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arXiv:0705.1608 · Physical Review E

Quantum transport on small-world networks: A continuous-time quantum walk approach

Oliver Mülken, Volker Pernice, A. Blumen

Published Nov 29, 2007• 90 citations
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Abstract

We consider the quantum mechanical transport of (coherent) excitons on small-world networks (SWNs). The SWNs are built from a one-dimensional ring of N nodes by randomly introducing B additional bonds between them. The exciton dynamics is modeled by continuous-time quantum walks, and we evaluate numerically the ensemble-averaged transition probability to reach any node of the network from the initially excited one. For sufficiently large B we find that the quantum mechanical transport through the SWNs is, first, very fast, given that the limiting value of the transition probability is reached very quickly, and second, that the transport does not lead to equipartition, given that on average the exciton is most likely to be found at the initial node.

DOI
10.1103/physreve.76.051125

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We consider the quantum mechanical transport of (coherent) excitons on small-world networks (SWNs). The SWNs are built from a one-dimensional ring of N nodes by randomly introducing B additional bonds between them. The exciton dynamics is modeled by continuous-time quantum walks, and we evaluate numerically the ensemble-averaged transition probability to reach any node of the network from the initially excited one. For sufficiently large B we find that the quantum mechanical transport through the SWNs is, first, very fast, given that the limiting value of the transition probability is reached very quickly, and second, that the transport does not lead to equipartition, given that on average the exciton is most likely to be found at the initial node.
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The coherent exciton transport on a class of deterministic and random scale-free networks (DSFNs and RSFNs) generated by simple rules is studied in this paper. The coherent exciton dynamics is modeled by continuous-time quantum walks, and we calculate the transition probabilities between two nodes of the networks. We find that the transport depends on the initial nodes of the excitation. For DSFNs, the probabilities of finding the excitation at the initial central nodes are nearly periodic, in contrast to the flat behavior found for RSFNs. In the long time limit, the transition probabilities on DSFNs show characteristic patterns with identical values. For RSFNs, we find that the excitation is most likely to be found at the initial nodes with high connectivity. All these features of quantum transport are significantly different from those of the classical transport modeled by continuous-time random walks.
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We model coherent exciton transport in dendrimers by continuous-time quantum walks. For dendrimers up to the second generation the coherent transport shows perfect recurrences when the initial excitation starts at the central node. For larger dendrimers, the recurrence ceases to be perfect, a fact which resembles results for discrete quantum carpets. Moreover, depending on the initial excitation site, we find that the coherent transport to certain nodes of the dendrimer has a very low probability. When the initial excitation starts from the central node, the problem can be mapped onto a line which simplifies the computational effort. Furthermore, the long time average of the quantum mechanical transition probabilities between pairs of nodes shows characteristic patterns and allows us to classify the nodes into clusters with identical limiting probabilities. For the (space) average of the quantum mechanical probability to be still or to be again at the initial site, we obtain, based on the Cauchy-Schwarz inequality, a simple lower bound which depends only on the eigenvalue spectrum of the Hamiltonian.
In this paper, we consider continuous-time quantum walks (CTQWs) on a one-dimensional ring lattice of N nodes in which every node is connected to its 2m nearest neighbors ( m on either side). In the framework of the Bloch function ansatz, we calculate the space-time transition probabilities between two nodes of the lattice. We find that the transport of CTQWs between two different nodes is faster than that of the classical continuous-time random walks (CTRWs). The transport speed, which is defined by the ratio of the shortest path length and propagating time, increases with the connectivity parameter m for both CTQWs and CTRWs. For fixed parameter m , the transport of CTRWs gets slower with the increase of the shortest distance while the transport (speed) of CTQWs turns out to be a constant value. In the long-time limit, depending on the network size N and connectivity parameter m , the limiting probability distributions of CTQWs show various patterns. When the network size N is an even number, the probability of being at the original node differs from that of being at the opposite node, which also depends on the precise value of parameter m .
We study the classical and quantum transport processes on some finite networks and model them by continuous-time random walks (CTRW) and continuous-time quantum walks (CTQW), respectively. We calculate the classical and quantum transition probabilities between two nodes of the network. We numerically show that there is a high probability to find the walker at the initial node for CTQWs on the underlying networks due to the interference phenomenon, even for long times. To get global information (independent of the starting node) about the transport efficiency, we average the return probability over all nodes of the network. We apply the decay rate and the asymptotic value of the average of the return probability to evaluate the transport efficiency. Our numerical results prove that the existence of the symmetry in the underlying networks makes quantum transport less efficient than the classical one. In addition, we find that the increasing of the symmetry of these networks decreases the efficiency of quantum transport on them.