Coherent exciton transport and trapping on long-range interacting cycles

We consider coherent exciton transport modeled by continuous-time quantum walks (CTQWs) on long-range interacting cycles (LRICs), which are constructed by connecting all the two nodes of distance $m$ in the cycle graph. LRIC has a symmetric structure and can be regarded as the extensions of the cycle graph (nearest-neighboring lattice). For small values of $m$, the classical and quantum return probabilities show power law behavior $p(t)\sim t^{-0.5}$ and $\pi(t)\sim t^{-1}$, respectively. However, for large values of $m$, the classical and quantum efficiency scales as $p(t)\sim t^{-1}$ and $\pi(t)\sim t^{-2}$. We give a theoretical explanation of this transition using the method of stationary phase approximation (SPA). In the long time limit, depending on the network size $N$ and parameter $m$, the limiting probability distributions of quantum transport show various patterns. When the network size $N$ is an even number, we find an asymmetric transition probability of quantum transport between the initial node and its opposite node. This asymmetry depends on the precise values of $N$ and $m$. Finally, we study the transport processes in the presence of traps and find that the survival probability decays faster on networks of large $m$.