Position-defect-induced reflection, trapping, transmission, and resonance in quantum walks

We investigate the scattering properties of quantum walks by considering single and double position defects on a one-dimensional line. This corresponds to introducing, at designated positions, delta potential defects for continuous-time quantum walks and phase-defect Hadamard coins for discrete time quantum walks. The delta potential defects can be readily considered as potential barriers in discrete position space, which affect the time evolution of the system in a similar way as the quantum wave-packet dynamics in a continuous position space governed by Schrödinger's equation. Although there is no direct analogy of potential barriers in the theoretical formulation of discrete time quantum walks, in this paper we show that the phase defects in the coin space can be utilized to provide similar scattering effects. This study provides means of controlling the scattering properties of quantum walks by introducing designated position-dependent defects.

Figure 1

Probability distribution at time $t=30$ for (a) CTQW with a defect potential ${\Gamma }_0=5$; (b) DTQW with a phase defect ${\varphi }_0=\pi$ and initial state $\left(\right|1\rangle +i|0\rangle )/\sqrt{2}$. The dashed line corresponds to a quantum walk without a defect.

Figure 2

Probability at the origin at time $t=30$ as a function of (a) the defect potential ${\Gamma }_0$ for CTQW and (b) phase defect ${\varphi }_0$ for DTQW. In (b), the solid line corresponds to the initial coin state $\left(\right|1\rangle +i|0\rangle )/\sqrt{2}$, while the dashed line corresponds to the coin initial state $|1\rangle$ or $|0\rangle$.

Figure 3

Probability distribution for (a) CTQW at time $t=30$ with a defect potential ${\Gamma }_{15}=5$, and (b) DTQW after $t=80$ steps with a phase defect ${\varphi }_{15}=\pi$ and initial coin state $\left(\right|1\rangle +i|0\rangle )/\sqrt{2}.$

Figure 4

Transmission probability as a function of (a) the defect potential ${\Gamma }_{15}$ for CTQW after time $t=30$ and (b) the phase defect $\varphi$ for DTQW after $t=80$ steps. In (b) the solid line corresponds to the coin initial state $\left(\right|1\rangle +i|0\rangle )/\sqrt{2}$, whilst the dotted and dashed lines correspond to the coin initial states $|1\rangle$ and $|0\rangle$, respectively.

Figure 5

Probability distribution for a walker initially localized at origin between two barriers for (a) CTQW at time $t=30$ with the defect potentials ${\Gamma }_{\pm 15}=5$, and (b) DTQW after $t=80$ steps with phase defects ${\varphi }_{\pm 15}=\pi$ and initial state $\left(\right|1\rangle +i|0\rangle )/\sqrt{2}$.

Figure 6

Trapping probability as a function of time $t$ for (a) CTQW with defect potential ${\Gamma }_{\pm 15}=5$, and (b) DTQW with phase defects ${\varphi }_{\pm 15}=\pi$ and initial state $\left(\right|1\rangle +i|0\rangle )/\sqrt{2}$.

Figure 7

Transmission coefficient vs momentum of a CTQW momentum eigenstate incident on two defects of amplitude $\alpha$ and $\beta$, separated by distances $L=0$ (solid, blue), $L=1$ (dashed, red), $L=2$ (dotted, black), and $L=5$ (dot-dashed, green). The results are shown for the cases (a) $\alpha =1,\beta =0$ (i.e., a single defect), (b) $\alpha =1,\beta =-1$, (c) $\alpha =\beta =1$, and (d) $\alpha \ne \left|\beta \right|$, $\alpha ,\beta \ne 0$.

Figure 8

Transmission coefficient $T\left(k\right)$ as a function of barrier separation $L$ for momentum eigenstate $|k\rangle$ incident on a double reflecting barrier. Top: $\alpha =\beta =1$; bottom: $\alpha =-\beta =1$.