Topics in Ultracold Fermi Gases and Topological Quantum Computing

该奖项支持理论研究和教育，将探索凝聚态理论的两个独立领域。第一个领域涉及到非常简并的费米或玻色系统理论中的各种问题，特别是涉及到它们可能接受实验输入的超冷碱气系统。具体地说，将研究(a)在含有磁性杂质的s波超导体中是否发生一阶相变；(b) pseudo-Couette ?偶极力作用下的流动；(c) a ?p+ip？费米超流体;(d)玻色凝聚系统中持续电流的衰减。第二个领域将评估拓扑量子计算在强层状超导体钌酸锶中实现的前景。要研究的问题是：(a)该系统中同质性中可能的域类型所引起的复杂性；(b)热力学稳定存在的一般条件？halfquantum漩涡?在p+ip费米超流体中，以及在超流体氦的a相中未能观测到它们的可能原因；(c)一个能维持的系统的多体基态波函数的性质？马约喇纳费米子吗?以及后者存在的一般条件。简并费米子系统中库珀对的形成问题早在50多年前就提出了。尽管在过去的半个世纪里在平均场水平上取得了进展，但许多问题，即使是在简单的s波配对的情况下，仍然没有答案。例如，在什么情况下（如果有的话），磁性杂质会在低温下诱导一阶相变？在各向异性配对的情况下，一些甚至更基本的问题，如被认为与超流氦的A相相对应的状态的固有角动量，仍然没有得到明确的答案。现在有一个真正的希望，在实验中回答这些问题中的一些在超冷碱气体，这个项目将为这样的实验提供一个理论框架。在过去几年中，量子信息领域出现了最有趣的想法之一，即拓扑量子计算，该项目也将讨论拓扑量子计算的具体实现。不仅对所讨论的拓扑非平凡类系统的基本激发，而且对多体基态的性质进行了明确的分析，同时还将进行更多的分析。螺母和螺栓？评估由于所选择的特定系统的物理特性而产生的约束。该研究项目将为研究生在凝聚态理论研究前沿的分析和数学方法方面的教育和培训提供极好的机会。该奖项支持两大领域的理论研究和教育。一种是利用被光波捕获的极低温原子的潜力来解决材料研究中提出的概念问题，例如超导体，它在足够低的温度下发展出一种合作电子状态，通常以其携带电流而不耗散的能力而闻名。这些基本问题很难在材料中解决，但可能会屈服于对被困冷原子系统的实验。第二个领域涉及对一种不寻常的超导体中的涡流的研究，理论上已经证明，人们可以通过相互移动涡流来执行基本的计算操作。PI将通过仔细研究超导电子状态和其他缺陷（除了涡旋）的影响，研究这种形式的量子计算（操纵量子力学状态来执行计算）在理论上可能实现的程度。该研究项目将为研究生在凝聚态理论研究前沿的分析和数学方法方面的教育和培训提供极好的机会。

TECHNICAL SUMMARYThis award supports theoretical research and education that will explore two separate realms of condensed matter theory. The first area concerns various questions in the theory of very degenerate Fermi or Bose systems, with particular reference to the ultracold alkali gas systems where they may receive experimental input. Specifically, studies will be made of (a) the occurrence or not of a first- order phase transition in an s-wave superconductor containing magnetic impurities; (b) ?pseudo-Couette? flow in the presence of dipolar forces; (c) the angular momentum of a ?p+ip? Fermi superfluid; and (d) the decay of persistent currrents in a Bose-condensed system.The second area will be evaluating the prospects for the implementation of topological quantum computation in the strongly layered superconductor strontium ruthenate. The questions to be investigated are (a) the complications caused by the probable domain-type in homogeneities in this system; (b) the general conditions for the existence of thermodynamically stable ?halfquantum vortices? in a p+ip Fermi superfluid, and possible reasons for the failure to observe them in the A phase of superfluid helium; and (c) the nature of the many-body groundstate wave function of a system that can sustain ?Majorana fermions? and the general conditions for the latter to exist.The problem of the formation of Cooper pairs in a system of degenerate fermions was posed more than 50 years ago. Although progress has been made at the mean-field level in the past half-century, many questions, even in the case of simple s-wave pairing, remain unanswered. For example, under what circumstances, if any, do magnetic impurities induce a first-order phase transition at low temperatures? In the case of anisotropic pairing, some even more basic questions, such as the intrinsic angular momentum of the state believed to correspond to the A phase of superfluid helium, have still not received a definitive answer. There is now a real hope of answering some of these questions experimentally in the ultracold alkali gases, and this project will provide a theoretical framework for such experiments.A specific implementation of one of the most intriguing ideas to have emerged in the field of quantum information over the last few years, namely topological quantum computation, will also be addressed in this project. An explicit analysis not only of the elementary excitations of the topologically nontrivial class of systems in question but also of the nature of the many-body groundstate will be undertaken, as well as a more ?nuts and bolts? assessment of the constraints due to the physical properties of the specific system of choice.This research program will provide excellent opportunities for the education and training of graduate students in analytical and mathematical methods at the cutting edge of condensed matter theory research. NON-TECHNICAL SUMMARYThis award supports theoretical research and education in two broad areas. One exploits the potential of atoms at very low temperature trapped by light waves to address conceptual issues raised by the study of materials, such as superconductors which develop a cooperative electronic state at sufficiently low temperatures often noted for its ability to carry electric current without dissipation. These fundamental questions are difficult to resolve in materials, but might succumb to experiments on systems of trapped cold atoms. The second area involves the study of vortices in an unusual kind of superconductor, where it has been theoretically shown that one can carry out the fundamental operations of computing by moving vortices around each other. The PI will study the extent to which this form of quantum computing, the manipulation of a quantum mechanical state to perform computation, is theoretically possible through a careful study of the superconducting electronic state and the effects of other defects, besides vortices. This research program will provide excellent opportunities for the education and training of graduate students in analytical and mathematical methods at the cutting edge of condensed matter theory research.