Modeling, Analysis, Computation and Experiments of Two-Component Bose-Einstein Condensates

该项目的目标是阐明双组分玻色-爱因斯坦凝聚物（BECs），以及它们与单组分凝聚物的区别。研究者和他的同事们的目标是通过引入一些基本的物理过程来重新定义在这些系统中建模和分析的方式，这些过程没有包括在目前使用的模型中。更具体地说，研究小组计划：(a)通过增加现有模型来解释原子间相互作用损失和高阶磁（塞曼）效应等过程，开发一种双组分bec的新模型；(b)对模型进行基准测试，在两种成分的原子总数或原子比例发生变化的各种情况下对模型进行测试，并将其与新开发的偏微分方程模型进行比较；(c)利用galerkin投影和Lyapunov-Schmidt约简对模型进行数学分析，以研究动力学的有限维近似，从而研究解的存在性和稳定性，并采用控制策略来稳定可能不稳定的解构型；(d)建立一个计算平台，能够研究非线性薛定谔方程的多组分、高维变体的存在性、稳定性和非线性动力学，这些方程是此类原子系统建模中的关键数学成分。在此过程中，产生了用于时间步进目的的空间/时间自适应和/或并行积分器，并开发了迭代方法，以分析围绕稳态解的线性稳定性问题。这个项目提出了一条在宇宙中出现的最低温度下系统量化量子动力学的途径，即最近创造的以玻色-爱因斯坦凝聚体为代表的新物质形式（其形成被授予2001年诺贝尔物理学奖，其性质，如超流动性，与2003年诺贝尔物理学奖密切相关）。研究者和他的同事组成了一个跨学科的团队，在实验室中直接监测物质的这种状态，在物理层面上对系统进行建模，在数学层面上探索由此产生的特征，最终将这种复杂系统的三维动力学完全可视化。在上述所有阶段之间的连续反馈旨在确定不仅是定性的而且是定量的理解这些原子物理系统，如铷、钠和其他碱蒸气的气体。研究中的多物种系统为未来的应用提供了丰富的机会，从超冷微观模式的可控形成（以“量子光刻”的形式）到量子门和开关的实现，反过来，旨在实现量子计算的长期目标。

CarreteroDMS-0806762 The goal of the project is to shed light on two-componentBose-Einstein Condensates (BECs) and how they differ from theirsingle-component counterparts. The investigator and hiscolleagues aim to redefine the way in which modeling and analysisare developed in such systems by introducing some fundamentalphysical processes that were not included in presently employedmodels. More specifically, the team of researchers plans to: (a) develop a new model for two-component BECs, by augmentingexisting models to account for processes such as inter-atomicinteraction losses and higher-order magnetic (Zeeman) effects; (b) benchmark the model, by testing it in a variety ofsituations where the total number of atoms or ratio of atomschanges between the two components and comparing it against thenewly developed partial differential equation model; (c) analyze the model mathematically by means of Galerkinprojections and Lyapunov-Schmidt reductions to study finitedimensional approximations of the dynamics, whereby the existenceand stability of solutions are studied and control strategies areemployed to stabilize potentially unstable solutionconfigurations; (d) produce a computational platform that enables the study ofexistence, stability and nonlinear dynamics of multi-component,high-dimensional variants of the nonlinear Schrodinger equationsthat are the key mathematical ingredient in the modeling of suchatomic systems. In the process, spatially/temporally adaptiveand/or parallel integrators are produced for time-steppingpurposes and iterative methods are developed in order to analyzethe linear stability problem around steady state solutions. This project presents a route to systematically quantify thequantum dynamics at the lowest temperatures that arise in theUniverse, namely in the recently created new form of matterrepresented by Bose-Einstein condensates (whose formation wasawarded with the 2001 Nobel prize in Physics and whoseproperties, such as superfluidity, were intimately connected tothe Nobel prize in Physics in 2003). The investigator and hiscolleagues form an interdisciplinary team to directly monitorthis state of matter in the laboratory, to model the system atthe physical level, to explore the resulting features at themathematical level, and finally to fully visualize thethree-dimensional dynamics of such complex systems. A continuousfeedback between all the above stages is intended to ascertainnot only a qualitative but also a quantitative understanding ofsuch atomic physics systems, such as gases of rubidium, sodiumand other alkali vapors. The multi-species systems under studypresent a wealth of opportunities for future applications,ranging from the controllable formation of ultracold microscopicpatterns (in a form of "quantum lithography") to the realizationof quantum gates and switches, that, in turn, aim towards thelonger term goal of enabling quantum computation.