The thick-thin decomposition and the bilipschitz classification of normal surface singularities

We describe a natural decomposition of a normal complex surface singularity $(X,0)$ into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts. By refining the thick-thin decomposition, we then give a complete description of the intrinsic bilipschitz geometry of $(X,0)$ in terms of its topology and a finite list of numerical bilipschitz invariants.Comment: Minor corrections. To appear in Acta Mathematic

Atomic wave packet dynamics in finite time-dependent optical lattices

Atomic wave packets in optical lattices which are both spatially finite and time-dependent exhibit many striking similarities with light pulses in photonic crystals. We analytically characterize the transmission properties of such a potential geometry for an ideal gas in terms of a position-dependent band structure. In particular, we find that at specific energies, wave packets at the center of the finite lattice may be enclosed by pairs of band gaps. These act as mirrors between which the atomic wave packet is reflected, thereby effectively yielding a matter wave cavity. We show that long trapping times may be obtained in such a resonator and investigate the collapse and revival dynamics of the atomic wave packet by numerical evaluation of the Schr\"odinger equation

Versatile transporter apparatus for experiments with optically trapped Bose-Einstein condensates

We describe a versatile and simple scheme for producing magnetically and optically-trapped Rb-87 Bose-Einstein condensates, based on a moving-coil transporter apparatus. The apparatus features a TOP trap that incorporates the movable quadrupole coils used for magneto-optical trapping and long-distance magnetic transport of atomic clouds. As a stand-alone device, this trap allows for the stable production of condensates containing up to one million atoms. In combination with an optical dipole trap, the TOP trap acts as a funnel for efficient loading, after which the quadrupole coils can be retracted, thereby maximizing optical access. The robustness of this scheme is illustrated by realizing the superfluid-to-Mott insulator transition in a three-dimensional optical lattice

Spin squeezing, entanglement and quantum metrology with Bose-Einstein condensates

Squeezed states, a special kind of entangled states, are known as a useful resource for quantum metrology. In interferometric sensors they allow to overcome the "classical" projection noise limit stemming from the independent nature of the individual photons or atoms within the interferometer. Motivated by the potential impact on metrology as wells as by fundamental questions in the context of entanglement, a lot of theoretical and experimental effort has been made to study squeezed states. The first squeezed states useful for quantum enhanced metrology have been proposed and generated in quantum optics, where the squeezed variables are the coherences of the light field. In this tutorial we focus on spin squeezing in atomic systems. We give an introduction to its concepts and discuss its generation in Bose-Einstein condensates. We discuss in detail the experimental requirements necessary for the generation and direct detection of coherent spin squeezing. Two exemplary experiments demonstrating adiabatically prepared spin squeezing based on motional degrees of freedom and diabatically realized spin squeezing based on internal hyperfine degrees of freedom are discussed.Comment: Phd tutorial, 23 pages, 17 figure

An introduction to Lipschitz geometry of complex singularities

The aim of this paper to introduce the reader to a recent point of view on the Lipschitz classifications of complex singularities. It presents the complete classification of Lipschitz geometry of complex plane curves singularities and in particular, it introduces the so-called bubble trick and bubble trick with jumps which are key tools to study Lipschitz geometry of germs. It describes also the thick-thin decomposition of a normal complex surface singularity and built two geometric decompositions of a normal surface germ into standard pieces which are invariant by respectively inner and outer bilipschitz homeomorphisms. This leads in particular to the complete classification of Lipschitz geometry for the inner metric.Comment: 50 pages, 36 figure

Extracting density-density correlations from in situ images of atomic quantum gases

We present a complete recipe to extract the density-density correlations and the static structure factor of a two-dimensional (2D) atomic quantum gas from in situ imaging. Using images of non-interacting thermal gases, we characterize and remove the systematic contributions of imaging aberrations to the measured density-density correlations of atomic samples. We determine the static structure factor and report results on weakly interacting 2D Bose gases, as well as strongly interacting gases in a 2D optical lattice. In the strongly interacting regime, we observe a strong suppression of the static structure factor at long wavelengths.Comment: 15 pages, 5 figure

Search reduction in hierarchical distributed problem solving

Knoblock and Korf have determined that abstraction can reduce search at a single agent from exponential to linear complexity (Knoblock 1991; Korf 1987). We extend their results by showing how concurrent problem solving among multiple agents using abstraction can further reduce search to logarithmic complexity. We empirically validate our formal analysis by showing that it correctly predicts performance for the Towers of Hanoi problem (which meets all of the assumptions of the analysis). Furthermore, a powerful form of abstraction for large multiagent systems is to group agents into teams, and teams of agents into larger teams, to form an organizational pyramid. We apply our analysis to such an organization of agents and demonstrate the results in a delivery task domain. Our predictions about abstraction's benefits can also be met in this more realistic domain, even though assumptions made in our analysis are violated. Our analytical results thus hold the promise for explaining in general terms many experimental observations made in specific distributed AI systems, and we demonstrate this ability with examples from prior research.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42828/1/10726_2005_Article_BF01384251.pd