Consistent Mori-Projector Theory in Two Dimensions

量子多体系统传统上是在热平衡中研究的，其中统计力学的方法在描述它们方面被证明是非常成功的。近年来，越来越多的实验在不同的技术，包括超导电路，半导体和原子里德伯介质，但是引起了一个新的，很大程度上是未开发的一类量子多体系统与相互作用的粒子在驱动和耗散，非平衡制度。这些都提出了一个新的科学挑战，因此需要开发新的技术，能够有效和准确地模拟它们。我最近开始开发一种新的技术，称为一致的森投影仪（c-MoP），它推导出耦合，但非线性的运动方程的减少密度矩阵的子系统的多体系统，从而导致一个有效的描述当地的数量。从我有希望的初步结果开始，我将在这个项目中推导出一个强大的理论，用于有效描述基于c-MoP概念的非平衡场景中的二维系统。我希望这一成就能够为二维量子多体系统的建模开辟可能性，迄今为止，这些系统一直无法进行简单但准确的描述。作为利用这些新的可能性的第一个应用程序，我将采用先进的技术来计算定态和它们的相图驱动耗散玻色-哈伯德和Jaynes-Cummings-哈伯德模型，其中起着核心作用的驱动耗散量子多体系统。该项目的结果将为我未来的研究计划提供基础，并将为未来利用量子效应的技术奠定基础。

Quantum many-body systems are traditionally studied in thermal equilibrium where the methodology of statistical mechanics has proven very successful in describing them. In recent years, an increasing number of experiments in diverse technologies including superconducting circuits, semiconductors and atomic Rydberg media are however giving rise to a new, largely unexplored class of quantum many-body systems with interacting particles in driven and dissipative, non-equilibrium regimes. These pose a new scientific challenge and thus generate a need to develop novel techniques that are capable of modelling them efficiently and accurately. I recently started to develop a new technique called consistent Mori Projectors (c-MoP) which derives coupled but non-linear equations of motion for the reduced density matrices of the subsystems of a many-body system and thus leads to an efficient description of local quantities. Starting from my promising initial results, I will in this project derive a powerful theory for the efficient description of two-dimensional systems in non-equilibrium scenarios that is based on the c-MoP concept. I expect this achievement to open up possibilities for modelling two-dimensional quantum many-body systems that hitherto defy a tractable but accurate description. As first applications that exploit these new possibilities I will employ the developed techniques for calculating stationary states and their phase diagram for driven-dissipative Bose-Hubbard and Jaynes-Cummings-Hubbard models, which play a central role among driven-dissipative quantum many-body systems. The results of the project will provide a basis for my future research plans and and will lay foundations for future technologies that exploit quantum effects.