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arXiv:1001.5131 · Physical Review E

Quantum persistence: A random-walk scenario

Sanchari Goswami, Parongama Sen, Arnab Das

Published Feb 18, 2010• 14 citations
arXivHTMLPDF

Abstract

In this paper we extend the concept of persistence, well defined for classical stochastic dynamics, to the context of quantum dynamics. We demonstrate the idea via quantum random walk and a successive measurement scheme, where persistence is defined as the time during which a given site remains unvisited by the walker. We also investigated the behavior of related quantities, e.g., the first-passage time and the succession probability (newly defined), etc. The study reveals power-law scaling behavior of these quantities with new exponents. Comparable features of the classical and the quantum walks are discussed.

DOI
10.1103/physreve.81.021121

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In this paper we extend the concept of persistence, well defined for classical stochastic dynamics, to the context of quantum dynamics. We demonstrate the idea via quantum random walk and a successive measurement scheme, where persistence is defined as the time during which a given site remains unvisited by the walker. We also investigated the behavior of related quantities, e.g., the first-passage time and the succession probability (newly defined), etc. The study reveals power-law scaling behavior of these quantities with new exponents. Comparable features of the classical and the quantum walks are discussed.
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=0x=0x=0 at time t=0t=0t=0, the persistence probability P(x,t)P(x,t)P(x,t) that a site at xxx has not been visited till time ttt has been calculated. P(x,t)P(x,t)P(x,t) behaves as (t/∣x∣−1)−α(t/|x|-1)^{-α}(t/∣x∣−1)−α with α∼0.3α\sim 0.3α∼0.3 while the global fraction P(t)=∑xP(x,t)/2t{\cal{P}}(t) = \sum_xP(x,t)/2tP(t)=∑x​P(x,t)/2t of sites remaining unvisited at time ttt attains a constant value. F(x,t)F(x,t)F(x,t), the probability that the site at xxx is visited for the first time at ttt behaves as (t/∣x∣−1)−β/∣x∣(t/|x|-1)^{-β}/|x|(t/∣x∣−1)−β/∣x∣ where β=1+αβ= 1+ αβ=1+α for t/∣x∣>>1t/|x|>> 1t/∣x∣>>1,and F(t)=∑xF(x,t)/2t∼1/t{\cal{F}}(t) =\sum_xF(x,t)/2t \sim 1/tF(t)=∑x​F(x,t)/2t∼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.
We analyze the asymptotic scaling of persistence of unvisited sites for quantum walks on a line. In contrast to the classical random walk, there is no connection between the behavior of persistence and the scaling of variance. In particular, we find that for a two-state quantum walk persistence follows an inverse power law where the exponent is determined solely by the coin parameter. Moreover, for a one-parameter family of three-state quantum walks containing the Grover walk, the scaling of persistence is given by two contributions. The first is the inverse power law. The second contribution to the asymptotic behavior of persistence is an exponential decay coming from the trapping nature of the studied family of quantum walks. In contrast to the two-state walks, both the exponent of the inverse power-law and the decay constant of the exponential decay depend also on the initial coin state and its coherence. Hence, one can achieve various regimes of persistence by altering the initial condition, ranging from purely exponential decay to purely inverse power-law behavior.
This work focuses on the study of quantum stochastic walks, which are a generalization of coherent, \ie unitary quantum walks. Our main goal is to present a measure of a coherence of the walk. To this end, we utilize the asymptotic scaling exponent of the second moment of the walk \ie of the mean squared distance covered by a walk. As the quantum stochastic walk model encompasses both classical random walks and quantum walks, we are interested how the continuous change from one regime to the other influences the asymptotic scaling exponent. Moreover this model allows for behavior which is not found in any of the previously mentioned model -- a model with global dissipation. We derive the probability distribution for the walker, and determine the asymptotic scaling exponent analytically, showing that ballistic regime of the walk is maintained even at large dissipation strength.
diffusion in disordered environments, random quantum spin chains Abstract. — We study the first passage time properties of an integrated Brownian curve both in homogeneous and disordered environments. In a disordered medium we relate the scaling properties of this center of mass persistence of a random walker to the average persistence, the latter being the probability P pr(t) that the expectation value 〈x(t) 〉 of the walker’s position after time t has not returned to the initial value. The average persistence is then connected to the statistics of extreme events of homogeneous random walks which can be computed exactly for moderate system sizes. As a result we obtain a logarithmic dependence P pr(t) ∼ ln(t) −θ with a new exponent θ = 0.191 ± 0.002. We note on a complete correspondence between the average persistence of random walks and the magnetization autocorrelation function of the transverse-field Ising chain, in the homogeneous and disordered case. First passage time or persistence problems have a long history in the physical literature [1]. Recently they gained a lot of interest, since the persistence exponents, describing the asymptotic behavior of first passage time probabilities, are shown to be independent dynamical critical exponents, which have been calculated for various models exactly[2, 3]. Not much is known about analogous quantities in systems with quenched disorder, for instance the random walk (or diffusion) in a disordered environment [4], which is in the one-dimensional case the Sinai-model[5]. For this model the first passage / persistence exponent for a single walker has been determined by us in a previous work [6]. In this letter we introduce and study the concept of average persistence of random walks both in homogeneous and random environments. We consider a random walk with nearest neighbor hopping in one dimension defined by the Master equation pi(t = 0) = δi,1,
This work focuses on the study of quantum stochastic walks, which are a generalization of coherent, i.e. unitary quantum walks. Our main goal is to present a measure of coherence of the walk. To this end, we utilize the Hurst exponent. As the quantum stochastic walk model encompasses both classical random walks and quantum walks, we are interested how the continuous change from one regime to the other influences the Hurst exponent. Moreover this model allows for behavior which is not found in any of the previously mentioned model -- the model with global dissipation. We derive the probability distribution for the walker, and determine the Hurst exponent analytically, showing that ballistic regime of the walk is maintained even at large dissipation strength.
This work focuses on the study of quantum stochastic walks, which are a generalization of coherent, i. e. unitary quantum walks. Our main goal is to present a measure of a coherence of the walk. To this end, we utilize the asymptotic scaling exponent of the second moment of the walk i. e. of the mean squared distance covered by a walk. As the quantum stochastic walk model encompasses both classical random walks and quantum walks, we are interested how the continuous change from one regime to the other influences the asymptotic scaling exponent. Moreover this model allows for behavior which is not found in any of the previously mentioned model -- the model with global dissipation. We derive the probability distribution for the walker, and determine the asymptotic scaling exponent analytically, showing that ballistic regime of the walk is maintained even at large dissipation strength.