Topological excitations in Bose-Einstein condensates: Existence, stability, dynamics, and interactions

非线性介质具有各种各样的局域相干结构(孤子、波列、涡旋、螺旋等)。具有复杂的内在属性和相互作用，进而产生具有非平凡动力学的紧急模式。这项研究的主题是对色散非线性介质中产生的涡旋的详细研究，主要集中在扩展和深化对涡旋结构、它们的存在和动力学稳定性以及它们在诸如玻色-爱因斯坦凝聚体等非线性介质和相关领域(如非线性光学)中的相互作用和介观晶格的理解。我们建议遵循循序渐进的方法来研究涡旋的产生、稳定性和动力学，从单个涡旋到少数涡旋，再到涡旋晶格。具体地说，我们计划研究：(1)通过动力学不稳定性和外部操纵，如相位印迹和通过快速移动的杂质(聚焦激光)驱动的不稳定性来产生涡旋。(2)单个涡旋的动力学和稳定性。(3)涡旋与涡偶极子的形成、稳定性和动力学之间的相互作用。分子/偶极子)。(4)最后，我们打算通过交叉受精材料科学的思想，研究大的涡旋阵列，它们的结晶和结构相变到规则的涡旋晶格(又名。在20世纪20年代，S、玻色和爱因斯坦预言，在低密度和超低温下的气体会经历一种转变，即现在所说的玻色-爱因斯坦凝聚态(BEC)。BEC最重要的特征是所有原子都占据相同的量子态，形成宏观的相干物质团块。BEC的重要性就像激光之于光一样。BEC允许在宏观层面上直接操纵和观察量子效应，提供对物质的最终控制。自从它们最近的实验实现(2001年诺贝尔物理学奖获得者)以来，BEC一直是密集且不断增长的实验和理论工作的焦点。涡旋是在BEC中出现的基本的相干、拓扑带电的非线性激发；但它也在超导和超流等令人兴奋的重要领域发挥着深远的作用(这是2003年诺贝尔物理学奖的主题)。有趣的是，它们也以水或空气中流体动力涡旋的形式出现在我们的日常生活中。在这些流体漩涡和我们计划研究和描绘的这些超冷超流体漩涡之间存在着强烈的相似之处(也有区别)。这项研究的结果将有助于揭示玻色-爱因斯坦凝聚体中这种涡旋结构的模式形成和相互作用。由于描述BEC的基本方程也描述了嵌入在非线性材料中的相干光的行为，因此本文提出的研究也将适用于光波导和光纤束、光子晶体和光陷阱中的光存储等问题，所有这些都是光学技术的前沿活跃研究领域。这项拟议的研究可能应用于量子光学存储和下一代计算机的量子计算。这项研究工作是PI和Co-PI正在进行的合作的一部分，涉及十多名同事，他们融合了动力系统、非线性光学、凝聚态、材料科学和科学计算等多个领域的专业知识。特别吸引人的是，我们的结果将部分驱动，并可能与目前在BEC实验中进行的实验研究相关。这一高度跨学科的研究计划还将通过研究生研究助理和博士后研究员的直接参与，涉及主要的教育组成部分。

Nonlinear media host a wide variety of localized coherent structures(solitons, wavetrains, vortices, spirals, etc.) with complex intrinsicproperties and interactions that, in turn, give rise to emergent patternswith nontrivial dynamics. The theme of the proposed research is a detailedexamination of vortices generated in dispersive nonlinear media.The main focus is to extend and deepen the understanding of vortexstructures, their existence and dynamical stability as well as theirinteractions and mesoscopic lattices in nonlinear media such asBose-Einstein condensates and related fields (such as nonlinear optics).We propose to follow a step-by-step methodology in studying thegeneration, stability and dynamics of vortices in a progression ofscenaria of increasing complexity, extending from single vortices, to few vortices, to vortex lattices. Specifically, we plan to study:(1) Vortex generation through dynamical instabilities and externalmanipulations such as phase imprinting and via instabilities driven byrapidly moving impurities (focused laser beams).(2) Dynamics and stability of single vortices (a.k.a. atoms) in thepresence of external traps and manipulation of vortices by translating theexternal traps or by using localized optical "tweezers".(3) Interactions between vortices and formation, stability and dynamicsof vortex dipoles (a.k.a. molecules/dipoles).(4) Finally, we intend to investigate, by cross-fertilizing ideas frommaterial science, large arrays of vortices, their crystallization andstructural phase transitions into regular vortex lattices (a.k.a. crystals).In the 1920's Bose and Einstein predicted that a gas at low density andultra-cold temperatures undergoes a transition towards what is nowadayscalled a Bose-Einstein condensate (BEC). The most important characteristicof a BEC is that all atoms occupy the same quantum state creating a macroscopiclump of coherent matter. BECs are to matter what laser is to light. BECs allowfor direct manipulation and observation of quantum effects at the macroscopiclevel, providing ultimate control over matter. Since their recentexperimental realization (for which the 2001 Physics Nobel prize wasgranted), BECs have been at the focus of an intensive and ever growingexperimental and theoretical effort. Vortices are fundamental coherent,topologically charged, nonlinear excitations that emerge in BECs; butwhich also play a profound role in exciting and important fields such assuperconductivity and superfluidity (which were the theme of the 2003Physics Nobel prize). Interestingly enough, they also arise in our dailylife in the form of hydrodynamic vortices in water or in air. There arestrong parallels (as well as differences) between such fluid vortices andthese ultra-cold, superfluid vortices that we plan to examine anddelineate. The outcome of this research will shed light into the patternformation and interaction of such vortex structures in Bose-Einsteincondensates. Since the underlying equation that describes the BECs alsodescribes the behavior of coherent light embedded in a nonlinearmaterial, the research hereby proposed will also be applicable toproblems of optical waveguides and fiber bundles, photonic crystals, andlight storage in optical traps, all of which are active research areas atthe forefront of optical technologies. The proposed research haspotential applications to quantum-optical storageand quantum computing for the next generation of computers.The research effort is part of an ongoing collaboration betweenthe PI and Co-PI and involves more than a dozen coworkers that blend inexpertise in fields as diverse as dynamical systems, nonlinear optics,condensed matter, materials science and scientific computing. Speciallyattractive is the prospect that our results will be partially driven andcould be relevant to current experimental research conducted in BECexperiments. This highly inter-disciplinary research program will alsoinvolve a major educational component through the direct involvement ofgraduate research assistants and postdoctoral fellows.