Collaborative Research: New Directions in Atomic Bose-Einstein Condensates

合作研究：原子玻色-爱因斯坦凝聚态的新方向

• 批准号: 1309035
• 负责人: Ricardo Carretero
• 金额: $ 5万
• 依托单位: San Diego State University Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1309035&HistoricalAwards=false
• 关键词: Collaborative; Research; New; Directions; Atomic

The aim of this proposal is to explore a number of novel, emerging directions in the context of atomic Bose-Einstein condensates (BECs). The project will extend the recent collaborative work of the PIs on the theme of vortex dynamics in trapped condensates. There, we will explore both connections with experimental results obtained by a collaborating experimental group at Amherst College and ones with other areas of mathematics and physics. These include most notably the spectral theory of such nonlinear coherent states, structural phase transitions thereof, reductions to particle-based dynamical models with interesting nonlinear bifurcation phenomena, and computational tools to monitor the existence, stability and dynamics of vortex clusters. Also, a multi-component generalization of these themes will be considered in the recently established direction of spin-orbit coupled BECs. These contain a more complex operator structure within the nonlinear problem, with a linear part featuring both a dispersive (Laplacian) and a Dirac-like part and the interplay of these terms and their impact on nonlinear states such as dark or bright solitons and vortices will be studied. The results in this direction will be compared to ongoing experiments by a collaborating experimental group at Washington State University. This research is expected to provide a new generation of both theoretical and computational tools for studying nonlinear coherent structures with a particular view towards the pristine atomic physics setup of Bose-Einstein condensates. Furthermore, the theoretical/mathematical tools developed here will bear broader impacts towards areas such as spectral theory and nonlinear ordinary and partial differential equations, among others. It is also envisioned that our findings will create connections with a number of areas of Physics such as Fluid Dynamics, Nonlinear Optics and Statistical Mechanics (of Phase Transitions). On the other hand, the developed computational tools will explore the interface between numerical bifurcation theory, dynamical system and even Monte-Carlo/Molecular Dynamics techniques and their potential use for the physical system at hand. Importantly, the research will be a genuine synergy in truly Applied Mathematics, involving not only the development of theoretical methods and computational techniques, but also their direct connection with physical experiments. Finally, the project will be critically focused on sustaining a dynamic and multi-disciplinary team with a strong and diverse core of graduate students and hence will be consistently geared towards having a significant set of broader impacts. The relevant research will be disseminated via high quality journal publications both in Mathematics and in Physics and will be presented at Nonlinear Science and Mathematical Physics conferences both in the US and abroad.

该提议的目的是在原子玻色 - 因斯坦冷凝物（BEC）的背景下探索许多新颖的新兴方向。该项目将扩大PI的最新合作工作，以捕获的冷凝水的涡流动力学主题。在那里，我们将探索这两种联系，并与阿默斯特学院的一个合作实验小组以及其他数学和物理学领域相关的实验结果。其中最著名的是这种非线性相干状态的光谱理论，其结构相变的结构相变，具有有趣的非线性分叉现象的基于粒子的动力学模型以及计算工具，以监视涡旋簇的存在，稳定性和动力学。同样，这些主题的多组分概括将在最近建立的自旋轨道耦合BEC的方向上考虑。这些包含非线性问题中更复杂的操作员结构，线性部分具有分散性（Laplacian）和类似狄拉克的部分，以及这些术语的相互作用及其对非线性状态的影响，例如黑暗或明亮的孤子和涡流等非线性状态被研究。华盛顿州立大学的一个合作实验小组正在将这个方向的结果与持续的实验进行比较。 预计这项研究将提供新一代的理论和计算工具，用于研究非线性相干结构，并具有特殊视图，以对原始原子物理学的bose-内施泰因凝结物设置。此外，这里开发的理论/数学工具将对诸如光谱理论和非线性普通和部分微分方程等领域产生更大的影响。还可以预见，我们的发现将与许多物理学区域（例如流体动力学，非线性光学和统计力学）（相变）建立联系。另一方面，开发的计算工具将探索数值分叉理论，动力学系统甚至蒙特卡洛/分子动力学技术之间的接口及其对手头物理系统的潜在用途。 重要的是，这项研究将是真正应用数学的真正协同作用，不仅涉及理论方法和计算技术的发展，而且还涉及它们与物理实验的直接联系。最后，该项目将非常集中于维持一个动态和多学科的团队，其研究生的核心强大而多样化，因此将始终如一地致力于产生更大的广泛影响。相关研究将通过数学和物理学的高质量期刊出版物进行传播，并将在美国和国外的非线性科学和数学物理学会议上介绍。