Rare and Exotic Nonlinear Effects in Cold Atomic Gases

对称既为不同的科学分支提供了统一的手段，也为科学家和非科学家之间的交流提供了语言。两种对称性在这个项目中是突出的：多维万花筒和量子力学的超对称性，它将运动与自由空间中的运动联系起来。通过这些对称性，一些真实的物理效应——禁止化学过程和减缓波导受限气体中的弛缓，孤立波对其他波的透明性等——与多边形、柏拉图式固体和猫眼反射器的平铺紧密地交织在一起。PI计划于2014年5月初在科学发现博物馆举行的“与科学家见面！”会议上发表题为“万花筒、瓷砖、牛顿的摇篮和绝对零度以上十亿分之一度的原子”的报告，其内容实际上与该项目的三个部分之一相同。本项目是培养本科生和硕士生不可或缺的教学工具。从李群到量子力学的超对称，这三个子项目中的每一个都涉及到一系列现代理论物理的先进方法，但它们都做得很温和，处理的对象就像常微分方程一样容易理解。子项目也是模块化的，可以分成几个为期一年的小项目，因此非常适合荣誉论文或硕士论文。在准备这个项目的过程中，许多定性的想法极大地促进了这本面向本科生的书的发展。非线性物理学以其未被探索或未被探索的小岛群而闻名。超冷原子气体是造访这些岛屿的理想工具。本项目考虑的非线性现象如下：在冷吸引玻色子波导中实验生成Gross-Pitaevskii呼吸子的协议。（进一步的研究将解决一维玻色气体的化学成分的可积性引起的鲁棒性。）利用先前在研究已知可积偏微分方程的Bogoliubov-de Gennes方程的透明度中获得的信息，分析Andreev-Bogoliubov-de Gennes BCS孤子对准粒子的透明度。目前，尚不清楚BCS孤子透明度应该能够实现什么物理现象，主要候选人是Andreev反射。系统地研究了与非晶体和特殊晶体根系统（万花筒镜排列）相关的无衍射的三体散射的量子和经典案例。将确定一个特定的质量三重体序列，其中（sqrt(5) + 2）： 1: （sqrt(5) + 2）是第一个与牛顿支架相似的非平凡成员。这里，动量分布在三体碰撞下保持不变。该计划包括对多体质量混合物进行系统的分子动力学研究，其中质量由光学晶格控制，并期望弛豫时间将在“扩展牛顿摇篮”质量比处显示峰值。

Symmetries provide both means for unification of otherwise disparate branches of science and a language for communication between scientists and non-scientists. Two symmetries are prominent in this project: multidimensional kaleidoscopes and the quantum-mechanical supersymmetry that relates the motion with obstacles to the motion in free space. Through these symmetries some real physical effects---prohibition of chemical processes and slowdown of relaxation in waveguide-confined gases, transparency of solitary waves for other waves, etc---are tightly intertwined with tiling with polygons, Platonic solids, and cat-eye reflectors. The PI's presentation at the "Meet the Scientists!" at the Science Discovery Museum, entitled "Kaleidoscopes, tilings, Newton's cradles, and atoms at a billionth degree above absolute zero", scheduled for the beginning of May 2014, has effectively the same content as one of the three parts of this project. This project constitutes an indispensable pedagogical tool for training undergraduate and masters-level students. Each of the three sub-projects, while touching upon a range of advanced methods of modern theoretical physics, from Lie groups to quantum mechanical supersymmetry, does so gently, dealing with objects as graspable as ordinary differential equations. The sub-projects are also modular, divisible into several year-long smaller projects, and so ideal for an honors or MS thesis. Many qualitative ideas developed in the course of preparation for the project greatly contributed to the development of an undergraduate-oriented book.Nonlinear physics is famous for its archipelagoes of small under- or unexplored islands. The ultracold atomic gases are the ideal vehicle for visiting such islands. The nonlinear phenomena considered in this project are as follows: a protocol for an experimental generation of a Gross-Pitaevskii breather, in a waveguide for cold attractive bosons. (A further study will address the integrability-induced robustness of the chemical composition of one-dimensional Bose gases.); an analysis of the transparency of the Andreev-Bogoliubov-de Gennes BCS soliton to quasi-particles, using information previously acquired in a study of transparency of the Bogoliubov-de Gennes equations for known integrable partial differential equations. At the moment, it is not clear what physical phenomenon the BCS soliton transparency is supposed to enable, with the primary candidate being the Andreev reflection. A systematic study of quantum and classical cases of the three body scattering without diffraction associated with non-crystallographic and exceptional crystallographic root systems (kaleidoscopic mirror arangements) will also be performed. A particular sequence of mass triplets-with ( sqrt(5) + 2) : 1 : (sqrt(5) + 2) being the first nontrivial member that behave similarly to the Newton cradle will be identified. Here, the momentum distribution remains invariant under three-body collisions. The plan includes a systematic molecular-dynamics study of many-body mass mixtures, with masses controlled by optical lattices, with an expectation that the relaxation time will show peaks at the "extended Newton cradle" mass ratios.