Topological Property Of Discrete-Time Quantum Walk

Quantum walk is an extension of the classical random walk to the field of quantum mechanics.Quantum walk displays many different characteristics from classical counterpart due to quantum coherence.On the one hand,much faster quantum walk is the basis to design the quantum algorithm.On the other hand,quantum walk can be used to model the dynamics of various physical systems.Discrete time quantum walk can show rich topological structure,which provides a wide platform for simulating novel physical phenomena.In this thesis,we first give the definitions of quantum walk,present the method of calculating the winding number in discrete-time quantum walk and introduce the topological structure of the Su-Schrieffer-Heeger model.Then,the topological protected edge states of a quantum walk with photons and the topological phase transition of a quantum walk with cold atoms are studied in detail.In the scheme of performing discrete time quantum walk with photons,we study theoretically the topological properties of the quantum walk by introducing two controllable parameters in coin operator and conditional shift operator respectively.Since the quasi-energy and quasi-momentum of the quantum walk has a unique dispersion relationship,the energy gap between the two bands can be opened or closed by changing the introduced parameters.We find that the quantum walk has a nontrivial topological structure by calculating the geometric phase,and the topological phase diagram in this parameter space is present.Furthermore,we consider an inhomogeneous quantum walk,and analyze the variation of its occupied probability on the boundary with time to demonstrate the existing of the topologically protected edge states.Finally,the effects of noise on edge states are studied.If the noise projected operator commutes with the chiral symmetry operator,the topological protected edge states is seldom to be affected.Inversely,if the two operators do not commute,the edge state is greatly affected.For the scheme realizing discrete time quantum walk with cold atoms,the cold atom is assumed to be a two level system,which is regard as the freedom of degree of coin space.The atom transmitting on the one-dimensional optical lattice is corresponding to the shift of quantum walk in position space.In order to demonstrate the feasibility of a quantum walk using cold atoms,the probability distribution in position space and the position standard deviation are calculated.The topological phase diagram in coin parameter space is present by the winding number.There are nontrivial topological phase and the trivial topological phase in different region of parameter space.Finally,the first and the second order moments of the quantum walk probability distribution are investigated.It is found that their inflection points on the curve of moments varying with coin parameter are corresponding to the transition points of topological phases.As a result,the first and the second order moments can reveal the topological characteristics of the quantum walk.