Continuous-time quantum walks on a defective lattice: boosting the spreading of delocalized states through Parrondo's strategy

TL;DR

This work analyzes continuous-time quantum walks on a one-dimensional lattice with a single complex-valued defect, focusing on delocalized Gaussian initial states. By modeling the defect with hopping amplitudes $\xi$ and phases $\theta_\pm$ and applying a time-periodic Parrondo protocol that alternates between two defective configurations, the authors show that both static defects and phase engineering can enhance ballistic spreading, while temporal alternation between two poor-spreading configurations can outperform the homogeneous case. They quantify transport via the standard deviation $\sigma(t)$ and defect-localization metrics such as the defect-site probability $P_0$ and the ratio $\sigma_d/\sigma$, uncovering phase-induced chirality, symmetry effects, and regimes where localization is suppressed by alternation. The results highlight how complex-phase defects and time-dependent control can be used to steer quantum transport in CTQWs, with potential implications for quantum state transfer and graph-engineered quantum networks.

Abstract

We investigate the quantum transport of delocalized states in continuous-time quantum walks (CTQWs) on a one-dimensional lattice containing a single defect. The defect is modeled by assigning complex-valued hopping amplitudes to the edges that connect the site corresponding to the mean position of the initial delocalized state to its nearest neighbors. We find that this single defective site is sufficient to enhance the ballistic spreading of an initially Gaussian wave packet. Extending these results, we implement a time-dependent alternation protocol between two distinct defect configurations, each individually yielding poor propagation of the state. The combination of these two unfavorable configurations improves the transport efficiency of the quantum walker, revealing a manifestation of Parrondo's paradox in CTQWs with delocalized initial states. This study provides new insights into the role of complex-phase defects and time-dependent protocols in CTQWs, demonstrating that the interplay between quantum interference and graph engineering can effectively enhance quantum transport in discrete lattices.

Figures (8)

• Figure 1: Schematic representation of a CTQW on a defective lattice. The hopping amplitudes from site $d$ to its nearest neighbors $d\pm1$ are multiplied by $\xi e^{i\theta_\pm}$, whereas all other hopping amplitudes are real and equal to $\gamma$. Two scenarios are considered: (i) both phases are equal, $\theta_-=\theta_+=\theta$, and (ii) the phases have opposite signs, $\theta_-=\theta$ and $\theta_+=-\theta$. Throughout this study from here, we assume $\epsilon=0$ from $\hat{H}$ in Eq. \ref{['H_d']} and take $d=0$ for convenience.
• Figure 2: Spreading rates $\alpha$ of the standard deviation versus $\sigma_0$ obtained for CTQWs evolved up to $\gamma t=1000$ (black empty circle) from Gaussian initial states. The analytical expression $\alpha_c$ is the corrected spreading rate using the Error function given by Eq. \ref{['alphacorr']} (black line), and $\alpha_a$ is the asymptotic spreading rate from Eq. \ref{['alphaasym']} (red dashed line). Both expressions take $\gamma=1$.
• Figure 3: Probability $P_0$ at the defect site versus the hopping amplitude $\xi$ for (a) equal phases and (b) opposite phases. Panel (a) exhibits phase-induced chirality for Gaussian states, whereas opposite phases in (b) restore reflection symmetry. The fully localized state (black) is $\ket{\psi_0}=\ket{0}$, and Gaussian states have initial standard deviations $\sigma_0=1$ (red), $5$ (blue), and $10$ (green). All simulations were run to $\gamma t=1000$.
• Figure 4: Rescaled probability $\gamma t\,P_0$ versus $\theta$ for $\theta_\pm=\theta$ with $\xi=1$. Phase effects decay as $P_0 \propto (\gamma t)^{-1}$ and vanish in the long-time limit. The fully localized case (black) is $\ket{\psi_0}=\ket{0}$, and initial Gaussian states have $\sigma_0=1$ (red), $5$ (blue), and $10$ (green). All simulations were run to $\gamma t=1000$.
• Figure 5: Ratio $\sigma_d/\sigma$ between the standard deviations in the defective and homogeneous lattices, plotted versus $\xi$ for (a) equal phases and (b) opposite phases, and versus $\theta$ for (c) $\theta_-=\theta_+=\theta$ and (d) $\theta_-=-\theta_+=\theta$, with $\xi=1$ in (c,d). Equal phases in (a) lead to asymmetric spreading for Gaussian states, whereas opposite phases in (b) eliminate this asymmetry. Gaussian states exhibit a stronger response to the defect than the fully localized state. The localized state (black) corresponds to $\ket{\psi_0}=\ket{0}$, and initial Gaussian states have $\sigma_0=1$ (red), $5$ (blue), and $10$ (green). All simulations were run to $\gamma t=1000$.