Work extraction from the Gibbs ensemble by a cyclic operation is impossible, as represented by the second law of thermodynamics. On the other hand, the eigenstate thermalization hypothesis (ETH) states that just a single energy eigenstate can describe a thermal equilibrium state. Here we attempt to unify these two perspectives and investigate the second law at the level of individual energy eigenstates, by examining the possibility of extracting work from a single energy eigenstate. Specifically, we performed numerical exact diagonalization of a quench protocol of local Hamiltonians and evaluated the number of work-extractable energy eigenstates. We found that it becomes exactly zero in a finite system size, implying that a positive amount of work cannot be extracted from any energy eigenstate, if one or both of the pre- and the post-quench Hamiltonians are non-integrable. We argue that the mechanism behind this numerical observation is based on the ETH for a non-local observable. Our result implies that quantum chaos, characterized by non-integrability, leads to a stronger version of the second law than the conventional formulation based on the statistical ensembles.
Eigenstate Fluctuation Theorem in the Short and Long Time Regimes
正则系综在平衡态和非平衡态的统计力学中起着至关重要的作用。例如，涨落定理的标准推导依赖于热浴的初始状态是正则系综的假设。另一方面，统计力学基础的最新进展揭示了热平衡态不一定由正则系综描述，而是可以是量子纯态，甚至是单能量本征态，如本征态热化假设（ETH）所表述的。然后，一个问题是这两个图片，正则系综和一个单一的能量本征态作为一个热平衡状态，是如何兼容的涨落定理。在本文中，我们从理论上和数值计算表明，涨落定理在长时间和短时间的制度，即使当浴的初始状态是一个单一的能量本征态的多体系统。我们在长时间范围内的波动定理的证明是基于ETH，而之前在短时间范围内基于Lieb-Robinson约束和ETH [Phys. Rev. Lett. 119，100601（2017）]。这些时间制度的证明在理论上是独立的和互补的，这意味着在整个时域的波动定理。我们还进行了系统的数值模拟的硬核玻色子的精确对角化和验证的涨落定理在这两个时间制度，通过专注于有限尺寸标度。我们的结果有助于理解涨落定理从量子多体系统的幺正动力学中出现的机制，并且可以通过实验来检验，例如，超冷原子。
Semiclassical scaling of eigenstate thermalization in single-particle chaotic systems
University of Science and Technology of China（中国科学技术大学）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Wilczek Quantum Center, Shanghai Institute for Advanced Studies（复旦张肇韦量子中心，上海高等研究院）；CAS Key Laboratory of Microscale Magnetic Resonance（中国科学院微观磁学共振重点实验室）
Semiclassical analytical solutions of the eigenstate thermalization hypothesis in a quantum billiard
University of Science and Technology of China（中国科学技术大学）；CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and Technology of China（中国科学院微观磁共振重点实验室，中国科学技术大学）
