The tripartite entanglement in a phase-mismatched frequency non-degenerated optical parametric amplifier

We present an analytical solution for Fokker–Planck equation of phase-mismatched frequency non-degenerated optical parametric amplifier (FNOPA). We investigate the continuous-variable (CV) tripartite entanglement characteristic of FNOPA working at phase-matched and phase-mismatched condition. From below threshold to above threshold, we numerically calculate the dependence of CV tripartite entanglement on relative loss parameter η and phase-mismatched factor Δω. The larger η and the smaller Δω can bring a good entanglement.     

Squeezed displaced fock states and non-diagonal P representation

Abstract

An R representation with a squeezed displaced-Fock-states (SDFSs) basis is introduced. A non-diagonal P representation for the density operator with SDFS basis is defined. The special cases for the fields, chaotic and laser, are calculated. The Pegg–Barnett phase distribution is given and compared with the radial integration of the generalized P function. The Fokker–Planck equation for the damped harmonic oscillator is considered, and its steady-state solution is given.     

A technique note of exact stationary solutions on nonlinear systems under stochastic parametric excitations

In this paper, a systematic procedure is developed to obtain the stationary probability density function for the response of a general nonlinear system under parametric and external Gaussian white noise excitations. In Ref. [11], nonlinear function of system was expressed to the polynomial formula. The nonlinear system described here has the following form: x¨ + g(x, x˙) = k ₁ξ₁(t)+ k ₂ xξ₂(t), where g(x,x˙) = ∑ _{i = 0} ^∞ g _i (x)x˙ ⁱ and ξ₁,ξ₂ are Gaussian white noises. Thus, this paper is a generalization for the results studied in Ref. [11]. The reduced Fokker–Planck (FP) equation is employed to get the governing equation of the probability density function. Based on this procedure, the exact stationary probability densities of many nonlinear stochastic systems are obtained, and it is shown that some of the exact stationary solutions described in the literature are only particular cases of the presented generalized results.     

Parametric oscillation with squeezed vacuum reservoirs

Employing the quantum Hamiltonian describing the interaction of a two-mode light (signal–idler modes) generated by a non-degenerate parametric oscillator (NDPO) with two uncorrelated squeezed vacuum reservoirs (USVR), we derive the master and the Fokker–Planck equations. The corresponding Fokker–Planck equation for the Q-function is then solved employing a propagator method developed by K. Fesseha [J. Math. Phys. 33 2179 (1992)]. Making use of this Q-function, we calculate the quadrature fluctuations of the optical system. From these results we infer that the signal–idler modes are in squeezed states. When the NDPO operates below threshold we show that, for a large squeezing parameter, a squeezing amounting to a noise suppression approaching 100% below the vacuum level in one of the quadratures can be achieved.     

The fundamental solution of a turbulent kinetic theory

Abstract

The fundamental solution of a Fokker‐Planck type kinetic equation governing the probability density function of fluid elements in velocity space was obtained. The kinetic equation was treated as linear by assuming that the ensemble means appeared in the kinetic equation as known quantities with respect to the pdf (probability density function). By utilizing subsidiary transformations, the kinetic equation was transformed into a simpler diffusion type equation. Further integral transformations were performed on this diffusion type equation and the fundamental solution satisfying the instantaneous point source conditions in phase space was then established.     

Nonlinear dynamics in superfluid films

We have calculated the first non-linear corrections to the period shift andQ factor of a torsional oscillator containing a film of helium. To do so we have solved the Fokker Planck equation for the distribution of vortex pairs in the presence of an oscillating superfluid velocity field. The non-linear response is predicted to be proportional tov _n ² for low values of the normal fluid velocity,v _n, which is in good agreement with the experimental results of Adams and Glaberson. For temperatures above the Kosterlitz Thouless transition temperature,T _KT, the ratio of the correlation length ₊ to the diffusion length becomes important in describing the temperature dependence of the non-linear response. We predict how the non-linear response depends on this ratio in the region where the ratio is large.     

