A number of experimental platforms relevant for quantum simulations, ranging from arrays of superconducting circuits hosting correlated states of light to ultracold atoms in optical lattices in the presence of controlled dissipative processes. Their theoretical understanding is hampered by the exponential scaling of their Hilbert space and by their intrinsic nonequilibrium nature, limiting the applicability of many traditional approaches. In this work, we extend the nonequilibrium bosonic dynamical mean-field theory (DMFT) to Markovian open quantum systems. Within DMFT, a Lindblad master equation describing a lattice of dissipative bosonic particles is mapped onto an impurity problem describing a single site embedded in its Markovian environment and coupled to a self-consistent field and to a non-Markovian bath, where the latter accounts for fluctuations beyond Gutzwiller mean-field theory due to the finite lattice connectivity. We develop a nonperturbative approach to solve this bosonic impurity problem, which dresses the impurity, featuring Markovian dissipative channels, with the non-Markovian bath, in a self-consistent scheme based on a resummation of noncrossing diagrams. As a first application of our approach, we address the steady state of a driven-dissipative Bose-Hubbard model with two-body losses and incoherent pump. We show that DMFT captures hopping-induced dissipative processes, completely missed in Gutzwiller mean-field theory, which crucially determine the properties of the normal phase, including the redistribution of steady-state populations, the suppression of local gain, and the emergence of a stationary quantum-Zeno regime. We argue that these processes compete with coherent hopping to determine the phase transition toward a nonequilibrium superfluid, leading to a strong renormalization of the phase boundary at finite connectivity. We show that this transition occurs as a finite-frequency instability, leading to an oscillating-in-time order parameter, that we connect with a quantum many-body synchronization transition of an array of quantum van der Pol oscillators.

中文翻译：

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马尔可夫开放量子多体系统的动态平均场理论

许多与量子模拟相关的实验平台，从承载相关光态的超导电路阵列到存在受控耗散过程的光晶格中的超冷原子。他们的理论理解受到希尔伯特空间的指数缩放及其内在非平衡性质的阻碍，从而限制了许多传统方法的适用性。在这项工作中，我们将非平衡玻色子动力学平均场理论 (DMFT) 扩展到马尔可夫开放量子系统。在 DMFT 中，描述耗散玻色子晶格的 Lindblad 主方程映射到描述嵌入其马尔可夫环境中的单个位点并耦合到自洽场和非马尔可夫浴的杂质问题上，由于有限的晶格连通性，后者解释了超出 Gutzwiller 平均场理论的波动。我们开发了一种非微扰方法来解决这个玻色子杂质问题，该方法以基于非交叉图的恢复的自洽方案来装饰具有马尔可夫耗散通道和非马尔可夫浴的杂质。作为我们方法的第一个应用，我们解决了具有双体损失和非相干泵的驱动耗散 Bose-Hubbard 模型的稳态。我们展示了 DMFT 捕获了跳跃引起的耗散过程，这在 Gutzwiller 平均场理论中完全被忽略，它关键地决定了正常相的特性，包括稳态种群的重新分布、局部增益的抑制以及稳态量子芝诺机制。我们认为这些过程与相干跳跃竞争以确定向非平衡超流体的相变，导致有限连接处相边界的强烈重整化。我们表明，这种转变是作为有限频率不稳定性发生的，导致时间顺序参数振荡，我们将其与一系列量子范德波尔振荡器的量子多体同步转变联系起来。