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Physical Review A

Decoherence induced by a chaotic enviroment: A quantum walker with a complex coin

FCEyN Departamento de Fisica

Published Jan 15, 2006• 0 citations
OpenAlex

Abstract

We study the differences between the processes of decoherence induced by chaotic and regular environments. For this we analyze a family of simple models that contain both regular and chaotic environments. In all cases the system of interest is a ''quantum walker,'' i.e., a quantum particle that can move on a lattice with a finite number of sites. The walker interacts with an environment which has a D-dimensional Hilbert space. The results we obtain suggest that regular and chaotic environments are not distinguishable from each other in a (short) time scale t*, which scales with the dimensionality of the environment as t*{proportional_to}log{sub 2}(D). However, chaotic environments continue to be effective over exponentially longer time scales while regular environments tend to reach saturation much sooner. We present both numerical and analytical results supporting this conclusion. The family of chaotic evolutions we consider includes the so-called quantum multibaker map as a particular case.

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We study the differences between the processes of decoherence induced by chaotic and regular environments. For this we analyze a family of simple models that contain both regular and chaotic environments. In all cases the system of interest is a ''quantum walker,'' i.e., a quantum particle that can move on a lattice with a finite number of sites. The walker interacts with an environment which has a D-dimensional Hilbert space. The results we obtain suggest that regular and chaotic environments are not distinguishable from each other in a (short) time scale t*, which scales with the dimensionality of the environment as t*{proportional_to}log{sub 2}(D). However, chaotic environments continue to be effective over exponentially longer time scales while regular environments tend to reach saturation much sooner. We present both numerical and analytical results supporting this conclusion. The family of chaotic evolutions we consider includes the so-called quantum multibaker map as a particular case.
We study the differences between the processes of decoherence induced by chaotic and regular environments. For this we analyze a family of simple models that contain both regular and chaotic environments. In all cases the system of interest is a ``quantum walker,'' i.e., a quantum particle that can move on a lattice with a finite number of sites. The walker interacts with an environment which has a DDD-dimensional Hilbert space. The results we obtain suggest that regular and chaotic environments are not distinguishable from each other in a (short) time scale t∗{t}^{*}t∗, which scales with the dimensionality of the environment as ${t}^{*}\ensuremath{\propto}{\mathrm{log}}_{2}(D)$. However, chaotic environments continue to be effective over exponentially longer time scales while regular environments tend to reach saturation much sooner. We present both numerical and analytical results supporting this conclusion. The family of chaotic evolutions we consider includes the so-called quantum multibaker map as a particular case.
Making use of appropriate quantum-classical correspondence we have examined the differential behavior of regular and chaotic trajectories in terms of quantum decoherence, the evolution of average quantities, and variances and entropy in a driven double-well oscillator in contact with the surroundings. Our numerical analysis has shown that even on a short time scale the decay of the average quantum coherence is multiexponential. While the onset of decoherence is much faster for a regular trajectory, the decay is faster asymptotically for a chaotic trajectory compared to a regular one. The coupling of the system to the environment turns chaotic in a regular evolution. The environment also affects the evolution of quantum variances for a regular trajectory almost from the beginning, while it has an insignificant effect on chaotic evolution up to a time after which, for both trajectories, noise in the quantum variances exhibits remarkable suppression, although fluctuations of the reservoir modes, in general, tend to increase the level of variances. We identify three stages of quantum evolution; a short decoherence regime followed by a Liouville flow, the latter regime being dominated by the classical curvature of the potential. This is the regime at which growth of entropy or quantum variances is exponential, the rate being determined by the classical largest Lyapunov exponent. The last stage is the irreversible flow dominated by diffusion which suppresses noises in the quantum variances. \textcopyright{} 1996 The American Physical Society.
In this work we study several models of decoherence and how different quantum maps and algorithms react when perturbed by them. Following closely Ref. [1], generalizations of the three paradigmatic one single qubit quantum channels (these are the depolarizing channel, the phase damping channel and the amplitude damping channel) for the case of an arbitrarily-sized finite-dimensional Hilbert space are presented, as well as other types of noise in phase space. More specifically, Grover's search algorithm's response to decoherence is analyzed; together with those of a family of quantum versions of chaotic and regular classical maps (the baker's map and the cat maps). A relationship between how sensitive to decoherence a quantum map is and the degree of complexity in the dynamics of its associated classical counterpart is observed; resulting in a clear tendency to react the more decoherently the more complex the associated classical dynamics is.
We investigate decoherence in quantum systems coupled via dephasing-type interactions to an arbitrary environment with chaotic underlying classical dynamics. The coherences of the reduced state of the central system written in the preferential energy eigenbasis are quantum Loschmidt echoes, which in the strong coupling regime are characterized at long time scales by fluctuations around a constant mean value. We show that due to the chaotic dynamics of the environment, the mean value and the width of the Loschmidt-echo fluctuations are inversely proportional to the quantity we define as the effective Hilbert-space dimension of the environment, which in general is smaller than the dimension of the entire available Hilbert space. Nevertheless, in the semiclassical regime this effective Hilbert-space dimension is in general large, in which case even a chaotic environment with few degrees of freedom produces decoherence without revivals. Moreover we show that in this regime the environment leads the central system to equilibrate to the time average of its reduced density matrix, which corresponds to a diagonal state in the preferential energy eigenbasis. For the case of two uncoupled, initially entangled central systems that interact with identical local quantum environments with chaotic underlying classical dynamics, we show that in the semiclassical limit the equilibration state is arbitrarily close to a separable state. We confirm our results with numerical simulations in which the environment is modeled by the quantum kicked rotor in the chaotic regime.
By means of numerical simulation, we reveal a phenomenon which is due to decoherence induced by chaotic motion of a large environment. We study the possibility of replacing the environmental components of the state vector of the total system by some randomly-chosen vectors in the Hilbert space of the environment when decoherence has happened. It is shown that, in the future evolution, the reduced density matrices obtained in this way are quite close to those computed from the exact, continuous Schrödinger evolution of the total system. Similar results are also found in the case that the Hamiltonian is changed at the time of replacing components.
Decoherence in quantum systems which are classically chaotic is studied. The Arnold cat map and the quantum kicked rotor are chosen as examples of linear and nonlinear chaotic systems. The Feynman-Vernon influence functional formalism is used to study the effect of the environment on the system. It is well known that quantum coherence can obliterate chaotic behavior in the corresponding classical system. But interaction with an environment can under general circumstances quickly diminish quantum coherence and reenact classical chaotic behavior. How effectively decoherence works to sustain chaos, and how the resultant behavior qualitatively differs from the quantum picture, depending on the coupling of the system with the environment and the spectral density and temperature of the environment. We show how recurrence in the quantum cat map is lost and classical ergodicity is recovered due to the effect of the environment. Quantum coherence and diffusion suppression are instrumental to dynamical localization for the kicked rotor. We show how environment-induced effects can destroy this localization. Such effects can also be understood as resulting from external noises driving the system. Peculiar to decohering chaotic systems is the apparent transition from reversible to irreversible dynamics. We show such transitions in the quantum cat map and the kicked rotor and distinguish it from apparent irreversibility originating from dynamical instability and imprecise measurements. By performing a time reversal on and following the quantum kicked rotor dynamics numerically, we show how the otherwise reversible quantum dynamics acquires an arrow of time upon the introduction of noise or interaction with an environment.