Topological Properties And Dynamics Of The Periodic Discrete-time Quantum Walk

Quantum walk is the generalization of classical random walk in quantum mechanics.It has many dynamic characteristics which are better than classical random walk,such as faster propagation speed.Quantum walk has become a powerful tool for many efficient quantum algorithms.In addition,quantum walk can act as a versatile quantum simulator of topological phases in condensed matter physics.The dynamics and topological properties of quantum walks with different configurations are now being explored.In this thesis,we first introduce the basis concepts of quantum walk,PT symmetry and the topological properties double-step quantum walk.Then we study the topological properties of unitary and non-unitary multi-period quantum walks,and the dynamics of time-dependent quantum walk.For discrete-time quantum walk on one-dimensional lattice,we introduce a new kind of step-dependent coin operator to study the probability distribution and entanglement dynamics.The discrete-time quantum walk is driven by a single coin operator and the direct product of two coin operators respectively.When we change the coin parameter,it is found that both of two driven-type quantum walks exhibit the dynamic behaviors including complete localization,periodic localization and diffusive distribution.Meanwhile,the dynamic periodical oscillation or rapidly reaching the maximum of the entanglement between the degrees of freedom of the coin and the position are displayed.In particular,the probability distribution with different characteristics are obtained by controlling the evolution steps for two-coin-operator driven quantum walk,which is significant to the quantum manipulation.Topological phases and the associated multiple edge states are studied by constructing a one-dimensional unitary multi-period quantum walk with chiral symmetry.Introducing gain and loss to coin operator,the case is generalized to an one-dimensional non-unitary multi-period quantum walk with parity time(PT)symmetry.It is shown that large topological numbers can be obtained when choosing the appropriate time frame.The maximum value of the winding number can reach the number of period in one-step evolution operator.The validity of the bulk-edge correspondence is confirmed,but for odd-period quantum walk and even-period quantum walk,they have different configuration of the 0-energy edge state and ?-energy edge state.On the boundary,two kinds of edge states always coexist in equal amount for odd-period quantum walk,but three case including equal amount,unequal amount or even only one type,may occur for even-period quantum walk.