发表机构
SISSA – International School for Advanced Studies; Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche（国际高等研究院; 复杂系统研究所，意大利国家研究委员会）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究一维慵懒量子行走在平带局域化下的重启效应，分析随机与尖锐重启对位移、占据及首次探测时间的影响，发现平带活跃态与暗态在重启下呈现不同标度行为。
AI 中文摘要
我们研究了具有自环权重 $\ell$ 的一维慵懒离散时间量子行走中的随机重启和尖锐重启。在没有重启的情况下，动力学具有一个负责内在局域化的平带和两个支持弹道传播的色散带。我们比较了两种初始局域基准态：具有有限平带重叠的平带活跃态和具有零平带重叠的平带暗态。对于每步重启概率为 $q$ 的几何随机重启，当 $q\to0$ 时，平稳均方位移按 $q^{-2}$ 标度。在同一极限下，对于平带活跃态，重启位点占据概率趋近于无重启的内在局域化值，而对于平带暗态，它按 $q\ln(1/q)$ 趋于零。对于幂律重启，其中 $p_m\propto m^{-s}$ 是到下一次重启的等待时间为 $m$ 步的概率，仅当 $s>2$ 时存在归一化的平稳位点占据分布，而仅当 $s>p+2$ 时，$p$ 阶平稳绝对空间矩有限。在 $1<s\leq2$ 的范围内，在每个固定格点上，平带活跃态占据收敛于内在平带轮廓，而平带暗态占据趋于零。我们还考虑了具有尖锐重启的监测首次探测，其中行走在固定次数 $r$ 的连续不成功测量后重新初始化。对于固定的 $r$，平带活跃态的平均首次探测通过时间在中间自环权重处呈现最小值，而平带暗态在 $\ell\to\infty$ 时趋近于弹道探测极限。
英文摘要
We study stochastic and sharp restart in a one-dimensional lackadaisical discrete-time quantum walk with self-loop weight $\ell$. In the absence of restart, the dynamics has a flat band responsible for intrinsic localization and two dispersive bands supporting ballistic propagation. We compare two initially localized benchmark states: a flat-band-active state with finite flat-band overlap and a flat-band-dark state with zero flat-band overlap. For geometric stochastic restart with per-step restart probability $q$, the stationary mean-squared displacement scales as $q^{-2}$ as $q\to0$. In the same limit, the restart-site occupation probability approaches the restart-free intrinsic localized value for the flat-band-active state, whereas for the flat-band-dark state it vanishes as $q\ln(1/q)$. For power-law restart, where $p_m\propto m^{-s}$ is the probability that the waiting time to the next restart is $m$ steps, a normalized stationary site-occupation distribution exists only for $s>2$, while the stationary absolute spatial moment of order $p$ is finite only for $s>p+2$. In the regime $1<s\leq2$, at every fixed lattice site, the flat-band-active occupation converges to the intrinsic flat-band profile, while the flat-band-dark occupation tends to zero. We also consider monitored first detection with sharp restart, in which the walk is reinitialized after a fixed number $r$ of consecutive unsuccessful measurements. For fixed $r$, the mean first-detected-passage time of the flat-band-active state exhibits a minimum at an intermediate self-loop weight, whereas the flat-band-dark state approaches a ballistic detection limit as $\ell\to\infty$.
Comments21 pages, 9 figures
