Markovianity and non-Markovianity in quantum and classical systems

Vacchini, Bassano; Smirne, Andrea; Laine, Elsi-Mari; Piilo, Jyrki; Breuer, Heinz‐Peter

doi:10.1088/1367-2630/13/9/093004

Cited by 180 publications

(205 citation statements)

References 36 publications

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“…This term refers to a well-known concept of the theory of classical stochastic processes and is used to loosely indicate the presence of memory effects in the time evolution of the open system. However, the classical notions of Markovianity and non-Markovianity cannot be transferred to the quantum regime in a natural way since they are based on a Kolmogorov hierarchy of joint probability distributions which does not exist in quantum theory (Accardi et al, 1982;Lindblad, 1979;Vacchini et al, 2011). Therefore, the concept of a quantum non-Markovian process requires a precise definition which cannot be based on classical notions only.…”

Section: Introductionmentioning

confidence: 99%

Colloquium: Non-Markovian dynamics in open quantum systems

et al. 2016

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The dynamical behavior of open quantum systems plays a key role in many applications of quantum mechanics, examples ranging from fundamental problems, such as the environment-induced decay of quantum coherence and relaxation in many-body systems, to applications in condensed matter theory, quantum transport, quantum chemistry and quantum information. In close analogy to a classical Markovian stochastic process, the interaction of an open quantum system with a noisy environment is often modeled phenomenologically by means of a dynamical semigroup with a corresponding time-independent generator in Lindblad form, which describes a memoryless dynamics of the open system typically leading to an irreversible loss of characteristic quantum features. However, in many applications open systems exhibit pronounced memory effects and a revival of genuine quantum properties such as quantum coherence, correlations and entanglement. Here, recent theoretical results on the rich non-Markovian quantum dynamics of open systems are discussed, paying particular attention to the rigorous mathematical definition, to the physical interpretation and classification, as well as to the quantification of quantum memory effects. The general theory is illustrated by a series of physical examples. The analysis reveals that memory effects of the open system dynamics reflect characteristic features of the environment which opens a new perspective for applications, namely to exploit a small open system as a quantum probe signifying nontrivial features of the environment it is interacting with. This article further explores the various physical sources of non-Markovian quantum dynamics, such as structured environmental spectral densities, nonlocal correlations between environmental degrees of freedom and correlations in the initial system-environment state, in addition to developing schemes for their local detection. Recent experiments addressing the detection, quantification and control of non-Markovian quantum dynamics are also briefly discussed.

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Section: Introductionmentioning

confidence: 99%

Colloquium: Non-Markovian dynamics in open quantum systems

et al. 2016

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“…Refs. [16][17][18], we add below another definition which is furthermore applicable to closed quantum systems. The definition is based on mathematical constructions which have already been used by Wigner [19] to quantify probabilities for the occurrence of subsequent events.…”

Section: Introductionmentioning

confidence: 99%

Numerical evidence for approximate consistency and Markovianity of some quantum histories in a class of finite closed spin systems

Schmidtke

Gemmer

2016

Phys. Rev. E

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Closed quantum systems obey the Schrödinger equation whereas nonequilibrium behavior of many systems is routinely described in terms of classical, Markovian stochastic processes. Evidently, there are fundamental differences between those two types of behavior. We discuss the conditions under which the unitary dynamics may be mapped onto pertinent classical stochastic processes. This is first principally addressed based on the notions of "consistency" and "Markovianity." Numerical data are presented that show that the above conditions are to good approximation fulfilled for Heisenberg-type spin models comprising 12-20 spins. The accuracy to which these conditions are met increases with system size.

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“…Its importance is dictated by the prospects of applications in quantum optics, quantum computing, quantum measurements and control [1,2], as well as by the necessity of a deeper understanding of the theory itself [3][4][5][6][7][8]. The dynamics of open quantum systems was studied in several aspects: (i) the effect of initial correlations between an open system and its environment has been investigated in [9,10]; (ii) a new viewpoint concerning the nature and the measure of non-Markovianity has been presented in [11,12]; (iii) the effect of non-equilibrium environment on quantum coherence and the level populations has been considered in [13,14].…”

Section: Introductionmentioning

confidence: 99%

Bath dynamics in an exactly solvable qubit model with initial qubit-environment correlations

Ignatyuk¹,

Morozov²

2013

Condens. Matter Phys.

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We study the bath dynamics in the dephasing model of a two-state quantum system (qubit) coupled to an environment of harmonic oscillators. This model was shown [Morozov et al., Phys. Rev. A, 2012 85, 022101] to admit the analytic solution for the qubit and environment dynamics. Using this solution, we derive the exact expression for the bath reduced density matrix in the presence of initial qubit-environment correlations. We obtain the non-equilibrium phonon distribution function and discuss in detail the time behavior of the bath energy. It is shown that only the inclusion of dynamic correlations between the qubit and the bath ensures the proper time behavior of the quantity which may be interpreted as the "environment energy".

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Markovianity and non-Markovianity in quantum and classical systems

Cited by 180 publications

References 36 publications

Colloquium: Non-Markovian dynamics in open quantum systems

Colloquium: Non-Markovian dynamics in open quantum systems

Numerical evidence for approximate consistency and Markovianity of some quantum histories in a class of finite closed spin systems

Bath dynamics in an exactly solvable qubit model with initial qubit-environment correlations

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