Persistence of unvisited sites in presence of a quantum random walker

Abstract

A study of persistence dynamics is made for the first time in a quantum system by considering the dynamics of a quantum random walk. For a discrete walk on a line starting at $x=0$ at time $t=0$, the persistence probability $P(x,t)$ that a site at $x$ has not been visited till time $t$ has been calculated. $P(x,t)$ behaves as $(t/|x|-1{)}^{-\alpha }$ with $\alpha \sim 0.3$ while the global fraction $\mathcal{P}(t)={\sum }_{x}P(x,t)/2t$ of sites remaining unvisited at time $t$ attains a constant value. $F(x,t)$, the probability that the site at $x$ is visited for the first time at $t$ behaves as $(t/|x|-1{)}^{-\beta }/|x|$ where $\beta =1+\alpha$ for $t/|x|>>1$,and $\mathcal{F}(t)={\sum }_{x}F(x,t)/2t\sim 1/t$. A few other properties related to the persistence and first passage times are studied and some fundamental differences between the classical and the quantum cases are observed.

5 pages, 6 figures, revtex4