In this paper, the resonance transmission of discrete time quantum walk is studied when it walks on one-dimensional lattice in which two-phase defects or a piece of phase defects exists. The quasi energy of discrete time quantum walk has a unique dispersion relation with the momentum, from which we first discuss the wave velocity direction versus the values of momentum, and distinguish the incident wave and the reflected wave. The gap between two energy bands depends on the parameters of coincident operator, so the phase defects, which break down the translation invariance of quantum walk on uniform lattices, can be regarded as an analogue of quantum potential. Then we use the condition of energy conversion at the boundary points to obtain the transmission rate and discuss its variation with the incident momentum for different strengths and widths of defects in detail. The multiple resonant peaks are observed due to the enhanced interference effect. Different resonant behaviors are shown when the strength of defect is less or greater than π/2, correspondingly the resonances occur in a wide region of incident momentum or the sharp resonant peaks appear at discrete values of momentum. Under the condition of strong defect strength, i.e., φ approaching to π, the qualitative relation between the number of resonant peaks and the widths of defect region is given. The number of resonant peaks is 2(N-1) when the two phase defects are located at ± N sites symmetric about the origin, while the number is 2N when a piece of phase defects is located at -N to N sites. In the case of a piece of phase defects, we also present the phase diagram in parameter space of (k, φ) to show the discrete time of quantum walk propagating or tunneling through the defect region. In terms of this phase diagram, the variations of transmission rate with the incident momentum are reasonably explained. One special phenomenon is that the quantum walk is almost totally reflected in the tunneling case except for φ =π/2 and k being slightly off -π/2. Moreover, this behavior seems little affecting the defect strength, just similar to a classical particle. As a result of this research, we hope to deepen the insight of the quantum walk and provide methods to control the spreading of quantum walk through artificial defects.

Project supported by the National Natural Science Foundation of China (Grant Nos. 10974124, 11274208) and the Shanxi Scholarship Council of China (2015-012).

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