We consider continuous-time quantum walks (CTQWs) on multilayer dendrimer networks (MDs) and their application to quantum transport. A detailed study of properties of CTQWs is presented and transport efficiency is determined in terms of the exact and average return probabilities. The latter depends only on the eigenvalues of the connectivity matrix, which even for very large structures allows a complete analytical solution for this particular choice of network. In the case of MDs we observe an interplay between strong localization effects, due to the dendrimer topology, and good efficiency from the linear segments. We show that quantum transport is enhanced by interconnecting more layers of dendrimers.
We consider continuous-time quantum walks (CTQWs) on multilayer dendrimer networks (MDs) and their application to quantum transport. A detailed study of properties of CTQWs is presented and transport efficiency is determined in terms of the exact and average return probabilities. The latter depends only on the eigenvalues of the connectivity matrix, which even for very large structures allows a complete analytical solution for this particular choice of network. In the case of MDs we observe an interplay between strong localization effects, due to the dendrimer topology, and good efficiency from the linear segments. We show that quantum transport is enhanced by interconnecting more layers of dendrimers.
Abstract We consider continuous-time quantum walks (CTQWs) on modified multilayered spiderwebs (MMSs) and their application to quantum transport. Quantum transport efficiency is determined in terms of the exact and the average return probabilities and it is systematically studied for three kind of MMSs. First, we consider a single layer of modified spiderwebs and we introduce the parameter p , which controls the transition from a pure Cayley tree ( p = 0.0) to a complete spiderweb ( p = 1.0). Topologically, this transition corresponds to adding with probability p more links between nearest neighboring nodes from the same generation, when we start from a Cayley tree. By doing so, we observe an increase in the quantum efficiency and we notice that for all generation numbers its highest value is encountered when only one link is missing from each generation. In the second case we stack more modified spiderwebs on top of each other and interconnect them, obtaining an MMS network. In this case the quantum efficiency is further increased by more than two orders of magnitude. In the third case, we remove some interconnecting links between nearest neighboring layers with probability 1 − q . For these kind of networks we encounter that the highest quantum efficiency does not correspond to q = 1.0, but to , more precise when we remove a link between successive layers. For these networks and for a proper choice of parameters the quantum efficiency is further increased by several orders of magnitude.
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 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.
We study the transport properties on honeycomb networks motivated by graphene structures by using the continuous-time quantum walk (CTQW) model. For various relevant topologies we consider the average return probability and its long-time average as measures for the transport efficiency. These quantities are fully determined by the eigenvalues and the eigenvectors of the connectivity matrix of the network. For all networks derived from graphene structures we notice a nontrivial interplay between good spreading and localization effects. Flat graphene with similar number of hexagons along both directions shows a decrease in transport efficiency compared to more one-dimensional structures. This loss can be overcome by increasing the number of layers, thus creating a graphite network, but it gets less efficient when rolling up the sheets so that a nanotube structure is considered. We found peculiar results for honeycomb networks constructed from square graphene, i.e. the same number of hexagons along both directions of the graphene sheet. For these kind of networks we encounter significant differences between networks with an even or odd number of hexagons along one of the axes.
We consider quantum transport on generalized scale-free networks (GSFNs) in the continuous-time quantum walk (CTQW) model. The efficiency of the transport is monitored through the exact and the average return probabilities. In this model these probabilities are fully determined by the eigenvalues and eigenvectors of the connectivity matrix. In the case of GSFNs we observe a nontrivial interplay between strong localization effects, due to starlike segments, and good spreading because of the linear segments. We show that the quantum transport on GSFNs can be increased by varying the minimum or the maximum allowed degrees, i.e., the limiting number of links emerging from every node. The same quantum efficiency is reached by considering various combinations of the construction parameters of the network, which normally show different topological features.
We show by general arguments that networks whose density of states contains few highly degenerate eigenvalues result in inefficient performances of continuous-time quantum walks (CTQW) over these networks, while systems whose eigenvalues all have the same degeneracy lead to very efficient transport. We exemplify our results by considering CTQW and, for comparison, its classical counterpart, continuous-time random walks, over simple structures, whose eigenvalues and eigenstates can be calculated analytically. Extensions to more complicated, hyper-branched networks are discussed.