Figure 1

The plots of absolute values of $x_{+}$ (dashed) and $x_{-}$ (solid) as a function of $\phi$. The shaded region denotes the allowed values of $x_{\pm }$. $|x_{-}|<1$ for $\phi \in (\frac{1}{4},1)$, whereas $|x_{+}|<1$ for $\phi \in (0,\frac{3}{4})$.Reuse & Permissions

Figure 2

The plots of overlap $F$ of initial states of the quantum walk and the localized stationary states of $U_{\phi }^2$. The solid line corresponds to the initial coin state $|0\rangle$, whereas the dashed line corresponds to the initial coin state $\frac{1}{\sqrt{2}}\left(\right|0\rangle +i|1\rangle )$.Reuse & Permissions

Figure 3

Probability distribution after 50 steps of a quantum walk with a single phase defect at the origin corresponding to $\phi =\frac{1}{6}$. Solid line: Lack of localization for the initial coin state $\frac{1}{\sqrt{2}}\left(\right|0\rangle -i|1\rangle )$. Dashed line: Localization at the origin for the initial coin state $\frac{1}{\sqrt{2}}\left(\right|0\rangle +i|1\rangle )$.Reuse & Permissions

Figure 4

(a) Experimental proposal for four steps of a quantum walk with a phase defect at position $n=0$. It employs a single photon in an array of optical elements and is based on a setup introduced in Ref. [13]. Bold (blue) extended rectangles represent calcite beam displacers which realize a coin-dependent position shift, thin (red) extended rectangles represent half-wave plates and quarter-wave plates which realize coin operations, and small (yellow) rectangles represent phase shifters which implement a phase shift at position $n=0$. Below we present expected probability distributions after four steps for a phase shift corresponding to $\phi =\frac{1}{6}$. (b) Initial coin state $\frac{1}{\sqrt{2}}\left(\right|0\rangle -i|1\rangle )$. (c) Initial coin state $\frac{1}{\sqrt{2}}\left(\right|0\rangle +i|1\rangle )$.Reuse & Permissions