Continuous-time quantum walks on multilayer dendrimer networks

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.

Figure 1

Multilayer dendrimer with generation number $G=2$ and $L=3$ layers.

Figure 2

Normal modes for a multilayer dendrimer network of $G=2$ and $L=2$.

Figure 3

Contour plots of the probability for a CTQW to start at node $k$ and to be encountered at node $j$ after times $t=10$ (left column), $t=20$ (middle column), and $t=30$ (right column) for the following sets of MD parameters: (a)–(c) $(G,L,N_D)=(1,11,4)$, (d)–(f) $(2,4,10)$, and (g)–(i) $(3,2,22)$.

Figure 4

Contour plot of the probability ${\mathrm{\Pi }}_{j,k}$ for a CTQW on some small MD, namely, (a)–(c) $(G,L,N_D)=(1,11,4)$, (d)–(f) $(2,4,10)$, and (g)–(i) $(3,2,22)$, as a function of time $t$. The starting node of the walker corresponds to the central node $k=1$ of the peripheral layer (left and right columns) and to the central node $k=21$ (middle column, top row), central node $k=11$ (middle column, middle row), and peripheral node $k=11$ (middle column, bottom row) of the middle layer.

Figure 5

Probability ${\mathrm{\Pi }}_{j,k}\left(t\right)$ for CTQW on three small multilayer dendrimer networks having as the starting node the origin $k=1$ and as the ending node (a) the origin and (b) the central node of the last layer.

Figure 6

Comparison between quantum (CTQW) and classical (CTRW) average return probabilities for multilayer dendrimers with almost the same number of nodes, but with totally different geometry: a lone large dendrimer ($G=15$), a linear chain with $N=L=98302$ nodes, and a mixed situation ($L=16$ layers of dendrimers with generation number $G=11$).

Figure 7

Long-time average $\chi$ for multilayer dendrimers with fixed generation number $G=6$ and a variable number of layer $L$. The inset shows $\chi$ for a fixed number of layers $L=3$ and different generation number.

Figure 8

Long-time average $\chi$ and its approximation ${\chi }^{*}$ for multilayer dendrimers.