A non-Hermitian PT-symmetric version of the kicked top is introduced to study the interplay of quantum chaos with balanced loss and gain. The classical dynamics arising from the quantum dynamics of the angular momentum expectation values are derived. The presence of PT-symmetry can lead to stable mixed regular chaotic behaviour without sinks or sources for subcritical values of the gain-loss parameter. This is an example of what is known in classical dynamical systems as reversible dynamical systems. For large values of the kicking strength a strange attractor is observed that also persists if PT-symmetry is broken. The intensity dynamics of the classical map is found to provide the main structure for the Husimi distributions of the subspaces of the quantum system belonging to certain ranges of the imaginary parts of the quasienergies. Classical structures are identified in the quantum dynamics. Finally, the statistics of the eigenvalues of the quantum system are analysed and it is shown that if most of the eigenvalues are complex (which is the case already for fairly small non-Hermiticity parameters) the nearest-neighbour distances of the (unfolded) quasienergies follow a two-dimensional Posisson distribution when the classical dynamics is regular. In the chaotic regime, they are in line with recently identified universal complex level spacing distributions for non-Hermitian systems, with transpose symmetry $A^T= A$. It is demonstrated how breaking this symmetry (by introducing an extra term in the Hamiltonian) recovers the more familiar universality class for non-Hermitian systems given by the complex Ginibre ensemble. Both universality classes display cubic level repulsion. The PT-symmetry of the system does not seem to influence the complex level spacings. Similar behaviour is also observed for the spectrum of a PT-symmetric extension of the triadic Baker map.

中文翻译：

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非 Hermitian PT 对称踢顶

引入了踢顶的非厄米 PT 对称版本来研究量子混沌与平衡损失和增益的相互作用。推导出由角动量期望值的量子动力学引起的经典动力学。PT 对称性的存在可以导致稳定的混合规则混沌行为，而没有增益损失参数的亚临界值的汇或源。这是经典动力系统中称为可逆动力系统的一个例子。对于较大的踢动强度值，观察到一个奇怪的吸引子，如果 PT 对称性被破坏，它也会持续存在。经典映射的强度动力学被发现为属于准能量虚部某些范围的量子系统子空间的 Husimi 分布提供了主要结构。经典结构在量子动力学中被识别。最后，对量子系统的特征值的统计进行了分析，结果表明，如果大多数特征值是复数（对于相当小的非厄米参数已经是这种情况），则（展开的）准能量的最近邻距离当经典动力学是规则的时，遵循二维泊西松分布。在混沌状态中，它们与最近确定的非厄米系统的通用复杂水平间距分布一致，具有转置对称性 $A^T=A$。证明了打破这种对称性（通过在哈密顿量中引入一个额外项）如何恢复由复 Ginibre 系综给出的非厄米系统的更熟悉的普遍性类。两个普遍性类别都显示立方级排斥。系统的 PT 对称性似乎不会影响复杂的水平间距。对于三元贝克图的 PT 对称扩展的频谱，也观察到类似的行为。