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arXiv:1510.02333 · Physical Review A

Energy backflow and non-Markovian dynamics

Giacomo Guarnieri, Chikako Uchiyama, Bassano Vacchini

Published Jan 22, 2016• 66 citations
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Abstract

We explore the behavior in time of the energy exchange between a system of interest and its environment, together with its relationship to the non-Markovianity of the system dynamics. In order to evaluate the energy exchange we rely on the full counting statistics formalism, which we use to evaluate the first moment of its probability distribution. We focus in particular on the energy backflow from environment to system, to which we associate a suitable condition and quantifier, which enables us to draw a connection with a recently introduced notion of non-Markovianity based on information backflow. This quantifier is then studied in detail in the case of the spin-boson model, described within a second-order time-convolutionless approximation, observing that non-Markovianity allows for the observation of energy backflow. This analysis allows us to identify the parameters region in which energy backflow is higher.

DOI
10.1103/physreva.93.012118

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We explore the behavior in time of the energy exchange between a system of interest and its environment, together with its relationship to the non-Markovianity of the system dynamics. In order to evaluate the energy exchange we rely on the full counting statistics formalism, which we use to evaluate the first moment of its probability distribution. We focus in particular on the energy backflow from environment to system, to which we associate a suitable condition and quantifier, which enables us to draw a connection with a recently introduced notion of non-Markovianity based on information backflow. This quantifier is then studied in detail in the case of the spin-boson model, described within a second-order time-convolutionless approximation, observing that non-Markovianity allows for the observation of energy backflow. This analysis allows us to identify the parameters region in which energy backflow is higher.
By employing the full counting statistics formalism, we characterize the first moment of energy that is exchanged during a generally non-Markovian evolution in nondriven continuous-variable systems. In particular, we focus on the evaluation of the energy flowing back from the environment into the open quantum system. We apply these results to the quantum Brownian motion, where these quantities are calculated both analytically, under the weak-coupling assumption, and numerically also in the strong-coupling regime. Finally, we characterize the non-Markovianity of the reduced dynamics through a recently introduced witness based on the so-called Gaussian interferometric power and we discuss its relationship with the energy backflow measure.
We explore signatures of the non-Markovianity in the time-resolved energy transfer processes for quantum open systems. Focusing on typical systems such as the exact solvable damped Jaynes-Cummings model and the general spin-boson model, we establish quantitative links between the time-resolved energy current and the symmetric logarithmic derivative quantum Fisher information (SLD-QFI) flow, one of measures quantifying the non-Markovianity, within the framework of non-Markovian master equations in time-local forms. From the relationships, we find in the damped Jaynes-Cummings model that the SLD-QFI backflow from the reservoir to the system always correlates with an energy backflow, thus we can directly witness the non-Markovianity from the dynamics of the energy current. In the spin-boson model, the relation is built on the rotating-wave approximation, calibrated against exact numerical results, and proven reliable in the weak coupling regime. We demonstrate that whether the non-Markovianity guarantees the occurrence of an energy backflow depends on the bath spectral function. For the Ohmic and sub-Ohmic cases, we show that no energy backflow occurs and the energy current always flow out of the system even in the non-Markovian regime. While in the super-Ohmic case, we observe that the non-Markovian dynamics can induce an energy backflow.
A Markovian process of a system is defined classically as a process in which the future state of the system is fully determined by only its present state, not by its previous history. There have been several measures of non-Markovianity to quantify the degrees of non-Markovian effect in a process of an open quantum system based on information backflow from the environment to the system. However, the condition for the witness of the system information backflow does not coincide with the classical definition of a Markovian process. Recently, a new measure with a condition that coincides with the classical definition in the relevant limit has been proposed. Here, we focus on the new definition (measure) for quantum non-Markovian processes, and characterize the Markovian condition as a quantum process that has no information backflow through the reduced environment state (IBTRES) and no system-environment correlation effect (SECE). The action of IBTRES produces non-Markovian effects by flowing the information of quantum operations performed by an experimenter at earlier times back to the system through the environment, while the SECE can produce non-Markovian effect without carrying any earlier quantum operation information. We give the necessary and sufficient conditions for no IBTRES and no SECE, respectively, and show that a process is Markovian if and only if it has no IBTRES and no SECE. The quantitative measures and algorithms for calculating non-Markovianity, IBTRES, and solely SECE are explicitly presented.
Non-Markovianity has recently attracted large interest due to significant advances in its characterization and its exploitation for quantum information processing. However, up to now, only non-Markovian regimes featuring environment to system backflow of information (strong non-Markovianity) have been experimentally simulated. In this work, using an all-optical setup we simulate and observe the so-called weak non-Markovian dynamics. Through full process tomography, we experimentally demonstrate that the dynamics of a qubit can be non-Markovian despite an always increasing correlation between the system and its environment which, in our case, denotes no information backflow. We also show the transition from the weak to the strong regime by changing a single parameter in the environmental state, leading us to a better understanding of the fundamental features of non-Markovianity.
Quantum memory effects can be qualitatively understood as a consequence of an environment-to-system backflow of information. Here, we analyze and compare how this concept is interpreted and implemented in different approaches to quantum non-Markovianity. We study a nonoperational approach, defined by the distinguishability between two system states characterized by different initial conditions, and an operational approach, which is defined by the correlation between different outcomes associated to successive measurement processes performed over the system of interest. The differences, limitations, and vantages of each approach are characterized in detail by considering diverse system-environment models and dynamics. As a specific example, we study a non-Markovian depolarizing map induced by the interaction of the system of interest with an environment characterized by incoherent and coherent self-dynamics.
Exchange of information between a quantum system and its surrounding environment plays a fundamental role in the study of the dynamics of open quantum systems. Here we discuss the role of the information exchange in the non-Markovian behavior of dynamical quantum processes following the decoherence approach, where we consider a quantum system that is initially correlated with its measurement apparatus, which in turn interacts with the environment. We introduce a way of looking at the information exchange between the system and environment using the quantum loss, which is shown to be closely related to the measure of non-Markovianity based on the quantum mutual information. We also extend the results of Fanchini et al. [Phys. Rev. Lett. 112, 210402 (2014)] in several directions, providing a more detailed investigation of the use of the accessible information for quantifying the backflow of information from the environment to the system. Moreover, we reveal a clear conceptual relation between the entanglement- and mutual-information-based measures of non-Markovianity in terms of the quantum loss and accessible information. We compare different ways of studying the information flow in two theoretical examples. We also present experimental results on the investigation of the quantum loss and accessible information for a two-level system undergoing a zero temperature amplitude damping process. We use an optical approach that allows full access to the state of the environment.