Figure 1

(a)-(d): Probability distributions for the DTQW with initial state \(\pmb{\vert {0} \rangle _{p}\otimes \vert {\phi} \rangle_{c}}\) [with \(\pmb{\vert {\phi} \rangle_{c}}\) as in Eq. ( 5 )] for varying coin parameter θ and \(\pmb{N=100}\) steps. As θ decreases, the walk spreads much wider on the lattice, thus enabling the walker to visit more sites on the line. For \(\theta={\pi}/{20}\) the walker manages to spread to all the sites on the line, reaching the end-site on the end-lattice positions \(\vert {-100} \rangle_{p}\) and \(\vert {100} \rangle_{p}\). Moreover, the probability to occupy sites different from the end ones is strongly reduced, thus manifesting a pronounced coherent localization effect of the walker. (e): Shown is the value of θ needed for the probability of the walker being at the end site of the lattice to be double (top line), triple (middle line) and four times (bottom line) the probability of being at the previous site. As the number of sites in the lattice increases, the critical value of θ needed to reach the end site decreases. Note that we compare the probability of being at end sites ±N with the probability of being at site \(\pm N \mp2\). This is due to the action of the shift operator, which allows us to fill only odd or even positions, depending on how many steps the walker takes. When the evolution occurs over an even number of steps, the walker can only occupy even sites on the lattice. Therefore, \(P(N \pm1)\) will always equal 0.