and topological effects. In analogy to the Berry phase, these effects appear when eigenstates are transported in the parameter space of the system. However in the NH case the transport around a closed circuit may generate more than just a phase. These effects appear when the time-independent Schrödinger Equation (TISE) is considered.


Naively, by virtue of the (Hermitian) adiabatic theorem, one would expect that the aforementioned TISE based geometrical/topological effects could also be observed in a very slow time evolution when solving the time-dependent Schrödinger Equation (TDSE). However, we find that this is not the case. We examine the flip effect and show that a certain aspect of it cannot be observed using the TDSE regardless of how slow the evolution is. This is explained by the appearance of relative 'gain modes', which amplify even the slightest non-adiabatic couplings. We analyze and predict new dynamical effects in the vicinity of NH degeneracy. In addition, a realization in optical waveguide was designed. Finally, we studied quantum evolution speed in NH systems. While in Hermitian evolution the evolution speed bound is given by the largest energy gap in the system, in NH systems this energy bound completely fails. We derived a new bound suited for both Hermitian and NH systems, and showed how to construct “efficient” Hamiltonians that reach this upper speed bound for any desired quantum evolution.