Dynamical Evolution And Scattering Properties Of The Continuous Time Quantum Walk

Quantum walk(QW) is the quantum version of classical random walk. It is not only important to design the quantum algorithms but also very useful for modeling the dynamic evolution of complex system. In the paper, the basic concept and features of classical random walk and quantum walk are given firstly,and then we discuss the dynamic evolution properties and the scattering characteristics of the continuous time quantum walk on the lattice ring or line with position defects or transfer defects.For lattice ring structure,the probability distribution of QW on an uniform lattice ring with the number of lattice points N=6 or N=8 shows the periodic evolution characteristic,and this characteristic is damaged by introducing one position defect.However,the situations are different for two cases of N=6 and N=8.When N=6,QW is reflected back and forth by two position defects and is bounded between them with greater occupied probability.With the increasing of the strength of the defect,the probability passing through the defect is much smaller.But for the case of N=8,it is interesting to find that the QW can pass through the position defects with great probability,and show oscillation between both sides of two defects.The periodic evolution of occupied probability on each lattice point revive again.Considering the transfer defects on the on-dimensional lattice line,that is to say the transfer rate between two lattice points is different from the others,we study the behaviors of QW and it is shown than if QW initially is far away from the transfer defect,once it encounters the defects,some parts continue to walk freely forward,while the others are reflected backward and interfere with the later forward part.Moreover,the stronger the defect,the greater of the reflect particle.If the quantum walk initially is located on the lattice point with transfer defect,it will be localizeddynamically near the defect point and the localized probability is greater with increasing of the strength of defects.For double transfer defect case,QW also can be confined between the two defects.For the case of two-dimensional glued tree graph,we make it as an effective one dimensional lattice line segment with position defect and firstly study the transfer of quantum walk from one root to the other root.Then imposing two semi-infinite lattice line to two root nodes of the glued tree respectively,we investigate the static scattering characteristics of QW on it and give the transmission probability as a function of initial momentum for different the generation of tree and the potential at the glued edge.It is shown that the decay and oscillation behaviors of transmission probability are appear with increasing of the incident momentum and by adjusting the graph generation. Some resonant phenomena of QW completely through the Glued tree are due to the oscillation of traveling wave, and others come from bound state in tunneling region.