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.

这个建议的目的是探索一些新的，新兴的方向在原子玻色-爱因斯坦凝聚体（BEC）的背景下。该项目将扩展PI最近在被困冷凝物中的涡旋动力学主题上的合作工作。在那里，我们将探索与阿默斯特学院的一个合作实验小组获得的实验结果的联系，以及与数学和物理学其他领域的联系。这些包括最值得注意的光谱理论的非线性相干态，结构相变，减少粒子为基础的动态模型与有趣的非线性分岔现象，和计算工具，以监测存在，稳定性和动态的涡团。此外，这些主题的多分量概括将被认为是在最近建立的自旋轨道耦合BEC的方向。这些包含一个更复杂的运营商结构内的非线性问题，具有色散（拉普拉斯）和Dirac-like部分和这些条款的相互作用和它们对非线性状态，如暗或亮孤子和涡旋的影响将被研究的线性部分。这一方向的结果将与华盛顿州立大学的一个合作实验小组正在进行的实验进行比较。 这项研究有望为研究非线性相干结构提供新一代的理论和计算工具，特别是对玻色-爱因斯坦凝聚体的原始原子物理设置的看法。此外，这里开发的理论/数学工具将对光谱理论和非线性常微分方程和偏微分方程等领域产生更广泛的影响。我们还设想，我们的研究结果将与许多物理学领域建立联系，如流体动力学，非线性光学和统计力学（相变）。另一方面，开发的计算工具将探索数值分叉理论，动力系统，甚至蒙特-卡罗/分子动力学技术和它们的潜在用途的物理系统手头之间的接口。 重要的是，这项研究将是真正的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，还涉及它们与物理实验的直接联系。最后，该项目将重点关注维持一个充满活力和多学科的团队，拥有强大而多样化的研究生核心，因此将始终面向具有广泛影响的重要群体。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会议上发表。