Semiclassical Approach to Many-Particle Interference: Quantum Signatures of Classical Chaos and Criticality

多粒子干涉的概念近年来在多体物理学的多体局域化、光子量子网络和光晶格中的冷原子等领域得到了越来越多的关注。在更近的意义上，多粒子干涉是指Fock空间中描述平均场之外的相关性的动力学干涉效应，并且自然地在Feynman传播子中实现。在半经典区域，它可以用福克空间中带有干扰振幅的“经典”路径的和来表示。作为半经典概念的补充，单粒子动力学在短波长极限下有效，这样的状态在N粒子系统中达到大粒子数N，通常被称为热力学极限。在这个项目中，我们设想进一步发展这样一种多体半经典方法，其中量子传播和多体态密度可以被认为是基于不同经典（平均场）路径之间的干涉。这提供了多体量子行为和相应经典系统的性质之间的联系，特别是打开了一个有趣的问题，即经典动力学中的不稳定性（接近临界点或混沌）如何影响量子多体特征。我们的项目的过度的目标是描述签名的经典奇异性的离散量子光谱的多体系统，重点是激发态量子相变和快速混乱的量子相关性接近临界或由于混沌。更具体地说，（i）我们将为各种多体模型的不同类型的经典奇点的能隙和临界相互作用强度的标度提供分析估计，并将我们的结果与数值模拟进行基准测试。(ii)我们将探索与多体系统中所谓的量子蝴蝶效应相关的李雅普诺夫指数的量子作用，目前被理解为由于经典不稳定性导致快速混乱和加速多体量子关联的产生。这种相关性是通过所谓的时间顺序外的修正器（OTOC）来测量的。在这里，我们的目标是找到一个严格的多体半经典推导的指数增长率的OTOC，解释所观察到的饱和OTOC的多体量子干涉，并获得洞察到Maldacena的普遍约束的指数增长的OTOC从量子动力学的角度来看。

The notion of many-particle interference has recently gained increasing attention in many-body physics in fields such as many-body localization, photonic quantum networks and cold atoms in optical lattices. In a closer sense many-particle interference refers to dynamical interference effects in Fock space describing correlations beyond mean field and is naturally implemented in the Feynman propagator. In a semiclassical regime it can be represented by a sum over "classical" paths in Fock space carrying interfering amplitudes. Complementary to semiclassical concepts form single-particle dynamics, valid in the short-wavelength limit, such a regime is reached in an N-particle system for large particle number N, usually referred to as thermodynamic limit. In this project we envisage to further develop such a many-body semiclassical approach, where quantum propagation and the many-body density of states can be regarded as being based on interference between different classical (mean-field) paths. This provides a link between many-body quantum behavior and properties of the corresponding classical system and, in particular, opens up the interesting question how instabilities in the classical dynamics (close to a critical point or if chaotic) affect quantum many-body features. The overreaching goal of our project is to describe signatures of classical singularities in the discrete quantum spectra of many-body systems with emphasis on excited state quantum phase transitions and fast scrambling of quantum correlations close to criticality or due to chaoticity. More specifically, (i) we will provide analytical estimates for the scaling of energy gaps and the critical interaction strength for different types of classical singularities for various many-body models and will benchmark our results against numerical simulations. (ii) We will explore the quantum role of the Lyapunov exponent, associated with the so-called quantum butterfly effect in many-body systems, currently understood as leading to fast scrambling and speeding up the generation of many-body quantum correlations due to classical instability. Such correlations are measured through so-called out-of-time-order correlators (OTOCs). Here our objectives are to find a rigorous many-body semiclassical derivation of the exponential growth rates of OTOCs, to explain the observed saturation of OTOCs in terms of many-body quantum interference, and to obtain insight into Maldacena’s universal bound on the exponential growth of OTOCs from a quantum dynamics perspective.