Topological Phenomena In Quantum Dyanmical Processes

Since the discovery of quantum Hall effect,the research on topological proper-ties of electronic materials is approaching perfection.Recent experimental progress has further led to the exciting possibility of studying novel topological phenomena in interacting,non-Hermitian,and dynamical settings using synthetic systems.Facilitated by the highly tunable parameters in these synthetic systems,the physical mechanism of topological phenomena can be explored in greater depth,thus promoting the develop-ment of related fields.These include Floquet,non-Hermitian,and systems undergoing quantum quench.The effective Hamiltonian of a Floquet quantum system is a periodic function of time,where the temporal degrees of freedom can greatly enrich topologi-cal properties of the system.As a typical Floquet system,discrete-time quantum walk,in its simplest form,can be used to explore,dynamics in a wide range of topological phases.Non-Hermitian physics are ubiquitous in open systems,which give rise to lots of novel physical phenomena,captured by non-Hermitian quantum mechanics.Quan-tum quenches study the non-equilibrium physics,whose understanding often involves concepts and tools developed within the context of condensed-matter physics,quantum information,and topological phases.This thesis studies the novel topological phenomena in Floquet quantum system,non-Hermitian quantum system,and quantum-quench system.The thesis covers con-texts such as non-unitary discrete-time quantum walks and their topological physics,and topological phenomena in both unitarily and non-unitarily quench dynamics of quantum systems.More concretely,we introduce non-unitary discrete-time quantum walks,where non-unitarity is achieved via partial measurements.We then study topo-logical properties of the quantum-walk dynamics,and analytically solve for topological edge states,which can be detected in experiments.In the quantum quench process of topological systems,we obtain the sufficient condition for the occurrence of dynami-cal quantum phase transitions,which is closely related to the topological phase tran-sitions.We then apply this theory to a two-dimensional Hermitian topological system and a one-dimensional non-Hermitian topological system.Taking account of the peri-odicity in time,we construct dynamical topological invariants on the momentum-time sub-manifolds,including the Hopf invariant for the two-dimensional topological sys-tem and the dynamical Chern number for the one-dimensional topological system.We also present experimental proposals to use unitary and non-unitary discrete-time quan-turn walks to simulate dynamical quantum phase transitions and dynamical topological invariants.