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Physical Review B

Iterative quantum local measurements and Anderson localization inhibition

Juan Cesar Flores

Published Jan 5, 2004• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.69.012201

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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.
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 $\ensuremath{\Delta}\stackrel{\ensuremath{\rightarrow}}{t}0$.
We address here the issue of quantum measurement in which the interaction between the system at hand and the measuring apparatus causes a collapse of the wave function or the density operator. The analysis is carried out for the prototypical tight-binding chain involving a quantum particle hopping between nearest-neighbour sites on a lattice, with the system subject to repeated projective measurements to a detector site. However, unlike previous work, the measurements are performed at random times. In the absence of measurement, the particle while starting from a site is known to spread out to farther sites as time progresses, leading to complete delocalization of the particle in time. In presence of measurements, we derive exact results for the probability at a given time for the particle to be found on different sites and averaged with respect to different realizations of the dynamics. We consider the representative case in which the time gap between successive measurements are sampled independently from an exponential distribution. The novel result of our findings is the revelation of the striking effect of Stochastic Localization, whereby the particle has at long times time-independent probabilities to be on different sites. This implies localization of the particle at long times, achieved through classical stochasticity involved with the random times for measurements. An extreme case of such localization is attained if the measurements are done very frequently, when the particle is completely localized at the detector site! We also point out the stochastic version of the Zeno effect, whereby dynamical evolution of a quantum system is arrested through repeated measurements at frequent-enough times. In the wake of recent experiments on projective measurements at random times achieved through randomly-pulsed sequences, our proposition is amenable to experimental realization.
Quantum walks are promising for information processing tasks because in regular graphs they spread quadratically more rapidly than random walks. Static disorder, however, can turn the tables: unlike random walks, quantum walks can suffer Anderson localization, with their wave function staying within a finite region even in the infinite time limit, with a probability exponentially close to 1. It is thus important to understand when a quantum walk will be Anderson localized and when we can expect it to spread to infinity even in the presence of disorder. In this work we analyze the response of a one-dimensional quantum walk---the split-step walk---to different forms of static disorder. We find that introducing static, symmetry-preserving disorder in the parameters of the walk leads to Anderson localization. In the completely disordered limit, however, a delocalization transition occurs, and the walk spreads subdiffusively to infinity. Using an efficient numerical algorithm, we calculate the bulk topological invariants of the disordered walk and find that the disorder-induced Anderson localization and delocalization transitions are governed by the topological phases of the quantum walk.
We study the dynamics of a monitored single particle in a one-dimensional Anderson-localized system. The time evolution is governed by Hamiltonian dynamics for fixed time intervals, interrupted by local, projective measurements. The competition between disorder-induced localization and measurement-induced jumps leads to interesting behavior of readout-averaged quantities. We find that measurements at random positions delocalize the average position, similar to a classical random walk. Along each quantum trajectory, the particle remains localized, however, with a modified localization length. In contrast to measurement-induced delocalization, controlled measurements can be used to introduce transport in the system and localize the particle at a chosen site. In this sense, the measurements provide a controlled environment for the particle.
The inhibition of the decay of a quantum system by frequent measurements is known as quantum Zeno effect. Beyond the limit of projective measurements, the interplay between the unitary dynamics of the system and the coupling to a measurement apparatus becomes relevant. We explore this interplay by considering a quantum particle moving on a one-dimensional chain. A local measurement by coupling to an apparatus with a two-dimensional Hilbert space detects the presence of the particle on a specific chain site. The decay of the population is studied analytically for a two-site chain and numerically for a larger system as a function of the measurement time and the time between subsequent measurements. Particular attention is given to the shift of the energy of the measured site due to the coupling to the apparatus. The decay of the initial population can be hindered or accelerated, depending on the chosen system and the coupling parameters.
We experimentally study the effects of coupling one-dimensional many-body localized systems with identical disorder. Using a gas of ultracold fermions in an optical lattice, we artificially prepare an initial charge density wave in an array of 1D tubes with quasirandom on-site disorder and monitor the subsequent dynamics over several thousand tunneling times. We find a strikingly different behavior between many-body localization and Anderson localization. While the noninteracting Anderson case remains localized, in the interacting case any coupling between the tubes leads to a delocalization of the entire system.