Non-Hermitian Quantum Walks In One-dimensional Lattice And Topological Phase Transition

Quantum walk,originated from classical random walk,has now become a universal quantum simulation platform.Physicists have realized quantum walks in more and more physical systems,such as optical resonators,cold atoms,superconducting qubits,single photons,trapped ions,coupled waveguide arrays,and nuclear magnetic resonance.Introducing interactions,disorder,defects,and hopping modulations into these systems,and studying their effects on the dynamic behavior of quantum walks,have become main focuses of the frontiers.Within the paradigm of quantum walks,one can investigate many physical properties of the system,such as topology,entanglement,and correlation properties.Characterizing topological properties of a system by implementing quantum walks,has vastly expanded the research scope of quantum walks.In recent years,non-Hermitian physics has attracted more and more research attentions,since gain and loss are usually quite common in many physical systems.There are many such physical systems,such as coupled quantum dots,optical lattices,optical waveguides,etc.For systems mentioned above,it is more natural and appropriate to describe them with non-Hermitian Hamiltonians.Up to now,many pioneer research works have revealed many exotic and important discoveries.Different from the traditional Hermitian systems,some non-Hermitian systems come out with fractional topological numbers,while the classical bulk-boundary correspondence still holds in some of the non-Hermitian systems.However,in most cases,the traditional bulk-boundary correspondence fails.In this context,many central concepts of non-Hermitian systems need to be carefully studied and reconsidered.A large number of fundamental physical problems of non-Hermitian topological systems have emerged nowadays.Some researchers have proposed the generalized Brillouin zone by generalizing the concept of Brillouin zone.Based on that,a non-Bloch topological band theory suitable for non-Hermitian systems has been established.With the aid of non-Bloch topological invariants constructed with Q matrix,one can accurately predict the region where the boundary state exists.In this way,the bulk-boundary correspondence of the non-Hermitian system can be restored.In the second part of this thesis,we take the non-Hermitian SSH model as an example to give a specific introduction to some of the characteristics of the non-Hermitian system.Based on rigorous numerical results,the failure and reconstruction process of the bulk-boundary correspondence is shown,and the numerical calculation methods of the generalized Brillouin zone in the relevant literature is reviewed in detail.So far,there have been many studies focusing on the static properties of non-Hermitian systems.Here,we mainly study the quantum dynamics of non-Hermitian systems.In Chapter 3,we studied the non-Hermitian quantum walks in a finite bipartite lattice with long-range hopping.In one sublattice,there is an equal amount of loss rate at each site and the quantum walker will leak out whenever it arrives there.In the other sublattice,there is no decay.When particles start out from a lattice site with no loss,our intuition tells us that over time,the walker will eventually disappear completely.In principle,the probability of dissipation through each unit cell can be obtained by experimental detection.Since each unit cell has the same dissipation rate,the dissipation probability through certain unit cell is expected to decrease as the distance from the unit cell to the starting point of the quantum walker increases.However,unexpectedly,in a specific parameter area,rigorous numerical simulation gives out a very counterintuitive profile of the decay probability.Contrary to the above intuitive image,the decay probability has a significant distribution on the boundary site that is the farthest from the starting point.In order to understand this anomaly,we calculated the energy spectrum of the non-Hermitian lattice system and studied the change of its topological properties along with the variation of intracell hopping amplitude.It is found that the singular dissipation probability distribution obtained after the quantum walk completed,is closely related to the topological properties of the system.Our results show that the parameter region where the singular dissipation probability distributions appear is intimately related with the topological non-trivial region of the non-Hermitian bipartite lattice.