Rigidity of escaping dynamics for transcendental entire functions

We prove an analog of Boettcher's theorem for transcendental entire functions in the Eremenko-Lyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are *quasiconformally equivalent* in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points which remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane. We also prove that this conjugacy is essentially unique. In particular, we show that an Eremenko-Lyubich class function f has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic Eremenko-Lyubich class functions f and g which belong to the same parameter space are conjugate on their sets of escaping points.Comment: 28 pages; 2 figures. Final version (October 2008). Various modificiations were made, including the introduction of Proposition 3.6, which was not formally stated previously, and the inclusion of a new figure. No major changes otherwis

Lossless State Detection of Single Neutral Atoms

We introduce lossless state detection of trapped neutral atoms based on cavity-enhanced fluorescence. In an experiment with a single 87-Rb atom, a hyperfine-state-detection fidelity of 99.4% is achieved in 85 microseconds. The quantum bit is interrogated many hundreds of times without loss of the atom while a result is obtained in every readout attempt. The fidelity proves robust against atomic frequency shifts induced by the trapping potential. Our scheme does not require strong coupling between the atom and cavity and can be generalized to other systems with an optically accessible quantum bit.Comment: 4 pages, 4 figure

Escape rate and Hausdorff measure for entire functions

The escaping set of an entire function is the set of points that tend to infinity under iteration. We consider subsets of the escaping set defined in terms of escape rates and obtain upper and lower bounds for the Hausdorff measure of these sets with respect to certain gauge functions.Comment: 24 pages; some errors corrected, proof of Theorem 2 shortene

Computational optimization of synthetic water channels.

Membranes for liquid and gas separations and ion transport are critical to water purification, osmotic energy generation, fuel cells, batteries, supercapacitors, and catalysis. Often these membranes lack pore uniformity and robustness under operating conditions, which can lead to a decrease in performance. The lack of uniformity means that many pores are non-functional. Traditional membranes overcome these limitations by using thick membrane materials that impede transport and selectivity, which results in decreased performance and increased operating costs. For example, limitations in membrane performance demand high applied pressures to deionize water using reverse osmosis. In contrast, cellular membranes combine high flux and selective transport using membrane-bound protein channels operating at small pressure differences. Pore size and chemistry in the cellular channels is defined uniformly and with sub-nanometer precision through protein folding. The thickness of these cellular membranes is limited to that of the cellular membrane bilayer, about 4 nm thick, which enhances transport. Pores in the cellular membranes are robust under operating conditions in the body. Recent efforts to mimic cellular water channels for efficient water deionization produced a significant advance in membrane function. The novel biomimetic design achieved a 10-fold increase in membrane permeability to water flow compared to commercial membranes and still maintained high salt rejection. Despite this success, there is a lack of understanding about why this membrane performs so well. To address this lack of knowledge, we used highperformance computing to interrogate the structural and chemical environments experienced by water and electrolytes in the newly created biomimetic membranes. We also compared the solvation environments between the biomimetic membrane and cellular water channels. These results will help inform future efforts to optimize and tune the performance of synthetic biomimetic membranes for applications in water purification, energy, and catalysis