Mathematical and Computational Studies on Bose-Einstein Superfluid

玻色-爱因斯坦凝聚（BEC）是由S.玻色和a .爱因斯坦在20世纪20年代早期预测的，是一种接近绝对零度的物质状态，在这种状态下，所有原子都失去了它们的个体性质，并凝聚成宏观的相干“超波”。自1995年首次实验实现以来（2001年诺贝尔物理学奖授予e.a. Cornell, w.c Ketterle和c.e.w ieman）， BEC一直是实验和理论研究的重点。它不仅为研究物质的量子特性提供了一个新的平台，而且为理解超导和超流体现象开辟了新的视角。2018年5月21日，冷原子实验室发射到空间站，再次引起人们对BEC超流体的关注。本项目重点研究玻色-爱因斯坦超流体研究中出现的基本数学和计算问题，以了解其在人工规范场和远程色散相互作用影响下的行为。它可能有利于先进技术的发展，例如超导量子干涉装置和基于原子干涉仪的传感器。本研究的主要目的是建立磁性薛定谔方程的数学和计算处理方法，促进对玻色-爱因斯坦超流体的认识，从而推进BEC的实验和应用。本课题拟对远距离色散相互作用下BEC超流体的数学建模和数值模拟进行系统研究。一方面，将详细研究远程色散所带来的新的数学和数值问题。首先，将建立有效的吸收边界条件，以避免波在计算边界处的人工反射，这是波模拟中一个长期存在的问题。然后设计出精确有效的数值方法来离散磁性薛定谔模型。另一方面，将对BEC的解性质进行分析和数值研究，以了解远程色散相互作用的影响及其与非线性相互作用的相互作用。本文将通过稳定性分析和数值模拟来研究由于色散竞争和非线性相互作用导致的调制不稳定性和波坍缩。此外，还将研究在人工规范场和非局部色散和/或非线性相互作用存在下的量子化涡的性质，从而促进对BEC超流动性的理解。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。

Bose-Einstein condensate (BEC), predicted by S. Bose and A. Einstein in the early 1920s, is a state of matter near absolute zero temperature for which all atoms lose their individual properties and condense into a macroscopic coherent 'super-wave'. Since its first experimental realization in 1995 (2001 Nobel Prize in Physics awarded to E. A. Cornell, W. Ketterle, and C. E. Wieman), BEC has been the focus of active research both experimentally and theoretically. It not only provides a new platform to investigate quantum properties of matter but also opens new perspectives for understanding the phenomena of superconductivity and superfluidity. The recent launch of the Cold Atom Laboratory to the space station on May 21, 2018 has once again drawn the spotlight to BEC superfluidity. This project focuses on the fundamental mathematical and computational issues arising in the study of Bose-Einstein superfluids to understand its behavior under the influence of artificial gauge fields and long-range dispersive interactions. It could potentially benefit the development of advanced technologies, for instance, superconducting quantum interference devices and atom interferometer-based sensors. The main objectives of this research are to build mathematical and computational treatments for the magnetic Schrodinger equations, to promote the understanding of Bose-Einstein superfluidity, so as to advance the experiments and applications of BEC. This project proposes systematic research on the mathematical modeling and numerical simulations of BEC superfluids in the presence of long-range dispersion interactions. On one hand, new mathematical and numerical issues introduced by the long-range dispersion will be investigated in detail. First, effective absorbing boundary conditions will be developed to avoid the artificial reflections of waves at the computational boundary, one persistent problem in wave simulations. Then accurate and efficient numerical methods will be designed to discretize the magnetic Schrodinger models. On the other hand, the solution properties of BEC will be studied analytically and numerically to understand the influence of long-range dispersion interaction and its interplay with nonlinear interactions. Both stability analysis and numerical simulations will be carried out to study the modulation instability and wave collapse due to the competition of dispersion and nonlinear interactions. Moreover, the properties of quantized vortices in the presence of artificial gauge field and nonlocal dispersive and/or nonlinear interactions will be investigated, so as to advance the understanding of BEC superfluidity.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.