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arXiv:1509.04231 · Journal of Physics B Atomic Molecular and Optical Physics

Discrete dynamics and non-Markovianity

Kimmo Luoma, Jyrki Piilo

Published May 18, 2016• 10 citations
arXivHTMLPDF

Abstract

Abstract We study discrete quantum dynamics where a single evolution step consists of unitary system transformation followed by decoherence via coupling to an environment. Often, non-Markovian memory effects are attributed to structured environments, whereas, here, we take a more general approach within a discrete setting. In addition of controlling the structure of the environment, we are interested in how local unitaries on the open system allow the appearance and control of memory effects. Our first simple qubit model where local unitary is followed by dephasing illustrates how memory effects arise, despite having no structure in the environment the system is coupled with. We, then, elaborate on this observation by constructing a model for an open quantum walk where the unitary coin and transfer operation is augmented with the dephasing of the coin. The results demonstrate tha,t in the limit of strong dephasing within each evolution step, the combined coin-position open system always displays memory effects, and their quantities are independent of the structure of the environment. Our construction makes possible an experimentally realizable open quantum walk with photons exhibiting non-Markovian features.

DOI
10.1088/0953-4075/49/12/125501

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Abstract We study discrete quantum dynamics where a single evolution step consists of unitary system transformation followed by decoherence via coupling to an environment. Often, non-Markovian memory effects are attributed to structured environments, whereas, here, we take a more general approach within a discrete setting. In addition of controlling the structure of the environment, we are interested in how local unitaries on the open system allow the appearance and control of memory effects. Our first simple qubit model where local unitary is followed by dephasing illustrates how memory effects arise, despite having no structure in the environment the system is coupled with. We, then, elaborate on this observation by constructing a model for an open quantum walk where the unitary coin and transfer operation is augmented with the dephasing of the coin. The results demonstrate tha,t in the limit of strong dephasing within each evolution step, the combined coin-position open system always displays memory effects, and their quantities are independent of the structure of the environment. Our construction makes possible an experimentally realizable open quantum walk with photons exhibiting non-Markovian features.
Non-Markovian quantum effects are typically observed in systems interacting with structured reservoirs. Discrete-time quantum walks are prime example of such systems in which, quantum memory arises due to the controlled interaction between the coin and position degrees of freedom. Here we show that the information backflow that quantifies memory effects can be enhanced when the particle is subjected to uncorrelated static or dynamic disorder. The presence of disorder in the system leads to localization effects in 1-dimensional quantum walks. We shown that it is possible to infer about the nature of localization in position space by monitoring the information backflow in the reduced system. Further, we study other useful properties of quantum walk such as entanglement, interference and its connection to quantum non-Markovianity.
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We explore the possibility to generate nonlocal dynamical maps of an open quantum system through local system-environment interactions. Employing a generic decoherence process induced by a local interaction Hamiltonian, we show that initial correlations in a composite environment can lead to nonlocal open system dynamics which exhibit strong memory effects, although the local dynamics is Markovian. In a model of two entangled photons interacting with two dephasing environments, we find a direct connection between the degree of memory effects and the amount of correlation in the initial environmental state. The results demonstrate that, contrary to conventional wisdom, enlarging an open system can change the dynamics from Markovian to non-Markovian.
A profound understanding of decoherence and dissipation in quantum dynamics is crucial for the realistic modeling of the evolution of quantum systems. In open quantum dynamics one distinguishes between a memoryless, so-called Markovian evolution and dynamics incorporating memory effects, termed non-Markovian. In this work we study how prevalent memory effects are in the set of all such dynamics. We thus investigate how often a Markovian description is applicable. This question is approached by investigating randomly generated two-step qubit dynamics with respect to different concepts and witnesses of non-Markovianity. We observe that almost all dynamics are non-Markovian, and only a small (yet finite) fraction is Markovian. Furthermore, we study how this proportion changes when considering certain subclasses such as lower rank or mixed-unitary dynamics. Importantly, our results shed light on the relative ratios of -- and interrelations between -- the sets of dynamics that are non-Markovian with respect to different criteria. Finally, we investigate the fraction of dynamics in which the memory effects are necessarily of quantum nature and establish a connection between two recently developed concepts that characterize the quantumness of memory in non-Markovian dynamics.
The quantum channels with memory, known as non-Markovian channels, are of crucial importance for a realistic description of a variety of physical systems, and pave ways for new methods of decoherence control by manipulating the properties of an environment such as its frequency spectrum. In this work, the reduced dynamics of a coin in a discrete-time quantum walk is characterized as a non-Markovian quantum channel. A general formalism is sketched to extract the Kraus operators for a ttt-step quantum walk. Non-Markovianity, in the sense of P indivisibility of the reduced coin dynamics, is inferred from the nonmonotonous behavior of distinguishability of two orthogonal states subjected to it. Furthermore, we study various quantum information-theoretic quantities of a qubit under the action of this channel, putting in perspective the role such channels can play in various quantum information processing tasks.
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