Topological Chaos for Atomic Characterization and Control

近几十年来，在制备、操纵和控制量子系统的能力方面取得了巨大的进步，在越来越小的时间尺度和越来越精细的空间分辨率上，量子系统的保真度不断提高。这开辟了新的科学和技术前沿，例如，在量子信息处理，量子气体的操纵，和里德伯（高激发）原子的控制。虽然这样的系统本质上是量子力学的，但关于这样的系统行为的很多直觉是基于这样的系统的潜在经典模型。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殊的困难。本研究旨在利用新开发的理论（例如拓扑）工具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特性（例如衰变率）和能级。这种拓扑工具也将应用于里德伯原子的控制问题。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园。加州大学默塞德分校成立于2005年，已经对加州农业中心大中央山谷的经济和教育产生了重大影响。本提案的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)基于同伦叶动力学的先前发展和计算机实现——一种用于对大振幅混沌运动进行分类的拓扑技术——衰减率和其他原子输运性质将通过对相对少量系统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与量子相位信息相结合，以产生单个混沌能级的半经典估计。（iii）在成功使用相空间旋转门控制准一维里德堡态电离的基础上，旋转门技术将应用于圆形类玻尔里德堡波包的更具挑战性的控制问题。周期轨道技术的强大力量尚未广泛应用于“现实世界”问题，这在很大程度上是由于描述和计算相关轨道的困难，特别是在混沌和规则混合的复杂系统中。因此，成功计算混合系统的经典输运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特征值将解决量子混沌中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状态，例如增加或减少主量子数。它还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.