Bose-Einstein Condensation Beyond Mean Field: A Partial Differential Equation Approach to Quantum Fluctuations

本研究用应用数学的分析方法解决了极冷原子物理中一系列具有挑战性的问题。在玻色-爱因斯坦凝聚中，某种类型的粒子（玻色子）在非常低的温度下占据一个称为凝聚的宏观量子态。这种凝聚态是迄今为止已知的最相干的物质状态之一，允许在实验室环境中精确控制原子系统。玻色-爱因斯坦凝聚已经在实验上在通过磁性和光学手段捕获的原子气体中观察到，并且已经引发了对量子信息和计算中具有根本重要性的有吸引力的应用的积极实验研究，例如，量子原子计算机的设计和建造，以及精密测量。该项目的研究人员将开发模型并进行分析，从而更好地了解玻色-爱因斯坦凝聚中的原子效应。这项研究的影响将被应用数学界以及一些应用的社区所感受到，包括原子和凝聚态物理学家，以及计算机科学家和量子信息专业的工程师。研究者将培养一名研究生，他们将获得跨学科的研究教育和现代科学前沿问题的观点。这项研究的目标是开发一个层次的模型和分析工具，以便将单个原子的运动和相互作用与玻色-爱因斯坦凝聚中超冷原子气体的宏观性质联系起来。在这个项目中调用的方法挑出对激发的影响，这导致多体波函数的系统偏离通常的平均场张量积的单粒子状态。研究者的方法包括：（i）多体算子的微扰理论，它将微观哈密顿量与低维偏微分方程系统联系起来;（ii）将导出的方程系统与实验相关的微观结构进行均匀化;（iii）所得到的偏微分方程的全局存在性和唯一性。这项研究工作将阐明通过多体波函数控制凝聚的基本机制，从而改善过去仅基于平均场理论的预测。在这个方向上获得的结果可以指导与复杂凝聚态系统相关的模拟和实验，以及量子信息和精密计量学新设备的设计。

This research addresses a series of challenging questions for the physics of very cold atoms by analytical methods of applied mathematics. In Bose-Einstein condensation, a certain type of particles (bosons) occupy a single macroscopic quantum state called condensate at very low temperatures. This condensate is one of the most coherent states of matter known to date, allowing for a precise control of atomic systems in laboratory settings. The Bose-Einstein condensation has been observed experimentally in atomic gases that are trapped through magnetic and optical means, and has sparked active experimental research into attractive applications of fundamental importance in quantum information and computation, e.g., the design and construction of quantum atomic computers, as well as in precision measurements. The investigator of this project will develop models and carry out analysis that will lead to a better and more fundamental understanding of atomic effects in Bose-Einstein condensation. The impact of this research will be felt by the applied mathematics community as well as by communities of some of the applications, including atomic and condensed-matter physicists, and computer scientists and engineers with specialty in quantum information. The investigator will train one graduate student who will gain interdisciplinary research education and perspective in cutting-edge problems of modern science. The goal of this research is to develop a hierarchy of models and analytical tools in order to link the motion and interactions of individual atoms to the macroscopic properties of ultra-cold atomic gases in Bose-Einstein condensation. The methods invoked in this project single out the effect of pair excitations, which cause the many-body wave function of such systems to deviate from the usual mean-field tensor product of one-particle states. The methods of the investigator include: (i) perturbation theory for many-body operators, which links the microscopic Hamiltonian to systems of low-dimensional partial differential equations; (ii) homogenization of the derived systems of equations for settings with microstructures of experimental relevance; and (iii) the global existence and uniqueness of the resulting partial differential equations. This research effort will elucidate fundamental mechanisms of controlling the condensate through the many-body wave function, hence improving past predictions based solely on mean-field theory. Results obtained in this direction can guide simulations of and experiments related to complex condensed-matter systems, and the design of new devices for quantum information and precision metrology.