CAREER: Solitons in Bose-Einstein Condensates: Generation, Manipulation and Pattern Formation

摘要：职业奖DMS-0349023 Panayotis Kevrekidis，马萨诸塞州大学阿默斯特分校标题：职业：玻色-爱因斯坦凝聚体中的孤子：产生，操纵和图案形成本项目的目的是研究原子物理（玻色-爱因斯坦凝聚体）中孤立波结构的行为。根据其结构鲁棒性和准弹性相互作用，孤子是过去用于光学设置中的信息传输的天然构建块，并且可以自然地扩展为这种新物质波设置中的信息载体。此外，这些结构可以被适当地操纵、波导或用于在微观（原子）水平上构建各种图案。研究将在三个不同的层次上进行：（1）利用不稳定性和/或实验可用机制产生这些孤立波的层次（如Feshbach共振）;（2）操纵波的一个（或者通过光镊拖动它们，或者通过结波导它们）;（3）最后，在将它们结合起来以创造模式并确定其稳定状态和结构转型的层面上。这些步骤将针对两种主要类型的相互作用进行：a）对于原子之间的吸引相互作用（例如，用于聚焦非线性和负散射长度（如锂的情况）; B）对于排斥相互作用（例如，对于散焦非线性和正散射长度，如铷和钠的情况）。将被检查的模型将是两个连续模型的偏微分方程与外部电位（线性或二次，或其组合），以及准离散的（相关的周期性的外部potentialas在所谓的光晶格的情况下）。将使用的技术将涉及正规和奇异摄动方法，线性和调制稳定性分析，正规和可能的指数渐近，数值分叉理论以及直接数值模拟和分子动力学技术（研究模式及其结构转变）。该研究项目的主要重点是详细研究在玻色-爱因斯坦凝聚体（BEC）的非常可控的超低温原子物理背景下产生的孤波。由于他们最近的实验实现（2001年诺贝尔物理学奖授予），BEC一直是一个密集的和不断增长的实验和理论工作在数学和物理界的中心。BEC的研究也与更深入地理解超导性和超流性（这是2003年诺贝尔物理学奖的主题）的令人兴奋和重要领域有着密切的联系。从数学物理学的角度来看，研究BEC最有趣和最吸引人的方面之一是它们丰富的非线性波动现象，各种各样的可能设置（一到三维）和详细的实验控制，允许精确的工程/操纵这些波动态演变的外部条件。这项研究工作的主要目的是扩展和加深我们对基本结构和波及其在BEC中的作用和重要性，而且更普遍地（由于类似的数学描述）在非线性光学（光纤和波导）以及波动物理学中。顺便说一句，应该指出，这项工作将严重依赖计算资源和随之使用的模拟这些现象的数字代码;还应该指出的是，对物质波的研究，从长远来看，其中一个前景是设想和建造新的装置，这些装置将引导和更普遍地控制物质波的运动，并有可能用于量子信息处理在纳米尺度上。这些方面使我们预期，在战略联邦利益的领域，如高性能计算和材料和制造，该项目的实施可能会带来重大利益。

Abstract: CAREER Award DMS-0349023Panayotis Kevrekidis, University of Massachusetts at AmherstTitle: CAREER: Solitons in Bose-Einstein Condensates: Generation, Manipulation and Pattern FormationThe aim of this project is to examine the behavior of solitary wavestructures in the setting of atomic physics (Bose-Einstein Condensates). Solitons as per their structural robustness and quasi-elastic interactions are natural building blocks that have been used for information transmission in optical settings in the past and could naturally be extended as information carriers in this new matter wave setup. Furthermore, these structures can be appropriately manipulated, waveguided or used to construct various patterns at this microscopic (atomic) level. The study will be undertaken at three different levels: (1) The level of creating these solitary waves by taking advantage of instabilities and/or experimentally available mechanisms (such as the Feshbach resonance); (2) The one of manipulating the waves (either dragging them by means of an optical tweezers or waveguiding them through junctions); (3) And, finally, at the level of combining them to create patterns and to identify their steady states and structural transitions. These steps will be carried through for the two principal types of interactions: a) For attractive interactions between the atoms (e.g., for focusing nonlinearities and negative scattering lengths as in the case of lithium); b) For repulsive interactions (e.g., for defocusing nonlinearities and positive scattering lengths as in the case of rubidium and sodium). The models that will be examined will be both continuum models of partial differential equations with external potentials (linear or quadratic, or combination thereof), as well as quasi-discrete ones (relevant for periodic external potentialsas in the case of the so-called optical lattice). The techniques that will be used will involve regular and singular perturbationmethods, linear and modulational stability analysis, regular and possibly exponential asymptotics, numerical bifurcation theory as well as direct numerical simulations and also molecular dynamics techniques (to study patterns and their structural transitions). The main focus of this research project is a detailed study of solitary waves generated in the very controllable, ultra-low temperature, atomic physics context of Bose-Einstein condensates (BECs). Since their recent experimental realization (for which the 2001 Physics Nobel prize was awarded), BECs have been the center of an intensive and ever growing experimental and theoretical effort in the Mathematics and Physics communities. The examination of BECs has also strong ties with a deeper understanding of the exciting and important fields of superconductivity and superfluidity (which were the theme of the Physics Nobel prize in 2003). From a Mathematical Physics perspective, one of the most interesting and appealing aspects of studying BECs is their rich nonlinear wave phenomenology, the wide variety of possible settings (one to three dimensions) and the detailed experimental control that permits a precise engineering/manipulation of the external conditions under which these waves dynamically evolve.The main purpose of this research effort is to extend and deepen our understanding of the fundamental structures and waves and their roleand importance in BECs, but also more generally (due to the similarmathematical description) in nonlinear optics (optical fibers andwaveguides) as well as wave physics. As an aside, it should be noted that this effort will heavily rely on computational resources and the concomitant use of numerical codes that model these phenomena; it should also be remarked that one of the longer term perspectives of this activity on matter waves is to conceive and construct novel devices that would guide and more generally control the motion of the matter waves and could potentially be used for quantum information processing at the nanoscale. These aspects lead us to expect that significant benefits may result from the implementation of this project in areas of strategic federal interest such as high performance computing and materials and manufacturing.