Topics in Ultracold Fermi Gases and Topological Quantum Computing

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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.