Quantum theory and applications

量子力学基本上支配着微观尺度及以下的所有现象。我打算研究量子力学及其更复杂的相关量子场论中的几个问题。我的工作可以被认为是在两个领域：量子场论和应用，以及量子信息（前者比后者更重要）。我正在做三个项目，大致属于量子场论的范畴。第一个是对约瑟夫森效应的场论描述，约瑟夫森效应会在被绝缘体隔开的两个超导体之间产生电流。这个描述很有趣，因为它可以很容易地推广到一个更复杂的设置（一个被称为非阿贝尔序参量），正如我和我的同事所证明的那样。我打算加深我们对这种描述的理解，并找到非阿贝尔效应在真实的物理系统中的应用。第二个是研究被称为阿贝尔希格斯模型的玩具场论中的可能阶段，并添加了所谓的陈-西蒙斯项。（“玩具”模型不是因为它描述了一个现实的物理系统而被研究，而是因为它比现实模型更简单，所以更容易进行数学分析，希望这样的分析能揭示更现实的模型。在我们的例子中，玩具模型只存在于两个空间维度中。）研究一个模型的相结构是很有趣的，因为不同的相具有非常不同的性质;一个例子是夸克（质子、中子和其他不太熟悉的粒子的组成部分）之间相互作用模型中的禁闭跃迁。第三个项目是继续研究拓扑缺陷对假真空衰变的影响。假真空是一种经典稳定但通过量子力学隧穿不稳定的真空。虽然这似乎与现实相去甚远，但实际上它很像任何相变中发生的事情，特别是在早期宇宙中已知发生的相变中。我最近一直在研究这个问题，还有很多工作要做。两个项目研究量子力学问题，着眼于量子信息。在一个方面，我正在研究量子力学中被称为绝热近似的近似的有效性，该近似适用于依赖于时间但缓慢演化的系统。控制系统对依赖于时间的外部效应的反应是游戏的名称。这种控制在量子力学中几乎是必需的，但在量子信息中尤其如此，量子比特（量子力学信息单位）的操纵可以说是掌握的最重要任务。第二个项目是研究自旋晶格作为研究量子力学基础的工具，特别是贝尔定理。虽然第一个领域我已经研究了好几年，但第二个领域我最近才开始思考。粗略的想法是考虑一个有限的自旋晶格，通常的主角爱丽丝和鲍勃可以在自旋链或梯子的末端获得自旋。在这个模型中，中间自旋起着隐变量的作用，目标是研究Alice和Bob在这种情况下测量之间的相关性。

Quantum mechanics governs essentially all phenomena at microscopic scales and below. I intend to examine several problems in quantum mechanics and its more sophisticated relative, quantum field theory. My work can be thought of as being in two fields: quantum field theory and applications, and quantum information (the former weighted more heavily than the latter). I am working on three projects falling roughly in the category of quantum field theory. The first is a field theoretic description of a phenomenon known as the Josephson effect, which gives rise to current flow between two superconductors separated by an insulator. This description is interesting because it can easily be generalized to a more complicated setting (one with what is known as a non-Abelian order parameter), as colleagues and I have demonstrated. I intend to deepen our understanding of this description, and to find applications of the non-Abelian effect to real physical systems. The second is a study of the possible phases in a toy field theory known as the Abelian Higgs model, with the addition of a so-called Chern-Simons term. (A "toy" model is one that is studied not because it describes a realistic physical system, but rather because it is simpler than realistic models, so it is easier to analyze mathematically, the hope being that such an analysis will shed light on more realistic models. In our case, the toy model is one that lives in only two space dimensions.) Studying the phase structure of a model is interesting because different phases have very different properties; one example is the confinement transition in the model of interactions among quarks (the constituents of protons, neutrons, and other less familiar particles). The third project is to continue studying the effect of topological defects on false vacuum decay. A false vacuum is one that is stable classically but unstable via tunnelling quantum mechanically. While this might seem far removed from reality, it is actually much like what happens in any phase transition, and particularly in the phase transitions that were known to occur in the early universe. I have worked on this problem in the recent past, and there is much still to be done. Two projects study quantum mechanical issues, with an eye towards quantum information. In one, I am studying the validity of an approximation in quantum mechanics known as the adiabatic approximation, which applies to time-dependent, yet slowly-evolving, systems. Controlling the reaction of the system to time-dependent external effects is the name of the game. This control is necessary almost across the board in quantum mechanics, but it is particularly at the forefront in quantum information, where manipulation of qubits (quantum-mechanical units of information) is arguably the most important task to master. The second project is to study spin lattices as a tool for investigation the foundations of quantum mechanics, and specifically Bell's Theorem. While the first of these is an area I have worked on for several years, the second is something I have begun to think about only recently. The rough idea is to consider a finite lattice of spins, the usual protagonists Alice and Bob having access to spins at the ends of a spin chain or ladder. In this model, the intermediate spins play the role of hidden variables, and the objective is to study correlations between measurements made by Alice and Bob in this setting.