Non-equilibrium dynamics and turbulence in disordered Bose gas

玻色-爱因斯坦凝聚（BEC）的存在最初是由Satyendra Nath Bose预测的，后来由阿尔伯特·爱因斯坦在理论上发展。在占据宇宙大部分的气体中，粒子之间的台球式碰撞主导着控制动力学。据预测，在非常低但有限的温度下，一类被称为玻色子的粒子将落入它们的最低能级。在这里，玻色子不再像单个粒子那样运动。这些粒子不是相互碰撞，而是进入相同的量子态，几百万个玻色子表现得像一个巨大的原子。1995年，康奈尔和威曼将2000个铷原子减少到比绝对零度高不到1000亿分之一度。这一突破引起了人们对BEC实验的巨大兴趣，2001年，Cornell， Ketterle和Wieman因其对BEC的研究获得了诺贝尔奖。目前，在全球范围内，有数百种新颖的实验正在进行，在这个丰富多样的领域中发现了大量的结果。直到今天，bec仍然是一个非常活跃的研究领域，因为它们在实验上是高度可控的，具有广泛的潜在应用，从干涉测量和量子计算到理解中子星的行为，并且是量子力学的一个令人兴奋的测试平台。弱相互作用bec的一个显著特性是它们没有任何粘性效应，这意味着这些流体可以在不失去动能的情况下流动。这导致BEC气体被称为“超流体”。强迫超流体旋转导致多个量子漩涡的形成，其中一个规则的、有序的漩涡晶格是旋转BEC的基态。大型系统有许多涡旋，在低温下，涡旋在能量上更有利于存在于紧密结合的涡旋反涡旋对中，从而在系统中形成长程有序。在高温下，涡旋对脱离束缚，导致长程序的破坏。这种转变被称为Berezinskii-Kosterlitz-Thouless （BKT）转变，描述了系统从束缚涡旋对急剧转变为非束缚涡旋的临界温度。这与3D中的平滑过渡形成鲜明对比。重要的是，这些关于BKT跃迁的结果只适用于均匀的二维BEC，而不一定适用于被困旋转系统的情况，后者是一个更复杂的问题，因为基态是涡晶格。我们将应用数值技术来实现这个问题的新颖模拟，在非常大的系统的限制下，为了确定旋转参照系中BKT模拟的性质。在完成旋转参照系中BKT转换的项目后，我们将转向非平衡涡旋动力学的项目。本文的目的是建立新的理论模型来描述冷BEC的涡动力学，以研究具有多声波的热BEC。在观察无序势中热BEC的行为时，我们的目标是找到一种改进的点涡模型，例如带有随机噪声项的点涡模型。无序势中玻色子之间的无序和相互作用之间的联系是一个有趣的理论挑战，并将导致多体物理学中一系列引人入胜的结果。在研究这一问题时发展起来的理论技术和数值代码将用于研究通过点状失序势的流动向湍流的过渡。

The existence of Bose Einstein Condensate (BEC) was originally predicted by Satyendra Nath Bose, and was later developed theoretically by Albert Einstein. In gases - which occupy the majority of the universe - billiard ball like collisions between particles dominate the governing dynamics. At very low but finite temperatures, it was predicted that a class of particles called bosons would fall into their lowest energy levels. Here the bosons cease to behave like individual particles. Rather than bouncing off each other, the particles enter the same quantum state, and the millions of bosons present are behaving as one, single, giant atom. In 1995, Cornell and Wieman reduced 2000 rubidium atoms to less than 100 billionths of a degree above absolute zero. This breakthrough lead to huge interest in BEC experiments and, in 2001, Cornell, Ketterle and Wieman receives a Nobel prize for their studies on BEC. Presently, around the globe, there are hundreds of novel experiments taking place, unearthing a plethora of results in this rich and diverse field.To this day, BECs remain a very active area of research because they are highly experimentally controllable, have widespread potential applications from interferometry and quantum computing to understanding the behaviour of Neutron stars, and are an exciting testbed for quantum mechanics. A remarkable property of weakly interacting BECs is that they lack any viscous effects, meaning that these fluids can flow without losing kinetic energy. This has led to BEC gases being dubbed "superfluids". Forcing a superfluid to rotate leads to the formation of multiple quantum vortices, where a regular, ordered vortex lattice is the ground state of the rotating BEC. Large systems have many vortices, and at low temperatures it becomes energetically more favourable for vortices to exist in tightly bound vortex anti-vortex pairs, creating long range order in the system. At high temperature, vortex pairs unbind, resulting in the destruction of long-range-order. This transition, known as the Berezinskii-Kosterlitz-Thouless (BKT) transition, describes a critical temperature at which the system sharply changes from bound vortex pairs to unbound vortices. This is in stark contrast to the smooth transition seen in 3D. Importantly, these results on the BKT transition apply only to a uniform 2D BEC, and do not necessarily apply to the case of a trapped, rotating system, which is a more complicated problem as the ground state is a vortex lattice. We will apply numerical techniques to realise novel simulations of this problem, in the limit of very large systems, in order to determine the nature of a BKT analogue in a rotating frame of reference.On completion of the project on BKT transitions in a rotating frame of reference, we will then move to a project on non-equilibrium vortex dynamics. The aim of this will be to develop novel theoretical models for vortex dynamics of cold BECs described by a point-vortex model to study a hot BEC with many sound waves. In looking at the behaviour of a hot BEC in a disordered potential, we aim to find an improved point vortex model, e.g., a point vortex model with stochastic noise terms. The link between disorder and interactions between bosons in a disordered potential is an interesting theoretical challenge, and will lead to a fascinating array of results in many-body physics. The theoretical techniques and numerical codes developed in looking at this problem will then be used to study the transition to turbulence in flow through point-like disorder potentials.