Determination of s-ordered moments and moment generating function from quadrature distributions

Abstract

The sampling functions needed to reconstruct from quadrature distributions both the s-ordered moments of the field and the generalized moment generating function are determined. The sampling function in the latter case turns out to be surprisingly simple. It is proportional to the well-known pattern function f _0k needed to reconstruct from homodyne data the density matrix element e _0k in the Fock basis. As a by-product a useful sum relation for the pattern functions f _mn is found.     

On tomographic reconstruction of the quantum state of a two-mode light field

Abstract

We consider the state reconstruction of an optical two-mode light field from sum quadrature distributions measured with a single balanced homodyne detector. New explicit formulas for the pattern functions necessary to reconstruct the density matrix of the two-mode field in the photon-number basis are derived. Moreover, an expression of the measured quadature distribution in terms of the two-mode normally ordered moments is given and the determination of the moments from it is discussed.     

Radon transform and pattern functions in quantum tomography

Abstract

The two-dimensional Radon transform of the Wigner quasiprobability is introduced in canonical form and the functions playing a role in its inversion are discussed. The transformation properties of this Radon transform with respect to displacement and squeezing of states are studied and it is shown that the latter is equivalent to a symplectic transformation of the variables of the Radon transform with the contragredient matrix to the transformation of the variables in the Wigner quasiprobability. The reconstruction of the density operator from the Radon transform and the direct reconstruction of its Fock-state matrix elements and of its normally ordered moments are discussed. It is found that for finite-order moments the integration over the angle can be reduced to a finite sum over a discrete set of angles. The reconstruction of the Fock-state matrix elements from the normally ordered moments leads to a new representation of the pattern functions by convergent series over even or odd Hermite polynomials which is appropriate for practical calculations. The structure of the pattern functions as first derivatives of the products of normalizable and non-normalizable eigenfunctions to the number operator is considered from the point of view of this new representation.     

The Whittaker-Shannon sampling theorem for experimental reconstruction of free-space wave packets

Abstract

A data analysis scheme is proposed for determining the quantum state of a one-dimensional wave packet describing atomic translational motion in free space. The Whittaker-Shannon sampling theorem is applied to reconstruct exactly a certain portion of the momentum-basis density matrix from Fourier transforms of measured position-probability densities. The sampling theorem is valid if the spatial wave correlation function equals zero ouside some finite range. It is most useful in cases where the range of available data is insufficient to allow a complete reconstruction of the state by phase-space tomography, as in recent experimental studies of atomic-beam state reconstruction by Pfau and Kurtsiefer.     

Stress distributions along a short fibre in fibre reinforced plastics

This paper develops an analysis for predicting the normal stress and interfacial shearing stress distribution along a single reinforcing fibre of a randomly oriented chopped-fibre composite, such as sheet moulding compound (SMC), from a knowledge of the constituent properties and the length-to-diameter ratio of the fibres. The analysis is useful in analysing the tensile strength of SMC, and as a guide to increasing the tensile strength by altering the elastic characteristics. The model is based on a generalized shear-lag analysis. Numerical values of the normal stress and interfacial shearing stress are presented as functions of various parameters. It is observed that the maximum normal stress occurs at the middle of the fibre and the maximum shear stress occurs at the end. The analysis is restricted to loading which does not result in buckling of the fibre; i.e., axial loads on the fibre can be at most only slightly compressive.List of symbols a _f Ratio of the fibre length to diameter (aspect ratio, l _f/d _f) - E _a Young's modulus of the composite (defined in Equation 21) - E _f Young's modulus of the fibre material - E _m Young's modulus of the matrix material - G _f Shear modulus of the fibre material - G _m Shear modulus of the matrix material - l Half the length of the matrix sheath which surrounds the fibre - l _f Half of the length of the fibre - Q Defined in Equation 14. - R Ratio of the length of the fibre to the matrix in a representative volume element; a parameter 0R[(1/V _f–1) ] - r _a Radius of the composite body (we assume r _ar _m, r _f) - r _f Radius of the fibre - r _m Radius of the matrix sheath which surrounds the fibre - u _a Displacement of the composite along the fibre direction - u _f Displacement of the fibre along the fibre direction - V _f Fibre volume fraction - (XYZ) Co-ordinate system with Z-axis parallel to the direction of the applied load (Fig. 1a) - (xyz) Co-ordinate system which is rotated by about the X-axis (Fig. 1a) - (¯x¯y¯z) Co-ordinate system which is rotated by about the z-axis (Fig. 1b) - Fibre orientation angle measured from the Z-axis - _m Engineering shear strain in the matrix - Defined in Equation 8 - Polar angle measured from the x–z plane - Defined in Equation 9 - Applied normal stress - _a Normal stress in the composite along the fibre axis - _f Normal stress in the fibre along the fibre axis - _m Normal stress in the matrix along the fibre axis - Shear stress on the fibre—matrix interface     

