Transport properties of continuous-time quantum walks on Sierpinski fractals

We model quantum transport, described by continuous-time quantum walks (CTQWs), on deterministic Sierpinski fractals, differentiating between Sierpinski gaskets and Sierpinski carpets, along with their dual structures. The transport efficiencies are defined in terms of the exact and the average return probabilities, as well as by the mean survival probability when absorbing traps are present. In the case of gaskets, localization can be identified already for small networks (generations). For carpets, our numerical results indicate a trend towards localization, but only for relatively large structures. The comparison of gaskets and carpets further implies that, distinct from the corresponding classical continuous-time random walk, the spectral dimension does not fully determine the evolution of the CTQW.

Figure 1

The graphs under study. The graphs are at third generation ($g=3$), except the DSC, for which $g=2$. We denoted the $g=1$ graphs with green, and the holes with a gray (striped) background. Traps are put either at the positions indicated by the small diamonds or at the positions indicated by the small squares.

Figure 2

The eigenvalue counting function $\mathcal{N}\left(x\right)$ [Eq. (18)] for several systems under study, compared to the simplest case of an infinite discrete square lattice; see text for details.

Figure 3

Quantum return probability ${\pi }_{1,1}\left(t\right)$ to the corner node $j=1$ (green solid line) along with its classical analog (red dashed line) for the SG of $g=7$. Inset: CTQW lower bound $|\overline{\alpha }{\left(t\right)|}^2$ of $\overline{\pi }\left(t\right)$ on the SG at $g=7$ (blue solid line) and ${\overline{\chi }}_{lb}$, the long-time value (black dashed line).

Figure 4

The average return amplitude $|\overline{\alpha }{\left(t\right)|}^2$ on the DSC of $g=5$ (blue solid line) and the long-time average (black dotted line). Inset: classical return probability $p_{1,1}\left(t\right)$ for DSCs of $g=2,3$, and 4 (dotted blue, dashed green, solid red line, respectively) as well as the fitted decay (solid straight black line), $0.4·t^{-1.8/2}$.

Figure 5

Average return amplitude on the SC at $g=6$ (blue solid line) and the long-time average ${\overline{\chi }}_{lb}$ (black dotted line). Inset: classical return probability to the corner node for the SC at $g=2$, 3, 4, and 5 (pink dash dotted, blue dotted, green dashed, red solid line, respectively) and the decay according to the spectral dimension $d_s$ (black straight solid line).

Figure 6

Long-time average ${\overline{\chi }}_{lb}$ of ${\pi }_{1,1}\left(t\right)$ and probability $1-{\Pi }_{\infty }$ for an excitation to get trapped as a function of the number of nodes.