Topological Properties In Multi-dimensional Discrete-time Quantum Walks

Due to the advantages of being insensitive to details,not afraid of noise,and anti-jamming,the study of novel topological states has being one of the popular research topics in modern physics in the recent 40 years since the first discovery of topological states in 1980.For the first time in 2010,T.Kitagawa proposed the discrete-time quantum random walk(discrete-time quantum walk)as a general and concise quantum simulation platform for topological physics.Because of the advantages of the discrete-time quantum walk,such as strong controllability,simple form and rich topological physics,using discrete-time quantum walks to simulate various novel topological physical phenomena has become a hot research field in recent ten years.On the one hand,the theoretical group led by J.K.Asboth has carried out a comprehensive study on the topological problems in discrete-time quantum walks,and has given the basic method to calculate the topology.On the other hand,since the discrete-time quantum walk holds simple forms,strong controllability and is easy to realize in experiments,the experimental research in discrete-time quantum walks has been developed rapidly in the past ten years.The experimental progress of the discrete-time quantum walk provides a good platform for the application of topological states in the fields of information processing,materials science and quantum computing.Based on the current experimental techniques,the discrete-time quantum walk can be realized in different systems(cold atoms,linear optics,trapped ions,waveguides,superconducting circuits,nuclear magnetic resonance)and different spaces(coordinate space,orbital angular momentum space,momentum space,coherent state space,timebins space),respectively.At the same time,the experimental observations of topological boundary states,topological phase transitions and topological invariants have been greatly developed.It is worth noting that the conclusions based on discrete-time quantum walks are universal and suitable for general topological systems.Therefore,it is of great significance to systematically study the topological problems in discrete-time quantum walks.In this thesis,we study exotic topological phenomena of multi-dimentional discrete-time quantum walks in three different systems.The specific studies are as follows:1.One-dimensional topological quantum walks in cavity-based quantum networksWe propose a scheme to realize one-dimensional discrete-time quantum walk in cavity-based quantum networks.Using multiple cavity input-output processes,we successfully realize a translation operation of the discrete-time quantum walk.Moreover,due to its own characteristics of the cavity input-output process,the above translation operator will have a phase of ?,which is different from the standard translation operator discussed in other literature.We study the topological properties of this discrete-time quantum walk and give the complete topological phase diagram.By numerically calculating the probability distribution,we demonstrate the existence of topological edge states.By numerically calculating the second-order displacement,we observe the topological phase transitions of the system.Finally,by considering the loss and disorder of the cavity input-output system in real experiments,we verify that the above topological properties of the system are robust.2.Probe of topological invariants using quantum walks in coherent state spaceWe present a protocol to detect the topological invariants using discrete-time quantum walks in coherent state space,which is irrelevant to the selection of the initial coin state.Based on the current experimental techniques,the protocol presented here can be implemented in the real system of trapped ions.In this protocol,we successfully implement a spin-dependent flipping translation operation,which is different from the translation operator discussed in other literatures.This spin-dependent flipping translation operation will make the system have a simple chiral symmetry operator ?=?z,which will further simplify the measurement processing of the topological invariants.We study the topological properties of this quantum walk,give the topological phase diagram,and propose that the topological invariants of the system can be measured through the average projected phonon numbers.Finally,by considering the disorder and decoherence in the system of trapped ions,we verify that the above topological properties are robust.3.Second-order topological insulator in a two-dimensional coinless discrete-time quantum walkWe present a scheme to simulate the second-order topology using the two-dimensional coinless discrete-time quantum walk,which is constructed through the Benalcazar-Bernevig-Hughes(BBH)model.By studying the topological properties of this coinless discrete-time quantum walk systematically,we demonstrate that this quantum walk can be used to simulate the second-order topological insulator.We also verify that the nontrivial topological properties(when the system is in a non-trivial second-order topological phase,four-degenerate gapless zero-energy corner states will emerge)can be observed experimentally through measuring the probability distributions.Finally,we propose an experimental scheme to realize this quantum walk in a three-dimensional waveguide system and discuss the effect of disorder on the second-order topological properties.