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arXiv:1011.5328 · Physical Review A

Comparing different non-Markovianity measures in a driven qubit system

Pinja Haikka, James D. Cresser, Sabrina Maniscalco

Published Jan 31, 2011• 118 citations
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Abstract

We consider two recently proposed measures of non-Markovianity applied to a particular quantum process describing the dynamics of a driven qubit in a structured reservoir. The motivation for this study is twofold: on one hand, we study the differences and analogies of the non-Markovianity measures, and on the other hand, we investigate the effect of the driving force on the dissipative dynamics of the qubit. In particular we ask if the driving force introduces new channels for energy and/or information transfer between the system and the environment or if it amplifies existing ones. We show under which conditions the presence of the driving force slows down the inevitable loss of quantum properties of the qubit.

DOI
10.1103/physreva.83.012112

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We consider two recently proposed measures of non-Markovianity applied to a particular quantum process describing the dynamics of a driven qubit in a structured reservoir. The motivation for this study is twofold: on one hand, we study the differences and analogies of the non-Markovianity measures, and on the other hand, we investigate the effect of the driving force on the dissipative dynamics of the qubit. In particular we ask if the driving force introduces new channels for energy and/or information transfer between the system and the environment or if it amplifies existing ones. We show under which conditions the presence of the driving force slows down the inevitable loss of quantum properties of the qubit.
In this Thesis I discuss the exact dynamics of simple non-Markovian systems. I focus on fundamental questions at the core of non-Markovian theory and investigate the dynamics of quantum correlations under non-Markovian decoherence. \n \nIn the first context I present the connection between two different non-Markovian approaches, and compare two distinct definitions of non-Markovianity. The general aim is to characterize in exemplary cases which part of the environment is responsible for the feedback of information typical of non- Markovian dynamics. I also show how such a feedback of information is not always described by certain types of master equations commonly used to tackle non-Markovian dynamics. \n \nIn the second context I characterize the dynamics of two qubits in a common non-Markovian reservoir, and introduce a new dynamical effect in a wellknown model, i.e., two qubits under depolarizing channels. In the first model the exact solution of the dynamics is found, and the entanglement behavior is extensively studied. The non-Markovianity of the reservoir and reservoirmediated-interaction between the qubits cause non-trivial dynamical features. \n \nThe dynamical interplay between different types of correlations is also investigated. In the second model the study of quantum and classical correlations demonstrates the existence of a new effect: the sudden transition between classical and quantum decoherence. This phenomenon involves the complete preservation of the initial quantum correlations for long intervals of time of the order of the relaxation time of the system.
We investigate signatures of non-Markovianity in the dynamics of a periodically driven qubit coupled to a dissipative bosonic environment. We propagate the dynamics of the reduced density matrix of the qubit by integrating the numerically exact hierarchical equations of motion. Non-Markovian features are quantified by comparing on an equal footing the predictions from diverse and complementary approaches to quantum dissipation. In particular, we analyze the distinguishability of quantum states, the decay of the volume accessible to the qubit on the Hilbert space, the negativity of the canonical rates in the generalized Lindblad equation, and the relaxation of the memory kernels in the Nakajima-Zwanzig generalized quantum master equation. We study the effects of controlled driving on the coherent dynamics of the system. We show that a suitable external field can offset the ergodic relaxation of time-correlation functions, increase distinguishability over time, and strengthen non-Markovian effects witnessed by the canonical dissipation channels. We furthermore observe the phenomenon of eternal non-Markovianity for sufficiently small system-bath coupling and we discuss how this can be enhanced by modulating the frequency of the external drive. This work provides a broad theoretical analysis of quantum dissipation in the framework of open quantum dynamics and quantum information.
The basic features of the dynamics of open quantum systems, such as the dissipation of energy, the decay of coherences, the relaxation to an equilibrium or non-equilibrium stationary state, and the transport of excitations in complex structures are of central importance in many applications of quantum mechanics. The theoretical description, analysis and control of non-Markovian quantum processes play an important role in this context. While in a Markovian process an open system irretrievably loses information to its surroundings, non-Markovian processes feature a flow of information from the environment back to the open system, which implies the presence of memory effects and represents the key property of non-Markovian quantum behavior. Here, we review recent ideas developing a general mathematical definition for non-Markoviantiy in the quantum regime and a measure for the degree of memory effects in the dynamics of open systems which are based on the exchange of information between system and environment. We further study the dynamical effects induced by the presence of system-environment correlations in the total initial state and design suitable methods to detect such correlations through local measurements on the open system.
By considering an exactly solvable model for a driven two non-interacting qubits, each coupled to a bosonic environment with zero temperature, under the non-Markovian dissipative process, we study the variation of coherence and correlations in terms of different physical parameters. We show the influence of the external classical driving field as well as the initial quantum states. Moreover, we highlight the relationship between the coherence, single-qubit population, and correlations according to the physical parameters of the whole system. We reveal that the preservation and enhancement of coherence and correlations may occur by adjusting the strength of the classical driving field, initial states, and non-Markovian effects.
In this paper we present a detailed critical study of several recently proposed non-Markovianity measures. We analyze their properties for single-qubit and two-qubit systems in both pure-dephasing and dissipative scenarios. More specifically we investigate and compare their computability, their physical meaning, their Markovian to non-Markovian crossover, and their additivity properties with respect to the number of qubits. The bottom-up approach that we pursue is aimed at identifying similarities and differences in the behavior of non-Markovianity indicators in several paradigmatic open system models. This, in turn, allows us to infer the leading traits of the variegated phenomenon known as non-Markovian dynamics.
Recent femtosecond processes [Phys. Rev. Lett. 66, 2464 (1991)] confirm the importance of non-Markovian effects since the dynamical characteristic times of the system become of the same time scale as the reservoir. We analyze here the memory effects in the density-matrix coherence for a driven two-level atom in the stationary regime. The coherence parameter \ensuremath{\eta} is written in terms of the detuning, the strength of the external driving force, and the temperature as the reservoir. We verify that the memory effects, present in the density matrix, are enhanced by the external driving force. We introduce the Shanon information entropy and we verify that, while in the Markovian approximation the entropy decreases monotonically as function of the detuning, in the non-Markovian approach it is characterized by a minimum, which enables us to differentiate between the two processes.