The effect of a homogeneous cylindrical inlay on cracks in the doubly-periodic complete plane strain problem

The effect of a homogeneous cylindrical inlay on cracks in the doubly-periodic complete plane strain (CPS) problem is investigated in this paper. By employing the solutions of doubly quasi-periodic and doubly-periodic Riemann boundary value problems for analytic functions we obtain the general solution in closed form. And approximate analytical expression of the stress intensity factors, which are identical with the known results, are obtained when there are no holes or gaskets in the periodic parallelogram and e ₃=0, F _k=0, T _k=0, k=1,2, with constant load applied at the edges of the crack.     

Potential flow along a wavy wall and transonic controversy

Summary The flow of an inviscid, compressible, perfect gas along a sinus-shaped wall is used as a model to shed light into the long-standing transonic controversy. The solution of the small-disturbance approximation for the velocity potential is developed as a formal series in the similarity parameterk. Forty terms ink were obtained, delegating the computational work to a computer. The coefficients of the maximal-speed series turn out to be moments with positive weight on a finite interval of support. This implies % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWraaSqabe% aacaWG1baaaGqacOGaa8xBaiaa-fgacaWF4bGaeyypa0ZaaabCaeaa% caWGHbWaaSbaaSqaaiaad6gaaeqaaOGaam4AamaaCaaaleqabaGaam% OBaaaakiabg2da9maapedabaWaaSaaaeaacaWGKbGaam4DaiaacIca% caWG4bGaaiykaaqaaiaaigdacqGHsislcaWGRbGaamiEaaaaaSqaai% aadggaaeaacaWGIbaaniabgUIiYdaaleaacaWGUbGaeyypa0JaaGim% aaqaaiabg6HiLcqdcqGHris5aaaa!5264!\[{}^umax = \sum\limits_{n = 0}^\infty {a_n k^n = \int_a^b {\frac{{dw(x)}}{{1 - kx}}} } \]The solution of the classical moment problem, i.e. the recovering of the weight distribution, shows that-a=b=1/k _c, withk _c as the critical value of the parameter, at which the flow first becomes sonic.k _chas been determined as 0.8253, the four digits being regarded as definite. It follows thatu _max as a function ofk is analytic on (-k _c, k_c) and has exactly two singularities located on the real axis atk=–k _c andk=k _c. It is known from the theory of moments that there does not exist an analytic continuation on the real axis exceeding the interval. This means that the velocity depends analytically on the parameter as long as the local velocity of sound is nowhere reached, but that the exceeding of it is marked by termination of analyticity and, to this degree, is critical. Thus, the view that there will not exist a transonic potential flow having neighbour solutions is supported. The smallness of the weight distribution at the interval ends (at least atk=–k _c it is even exponentially small, giving an exponentially small singularity ofu _max) is the key to explaining, within the scope of the inviscid model, the shockless exceeding of the critical value by some per cent seen in numerical calculations and experiments.     

Squeezing spectra from s -ordered quasiprobability distributions: application to dispersive optical bistability

It is well known that the squeezing spectrum of the field exiting a nonlinear cavity can be directly obtained from the fluctuation spectrum of normally ordered products of creation and annihilation operators of the cavity mode. In this article we show that the output field squeezing spectrum can be derived also by combining the fluctuation spectra of any pair of s -ordered products of creation and annihilation operators. The interesting result is that the spectrum obtained in this way from the linearized Langevin equations is exact, and this occurs in spite of the fact that no s -ordered quasiprobability distribution verifies a true Fokker–Planck equation; that is, the Langevin equations used for deriving the squeezing spectrum are not exact. The (linearized) intracavity squeezing obtained from any s -ordered distribution is also exact. These results are exemplified in the problem of dispersive optical bistability.     

