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Physical Review E

Fractional quantum mechanics on networks: Long-range dynamics and quantum transport

Alejandro P. Riascos, José L. Mateos

Published Nov 30, 2015• 18 citations
Publisher

Abstract

In this paper we study the quantum transport on networks with a temporal evolution governed by the fractional Schrödinger equation. We generalize the dynamics based on continuous-time quantum walks, with transitions to nearest neighbors on the network, to the fractional case that allows long-range displacements. By using the fractional Laplacian matrix of a network, we establish a formalism that combines a long-range dynamics with the quantum superposition of states; this general approach applies to any type of connected undirected networks, including regular, random, and complex networks, and can be implemented from the spectral properties of the Laplacian matrix. We study the fractional dynamics and its capacity to explore the network by means of the transition probability, the average probability of return, and global quantities that characterize the efficiency of this quantum process. As a particular case, we explore analytically these quantities for circulant networks such as rings, interacting cycles, and complete graphs.

DOI
10.1103/physreve.92.052814

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In this paper we study the quantum transport on networks with a temporal evolution governed by the fractional Schrödinger equation. We generalize the dynamics based on continuous-time quantum walks, with transitions to nearest neighbors on the network, to the fractional case that allows long-range displacements. By using the fractional Laplacian matrix of a network, we establish a formalism that combines a long-range dynamics with the quantum superposition of states; this general approach applies to any type of connected undirected networks, including regular, random, and complex networks, and can be implemented from the spectral properties of the Laplacian matrix. We study the fractional dynamics and its capacity to explore the network by means of the transition probability, the average probability of return, and global quantities that characterize the efficiency of this quantum process. As a particular case, we explore analytically these quantities for circulant networks such as rings, interacting cycles, and complete graphs.
We generalize the Schrödinger equation on graphs to include long-range interactions (LRI) by means of the Mellin-transformed d-path Laplacian operators. We find analytical expressions for the transition and return probabilities of a quantum particle at the nodes of a ring graph. We show that the average return probability in ring graphs decays as a power law with time when LRI is present. In contrast, we prove analytically that the transition and return probabilities on a complete and start graphs oscillate around a constant value. This allowed us to infer that in a barbell graph—a graph consisting of two cliques separated by a path—the quantum particle get trapped and oscillates across the nodes of the path without visiting the nodes of the cliques. We then compare the use of the Mellin-transformed d-path Laplacian operators versus the use of fractional powers of the combinatorial Laplacian to account for LRI. Apart from some important differences observed at the limit of the strongest LRI, the d-path Laplacian operators produces the emergence of new phenomena related to the location of the wave packet in graphs with barriers, which are not observed neither for the Schrödinger equation without LRI nor for the one using fractional powers of the Laplacian.
We study the coherent exciton transport of continuous-time quantum walks (CTQWs) on Erdos-Renyi networks. The Erdos-Renyi network of N nodes is constructed by connecting every pair of nodes with probability ppp. We numerically calculate the ensemble averaged transition probability of quantum transport between two nodes of the networks. For finite networks, we find that the limiting transition probability is reached very quickly. For infinite networks whose spectral density follows the semicircle law, the efficiencies of the classical and quantum-mechanical transport are compared on networks of different average degree. In the long time limiting, we consider the distribution of the ensemble averaged transition probabilities, and show that there is a high probability to find the exciton at the initial node. Such high return probability almost do not alter in a wide range of connection probability p but increases rapidly when the network approaches to be fully connected. For networks whose topology is not extremely connected, the return probability is inversely proportional to the network size N. Furthermore, the transport dynamics are compared with that on a random graph model in which the degree of each node equals to the average degree of the Erdos-Renyi networks.
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.
Abstract We study single-particle quantum transport on multilayer generalized scale-free networks using the continuous-time quantum walk model. Our focus is directed at the average return probability and its long-time average value as measures for the transport efficiency. In the continuous-time model these quantities are completely determined by all the eigenvalues and eigenvectors of the connectivity matrix. For all multilayer networks a nontrivial interplay between good spreading and localization effects is observed. The spreading is enhanced by increasing the number of layers L or the power-law exponent γ of the degree distribution. For our choice of the parameters, namely L (1 ≤ L ≤ 50) or γ (1 ≤ γ ≤ 4), the quantum efficiency is increased by at least one order of magnitude. The topological transition between networks without loops, which corresponds to a single scale-free network layer (L = 1), and networks with loops (L = 2) is the most impactful. Another important change occurs when L gets higher than the average diameter d of the layers, namely a new scaling behavior for random walks and lower fluctuations around the long-time average value for quantum walks. The quantum transport is more sensitive to changes of the minimum allowed degree, K min, than to the maximum allowed degree, K max. The same quantum efficiency is found by varying at least one of the parameters: L, γ, K min, or K max, although the network’s topology is different. The quantum efficiency of all multilayer scale-free networks shows a universal behavior for any size of the layers, more precise, is inversely proportional to the number of layers.
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.
Abstract. We study quantum transport on finite discrete structures and we model the process by means of continuous-time quantum walks. A direct and effective comparison between quantum and classical walks can be attained based on the average displacement of the walker as a function of time. Indeed, a fast growth of the average displacement can be advantageously exploited to build up efficient search algorithms. By means of analytical and numerical investigations, we show that the finiteness and the inhomogeneity of the substrate jointly weaken the quantum walk performance. We further highlight the interplay between the quantum-walk dynamics and the underlying topology by studying the temporal evolution of the transfer probability distribution and the lower bound of long time averages. PACS numbers: 05.60Gg, 71.35.-y, 05.60CdDynamics of continuous-time quantum walks in restricted geometries 2 1.