Resonance Transmission And Topological Properties In Quantum Walk

As an extension of classical random walk,quantum walks have been extensivelyinvestigated over the past decades.Due to quantum coherence,quantum walks showmany novel transmission characteristics,which is not only important to designing thequantum algorithms but also very useful for modeling the quantum dynamics.In thisthesis,firstly present are the concepts of quantum walks,the relationship of discretetime quantum walk and the dynamic equation in relativistic quantum mechanics,andthe topological description of discrete time quantum walk.Then,we in detail studythe resonance transmission of one-dimensional quantum walk with phase defects andthe topological properties of split step quantum walk respectively.Our results may beuseful for high fidelity transfer of quantum states and synthetic topological insulator.Regarding the phase parameters of coin operators in discrete time quantum walkas the defect potential,the resonance transmissions of discrete time quantum walk arestudied when the one-dimensional lattice includes two phase defects or a piece ofphase defects.The dependence of the wave velocity direction on the moment andenergy is analyzed through the unique dispersion relation of discrete time quantumwalk.Then we use the condition of energy conversation at the boundary points toobtain the transmission rate and discuss its variation with the incident momentum fordifferent strengths and widths of defects in detail.The different resonant behaviors areshown when the strength of defect is less or greater than π/2.Correspondingly theresonances occur in the wide region of incident momentum or the sharp resonantpeaks appear at the discrete values of momentum.Under the condition of strongdefect strength,i.e.,Φ approaching to π,the qualitative relation of the number ofresonant peaks and the widths of defect region is present.The split-step quantum walk has rich topological structures.We obtain the phasediagram in coin parameter space by calculating the energy band and the windingnumber.When the energy gap closes,the corresponding parameters are just theboundary between two different topological structures.The second order moment ofposition and the entanglement between the freedom degree of position and thefreedom degree of coin have the similar variation with the coin parameter.Both of them show different change behaviors in different topological phase regions.In particular,the curve of the second order moment of position has inflection points which are corresponding to the boundary of different topological phase.As a result,the topological transition can be described by these observable quantities.As prediction by the bulk-boundary correspondence,the bound state indeed exists at the boundary of topological phase,which is illustrated by the variation of probability at the boundary with the time evolution.