Non-equilibrium dynamics and turbulence in disordered Bose gas

玻色爱因斯坦凝聚体(BEC)的存在最早是由Satyendra Nath Bose预言的，后来由阿尔伯特·爱因斯坦从理论上提出。在气体中--它占据了宇宙的大部分--粒子之间类似台球的碰撞主导着主导动力学。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降至最低能级。在这里，玻色子的行为不再像单个粒子一样。这些粒子不是相互反弹，而是进入相同的量子状态，存在的数百万玻色子表现为一个单一的巨大原子。1995年，康奈尔和威曼将2000个Rb原子减少到绝对零度以上不到千亿分之一。这一突破引发了人们对BEC实验的巨大兴趣，2001年，康奈尔、凯特勒和威曼因对BEC的研究而获得诺贝尔奖。目前，在全球范围内，有数百个新的实验正在进行，在这个丰富而多样的领域发现了大量的结果。直到今天，BEC仍然是一个非常活跃的研究领域，因为它们具有高度的实验可控性，从干涉测量和量子计算到理解中子星的行为具有广泛的潜在应用，并且是量子力学的一个令人兴奋的试验台。弱相互作用的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损失动能的情况下流动。这导致BEC气体被称为“超流体”。强迫超流体旋转会导致多量子涡旋的形成，其中规则有序的涡旋晶格是旋转的BEC的基态。大系统有许多涡旋，在低温下，涡旋以紧密结合的涡反涡对的形式存在在能量上更有利，从而在系统中产生长程有序。在高温下，涡对解开，导致长程有序的破坏。这种转变被称为Berezinskii-Kosterlitz-Thouless(BKT)转变，描述了系统从束缚涡对急剧变化到非束缚涡的临界温度。这与3D中看到的平稳过渡形成了鲜明的对比。重要的是，这些关于BKT跃迁的结果只适用于均匀的2DBEC，而不一定适用于囚禁、旋转系统的情况，这是一个更复杂的问题，因为基态是涡旋晶格。我们将应用数值技术在超大型系统的限制下实现对这个问题的新颖模拟，以确定旋转坐标系中BKT模拟的性质。在旋转坐标系中BKT转变的项目完成后，我们将转移到非平衡涡旋动力学项目。其目的是发展一种新的理论模型来描述冷BEC的旋涡动力学，以研究具有多个声波的热BEC。在研究热BEC在无序势中的行为时，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。无序和无序势中玻色子之间的相互作用之间的联系是一个有趣的理论挑战，将导致多体物理学中一系列迷人的结果。在研究这一问题时发展的理论技术和数值代码将被用来研究流动中通过点状无序势向湍流的转变。

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.