One-dimensional quantum walks with single-point phase defects

We observe the localization effect of one-dimensional quantum walks with single-point phase defects. The walker's spread velocity is dramatically suppressed by interference effects due to the phase defect. We show that the localization effect depends on four factors: the value and the position of the phase defect, the parameter of coin flipping, and the initial state of the walker + coin system.

Figure 1

Probability distributions of the 1D QWs with an SPPD $\varphi$ located at the original position of the QW for different initial coin states. (a) $\varphi =\pi /4$, (b) $\varphi =\pi$. The solid black lines correspond to $|{\psi }_s{\rangle }_c$, the dashed red lines correspond to $|{\psi }_a{\rangle }_c$, and the dotted blue lines correspond to $|{\psi }_{as}{\rangle }_c$.

Figure 2

Plot of ratio of variance ${\sigma }^2\left(\varphi \right)/{\sigma }^2(\varphi =0)$ of 1D QWs with the different value $\varphi$ of SPPD at origin and different initial coin states as a function of $n$, number of steps.

Figure 3

(a) The distribution of the 1D QW with SPPD $\varphi =\pi /2$ at $x=1$, the initial coin state $|{\psi }_a{\rangle }_c$ after 21 steps. (b) The distribution of the 1D QW with SPPD $\varphi =\pi /2$ at both $x=1$ and $x=-1$ with $|{\psi }_a{\rangle }_c$ after 21 steps.

Figure 4

The profiles of all eigenstates of $U_{\varphi }^2$ for 1D QW with different value of SPPD in the position $x=0$ (a) $\varphi =\pi /4$ and (c) $\varphi =\pi$ where $j$ labels the eigenstate and we have truncated to 41 spatial lattice sites. The corresponding localized eigenstates are shown in (b) and (d), respectively.

Figure 5

The plots of the profiles of all eigenstates of $U_{\varphi }^2$ for 1D QW with the SPPD $\varphi =\pi /2$ at $x=1$ are shown in (a). The corresponding nondegenerate localized eigenstates are shown in (b).

Figure 6

The overlaps $F$ between the initial states and localized eigenstates as a function of the value of SPPD $\varphi$. The solid lines correspond to analytical results and the dotted lines correspond to the simulated results. The black line and circle dot correspond to the initial coin state $|{\psi }_s{\rangle }_c$, the blue line and triangle dot correspond to the initial coin state $|{\psi }_a{\rangle }_c$, and the red line and pentagonal dot correspond to the initial coin state $|{\psi }_{as}{\rangle }_c$.

Figure 7

The overlap $F$ between the asymmetric initial state $|{\psi }_a{\rangle }_c$ and the localized eigenstates of $U$ as a function of the position $n$ of SPPD with the value of the SPPD $\varphi =\pi /4$ (black line and circle dot), $\varphi =\pi /2$ (blue line and triangle dot), and $\varphi =\pi$ (red line and pentagonal dot). Solid lines correspond to the analytical results and dot lines correspond to the simulated results.