Despite its long history, a canonical formulation of quantum ergodicity that applies to general classes of quantum dynamics, including driven systems, has not been fully established. Here we introduce and study a notion of quantum ergodicity for closed systems with time-dependent Hamiltonians, defined as statistical randomness exhibited in their longtime dynamics. Concretely, we consider the temporal ensemble of quantum states (time-evolution operators) generated by the evolution, and investigate the conditions necessary for them to be statistically indistinguishable from uniformly random states (operators) in the Hilbert space (space of unitaries). We find that the number of driving frequencies underlying the Hamiltonian needs to be sufficiently large for this to occur. Conversely, we show that statistical —indistinguishability up to some large but finite moment—can already be achieved by a quantum system driven with a single frequency, i.e., a Floquet system, as long as the driving period is sufficiently long. Our work relates the complexity of a time-dependent Hamiltonian and that of the resulting quantum dynamics, and offers a fresh perspective to the established topics of quantum ergodicity and chaos from the lens of quantum information. Published by the American Physical Society 2024

中文翻译：

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驱动量子系统中的希尔伯特空间遍历性：障碍和设计


尽管量子遍历率的规范表述历史悠久，但适用于一般量子动力学类别（包括驱动系统）的规范公式尚未完全建立。在这里，我们介绍并研究了具有瞬态哈密顿量的封闭系统的量子遍历性概念，该概念定义为在其长期动力学中表现出的统计随机性。具体来说，我们考虑了进化产生的量子态（时间演化算子）的时间集合，并研究了它们在统计上与希尔伯特空间（幺正空间）中的均匀随机态（算子）没有区别的必要条件。我们发现哈密顿量背后的驱动频率数量需要足够大才能发生这种情况。相反，我们表明，只要驱动周期足够长，统计——在某个大但有限的时刻内无法区分——已经可以通过以单一频率驱动的量子系统（即 Floquet 系统）来实现。我们的工作将瞬态哈密顿量的复杂性与由此产生的量子动力学的复杂性联系起来，并从量子信息的角度为量子遍历性和混沌的既定主题提供了新的视角。 美国物理学会 2024 年出版