CAREER: Chaotic transport -- from fundamental theory to applications in atomic physics

历史上，非线性动力学在解释多样而复杂的原子过程中发挥了重要作用，这一作用反过来又刺激了经典和量子混沌的许多理论进展。随着原子和光学技术的进步，为原子系统中混沌的实验研究提供了前所未有的控制和精确度，这一趋势仍在继续。例如，高度局域化的初始状态（如超短光脉冲，里德堡波包，和超冷原子的局域系综）的时间演化，可以测量它们在混沌势中的演化和分散，从而探测混沌相空间中的详细分形结构。然而，这种精确的工具突出了理论非线性动力学的一个基本缺陷：不存在通用框架（在符号动力学的意义上）能够对真实的物理系统中的传输所表现出的混沌行为的多样性进行分类。（i）它将利用混沌动力学的新进展来激励，引导，and interpret解释experiments实验capable能力of probing探测chaotic混沌phase相space空间with unprecedented空前resolution分辨率.这些研究包括里德堡波包的混沌输运、电离和控制，以及超冷原子在光阱中的混合和损失的新混沌途径的阐明。(ii)它将开发一个非线性动力学工具箱，可以提取一个精确的（符号动力学）模型的结构混沌输运的哈密顿系统与两个自由度。这个模型可以任意精确，甚至对于表现出混沌和规律性的混合物的系统。之前的成功？同伦叶动力学在描述阶段空间运输将作为这个计划的一个关键组成部分。最后，认识到同伦叶动力学的自然拓扑扩展到高维空间，这个建议将解决在超过两个自由度的混沌运输的挑战，目前存在的技术少得多。

Nonlinear dynamics has historically played a fundamental role in explaining diverse andcomplex atomic processes, a role which in turn has stimulated numerous theoreticaladvances in classical and quantum chaos. This trend continues as advances in atomicand optical techniques provide an unprecedented level of control and precision forexperimental studies of chaos in atomic systems. For example, the time evolution ofhighly localized initial states (e.g. ultrashort optical pulses, Rydberg wavepackets, andlocalized ensembles of ultracold atoms), can be measured as they evolve and dispersewithin a chaotic potential, thereby probing the detailed fractal structure in the chaoticphase space. Such precision tools, however, highlight a fundamental deficiency intheoretical nonlinear dynamics; there exists no general framework (in the sense ofsymbolic dynamics) capable of classifying the diversity of chaotic behavior exhibited bytransport in real physical systems.The focus of this proposal is twofold: (i) It will exploit new advances in chaotic dynamicsto motivate, guide, and interpret experiments capable of probing chaotic phase spacewith unprecedented resolution. These include the chaotic transport, ionization, andcontrol of Rydberg wavepackets as well as the elucidation of novel chaotic pathways forthe mixing and loss of ultracold atoms in optical traps. (ii) It will develop a nonlineardynamics toolbox that can extract an accurate (symbolic dynamics) model for thestructure of chaotic transport in Hamiltonian systems with two degrees of freedom. Thismodel can be made arbitrarily precise, even for systems exhibiting a mixture of chaosand regularity. The prior success of ?homotopic lobe dynamics? in describing phasespace transport will serve as a key ingredient of this program. Finally, recognizing thenatural topological extension of homotopic lobe dynamics to higher dimensional spaces,this proposal will address the challenge of chaotic transport in more than two degrees offreedom, for which far fewer techniques currently exist.