OP: Collaborative research: Nonlinear theory of slow light

这一合作项目扩展了首席研究人员在光学现象数学模型方面的研究计划。这是对美国国家科学基金会“光学和光子学”倡议的回应。光与光学介质之间的相互作用是应用物理学中最富有成效的研究领域之一，它为激光和光放大等器件提供了基本的机制。几十年来，它一直在为新的物理现象提供丰富的来源，其中最新的是“慢光”，即最近观察到的光脉冲减速到自行车的速度。慢光有可能用于光学存储器等设备中。这个项目的目的是通过使用一个显著的、高精度的数学模型来理解慢光现象背后的物理机制，该模型可以用显式求解。该模型及其显式解的有效性将通过更真实的模型的数值模拟和与实验的仔细比较来验证。将为研究生和本科生提供应用数学和非线性光学的跨学科培训，并为两个学生群体建立一个活跃的、具有挑战性的研究和培训环境。光脉冲的减慢被模拟为光脉冲与具有两个或三个工作能级的主动介质之间的相互作用，后者是被称为兰姆达结构的典型案例。这种相互作用用完全可积的麦克斯韦-布洛赫方程来描述，边界条件不为零，这是一种新的扭曲。这个项目是对光与二能级介质和Lambda组态介质相互作用产生的新动力学的数学研究，包括：(I)发展具有非零边界条件的二能级和Lambda组态Maxwell-Bloch方程的系统的、完全可积的动力学理论；(Ii)使用步骤(I)的分析结果来描述与慢光有关的现象；(Iii)在更一般的情况下，即二能级和Lambda组态的Maxwell-Bloch方程不可积的情况下，对动力学现象进行数值研究。慢光的完全可积描述涉及两个新的方面：(1)非零边界条件；(2)光谱数据的非平凡演化。通过研究黎曼表面上具有谱参数的散射和逆散射问题，第一方面的理解将从非线性薛定谔方程扩展到麦克斯韦-布洛赫方程。第二个方面由于前者的存在而变得复杂，包括仔细推导光谱数据是如何从介质的初始状态演变而来的，并找到高振荡项的正确抵消。除了产生光与活性光学介质相互作用所表现出的动力学的新模型和描述外，该项目还将推进完全可积系统的理论。

This collaborative project expands the research programs of the Principal Investigators on mathematical models of optical phenomena. It comes in response to the NSF initiative on "Optics and Photonics". The interaction between light and optical media is one of the most fruitful areas of study in applied physics and provides the basic mechanism underlying devices such as lasers and optical amplifiers. For decades, it has been providing a rich source of new physical phenomena, among the latest being "slow light", the recently-observed slowing-down of light pulses to the speed of a bicycle. Slow light can potentially be used in devices such as optical memory. This project is aimed at understanding the physical mechanisms underlying the slow light phenomenon by using a remarkable, highly accurate mathematical model that can be solved with explicit formulas. The validity of this model and its explicit solutions will be verified using numerical simulations of more realistic models and careful comparisons with experiments. Interdisciplinary training in applied mathematics and nonlinear optics will be provided to graduate and undergraduate students, and a lively, challenging research and training environment for both student groups will be established.The slowdown of light pulses is modeled as the interaction between an optical pulse and an active medium with two or three working levels, the latter being a prototypical case known as the Lambda configuration. This interaction is described by completely integrable Maxwell-Bloch equations with non-vanishing boundary conditions, a new twist. This project is a mathematical study of novel dynamics generated by the interaction of light with two-level media and the Lambda-configuration medium, and includes: (i) developing a systematic, completely integrable theory of the dynamics for the two-level and Lambda-configuration Maxwell-Bloch equations with non-zero boundary conditions, (ii) using the analytical results of step (i) to describe phenomena related to slow light, (iii) numerical studies of dynamical phenomena in more general cases in which the two-level and Lambda-configuration Maxwell-Bloch equations are not integrable. The completely-integrable description of slow light involves two new aspects: (1) non-zero boundary conditions, (2) non-trivial evolution of the spectral data. The understanding of the first aspect will be extended from the Nonlinear Schroedinger equation to the Maxwell-Bloch equations by studying scattering and inverse-scattering problems with the spectral parameter on a Riemann surface. The second aspect is complicated by the presence of the former and involves a careful derivation of how spectral data evolves from the initial state of the medium and finding correct cancellations of highly oscillatory terms. In addition to generating new models and descriptions of the dynamics exhibited by light interacting with active optical media, the project will advance the theory of completely integrable systems.