Non-equilibrium dynamics and turbulence in disordered Bose gas

玻色-爱因斯坦凝聚体（BEC）的存在最初由Satyendra Nath Bose预言，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主导动力学。在非常低但有限的温度下，预测一类称为玻色子的粒子将落入它们的最低能级。在这里，玻色子不再表现得像单个粒子。而不是相互反弹，粒子进入相同的量子态，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个铷原子减少到绝对零度以上不到100亿分之一度。这一突破引起了人们对BEC实验的极大兴趣，2001年，康奈尔大学、凯特尔和威曼因对BEC的研究而获得诺贝尔奖。目前，在地球仪上，有数百个新的实验正在进行，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，因为它们是高度实验可控的，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试验平台。弱相互作用BEC的一个显着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”。迫使超流体旋转会导致多个量子涡旋的形成，其中规则、有序的涡旋晶格是旋转BEC的基态。大型系统有许多涡旋，在低温下，涡旋以紧密结合的涡旋反涡旋对的形式存在，在系统中产生长程有序，这在能量上变得更有利。在高温下，涡旋对解束缚，导致长程有序的破坏。这种转变被称为Berezinskiii-Kosterlitz-无涡（BKT）转变，描述了系统从束缚涡对急剧变化为非束缚涡的临界温度。这与3D中看到的平滑过渡形成鲜明对比。重要的是，BKT跃迁的这些结果仅适用于均匀的2D BEC，并且不一定适用于被捕获的旋转系统的情况，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来实现这个问题的新的模拟，在非常大的系统的限制，以确定BKT类似物在旋转的参考框架的性质。在旋转的参考框架中的BKT过渡项目完成后，我们将移动到一个项目的非平衡涡动力学。这将是一个点涡模型描述的冷BEC的涡动力学发展的新的理论模型，以研究一个热的BEC与许多声波。在研究无序势中热BEC的行为时，我们的目标是找到一个改进的点涡模型，例如，带随机噪声项的点涡模型。无序势中的无序和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，并将导致多体物理学中一系列迷人的结果。在这个问题上的理论技术和数值代码开发，然后将被用来研究通过点状无序势流的湍流过渡。

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.