【作者】 张晓霞；

【导师】 李志坚；

【作者基本信息】 山西大学 ， 凝聚态物理， 2017， 硕士

【摘要】 量子行走作为经典随机行走在量子力学中的推广,因量子干涉效应的存在,表现出许多与经典随机行走不同特性。一方面,其在扩散速度上远远快于经典随机行走,为设计高效的量子计算提供重要基础;另一方面,量子行走可以用来模拟各种物理系统的动力学行为,分离时间量子行走展示出丰富的拓扑相图,为人工模拟一些新奇的物理效应提供了广阔的平台。本文首先给出了量子行走的基本概念,并对分离时间量子行走中绕数的计算方法及Su-Schrieffer-Heeger模型的拓扑结构作了简单的介绍;接着就光学系统中的量子行走的拓扑特性和边界态及利用冷原子实现的量子行走的拓扑相变进行了研究。光学系统中的量子行走是操控一个粒子的量子动力学方案,我们理论上通过在该量子行走的旋转算符和条件平移算符中各引入一个可控参数,研究其随着这些参数变化的拓扑特性。该量子行走的准能量与准动量间具有独特的色散关系,可以通过改变引进的参数使得两支能带间的能隙打开或闭合。通过计算系统的几何相位,发现该量子行走具有非平庸的拓扑结构,并给出了系统在可调参数空间的拓扑相图。进而考虑空间不均匀的一维分离时间量子行走,分析其在边界上的占有几率随时间的变化,验证了拓扑保护边界态的存在。最后研究了不同种类噪声对边界态的影响,当在系统中引入噪声的投影算符与系统手征算符对易时,系统的边界态只受到微小的影响,而当这两个算符不对易时,边界态会受到较大的影响。在利用冷原子实现量子行走的方案中,将冷原子近似为二能级系统,作为量子行走中的硬币自由度,其在一维光学晶格上发生跃迁作为位置空间中的平移操作。通过计算冷原子在位置空间的几率分布,以及位置标准偏差随着时间的变化,说明利用冷原子实现量子行走的可行性。接着计算该系统的拓扑不变量—绕数,发现在不同参数区域具有平庸拓扑相和非平庸拓扑相,由此给出系统在可调参数空间的拓扑相图。对比拓扑相图与一阶位置矩和二阶位置矩在参数空间的变化特征,发现可观测量一阶位置矩和二阶位置矩在不同的拓扑相区域会表现出两种完全不同的变化形式,变化曲线的拐点刚好对应两支能带闭合时的相变点,由此可以通过一阶位置矩和二阶位置矩的变化特征来揭示系统的拓扑相变。更多还原

【Abstract】 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.更多还原