Directionality and quantum backfire in continuous-time quantum walks from delocalized states: Exact results

TL;DR

The paper addresses transport control in continuous-time quantum walks with complex hopping by introducing a tunable delocalization parameter $D$ in the initial state. The authors derive exact, closed-form expressions for the wavefunction, mean position, mean-square displacement, and survival probability, revealing three main outcomes: (i) directed transport from an unbiased initial setup at intermediate $D$, (ii) a quantum backfire where larger $D$ enhances short-time spread but reduces long-time spreading after a crossing time, and (iii) a finely tuned $t^{-3}$ decay of survival probability only for $D=1$ and specific phases $\alpha$. The findings provide a framework to steer quantum transport through the interplay of initial delocalization and Hamiltonian phase, with potential experimental relevance and avenues for extension to higher dimensions. These results deepen understanding of how initial-state structure and complex hopping shape dynamical transport in quantum networks.

Abstract

We derive analytical results for continuous-time quantum walks from a new class of initial states with tunable delocalization. The dynamics are governed by a Hamiltonian with complex hopping amplitudes. We provide closed-form equations for key observables, revealing three notable findings: (1) the emergence of directed quantum transport from completely unbiased initial conditions; (2) a quantum backfire effect, where greater initial delocalization enhances short-time spreading but counterintuitively induces a comparatively smaller long-time spreading after a crossing time $t_{\mathrm{cross}}$; and (3) an exact characterization of survival probability, showing that the transition to an enhanced $t^{-3}$ decay is a fine-tuned effect. Our work establishes a comprehensive framework for controlling quantum transport through the interplay between intermediate initial delocalization and Hamiltonian phase.

Figures (5)

• Figure 1: Probability distributions for $\alpha=\pi/2$ at $\gamma t=50$. Our results show that while both localized ($D=0$, bottom) and fully delocalized ($D=1$, top) initial states yield symmetric spreading, intermediate delocalization ($D=0.5$, middle) generates a pronounced bias. The analytical results were obtained with Eq. (\ref{['eq:wave_function_main']}) and $P(x,t) = \left| \psi(x,t) \right|^2$.
• Figure 2: Absolute value of the average group velocity $\abs{\langle v_g \rangle/\gamma}$, Eq. (\ref{['eq:average_main']}), as a function of the delocalization parameter $D$ for various phases $\alpha$. While the extreme states ($D=0$ and $D=1$) yield zero net velocity for any $\alpha$, a maximum bias emerges at intermediate delocalization ($D=0.5$), demonstrating tunable directed transport.
• Figure 3: Time evolution of the mean square displacement (MSD) obtained with Eq. (\ref{['eq:MSD_main']}). The top panel ($\alpha = 0$) shows the quantum backfire effect: (i) for $t < t_{\mathrm{cross}}$, a larger initial MSD(0) promotes a larger MSD($t$); (ii) for $t > t_{\mathrm{cross}}$, this relationship inverts: a larger initial MSD(0) produces a smaller MSD($t$). The dashed vertical line marks $\gamma t_{\mathrm{cross}}$. The bottom panel ($\alpha = \pi/2$) exhibits no-crossing behavior: The ordering of MSD curves with respect to $D$ is preserved for all time.
• Figure 4: Dependence of the MSD crossing time $\gamma t_{\mathrm{cross}}$ on the phase $\alpha$, Eq. (\ref{['eq:t_cross']}). Two regimes are shown. 1) No-crossing (yellow, $\sin^{2}\alpha \ge 1/2$): MSD curves remain ordered for all time. 2) Crossing (blue, $\sin^{2}\alpha < 1/2$): MSD curves intersect at $t = t_{\mathrm{cross}}$. For $t > t_{\mathrm{cross}}$, the ordering inverts, demonstrating the quantum backfire effect where a greater initial delocalization ($D$) boosts short-time spreading, but is detrimental to the long-time propagation.
• Figure 5: Time evolution of $P_{\text{surv}}$ on a log-log scale for $\alpha=\pi/2$, Eq. (\ref{['eq:survival_main']}). The $D=1$ case exhibits enhanced decay ($\sim t^{-3}$), while any partially delocalized state ($D<1$) shows the standard scaling ($\sim t^{-1}$). Dashed lines show the analytical asymptotic predictions.