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

近年来，在多体物理中，多粒子干涉的概念在多体局域化、光子量子网络和光晶格中的冷原子等领域受到越来越多的关注。在更紧密的意义上，多粒子干涉是指在Fock空间中描述超越平均场的相关性的动态干涉效应，并且自然地在费曼传播子中实现。在半经典状态下，它可以表示为在Fock空间中携带干扰幅度的“经典”路径的和。补充半经典概念形成单粒子动力学，有效于短波长极限，这样的制度是在一个大粒子数N的N粒子系统，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属性之间的联系，特别是，开辟了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的有趣问题。我们项目的目标是描述多体系统离散量子光谱中的经典奇点特征，重点是激发态量子相变和接近临界或由于混沌性而导致的量子相关的快速置乱。更具体地说，(i)我们将对各种多体模型中不同类型的经典奇点的能量间隙缩放和临界相互作用强度提供分析估计，并将我们的结果与数值模拟相比较。（ii）我们将探索李雅普诺夫指数的量子作用，它与多体系统中所谓的量子蝴蝶效应有关，目前被理解为由于经典不稳定性导致快速乱置和加速多体量子相关的产生。这种相关性是通过所谓的超时间顺序相关器（OTOCs）来测量的。本文的目标是找到OTOCs指数增长率的严格多体半经典推导，从多体量子干涉的角度解释观察到的OTOCs饱和，并从量子动力学的角度深入了解Maldacena关于OTOCs指数增长的通用界。

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.