Localization of quantum walks on a deterministic recursive tree
Localization of quantum walks can be characterized by the return probability, i.e., the probability for the walker returning to its original site. In this paper, we consider localization of continuous-time quantum walks in terms of return probability on a deterministic recursive tree, which is generated by adding one node and connecting it to each node of the existing tree recursively. We obtain an approximate form for the return probability using the complete eigenvalues and eigenstates of Laplace matrix of the structure. It is found that the return probability depends on the initial node of the excitation. When the walk starts at the central nodes, the return probability converges to a constant value even in the limit of infinite system, in contrast to an exponential decay of the return probability if the walk starts at the outlying nodes. We also observe a bipartite structure for the distribution of return probability, and provide theoretical interpretation for all our findings. Our results suggest that quantum walks display significant localization and well-bedded structure of return probability on heterogeneous trees.