Intracavity Optical Bistability Driven by Squeezed Light from a Parametric Amplifier

Abstract

We describe the theory of optical bistability when atoms are collectively excited within the cavity of a parametric oscillator. Both optical bistability and parametric amplification can squeeze significantly the cavity-field quantum noise. When they are coupled together we find significant changes both on the mean value bistability and on the spectrum of squeezing as the parametric coupling increases. These are calculated directly from the appropriate master equation for the density matrix using a quantum distribution function (positive P) to develop Fokker—Planck and Ito equations.     

The three isentropic exponents of wet steam

Real gas isentropic changes may be described using the three well known ideal gas relations,pv ^k =const, p^(1−k)T^k=const andTv ^k-1=const, where exponent k has for each equation a different value k_{p, ν}, k_{p, T} and k_{T, ν} respectively. In this paper the three isentropic exponents theory for real gases is extended to the two phase region. As an application the numerical values of the three exponents are calculated for wet steam.     

Transient Fokker–Planck–Kolmogorov equation solved with smoothed particle hydrodynamics method

Probabilistic theories aim at describing the properties of systems subjected to random excitations by means of statistical characteristics such as the probability density function ψ (pdf). The time evolution of the pdf of the response of a randomly excited deterministic system is commonly described with the transient Fokker–Planck–Kolmogorov (FPK) equation. The FPK equation is a conservation equation of a hypothetical or abstract fluid, which models the transport of probability. This paper presents a generalized formalism for the resolution of the transient FPK equation by using the well‐known mesh‐free Lagrangian method, smoothed particle hydrodynamics). Numerical implementation shows notable advantages of this method in an unbounded state space: (1) the conservation of total probability in the state space is explicitly written; (2) no artifact is required to manage far‐field boundary conditions; (3) the positivity of the pdf is ensured; and (4) the extension to higher dimensions is straightforward. Furthermore, thanks to the moving particles, this method is adapted for a large kind of initial conditions, even slightly dispersed distributions. The FPK equation is solved without any a priori knowledge of the stationary distribution, just a precise representation of the initial distribution is required.Copyright © 2013 John Wiley & Sons, Ltd.     

On the optimal design of radial basis function neural networks for the analysis of nonlinear stochastic systems

In this paper, an iterative selection strategy of Gaussian neurons for radial basis function neural networks (RBFNN) is proposed when the RBFNN method is applied to obtain the steady-state solution of the Fokker–Planck–Kolmogorov (FPK) equation. A performance index is introduced to rank neurons. Top rank neurons are selected, leading to a RBFNN with optimal number and locations of Gaussian neurons for the FPK equation under consideration. The statistical properties of the performance index are studied. It is found that the index assigned to the $j$th neuron is proportional to the probability of the system falling into the small neighborhood of the mean of this neuron as well as proportional to the weight of the neuron. The RBFNN method with the optimally selected neurons is then applied to several challenging examples of nonlinear stochastic systems in 2, 3 and 4 dimensional state space. The RBFNN solutions are also compared with the results of extensive Monte Carlo simulations. It is observed that the RBFNN method with optimally selected neurons by the proposed iterative algorithm is much more efficient than the RBFNN method with uniformly distributed neurons, and is very accurate in terms of the root mean squared (RMS) errors of the FPK equation or the RMS errors of the PDF solution compared with simulation results.     

Decomposition of singular large sparse matrix by adding dummy links and dummy degrees

Abstract

This paper proposes a simple scheme to decompose an n×n nonpositive definite matrix, A, associated with simultaneous equations, A X = B, into a triple‐factors (lower triangular, diagonal, and upper triangular matrices), i.e., Å = L D U, without interchanging rows or columns of A, but with A expanded with new rows and new columns to an m×m matrix Å. Whenever a near‐zero diagonal element, say ā_ii , is encountered and used as a pivoting element, an appropriate positive real number, say p, is added to this diagonal element, and a new term —px_k is also added to the i‐th equation, where x_k is a new variable called “dummy variable'’. If we also add a new equation —px_i + px_k = 0 to enforce the new added variable x_k equal to x_i then the modified i‐th equation has the same effect as the original equation. Therefore, the original solution X can be found directly from the expanded solution of the modified expanded equation. The method is very useful in solving the following problems: (1) nonlinear problems near the limit state, (2) postbuckling analysis, (3) system equations with constraint conditions, and (4) getting eigenvectors from eigenvalues.     

