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arXiv:1108.3763 · Physical Review A

Non-Markovian open quantum systems: Input-output fields, memory, and monitoring

Lajos Diósi

Published Mar 16, 2012• 33 citations
arXivHTMLPDF

Abstract

Principles of monitoring non-Markovian open quantum systems are analyzed. We use a field representation of the environment [C. W. Gardiner and M. J. Collett, Phys. Rev. A 31, 3761 (1985)] for the separation of its memory and detector part, respectively. We claim that the system-plus-memory compound becomes Markovian; the detector part is tractable by standard Markovian monitoring. Because of non-Markovianity, only the mixed state of the system can be predicted; the pure state of the system can be retrodicted. We present the corresponding non-Markovian stochastic Schr\"odinger equation.

DOI
10.1103/physreva.85.034101

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Principles of monitoring non-Markovian open quantum systems are analyzed. We use a field representation of the environment [C. W. Gardiner and M. J. Collett, Phys. Rev. A 31, 3761 (1985)] for the separation of its memory and detector part, respectively. We claim that the system-plus-memory compound becomes Markovian; the detector part is tractable by standard Markovian monitoring. Because of non-Markovianity, only the mixed state of the system can be predicted; the pure state of the system can be retrodicted. We present the corresponding non-Markovian stochastic Schr"odinger equation.
Although a number of measures for quantum non-Markovianity have been proposed recently, it is still an open question whether these measures directly characterize the memory effect of the environment, i.e., the dependence of a quantum state on its past in a time evolution. In this paper, we present a criterion and propose a measure for non-Markovianity with clear physical interpretations of the memory effect. The non-Markovianity is defined by the inequality $T({t}_{2},{t}_{0})\ensuremath{\ne}T({t}_{2},{t}_{1})T({t}_{1},{t}_{0})$ in terms of memoryless dynamical map TTT introduced in this paper. This definition is conceptually distinct from that based on divisibility used by Rivas et al. [Phys. Rev. Lett. 105, 050403 (2010)], whose violation is manifested by noncomplete positivity of the dynamical map. We demonstrate via a typical quantum process that without Markovian approximation, nonzero memory effects (non-Markovianity) always exist even if the non-Markovianity is zero by the other non-Markovianity measures.
Quantum Markov models are employed ubiqui-tously in quantum physics and in quantum information theory due to their relative simplicity and analytical tractability. In particular, these models are known to give accurate approxi-mations for a wide range of quantum optical and meso scopic systems. However, in general, the validity of the Markov approximation entails assumptions regarding properties of the system of interest and its environment, which may not be satisfied or accurate in arbitrary physical systems. Therefore, developing useful modelling tools for general non-Markovian quantum systems for which the Markov approximation is inappropriate or deficient is an undertaking of significant importance. This work considers non-Markovian principal quantum systems that can be embedded in a larger Markovian quantum system with one or more compound baths consisting of an auxiliary quantum system and a quantum white noise field, and derives a set of coupled stochastic and quantum master equations for embedded non-Markovian quantum systems. The case of a purely Hamiltonian coupling between the principal and auxiliary systems as a closed system without coupling to white noises is included as a special case. The results are expected to be of interest for (open-loop and feedback) control of continuous-time non-Markovian systems and studying reduced models for numerical simulation of such systems. They may also shed more light on the general structure of continuous-time non-Markovian quantum systems.
In the present Thesis we study the behavior of multi-time correlation functions and of thermodynamical quantities such as heat in open quantum systems undergoing an evolution generally affected by the presence of memory effects, i.e. non-Markovian. In the last decade, a large part of the scientific community in this field has dedicated its efforts to the understanding, precise definition and quantification of non-Markovianity in the quantum realm and now we have at our disposal several benchmark results and a plethora of different estimators that allow to determine the degree of non-Markovianity of a given dynamics. It comes therefore natural to investigate how other different dynamical quantities relate to such estimators also in order to understand the physical implications of memory effects on the statistics of observable quantities. In the first part of this work, a quantitative test of the violation of the so-called quantum regression theorem in presence of a non-Markovian dynamical regime is investigated. The quantum regression theorem represents a procedure that, whenever valid, allows to reconstruct two-time correlation functions of system's operators from the sole knowledge of the dynamics of mean values. It is worth stressing that two-time correlation functions are necessary in order to fully characterize the statistical properties of a quantum system, since they are able to catch aspects of the dynamics, such as fluorescence spectrum, in general not accessible looking at mean values. Despite their relevance however, obtaining two-time correlation functions often represents a formidable task, since the knowledge of the full "system+environment" dynamics is required, a generally too demanding request in the context of open quantum systems theory. The quantum regression theorem represents in this regard the easiest route to determine two-time correlation functions, this highlighting its importance. In this work