Robustness of optimal transport in one-dimensional particle quantum networks

We study the robustness of quantum transport in four-site one-dimensional (1D) particle quantum networks irreversibly connected to a target site in the presence of environmental dephasing noise. Under conditions of Markovian decoherence we define a quantity—which we denote as geometrical robustness—to quantify the robustness of transport with regard to the geometrical arrangement of the spin network. We examine the behavior of this geometrical robustness for both nearest and non-nearest site interactions and show that for some values of dephasing noise and other network parameters the robustness undergoes revivals, i.e., the robustness periodically increases with dephasing. In addition, we find that for high amounts of dephasing noise, the pattern of optimal network configurations changes noticeably from central symmetric configurations to laterally gathered ones, and we show that this change of pattern seems to occur for any number of network sites in the 1D chain.

Figure 1

This figure shows the excitation transfer to the target for all configurations of four-site networks in graph (a) (type 1) and graph (b) (type 2). The sites are realized to be at $(x_1=-0.5,x_2,x_3,x_4=0.5)$, respectively. Network parameters are $\Gamma t=100,V{L}^{-3}/\Gamma =1$, and $\gamma =0$. The color bar shows the population of excitation accumulated in the target at time $t$. In both types (a) and (b), the symmetric positions of middle sites can efficiently transfer the initial excitation to the target (bright regions); however, joint configurations of middle sites are only efficient for interaction type 1. In all dark regions the fourth site of the network is isolated so that the efficiency of the excitation transfer is very small.

Figure 2

Graph (a) shows the robustness $\mu$ against the logarithm of normalized dephasing noise ${\log }_{10}(\gamma /\Gamma )$ for interaction types 1 and 2, while network parameters are $V{L}^{-3}/\Gamma =1$ and $\Gamma .T=100$. The upward trend of the first peak for both interaction types is due to the symmetry breaking of trapping configurations, and the downward slope is due to a kind of quantum Zeno effect. Graph (b) shows the efficient configurations of type 1 networks for three different dephasing noise levels during the second peak of $\mu$. According to these the second peak is due to the change of optimal configurations or stable energy levels of the system. The main difference of the behavior of robustness in type 1 and 2 interactions is within the first peak (the additional subpeak of type 1) due to the different patterns of efficient configurations shown in Fig. 1. During this subpeak the joint configurations of type 1 gradually disappear with increasing dephasing.

Figure 3

Graph (a) shows how the robustness $\mu$ is influenced by the strength of tunnel coupling $V{L}^{-3}/\Gamma$ and dephasing noise $\gamma /\Gamma$, while $\Gamma t=100$. According to the sharp linear border the high values of robustness are in the regime of $V{L}^{-3}/\sqrt{\gamma \Gamma }\gtrsim 0.0412$, and the low values [starting from the first peak of Fig. 2] are in the complementary region of $V{L}^{-3}/\sqrt{\gamma \Gamma }\lesssim 0.0412$. The high value regime of robustness which includes symmetric configurations are growing by increasing dephasing. The position of the main peak of robustness is around $V{L}^{-3}/\sqrt{\gamma \Gamma }\simeq 0.0412$. Graph (b) shows the dependency of robustness on coupling rates $\left[J=V/{\left(L{r}_{ij}\right)}^{\alpha }\right]$ by changing $\alpha$ possibly represent different physical implementations. It seems that increasing $\alpha$ shifts the position of the second peak to larger $\gamma /\Gamma$. Moreover, $\alpha$ does not change the position of the first subpeak and so efficient pair levels of middle sites are not affected by the model of physical tunnel coupling.

Figure 4

The figure shows how robustness $\mu$ is influenced by $\Gamma t$ for different dephasing noise rates $\left(\gamma /\Gamma \right)$, while $V{L}^{-3}/\Gamma =1$. It is shown that by increasing time both peaks shift linearly forward. In addition, in the range of ${\Gamma }^{2}T/\gamma \gtrsim 0.0463$ robustness during the first peak has large values.

Figure 5

Graph (a) shows the dependence of the robustness $\mu$ on dephasing noise for three different numbers of network sites $\left(N=3,4,5\right)$ for interaction type 1 with network parameters $V{L}^{-3}/\Gamma =1$, and $\Gamma T=100$. It can be seen that adding one site to the system does not have a significant effect on the average amount of robustness, however, the peaks move forward down. Graph (b) shows the efficient network configurations for different dephasing noise for $N=3,4$, and 5 which could show the general behavior of optimal configurations of arbitrary N-site networks in different regimes of dephasing noise.

Figure 6

Evidence of quantum entanglement in the dephased networks $\left(\gamma /\Gamma =2\times {10}^{-2},10,{10}^3\right)$. Logarithmic negativity between site 1 and the rest of the network for different site couplings or network configurations are shown when the network environment parameters are considered as $\Gamma t=100,V{L}^{-3}/\Gamma =1$. The rapid oscillations of entanglement dominate mostly for poor transport network configurations in small $\gamma /\Gamma$, indicating the existence of trapping network geometries where the excitation is trapped in correlations within the chain. With higher dephasing noise, the log negativity does not oscillate and is predominant for configurations which are nearer to the last site. The black regions in symmetric areas and diagonal regions are the efficient configurations of such networks passing the excitation successfully. For large dephasing this area is shown by the gray triangular dashed lines in part (d). The change of patterns shows the strong correlation between the dephasing noise and the amount of entanglement for different classes of network coupling denoted by different network configurations.

Figure 7

This figure shows the increase of geo-robustness due to the line broadening of the energy levels analyzed by the Pauli master equation. (a) Shows the variation of geo-robustness of a network of four nearest-neighbor interacting dipoles with a Lorentzian hopping rate of $1/\left(4\pi \right)1/(\Delta E_{k,l}^2+1/16)$, initial interaction energies corresponding to $5\times {10}^{-3}/r_{ij}^3$, at $t=100$, and $\Gamma =1$, while suffering from different broadened energy levels of width ELW due to environmental collisions. However, the average amounts of $\mu$ in very large energy level widths $\left(\mathrm{ELW}>10\right)$ are showing a peak; due to some numerical errors the decrease of geo-robustness in this range is not deterministic. (b) Shows the optimal patterns of configurations of the same network in the presence of four different values of line broadening. It can be seen that for the chosen network parameters the increase of geo-robustness in the presence of dephasing noise qualitatively matches that observed in the quantum case.