Large deviations and dynamical phase transitions in open quantum systems: from mathematical theory to physical applications

我们目前正在进入一个新的技术时代，在这个时代，量子力学不仅被用来预测物理行为，而且越来越多地被用来在量子通信和计算以及量子计量学等应用中开发量子资源。在过去的几十年里，量子系统的实验实现经历了一场革命。如今，超冷原子气体经常被创造出来，并用于研究复杂的多体现象，如量子相变，从而揭示凝聚态物理中的公开问题。真正的量子系统是“开放的”，因为它们与环境(无论是热浴还是更大量子系统的其他组件)相互作用。这将导致不可逆转的一致性损失和能量耗散。对这类问题最简单的分析是基于马尔可夫近似，即环境没有记忆，与系统的相互作用很弱。马尔可夫公式已经成功地应用于许多物理问题，如连续时间测量的处理和“量子跳跃”的存在。这导致了描述量子系统的随机动力学的“量子轨迹”理论的发展，该理论以间接观测的随机结果为条件。当前量子工程的进展推动了理论和实验研究朝着量子动力学和经典控制工程的新综合方向发展，称为量子控制理论。马尔可夫设置非常适合应用反馈控制技术，其中经典或量子输出信号被实时处理，并用于根据控制策略对系统进行反馈。量子轨道形式论与经典的非平衡统计力学有着天然的联系。近年来，我们对非平衡系统的认识有了很大的进步，许多进展源于对玻璃等复杂软凝聚态系统的研究。统计力学的数学语言是大偏差(LD)理论，该理论传统上用于研究许多物体系统的平衡相和相变。LD公式现在被用来研究非平衡系统的动力学相，它用平衡统计力学处理组态的系综的方式来处理轨道系综。利用这个类比可以揭示经典非平衡系统的重要性质。这套新的想法和技术还没有在量子领域得到充分的利用，因此对量子非平衡动力学的了解更少。在这个提议中，我们的目标是建立数学基础并确定基本原理，最终使我们能够构建一个框架，详细而彻底地描述脱离平衡的量子物质。这个提议的中心目标是发展开放量子系统的LD理论，并用它来探索亚稳态和动态相变现象。我们的目标是通过应用和推广非平衡统计力学、量子马尔可夫过程理论和随机薛定谔方程中最新的方法，弥合经典非平衡系统和量子非平衡系统之间存在的理解差距。在发展数学理论的同时，我们将对物理上相关的模型进行详细分析，例如驱动多体系统和微脉塞。通过将马尔可夫过程的统计和概率方法与量子反馈控制理论相结合，我们将研究带有经典控制器或量子控制器的系统的大偏差和中偏差行为，并将该理论应用于系统辨识和量子计量学等主题。

We are currently entering a new technological era where quantum mechanics is used not only to predict physical behavior, but increasingly to exploit quantum resources in applications such as quantum communication and computation and quantum metrology. The last couple of decades have witnessed a revolution in the experimental realisation of quantum systems. Ultracold atomic gases are nowadays routinely created and used for the study of complex many body phenomena such as quantum phase transitions shedding light on open problems in condensed-matter physics.Real quantum systems are "open" in the sense that they interact with their environment (be it a thermal bath or other components of a larger quantum system). This leads to an irreversible loss of coherence and to energy dissipation. The simplest analysis applied to this kind of problems is based on the Markov approximation in which the environment possesses no memory and interacts weakly with the system. The Markov formalism has been successfully applied to many physical problems, such as the treatment of continuous-time measurements and the existence of "quantum jumps". This lead to the development of the "quantum trajectories" theory describing the stochastic dynamics of quantum systems, conditioned on the random outcomes of indirect observations. Current advances in quantum engineering drive theoretical and experimental research towards a new synthesis of quantum dynamics and classical control engineering, called quantum control theory. The Markov set-up is perfectly suited for applying feedback control techniques, where a classical or quantum output signal is processed in real time and used to act back on the system according to a control strategy. The quantum trajectories formalism is naturally connected to classical non-equilibrium statistical mechanics. Recently there has been much progress in our understanding of non-equilibrium systems, with many advances originating from the study of complex soft condensed-matter systems such as glasses. The mathematical language of statistical mechanics is large deviations (LD) theory, which was traditionally applied to the study of equilibrium phases and phase changes in many body systems. The LD formalism is now used to investigate the dynamical phases in non-equilibrium systems by treating ensembles of trajectories in the same way that equilibrium statistical mechanics treats ensembles of configurations. Important properties of classical non-equilibrium systems can be uncovered by exploiting this analogy. This new set of ideas and techniques has not yet been fully exploited in the quantum realm, and therefore much less is understood of quantum non-equilibrium dynamics. In this proposal we aim at building the mathematical foundation and identifying fundamental principles that will eventually allow us to construct a framework for a detailed and thorough description of quantum matter out of equilibrium. The central goal of this proposal is to develop the LD theory of open quantum systems and use it to explore the phenomena of metastability and dynamical phase transitions. We aim to bridge the gap in understanding that exists between classical and quantum non-equilibrium systems, by applying and extending the most novel methods developed in non-equilibrium statistical mechanics, the theory of quantum Markov processes and stochastic Schrödinger equations. In parallel to developing the mathematical theory we will perform a detailed analysis of physically relevant models such as driven many-body systems and the micromaser. By combining statistical and probabilistic methods of Markov processes with quantum feedback control theory, we will study the large and moderate deviations behaviour of systems coupled with classical or quantum controllers, and apply the theory to topics such as system identification and quantum metrology.