Optoelectromechanical transducer: reversible conversion between microwave and optical photons

Quantum states encoded in microwave photons or qubits can be effectively manipulated, whereas optical photons can be coherently transferred via optical fibre and waveguide. The reversible conversion of quantum states between microwave and optical photons will hence enable the distribution of quantum information over long distance and significantly improve the scalability of hybrid quantum systems. Owning to technological advances, mechanical resonators couple to quantum devices in distinctly different spectral range with tunable coupling, and can serve as a powerful interface to connect those devices. In this review, we summarize recent theory and experimental progress in the coherent conversion between microwave and optical fields via optoelectromechanical transducers. The challenges and perspectives in achieving single-photon-level quantum state conversion will also be discussed.Comment: Review article, to appear in Annalen der Physik; 7 figure

The Crucial Constants in the Exponential-type Error Estimates for Multiquadric Interpolations

It's well known that in the high-level error bound for multiquadric interpolation there is a crucial constant lambda lying between 0 and 1 which connot be calculated or even approximated. The purpose of this paper is to answer this question.Comment: 14 pages, approximation theory, radial basis functio

The Shape Parameter in the Shifted Surface Spline

There is a constant c contained in the famous radial basis function shifted surface spline. It's called shape parameter. RBF people only know that this constant is very influential, while its optimal choice is unknown. This paper presents criteria of its optimal choice.Comment: 10 page

An extension of gas-kinetic BGK Navier-Stokes scheme to multidimensional astrophysical magnetohydrodynamics

The multidimensional gas-kinetic scheme for the Navier-Stokes equations under gravitational fields [J. Comput. Phys. 226 (2007) 2003-2027] is extended to resistive magnetic flows. The non-magnetic part of the magnetohydrodynamics equations is calculated by a BGK solver modified due to magnetic field. The magnetic part is treated by the flux splitting method based gas-kinetic theory [J. Comput. Phys. 153 (1999) 334-352 ], using a particle distribution function constructed in the BGK solver. To include Lorentz force effects into gas evolution stage is very important to improve the accuracy of the scheme. For some multidimensional problems, the deviations tangential to the cell interface from equilibrium distribution are essential to keep the scheme robust and accurate. Besides implementation of a TVD time discretization scheme, enhancing the dynamic dissipation a little bit is a simply and efficient way to stabilize the calculation. One-dimensional and two-dimensional shock waves tests are calculated to validate this new scheme. A three-dimensional turbulent magneto-convection simulation is used to show the applicability of current scheme to complicated astrophysical flows.Comment: 24 pages, 13 figures. submitted to JC

The high-level error bound for shifted surface spline interpolation

Radial function interpolation of scattered data is a frequently used method for multivariate data fitting. One of the most frequently used radial functions is called shifted surface spline, introduced by Dyn, Levin and Rippa in \cite{Dy1} for $R^{2}$. Then it's extended to $R^{n}$ for $n\geq 1$. Many articles have studied its properties, as can be seen in \cite{Bu,Du,Dy2,Po,Ri,Yo1,Yo2,Yo3,Yo4}. When dealing with this function, the most commonly used error bounds are the one raised by Wu and Schaback in \cite{WS}, and the one raised by Madych and Nelson in \cite{MN2}. Both are $O(d^{l})$ as $d\to 0$, where $l$ is a positive integer and $d$ is the fill-distance. In this paper we present an improved error bound which is $O(\omega^{1/d})$ as $d\to 0$, where $0<\omega <1$ is a constant which can be accurately calculated.Comment: 14 pages, radial basis functions, approximation theory. arXiv admin note: text overlap with arXiv:math/060115

The Mystery of the Shape Parameter II

In this paper we present criteria for the choice of the shape parameter c contained in the famous radial function multiquadric. It may be of interest to RBF people and all people using radial basis functions to do approximation.Comment: 15 figure

The Mystery of the Shape Parameter IV

In this paper we present a set of criteria for the choice of the shape parameter c contained in multiquadrics.Comment: 12 pages, 15 figure

A Smooth and Compactly Supported Radial Function

In the field of radial basis functions mathematicians have been endeavouring to find infinitely differentiable and compactly supported radial functions. This kind of functions are extremely important for some reasons. First, its computational properties will be very good since it's compactly supported. Second, its error bound will converge very fast since it's infinitely differentiable. However there is hitherto no such functions which can be expressed in a simple form. This is a famous question. The purpose of this paper is to answer this question.Comment: 4 pages, radial basis functions, approximation theor

On the High-Level Error Bound for Gaussian Interpolation

It's well-known that there is a very powerful error bound for Gaussians put forward by Madych and Nelson in 1992. It's of the form$(Cd)^{\frac{c}{d}}\left\Vert f\right\Vert_{h}$ where $C,c$ are constants, $h$ is the Gaussian function, $$ is the interpolating function, and d is called fill distance which, roughly speaking, measures the spacing of the points at which interpolation occurs. This error bound gets small very fast as $d\to 0$. The constants $C$ and $c$ are very sensitive. A slight change of them will result in a huge change of the error bound. The number $c$ can be calculated as shown in [9]. However, $C$ cannot be calculated, or even approximated. This is a famous question in the theory of radial basis functions. The purpose of this paper is to answer this question.Comment: approximation theory,radial basis function

Solving Poisson equations by the MN-curve approach

In this paper we apply the newly born choice theory of the shape parameters contained in the smooth radial basis functions to solve Poisson equations. Some people complain that Luh's choice theory, based on harmonic analysis, is mathematically complicated and applies only to function interpolations. Here we aim at presenting an easily accessible approach to solving differential equations with the choice theory which proves to be successful, not only by its easy accessibility, but also by its striking accuracy and efficiency.Comment: 9 pages, 11 figure