Ergodic Quantum Processes: Localization, Diffusion, and Steady States

本项目的重点是研究时空依赖紊乱的量子系统。虽然基本的科学问题“无序的影响是什么？”总是相关的，但由于最近出现了嘈杂的中尺度量子设备，在量子系统的背景下，这一问题在今天尤为紧迫。该项目的目的是加深我们对具有不可忽略的环境相互作用的“开放”量子系统量子演化关键模型中的无序效应的理论和数学理解。本项目研究的方程起源于无序材料理论和量子信息科学，但由于波动和无序的基本性质而引起普遍兴趣。理解这些方程的解的进展将提高对理论物理和应用数学模型的基本理解。该项目将为本科生和研究生提供广泛的机会。本项目是一个研究量子系统模型中无序效应的项目，主要研究三个方面：1)遍历量子过程的演化。基于PI和他的合作者最近发现的开放量子系统背景下的一般遍历定理，这项研究的目标是将概率论和遍历理论的工具应用于与量子信息科学相关的问题。2)开放、噪声、广泛系统中扩散的定量分析。本研究的主要目的是发展定量方法来估计受热噪声影响的量子粒子的扩散率。3)许多身体系统和相关随机性系统的安德森定位研究。众所周知，由于强静态失序引起的干扰效应可以捕获波函数，这种现象称为安德森局域化。然而，在许多身体和开放量子系统中，定位的确切性质尚不清楚。本研究的目的是证明多体系统和具有强相关随机性的系统中存在安德森局域化，并阐明其本质。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The focus of this project is the study of quantum systems with time and space dependent disorder. While the basic scientific question "What are the effects of disorder?" is always relevant, it is particularly urgent today in the context of quantum systems due to the recent advent of noisy, intermediate scale quantum devices. The aim of this project is to deepen our theoretical and mathematical understanding of disorder effects in key models of the quantum evolution of "open" quantum systems, with non-negligible environment interactions. The equations studied in this project arise in the theory of disordered materials and quantum information science but are of general interest because of the fundamental nature of both wave motion and disorder. Progress in understanding the solutions to these equations will improve basic understanding of models of theoretical physics and applied mathematics. The project will entail extensive opportunities for undergraduate and graduate students.The project is a program of research on the effects of disorder in models of quantum system models, and will focus on three main investigations: 1) Evolution of ergodic quantum processes. Building on the recent discovery, by the PI and his collaborator, of a general ergodic theorem in the context of open quantum systems, this investigation will aim to bring the tools of probability and ergodic theory to bear on questions relevant to quantum information science. 2) Quantitative analysis of diffusion in open and noisy, extensive systems. The main goal of this investigation is to develop quantitative methods to estimate the diffusivity for a quantum particle subject to thermal noise. 3) Study of Anderson localization in many body systems and systems with correlated randomness. It is known that interference effects due to strong static disorder can trap the wave function, a phenomenon called Anderson localization. However, the exact nature of localization in many body and open quantum systems is poorly understood. The goal of this investigation is to prove the existence of, and clarify the nature of, Anderson localization in many-body systems and systems with strongly correlated randomness.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.