Collaborative Research: From Quantum Droplets & Spinor Solitons to Vortex Knots & Topological States: Beyond the Standard Mean-Field in Atomic BECs

玻色-爱因斯坦凝聚（BECs）领域最初是玻色和爱因斯坦在20世纪20年代提出的，作为具有整数自旋的原子粒子的统计特性的一个奇怪特征。这包括将激发态粒子凝聚到系统的基态，并在其中形成宏观的、相干的“超波”，从而允许对微观尺度之外的量子力学特性进行研究和观察。然而，它的实验实现所需的温度是如此之低，以至于E.A.康奈尔，W. Ketterle和C.E. Wieman花了大约70年的时间才在实验室中实现了BECs。这一成就的重要性在几年后的2001年诺贝尔物理学奖上才得到认可。这反过来又提供了一个原始的平台，在这里可以研究和实验观察到波浪和相干结构的非线性动力学的许多令人兴奋的特征。重要的是，这些相干结构也广泛适用于许多其他物理领域，包括最著名的非线性光学、等离子体物理和水波。在原子物理学中，bec也是研究超导性和超流动性等显著量子特征的基础，在这方面，它们一直是2003年诺贝尔物理学奖中引用的漩涡及其晶格的实验发现的前沿和中心，以及与2016年诺贝尔物理学奖相关的拓扑相及其跃迁。该项目的目标是在原子物理理论、物理BEC实验、应用数学分析和科学计算前沿的令人兴奋的联系中推进最先进的技术，同时在这个科学界面和超越学科界限的领域培养新一代科学家和数学家。与过去的pi轨迹一致，将在这项研究工作中寻求对代表性不足群体的多样性、公平性和包容性的强调。更具体地说，本项目的主要内容包括对标准BEC设置的非平凡扩展的研究。具体而言，提案的主轴考虑了以下主题。(1)双组分相互吸引的BECs，通过量子修正和著名的李黄杨（Lee-Huang-Yang， LHY）贡献，可以非常及时地形成所谓的量子液滴。这种液滴的关键认识是，它们的出现源于排斥性平均场和吸引力超越平均场贡献之间的相互作用。(2) 3个（F= 1）和5个（F=2）自旋分量设置支持具有前所未有的可积或弱不可积性质的共生（暗-反暗和暗-亮）孤波结构。(3)单分量和多分量/旋量设置下的三维涡结结构。旋涡节是一种最难以捉摸的旋涡结构，其实验和理论分析都很有限。pi还将在旋量设置中探索复杂的非平凡拓扑模式，如爱丽丝环和狄拉克单极子。(4)拓扑上的非平凡环面捕获设置，其中系统的内在度量和曲率与有效非线性的相互作用可以产生前所未有的相干结构和动力学。更广泛地说，在这个主题中，pi将研究非线性波，如限制在不同类型曲面上的孤子和漩涡。这个雄心勃勃的计划应该推动最先进的平均场理论理解的边界，提供许多超越平均场的见解，并阐明其有效性范围，以及非线性与量子的相互作用，以及热力学效应。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 realm of Bose-Einstein condensates (BECs) was originally proposed as a curious feature of the statistical properties of atomic particles with integer spin by Bose and Einstein in the 1920's. This consisted of the condensation of the excited states particles into the ground state of the system and the formation of a macroscopic, coherent “super-wave” therein, allowing the study and observation of quantum mechanical properties beyond microscopic scales. However, the temperatures needed for its experimental realization were so low that it took about 70 years for E.A. Cornell, W. Ketterle, and C.E. Wieman to realize BECs in the lab. The importance of this feat was recognized only a few years later via the 2001 Nobel Prize in Physics. This has, in turn, enabled a pristine platform where numerous exciting features of nonlinear dynamics of waves and coherent structures can be studied and experimentally observed. Importantly, these coherent structures are also of wide applicability in numerous other areas of physics including, most notably, nonlinear optics, plasma physics, and water waves. Within atomic physics, BECs have also been fundamental toward the study of remarkable quantum features such as superconductivity and superfluidity and, in that capacity, they have been front and center toward the experimental discoveries connected to the vortices and their lattices cited in the 2003 Nobel Prize in Physics and the topological phases and their transitions associated with the 2016 Nobel Prize in Physics. The aim of this project is to advance the state-of-the-art at this exciting nexus of atomic physics theory, physical BEC experiments, applied mathematical analysis, and the forefront of scientific computing, while at the same time training a new generation of scientists and mathematicians at this scientific interface and transcending disciplinary boundaries. In line with the past trajectory of the PIs, an emphasis on the diversity, equity and inclusion of under-represented groups will be sought within this research effort.More concretely, the principal thrust of the present project consists of the study of non-trivial extensions of standard BEC settings. In particular, the main axes of the proposal consider the following themes. (1) Two-component mutually attractive BECs that allow, through quantum corrections and the famous Lee-Huang-Yang (LHY) contribution, for the highly timely formation of so-called quantum droplets. The key realization for such droplets is that their emergence stems from the interplay between repulsive mean-field and attractive beyond-mean-field contributions. (2) Three (F =1) and five (F=2) spin component settings supporting symbiotic (dark-antidark and dark-bright) solitary wave structures with unprecedented integrable or weakly non-integrable properties. (3) 3D vortex knot structures in one and multi-component/spinor settings. Vortex knots constitute one of the most elusive types of vortical structures for which limited experimental and theoretical analysis exists. The PIs will also explore in the spinor settings complex non-trivial topological patterns such as Alice rings and Dirac monopoles. (4) Topologically nontrivial toroidal trapping settings, where the interplay of the intrinsic metric and curvature of the system with the effective nonlinearity can yield unprecedented coherent structures and dynamics thereof. More broadly within this theme, the PIs will study nonlinear waves such as solitons and vortices confined on different types of curved surfaces. This ambitious program should push the boundaries of the state-of-the-art mean-field-theoretic understanding, offering numerous beyond-mean-field insights and elucidating their range of validity as well as the interplay of nonlinearity with quantum, as well as thermodynamic effects.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.