We analyze the origin and properties of the chaotic dynamics of two atomic ensembles in a driven-dissipative experimental setup, where they are collectively damped by a bad cavity mode and incoherently pumped by a Raman laser. Starting from the mean-field equations, we explain the emergence of chaos by way of quasiperiodicity -- presence of two or more incommensurate frequencies. This is known as the Ruelle-Takens-Newhouse route to chaos. The equations of motion have a $\mathbb{Z}_{2}$-symmetry with respect to the interchange of the two ensembles. However, some of the attractors of these equations spontaneously break this symmetry. To understand the emergence and subsequent properties of various attractors, we concurrently study the mean-field trajectories, Poincare sections, maximum and conditional Lyapunov exponents, and power spectra. Using Floquet analysis, we show that quasiperiodicity is born out of non $\mathbb{Z}_{2}$-symmetric oscillations via a supercritical Neimark-Sacker bifurcation. Changing the detuning between the level spacings in the two ensembles and the repump rate results in the synchronization of the two chaotic ensembles. In this regime, the chaotic intensity fluctuations of the light radiated by the two ensembles are identical. Identifying the synchronization manifold, we understand the origin of synchronized chaos as a tangent bifurcation intermittency of the $\mathbb{Z}_{2}$-symmetric oscillations. At its birth, synchronized chaos is unstable. The interaction of this attractor with other attractors causes on-off intermittency until the synchronization manifold becomes sufficiently attractive. We also show coexistence of different phases in small pockets near the boundaries.

中文翻译：

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谐振腔中原子系综的驱动耗散动力学：混沌和混沌同步的准周期路径

我们在驱动耗散实验装置中分析了两个原子系综的混沌动力学的起源和性质，其中它们被坏腔模式共同阻尼并由拉曼激光器非相干泵浦。从平均场方程开始，我们通过准周期性来解释混沌的出现——存在两个或多个不相称的频率。这被称为通往混乱的 Ruelle-Takens-Newhouse 路线。运动方程相对于两个系综的交换具有 $\mathbb{Z}_{2}$ 对称性。然而，这些方程的一些吸引子自发地打破了这种对称性。为了理解各种吸引子的出现和后续特性，我们同时研究了平均场轨迹、庞加莱截面、最大和条件李雅普诺夫指数以及功率谱。使用 Floquet 分析，我们表明准周期性是通过超临界 Neimark-Sacker 分岔产生于非 $\mathbb{Z}_{2}$ 对称振荡的。改变两个集合中的电平间距和重泵率之间的失谐会导致两个混沌集合的同步。在这种情况下，两个系综发出的光的混沌强度波动是相同的。识别同步流形，我们将同步混沌的起源理解为 $\mathbb{Z}_{2}$ 对称振荡的切线分岔间歇性。在诞生之初，同步混沌是不稳定的。这个吸引子与其他吸引子的相互作用导致开关间歇，直到同步流形变得足够有吸引力。