Classical and quantum non-Markovianity

在复杂系统中，消除不相关的自由度往往会导致一个具有记忆效应的非马尔可夫过程，这种非马尔可夫过程是由一组减少的相关变量组成的开放系统。近年来，关于非马尔可夫过程的定义、检测和控制的理论和实验研究在经典和量子非平衡系统中引起了极大的兴趣，并在各个领域得到了广泛的应用。对于经典系统，马尔可夫动力学是由底层随机过程的联合概率分布层次的标准马尔可夫条件唯一定义的。虽然这个定义不能转移到量子状态，但可以证明，量子马尔可夫性的一致表征可以基于开放系统（相关自由度）与其环境（不相关自由度）之间信息流的性质。作为“降低非平衡系统复杂性”研究单元的一部分，本项目涉及远离平衡的多体系统中记忆效应和非马尔可夫行为的基本性质。基于一个复杂动力系统的典型例子，Fermi-Pasta-Ulam模型，我们将构建适合于表征和量化系统与环境之间的信息流以及随之而来的记忆效应的可观测量。特别是，量子和经典记忆效应之间的关系以及量子和经典相关性对动力学的影响在我们的研究中是至关重要的。在这个项目中，我们将采用开放系统和非平衡统计物理理论的分析方法，以及使用Wigner表示和多组态时相关Hartree方法的数值技术。

The elimination of irrelevant degrees of freedom in complex systems often leads to a non-Markovian process featuring memory effects in the dynamics of a reduced set of relevant variables which form an open system. In recent years theoretical and experimental studies on the definition, detection and control of non-Markovian processes have attracted a huge amount of interest both in classical and quantum nonequilibrium systems, and havefound a multitude of applications in diverse fields. For classical systems Markovian dynamics is uniquely defined by the standard Markov condition for the hierarchy of joint probability distributions of the underlying stochastic process. While this definition cannot be transferred to the quantum regime, it can be shown that a consistent characterization of quantum Markovianity can be based on the properties of the flow of information between the open system (relevant degrees of freedom) and its environment (irrelevant degrees of freedom). As part of the Research Unit ''Reducing complexity of nonequilibrium system'' the present project is concerned with fundamental properties of memory effects and non-Markovian behavior in many-body systems far from equilibrium. Based on a paradigmatic example of a complex dynamical system, the Fermi-Pasta-Ulam model, we will construct observable quantities which are suitable for the characterization and quantification of the information flow between system and environment and the ensuing memory effects. In particular, the relation between quantum and classical memory effects as well as the impact of quantum and classical correlations on the dynamics are of central importance in our investigations. In this project we will employ analytical methods from the theory of open system and nonequilibrium statistical physics, as well as numerical techniques using the Wigner representation and the multiconfiguration time-dependent Hartree method.