Universal aspects of quantum dynamics: quantum catastrophes and long-range interactions

物理学的最大目标之一是确定自然现象中的普遍行为模式。一些最著名的例子是相变，它是平衡状态的变化。在相变附近，具有不同微观组成的系统以一种普遍的方式表现，其量根据共同的临界指数集而发散。这一提议涉及非平衡动力学的普适性，特别是量子动力学。特别是，我们将研究冷原子气体中的非平衡动力学，这是一种多功能系统，可以以一种可控和可实验访问的方式模拟大量凝聚态物质、核和天体物理模型。*到目前为止，寻找量子动力学普适性的尝试主要集中在相变附近的动力学，这些相变继承了相变本身的普适性(例如Kibble-Zurek机制)。这一提议采取了一种截然不同的方法，我称之为与奇点相关的普遍性，灵感来自于自然聚焦现象，即波中的普遍模式(彩虹、引力透镜、潮汐孔、反常波)可以在没有相变的情况下发生。事实上，虽然不是一目了然，但通过奇点的存在与相变有着深刻的联系。这项研究旨在了解这些联系。突变理论的一个显著结果是，一般奇点(那些没有特殊对称性的奇点)只能呈现出特定的几何形状。这就是奇点具有普遍性的原因。我们将使用量子波中的突变作为一种手段来分类它们的动力学，就像相变一样，每类突变遵循一组具有标度指数的标度定律。*我们将应用突变理论来寻找捕获在光学晶格中的原子、自旋系统中的原子以及通过模拟引力的长程力相互作用的原子的动力学中的共同模式。为了实现后一个目标，我们将设计在实验上可行的方案，在这些方案中，光学腔中的原子通过光中介的相互作用进行相互作用，这种相互作用延伸到整个腔的长度。实现如此远距离的原子间相互作用将为实验室天体物理学研究引力相互作用粒子的集体行为打开大门，这是目前所缺乏的。我们将进行的研究是基础的，适用于与波动动力学相关的广泛学科。一些将要研究的量子模型，如自旋链和耦合约瑟夫森结，已经在积极研究它们作为未来量子计算机的基础的潜力。加拿大在这一领域处于世界领先地位。了解它们行为的普遍模式将有助于这一努力。

One of the greatest goals of physics is to identify universal patterns of behaviour in natural phenomena. Some of the best known examples are phase transitions which are changes in an equilibrium state. Near the transition, systems with different microscopic compositions behave in a universal way, with quantities diverging according to common set of critical exponents. This proposal concerns universality in non-equilibrium dynamics, and specifically in quantum dynamics. In particular, we will investigate non-equilibrium dynamics in cold atomic gases which are versatile systems that can mimic a large number of condensed matter, nuclear and astrophysical models in a controllable and experimentally accessible way. ******Thus far, attempts to find universality in quantum dynamics have centred on dynamics near phase transitions that inherit their universality from the phase transition itself (e.g. Kibble-Zurek mechanism). This proposal takes a radically different approach, what I call universality associated with singularities, and is inspired by the phenomenon of natural focusing where universal patterns in waves (rainbows, gravitational lensing, tidal bores, freak waves) can occur without a phase transition. In fact, although not immediately apparent, there are deep connections with phase transitions through the presence of singularities. This research aims to understand these connections. It is a remarkable result of Catastrophe Theory that generic singularities (those that occur without special symmetries) can only take on certain geometric shapes. This is what gives singularities their universality. We will use catastrophes in quantum waves as a means to categorize their dynamics as, like phase transitions, each class of catastrophe obeys a set of scaling laws with scaling exponents.******We will apply catastrophe theory to look for common patterns in the dynamics of atoms trapped in optical lattices, spin systems, and atoms interacting via long range forces which mimic gravity. To achieve this latter goal we will design experimentally feasible schemes where atoms in optical cavities interact via light-mediated interactions that extend over the entire length of the cavity. Realizing such long range interatomic interactions would open the door to laboratory astrophysics studies of the collective behaviour of gravitationally interacting particles, something which is currently lacking.******The research we will undertake is fundamental and applicable across a broad range of disciplines related to wave dynamics. A number of the quantum models that will be studied, such as spin chains and coupled Josephson junctions, are already under active investigation for their potential to be the basis of a future quantum computer. Canada is a world leader in this field. Understanding universal patterns of their behaviour will aid this effort.