Topological Chaos for Atomic Characterization and Control

近几十年来，在越来越小的时间尺度和越来越精细的空间分辨率下，制备、操作和控制量子系统的能力有了显著的提高，保真度越来越高。这开辟了新的科技前沿，例如在量子信息处理、量子气体操纵和里德堡(高激发)原子控制方面。尽管这类系统本质上是量子力学的，但对这类系统行为的很多直觉都是基于这类系统的基本经典模型。由于经典轨迹的极端复杂性和数量，经典混沌模型呈现出特别的困难。这项研究试图利用新发展的理论(如拓扑学)工具来对原子系统中的混沌轨迹进行分类，并使用这些轨迹来计算此类系统的输运性质(例如衰减率)和能级。这样的拓扑工具也将被应用于里德堡原子控制的问题。通过研究生和本科生的研究经验，这项工作还将有助于培养下一代科学家，并有助于培育加州大学默塞德分校新成立的校区。加州大学默塞德分校于2005年开业，已经对加州农业中心加州大中央山谷的经济和教育产生了重大影响。这项提议的广泛主题是通过对混沌轨迹的深刻结构理解来分析和控制原子系统。这体现在以下具体目标中。(I)在同伦波瓣动力学以前的发展和计算机实现的基础上-一种用于对大幅度混沌运动进行分类的拓扑技术-衰减率和其他原子传输特性将通过对系统相对较少的周期轨道的详细分析来计算。(Ii)这些经典的周期轨道计算将增加量子相位信息，以产生对单个混沌能级的半经典估计。(Iii)在成功利用相空间旋转门控制准一维里德堡态电离的基础上，旋转门技术将被应用于更具挑战性的类玻尔圆形里德堡波包的控制问题。由于很大程度上由于描述和计算相关轨道的困难，周期轨道技术的成熟能力尚未被广泛应用于“现实世界”问题，特别是在混杂着混沌和规则的复杂系统中。因此，成功地计算混合系统的经典传输率将极大地扩展周期轨道技术的适用性。此外，使用这些技术来估计单个量子本征值将解决量子混沌中的一个长期存在的问题：是否有可能半经典地计算一般混沌系统的量子谱？最后，将相空间旋转门应用于三维类玻尔波包，将提供一种新的实验室机制来快速控制和设计这种态，例如增加或减少主量子数。这也将提高对高维系统中混沌输运的一般理解。

Recent decades have seen dramatic improvement in the ability to prepare, manipulate, and control quantum systems with ever increasing fidelity on smaller and smaller time scale and with finer and finer spatial resolution. This has opened new scientific and technological frontiers, for example, in quantum information processing, manipulation of quantum gases, and the control of Rydberg (highly excited) atoms. Though such systems are inherently quantum mechanical, much intuition about the behavior of such systems is based on the underlying classical models for such systems. Classically chaotic models present particular difficulties, due to the extreme complexity and number of classical trajectories. This research seeks to utilize newly developed theoretical (e.g. topological) tools to classify chaotic trajectories in atomic systems and to use these trajectories to compute transport properties (e.g. decay rates) and energy levels of such systems. Such topological tools will also be applied to problems in the control of Rydberg atoms. Through graduate and undergraduate research experiences, this work will also help train the next generation of scientists and help to nurture the newly established campus of the University of California at Merced. UC Merced was opened in 2005 and has already made a significant impact on the economy and education in California's Great Central Valley, the agricultural heart of the state.The broad theme of this proposal is the analysis and control of atomic systems through a deep structural understanding of chaotic trajectories. This is manifest in the following specific objectives. (i) Building on the prior development and computer implementation of homotopic lobe dynamics---a topological technique for classifying large-amplitude chaotic motion---decay rates and other atomic transport properties will be computed from a detailed analysis of a relatively small number of a system's periodic orbits. (ii) These classical periodic orbit computations will be augmented with quantum phase information to generate semiclassical estimates of individual chaotic energy levels. (iii) Building on the successful use of phase space turnstiles to control the ionization of quasi-1D Rydberg states, the turnstile technique will be applied to the more challenging control problem of circular Bohr-like Rydberg wavepackets.The well-established power of periodic orbit techniques has not been widely applied to "real world" problems due in large part to the difficulty of characterizing and computing the relevant orbits, especially in complex systems with a mixture of chaos and regularity. Successfully computing classical transport rates for mixed systems would thus dramatically expand the applicability of periodic orbit techniques. Furthermore, using these techniques to estimate individual quantum eigenvalues would address a long-standing question in quantum chaos: is it possible to semiclassically compute quantum spectra for general chaotic systems? Finally, applying phase-space turnstiles to 3D Bohr-like wavepackets would provide a new laboratory mechanism to rapidly control and engineer such states, e.g. increase or decrease the principle quantum number. It would also improve the general understanding of chaotic transport in higher dimensional systems.