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arXiv:cond-mat/9904044 · Physical review. B, Condensed matter

Diffusion in disordered systems under iterative measurement

Juan Cesar Flores

Published Jul 1, 1999• 33 citations
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Abstract

We consider a sequence of idealized measurements of time separation $\ensuremath{\Delta}t$ onto a discrete one-dimensional disordered system. A connection with Markov chains is found. For a rapid sequence of measurements, a diffusive regime occurs and the diffusion coefficient D is analytically calculated. In a general point of view, this result suggests the possibility to break the Anderson localization due to decoherence effects. Quantum Zeno effect emerges because the diffusion coefficient D vanishes at the limit $\ensuremath{\Delta}\stackrel{\ensuremath{\rightarrow}}{t}0$.

DOI
10.1103/physrevb.60.30

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We consider a sequence of idealized measurements of time separation \ensuremathΔt\ensuremath{\Delta}t\ensuremathΔt onto a discrete one-dimensional disordered system. A connection with Markov chains is found. For a rapid sequence of measurements, a diffusive regime occurs and the diffusion coefficient D is analytically calculated. In a general point of view, this result suggests the possibility to break the Anderson localization due to decoherence effects. Quantum Zeno effect emerges because the diffusion coefficient D vanishes at the limit \ensuremathΔt\ensuremath→0\ensuremath{\Delta}\stackrel{\ensuremath{\rightarrow}}{t}0\ensuremathΔt\ensuremath→​0.
Using locally the projector postulate (wave-function collapse) we consider a one-dimensional disordered system under local (one-site) iterative measurements. We found that Anderson localization is broken due to these decoherence effects. Explicitly, the probability of return falls exponentially to zero and the delocalization time is computed. For infinite size systems the delocalization time remains finite. Quantum Zeno effect emerges in the corresponding limit of a rapid sequence of measurements including off-diagonal disorder.
Quantum walks are considered in a one-dimensional random medium characterized by static or dynamic disorder. Quantum interference for static disorder can lead to Anderson localization which completely hinders the quantum walk and it is contrasted with the decoherence effect of dynamic disorder having strength WWW, where a quantum to classical crossover at time ${t}_{c}\ensuremath{\propto}{W}^{\ensuremath{-}2}$ transforms the quantum walk into an ordinary random walk with diffusive spreading. We demonstrate these localization and decoherence phenomena in quantum carpets of the observed time evolution, we relate our results to previously studied models of decoherence for quantum walks, and examine in detail a dimer lattice which corresponds to a single qubit subject to randomness.
We investigate the quantum walk on the line when decoherence is introduced through simultaneous measurements of the chirality and the particle position. This decoherence drives the system to a classical diffusive behavior, for which we have obtained analytical expressions for its diffusion coefficient. We show that this coefficient is proportional to the variance of the initially localized quantum random walker just before the first measurement. Through an association of the measurement processes to the collisions in Brownian motion, we show that this treatment provides a smooth connection between the pure classical and quantum limits.
We study the spreading of a quantum-mechanical wave packet in a tight-binding model with a noisy potential and analyze the emergence of classical diffusion from the quantum dynamics due to decoherence. We consider a finite correlation time of the noisy environment and treat the system by utilizing the separation of fast (dephasing) and slow (diffusion) processes. We show that classical diffusive behavior emerges at long times and we calculate analytically the dependence of the classical diffusion coefficient on the noise magnitude and correlation time. This method provides a general solution to this problem for arbitrary conditions of the noisy environment. The calculation can be done in any dimension, but we demonstrate it in one dimension for clarity of representation. The results are relevant to a large variety of physical systems, from electronic transport in solid-state physics to light transmission in optical devices, diffusion of excitons, and quantum computation.
Repeated quantum measurements can generate effective new nonequilibrium dynamics in matter. Here we combine such a measurement driven system with disorder. In particular, we investigate the diffusive behavior in the system and the effect of various types of disorder on the measurement induced chiral transport protocol. We begin by characterizing the diffusive behavior produced by the measurements themselves in a clean system. We then examine the edge flow of particles per measurement cycle for three different types of disorder: site dilution, lattice distortion, and disorder in on-site chemical potential. In the quantum Zeno limit, the effective descriptions for the disordered measurement system with lattice distortions and random on-site potential can be modeled as a classical stochastic model, and the overall effect of increasing these disorders induces a crossover from perfect flow to zero transport. On the other hand if vacancies are present in the lattice the flow of particles per measurement cycle undergoes a percolation phase transition from unity to zero with percolation threshold ${p}_{c}\ensuremath{\approx}0.26$, with critical exponent $\ensuremath{\nu}\ensuremath{\approx}1.35$. We also present numerical results away from Zeno limit and note that the overall effect of moving away from the Zeno effect is to reduce particle flow per cycle when the measurement frequency in our protocol is reduced.
We consider the phase coherent transport of a quasi-one-dimensional beam of Bose-Einstein condensed particles through a disordered potential of length LLL. Among the possible different types of flow we identified [T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, Phys. Rev. Lett. 98, 210602 (2007)], we focus here on the supersonic stationary regime where Anderson localization exists. We generalize the diffusion formalism of Dorokhov-Mello-Pereyra-Kumar to include interaction effects. It is shown that interactions modify the localization length and also introduce a length scale ${L}^{\ensuremath{\ast}}$ for the disordered region, above which most of the realizations of the random potential lead to time-dependent flows. A Fokker-Planck equation for the probability density of the transmission coefficient that takes this effect into account is introduced and solved. The theoretical predictions are verified numerically for different types of disordered potentials. Experimental scenarios for observing our predictions are discussed.