Second and third order perturbation solutions of a generalized Burgers' equation

Summary The differential equationu –uu_x=k(u_xx+cu_x) with initial values on =0 is considered. Whenc0 this represents a hyperbolic generalization of Burgers' equation. Fork1 perturbation solutions are obtained, the outer solution being given completely up to third order, the inner solution (i.e. close to the shock) being given to second. The determination of the unknown functions in the second order inner solution is completed using an integral conservation technique. While the third order inner solution is not explicitly determined, it is shown that matching of the inner and outer solutions at third order is satisfied.With 1 Figure     

Effect of filler content on the dielectric properties of anisotropic conductive adhesives materials for high-frequency flip–chip interconnection

The dielectric property of anisotropic conductive film (ACF) as an interconnect materials in the flip–chip joints is becoming important concern for device packaging solution at high-frequency due to low parasitic effect on the signal transfer. The effects of non-conductive, dielectric filler content on dielectric properties of ACA materials, like dielectric constant, loss factor and loss tangent, and conductivity at high-frequency were investigated. Frequency is dominating factor in determining dielectric constant, loss factor, and conductivity. However, the filler content is dominant only on dielectric constant, not on the loss factor, and conductivity at low-frequency range. The effect of low dielectric constant (low-k) filler addition on high-frequency behavior of ACF interconnection in flip–chip assembly was also investigated. Impedance parameters of low-k ACF with Ni filler and low-k SiO₂ filler extracted from measurement were compared with that of conventional ACF with only Ni filler. The resonant frequency of conventional ACF flip–chip interconnect was 13 GHz, while the resonant frequency of low-k ACF including low-k SiO₂ filler was found at 15 GHz. This difference is originated from capacitance decrease of polymer matrix between bump and substrate pad due to change in dielectric constant of polymer matrix, which was verified by measurement-based modeling. The high-frequency property of the conductive adhesive flip–chip joint, such as resonant frequency can be enhanced by low-k polymer matrix.     

Stationary response of bilinear hysteretic system driven by Poisson white noise

Stationary response of single-degree-of-freedom (SDOF) bilinear hysteretic system driven by Poisson white noise is investigated via stochastic averaging of energy envelope in this paper. The averaged generalized Fokker–Planck–Kolmogorov (GFPK) equation for SDOF bilinear hysteretic system driven by Poisson white noise is derived and the approximate stationary solutions of the averaged GFPK equation are obtain by using a modified exponential polynomial closure method. The effectiveness and accuracy of the approximate solution are assessed by performing appropriate Monte Carlo simulations. It is found that analytical and numerical results agree well and the effect of non-Gaussianity of the excitation process on stationary probability densities of total energy and displacement of bilinear hysteretic system is predicted successfully via stochastic averaging of energy envelope.     

Superconvergence of quadratic optimal control problems by triangular mixed finite element methods

The aim of this work is to investigate the discretization of a quadratic convex optimal control problem using the mixed finite element method. The state and co‐state are approximated by the order k?1 Raviart–Thomas mixed finite element spaces, and the control is approximated by piecewise constant functions. We construct an interpolation of the exact control and a projection of the discrete scalar co‐state to be the approximated solution of the control variable for the continuous optimal control problem. As a result, it can be proved that the difference between the interpolation and the piecewise constant approximation has superconvergence property for the control of order h^3/2 for k=0 and of order h² for k=1. Moreover, only for the order k=1 Raviart–Thomas mixed finite element approximation does the postprocessing technique possess the superconvergence property of order h². Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence. Copyright © 2008 John Wiley & Sons, Ltd.