we show that, in a pure-dephasing spin-boson model, the quantum regression theorem represents a stronger condition than non-Markovianity, in the sense that any presence of memory effects in the reduced dynamics inevitably results in violations to the former. These results have been published in [G.Guarnieri, A. Smirne, B. Vacchini, Phys. Rev. A 90, 022110 (2014)]. The second part of the Thesis is devoted to the characterization of heat ow at the microscopic level in open quantum systems, both finite and infinite dimensional. In particular we begin by studying the time behavior of its mean value in a non-Markovian dynamical regime, showing that, at variance with what happens in the Born-Markov semigroup limiting case, heat can backflow from the environment to the system. After providing a condition for the occurrence of such phenomenon and a measure for its amount for a given dynamics, the relationship with suitable non-Markovianity estimators is sought in two paradigmatic models, namely the spin-boson and the quantum brownian motion. The results, collected in [ G. Guarnieri, C. Uchiyama, B. Vacchini, Phys. Rev. A 93, 012118 (2016); G. Guarnieri, J. Nokkala, R. Schmidt, S. Maniscalco, B. Vacchini, Phys. Rev. A 94, 062101 (2016)], on the one hand allow for the identification of parameter-regions where the heat backflow is absent or maximum. On the other hand they show that the occurrence of heat backflow represents a stricter condition than non-Markovianity, in the sense that non-Markovianity allows for the observation of heat flowing back from the environment to the system and, vice versa, a Markovian dynamics prevents its occurrence. This Thesis concludes with the formulation of a new family of lower bounds to the mean dissipated heat in an environmental-assisted erasure-protocol scenario where Landauer's principle applies. As originally conceived for classical systems, this principle states that every irreversible erasure of information stored in a system inevitably carries along an amount of heat dissipated into the environment which is expended to perform the action. Within the framework recently put forward in [D. Reeb, M. M. Wolf, New J. Phys. 16, 103011 (2014)], which guarantees the validity of Landauer's principle in an open quantum systems scenario, we provide an asymptotically tight family of lower bounds to the dissipated heat which are also valid in the non-equilibrium setting. This construction is applied to an open system consisting of a three-level V-system, in which one transition is externally pumped by a laser field while the other is coupled through an XX-interaction to an environment consisting of a spin chain. Beside calculating all these quantities, an exact solution for the dynamics of such system is also provided. These results are collected in [G. Guarnieri, S. Campbell, J. Goold, S. Pigeon, M. Paternostro, B. Vacchini, in preparation].
The dynamics of an open quantum system is non-Markovian when memory effect is non-negligible. The quantum memory effect is a very complex issue and presently there is no universally accepted method to quantify it. By examining the memory kernel of the Nakajima-Zwanzig master equation, we study the memory effect in a theoretical model consisting of a spin coupled to an environment of an Ising spin chain subject to a transverse field. We find that it can be dictated by either the time correlation in the environment, or the system-environment correlation, depending on the strength of the transverse field and the state of the environment. We analyze the information loss to the environment and backflow from it, and discover that the information flow is not always a reliable indicator for the nature of the dynamics.
Do diffusive non-Markovian stochastic Schr\"odinger equations (SSEs) for open quantum systems have a physical interpretation? In a recent paper [Phys. Rev. A 66, 012108 (2002)] we investigated this question using the orthodox interpretation of quantum mechanics. We found that the solution of a non-Markovian SSE represents the state the system would be in at that time if a measurement was performed on the environment at that time, and yielded a particular result. However, the linking of solutions at different times to make a trajectory is, we concluded, a fiction. In this paper we investigate this question using the modal (hidden variable) interpretation of quantum mechanics. We find that the noise function z(t)z(t)z(t) appearing in the non-Markovian SSE can be interpreted as a hidden variable for the environment. That is, some chosen property (beable) of the environment has a definite value z(t)z(t)z(t) even in the absence of measurement on the environment. The non-Markovian SSE gives the evolution of the state of the system ``conditioned'' on this environment hidden variable. We present the theory for diffusive non-Markovian SSEs that have as their Markovian limit SSEs corresponding to homodyne and heterodyne detection, as well as one which has no Markovian limit.
The modeling and analysis of the dynamics of complex systems often requires to employ non-Markovian stochastic processes. While there is a clear and well-established mathematical definition for non-Markovianity in the case of classical systems, the extension to the quantum regime recently caused a vivid debate, leading to many different proposals for the characterization and quantification of memory effects in the dynamics of open quantum systems. Here, we derive a mathematical representation for the non-Markovianity measure based on the exchange of information between the open system and its environment, which reveals the locality and universality of non-Markovianity in the quantum state space and substantially simplifies its numerical and experimental determination. We further illustrate the application of this representation by means of an all-optical experiment which allows the measurement of the degree of memory effects in a photonic quantum process with high accuracy.