Optimal time decay of the non cut-off Boltzmann equation in the whole space

In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space \threed_x with \DgE. We use the existence theory of global in time nearby Maxwellian solutions from \cite{gsNonCutA,gsNonCut0}. It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption \cite{MR677262,MR2847536}. For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of O(t^{-\frac{\Ndim}{2}+\frac{\Ndim}{2r}}) in the L^2_\vel(L^r_x)-norm for any $2\leq r\leq \infty$.Comment: 31 pages, final version to appear in KR

Comment on "Quantum Phase Slips and Transport in Ultrathin Superconducting Wires"

In a recent Letter (Phys. Rev. Lett.78, 1552 (1997) ), Zaikin, Golubev, van Otterlo, and Zimanyi criticized the phenomenological time-dependent Ginzburg-Laudau model which I used to study the quantum phase-slippage rate for superconducting wires. They claimed that they developed a "microscopic" model, made qualitative improvement on my overestimate of the tunnelling barrier due to electromagnetic field. In this comment, I want to point out that, i), ZGVZ's result on EM barrier is expected in my paper; ii), their work is also phenomenological; iii), their renormalization scheme is fundamentally flawed; iv), they underestimated the barrier for ultrathin wires; v), their comparison with experiments is incorrect.Comment: Substantial changes made. Zaikin et al's main result was expected from my work. They underestimated tunneling barrier for ultrathin wires by one order of magnitude in the exponen

Correlations in interference and diffraction

Quantum formalism of Fraunhofer diffraction is obtained. The state of the diffraction optical field is connected with the state of the incident optical field by a diffraction factor. Based on this formalism, correlations of the diffraction modes are calculated with different kinds of incident optical fields. Influence of correlations of the incident modes on the diffraction pattern is analyzed and an explanation of the ''ghost'' diffraction is proposed.Comment: 16 pages, 2 figures, Latex, to appear in J. Mod. Op

Topological Properties of Spatial Coherence Function

Topology of the spatial coherence function is considered in details. The phase singularity (coherence vortices) structures of coherence function are classified by Hopf index and Brouwer degree in topology. The coherence flux quantization and the linking of the closed coherence vortices are also studied from the topological properties of the spatial coherence function.Comment: 9 page

Anharmonicity Induced Resonances for Ultracold Atoms and their Detection

When two atoms interact in the presence of an anharmonic potential, such as an optical lattice, the center of mass motion cannot be separated from the relative motion. In addition to generating a confinement-induced resonance (or shifting the position of an existing Feshbach resonance), the external potential changes the resonance picture qualitatively by introducing new resonances where molecular excited center of mass states cross the scattering threshold. We demonstrate the existence of these resonances, give their quantitative characterization in an optical superlattice, and propose an experimental scheme to detect them through controlled sweeping of the magnetic field.Comment: 6 pages, 5 figures; expanded presentatio

A scheme for demonstration of fractional statistics of anyons in an exactly solvable model

We propose a scheme to demonstrate fractional statistics of anyons in an exactly solvable lattice model proposed by Kitaev that involves four-body interactions. The required many-body ground state, as well as the anyon excitations and their braiding operations, can be conveniently realized through \textit{dynamic}laser manipulation of cold atoms in an optical lattice. Due to the perfect localization of anyons in this model, we show that a quantum circuit with only six qubits is enough for demonstration of the basic braiding statistics of anyons. This opens up the immediate possibility of proof-of-principle experiments with trapped ions, photons, or nuclear magnetic resonance systems.Comment: 4 pages, 3 figure

Evolution of the Chern-Simons Vortices

Based on the gauge potential decomposition theory and the $\phi$-mapping theory, the topological inner structure of the Chern-Simons-Higgs vortex has been showed in detail. The evolution of CSH vortices is studied from the topological properties of the Higgs scalar field. The vortices are found generating or annihilating at the limit points and encountering, splitting or merging at the bifurcation points of the scalar field $\phi .$Comment: 10 pages, 10 figure

Suppression of Phase Decoherence in a Single Atomic Qubit

We study the suppression of noise-induced phase decoherence in a single atomic qubit by employing pulse sequences. The atomic qubit is composed of a single neutral atom in a far-detuned optical dipole trap and the phase decoherence may originate from the laser intensity and beam pointing fluctuations as well as magnetic field fluctuations. We show that suitable pulse sequences may prolongate the qubit coherence time substantially as comparing to the conventional spin echo pulse.Comment: 4 pages, 3 figure

A new topological aspect of the arbitrary dimensional topological defects

We present a new generalized topological current in terms of the order parameter field $\vec \phi$ to describe the arbitrary dimensional topological defects. By virtue of the $$-mapping method, we show that the topological defects are generated from the zero points of the order parameter field $\vec \phi$, and the topological charges of these topological defects are topological quantized in terms of the Hopf indices and Brouwer degrees of $\phi$-mapping under the condition that the Jacobian $$. When $J(\frac \phi v)=0$, it is shown that there exist the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, we detail the bifurcation of generalized topological current and find different directions of the bifurcation. The arbitrary dimensional topological defects are found splitting or merging at the degenerate point of field function $\vec \phi$ but the total charge of the topological defects is still unchanged.Comment: 24 pages, 10 figures, Revte