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arXiv:1708.06208 · Physical Review A

Quantum non-Markovianity and localization

David Davalos, Carlos Pineda

Published Dec 21, 2017• 3 citations
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Abstract

We study the behavior of non-Markovianity with respect to the localization of the initial environmental state. The ``amount'' of non-Markovianity is measured using divisibility and distinguishability as indicators, employing several schemes to construct the measures. The system used is a qubit coupled to an environment modeled by an Ising spin chain kicked by ultrashort pulses of a magnetic field. In the integrable regime, non-Markovianity and localization do not have a simple relation, but as the chaotic regime is approached, simple relations emerge, which we explore in detail. We also study the non-Markovianity measures in the space of the parameters of the spin coherent states and point out that the pattern that appears is robust under the choice of the interaction Hamiltonian but does not have a classical-like phase-space structure.

DOI
10.1103/physreva.96.062127

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We study the behavior of non-Markovianity with respect to the localization of the initial environmental state. The ``amount'' of non-Markovianity is measured using divisibility and distinguishability as indicators, employing several schemes to construct the measures. The system used is a qubit coupled to an environment modeled by an Ising spin chain kicked by ultrashort pulses of a magnetic field. In the integrable regime, non-Markovianity and localization do not have a simple relation, but as the chaotic regime is approached, simple relations emerge, which we explore in detail. We also study the non-Markovianity measures in the space of the parameters of the spin coherent states and point out that the pattern that appears is robust under the choice of the interaction Hamiltonian but does not have a classical-like phase-space structure.
In this work we study the non-Markovian behavior of a qubit coupled to an environment in which the corresponding classical dynamics change from integrable to chaotic. We show that in the transition region, where the dynamics has both regular islands and chaotic areas, the average non-Markovian behavior is enhanced to values even larger than those in the regular regime. This effect can be related to the non-Markovian behavior as a function of the initial state of the environment, where maxima are attained at the regions dividing separate areas in classical phase space, particularly at the borders between chaotic and regular regions. Moreover, we show that the fluctuations of the fidelity of the environment?which determine the non-Markovianity measure?give a precise image of the classical phase portrait.
In this work we study the non-Markovian behavior of a qubit coupled to an environment in which the corresponding classical dynamics change from integrable to chaotic. We show that in the transition region, where the dynamics has both regular islands and chaotic areas, the average non-Markovian behavior is enhanced to values even larger than those in the regular regime. This effect can be related to the non-Markovian behavior as a function of the initial state of the environment, where maxima are attained at the regions dividing separate areas in classical phase space, particularly at the borders between chaotic and regular regions. Moreover, we show that the fluctuations of the fidelity of the environment—which determine the non-Markovianity measure—give a precise image of the classical phase portrait.
We discuss the conceptually different definitions used for the non-Markovianity of classical and quantum processes. The well-established definition for non-Markovianity of a classical stochastic process represents a condition on the Kolmogorov hierarchy of the n-point joint probability distributions. Since this definition cannot be transferred to the quantum regime, quantum non-Markovianity has recently been defined and quantified in terms of the underlying quantum dynamical map, using either its divisibility properties or the behavior of the trace distance between pairs of initial states. Here, we investigate and compare these definitions and their relations to the classical notion of non-Markovianity by employing a large class of non-Markovian processes, known as semi-Markov processes, which admit a natural extension to the quantum case. A number of specific physical examples is constructed which allow to study the basic features of the classical and the quantum definitions and to evaluate explicitly the measures for quantum non-Markovianity. Our results clearly demonstrate several fundamental distinctions between the classical and the quantum notion of non-Markovianity, as well as between the various quantum measures for non-Markovianity.
It is believed that an isolated and far-from-equilibrium quantum many-body system should try to attain equilibrium via a mechanism whereby any given subsystem acts as an open quantum system that is coupled to an environment, which is the complementary part of the full system, and undergoes a complicated equilibration process such that all the subsystems in the long-time limit attain equilibrium states compatible with the global equilibrium state. This picture begs the question whether the dynamics of any given subsystem is Markovian (monotonic loss of information and memory) or non-Markovian. In this work, by numerically probing the dynamical behaviour of the quantum distances between temporally-separated\textit{temporally-separated}temporally-separated states of small subsystems, we reveal the telltale signatures of (non-)Markovianity of the dynamics of subsystems of an isolated quantum spin system brought in the far-from-equilibrium regime, exemplified with the mixed-field Ising spin chain quenched between parameter regimes deep inside its magnetically ordered and disordered regimes. Additionally, remarkably systematic behaviour is seen in a measure of classical distances between the quantum states of the considered subsystems. These features strongly depend on the direction of quenching in the parameter space, with paramagnetic-to-ferromagnetic quenches offering considerably stronger signatures of subsystem non-Markovianity, for which we offer heuristic arguments.
A simple relationship between recently proposed measures of non-Markovianity and the Loschmidt echo is established, holding for a two-level system (qubit) undergoing pure dephasing due to a coupling with a many-body environment. We show that the Loschmidt echo is intimately related to the information flowing out from and occasionally back into the system. This, in turn, determines the non-Markovianity of the reduced dynamics. In particular, we consider a central qubit coupled to a quantum Ising ring in the transverse field. In this context, the information flux between system and environment is strongly affected by the environmental criticality; the qubit dynamics is shown to be Markovian exactly and only at the critical point. Therefore, non-Markovianity is an indicator of criticality in the model considered here.
ABSTRACT: We use the divisibility approach to describe non-Markovian dynamics (memory effects) in open quantum systems coupled to an environment (reservoir). Specifically, sufficient and necessary criteria were incorporated in the energy domain that play a relevant role in dynamics of the system. In this case, we adopt the Feshbach partition in the space of (Hilbert-Liouville). On the other hand, to investigate both Markovianity and non-Markovianity, we study the divisibility of the Wigner function propagator. For this, within the framework of path integrals, we carry out an analysis of non-classical paths in phase space. Furthermore, we have used tools of topology to propose theorems intimately related to measures of non-Markovianity that are experementally accessible.