Understanding spectral statistics and dynamics in strongly-interacting quantum many-body systems

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The dynamics of quantum many-body systems is a fundamental yet notoriously difficult subject due to the nature of strong interactions between macroscopic number of constituents in the systems. Consider setting up a many-body system in a "simple" quantum state, one that does not have much non-local correlation between different subsystems. What are the fates of the system as it evolves in time? Does the system thermalize and exhibit chaotic behaviour, or does it localize and retain information of its initial state?A simple and elegant way of tackling these questions is to investigate the spectral statistics of the quantum many-body systems. A physical system can often be represented by a Hamiltonian - a matrix with a spectrum of energy levels which the system can occupy. The study of spectral statistics asks, what generic features does the correlation among the energy levels in the spectrum capture? Spectral statistics is a fundamental subject in physics due to its role as a robust diagnostic of quantum chaos, and due to universality - generic systems exhibit identical spectral statistics depending only on symmetry classes and dimensionality. In the last five years, spectral statistics has been utilized in multiple frontiers of modern physics, including the demonstration that black holes behave like random matrices in sufficiently late time; a debate concerning the existence of an important dynamical phase called the many-body localization; and the discovery of universal spectral signatures in quantum many-body chaotic systems, as described below.A recent discovery shows that the spectrum of generic quantum many-body chaotic systems has an extended region in which the spectral correlation deviates from known behaviour derived from random matrices. This region grows as the system size increases, and therefore presents a significant gap in our understanding of spectral statistics in the presence of many-body interaction. How does the existence of anomalous spectral correlation affect the scrambling of quantum information? This proposal aims to address such a question, and analytically extract novel signatures of spectral statistics and dynamics in isolated and open quantum many-body systems. Furthermore, despite its importance, spectral statistics in quantum many-body systems has not been experimentally measured, owing to the difficulties of resolving the tight spacing in the spectrum. The second aim of this fellowship is to experimentally measure, in collaboration with experimentalist partners, key signatures of spectral statistics in quantum many-body simulators in the lab for the first time.This project is especially timely, as it deepens and sharpens the understanding of the roles of many-body interaction in the information scrambling and processing in quantum systems, responding to the recent revival in quantum chaos, and to the rapid developments in quantum simulations of quantum many-body systems. Achieving these goals will deliver significant impacts in the constructions of broadly applicable analytical frameworks; in the first experimental measurement of spectral statistics in quantum many-body simulators; and in establishing new connections between communities in condensed matter, quantum information, and high energy physics.