A new view on the Peřina-Mišta form of the Glauber-Sudershan P function

Abstract

The ‘regularized’ form of the Glauber-Sudershan P function in terms of a series of Laguerre polynomials proposed by Perina and Mista is reconsidered. It is shown that the corresponding expansion coefficients result from averaging sampling functions well known from optical homodyne tomography with respect to the quadrature distribution of the signal field. An illustrative example of a nonclassical state is considered.     

A Note on Estimation from a Type I Extreme-Value Distribution

If the random variable T has the ta-o-parameter Weibull distribution with cumulative distribution function F(t; θ, K) = 1 – exp[–(t/θ) ^k ], where θ is the scale parameter and K is the shape parameter, then the random vatiable X = In T has the Type I extreme-value distribution of smallest values with cumulative distribution function F(x; u, b) = 1 – exp {–exp [(x – u)/b}, where u = In θ is the location parameter (mode) and b = 1/K is the scale parameter. It is therefore possible to obtain the maximum-likelihood estimator û _mn | b of u, based on the first m order statistics of a sample of size n, when b is known, by a simple transformation of the corresponding estimator of θ when K is known. Use is made of the fact that û _mn | b = In _mn | K, where 2m( _mn | K) ^k /θ ^k has the chi-square distribution with 2m degrees of freedom, to set confidence bounds on u. The probability density function of û _mn | b which for given m is the same for any n ≥ m, is obtained by a simple transformation of that of _mn | K. Integration yields expressions, involving digamma and trigamma functions, for the bias E = E[(û _mn |b) – u] and the variance V = V(û _mn | b). By subtracting the bias E](û _mn |b) – u] from û _mn |b, one obtains an unbiased estimator û|b which has the same variance as the maximum-likelihood estimator. Values of E/b(6DP) and of V/b ²(6DP) are tabulated for m = 1(1)100. The use of the table is discussed and illustrated by a numerical example.     

Reconstruction of Scattering Potentials from Incomplete Data

Abstract

In many situations encountered in physics and in other fields, one can frequently experimentally determine some but not all the Fourier components of a scattering potential. In this paper we present an integral equation which makes it possible to reconstruct any square-integrable function f(r) of finite support from the knowledge of its Fourier transform j (K) over any finite three-dimensional domain of K space. We illustrate the use of this integral equation by application to potential scattering at fixed energy and we show how it can be used to reconstruct details of the scattering potential beyond the usual resolution limit from measurements of the scattered field in the far zone of the scatterer.     

Periodic Squeezing in the Tavis-Cummings Model

Abstract

By utilizing our previous operator solution [17], we have investigated the squeezing in the radiation field of the Tavis-Cummings model (collective N ? 1 two-level atoms interacting with a resonant single cavity quantized mode). With field and atoms initially in coherent field state strong or weak and atomic coherent state (of few excited atoms), periodic time-dependent squeezing in the field and the macroscopic polarization is expressed in terms of Jacobian elliptic functions of the first kind. The statistical investigations are carried out for the quasiprobability distribution functions (Wigner function and Q function). The distribution function of the field quadrature has a variance less (greater) than that for a coherent state if this quadrature is squeezed (unsqueezed).     

Numerical solution of Cauchy type singular integral equations with logarithmic weight,based on arbitrary collocation points

Singular integral equations with a Cauchy type kernel and a logarithmic weight function can be solved numerically by integrating them by a Gauss-type quadrature rule and, further, by reducing the resulting equation to a linear system by applying this equation at an appropriate number of collocation points x _k. Until now these x _k have been chosen as roots of special functions. In this paper, an appropriate modification of the original method permits the arbitrary choice of x _k without any loss in the accuracy. The performance of the method is examined by applying it to a numerical example and a plane crack problem.     

Solving initial value problems by differential quadrature method—part 1: first‐order equations

In this paper, the differential quadrature method is used to solve first‐order initial value problems. The initial condition is given at the beginning of a time interval. The time derivative at a sampling grid point within the time interval can be expressed as a weighted linear sum of the given initial condition and the function values at the sampling grid points within the time interval. The order of accuracy and the stability property of the quadrature solutions depend on the locations of the sampling grid points. It is shown that the order of accuracy of the quadrature solutions at the end of a time interval can be improved to 2n–1 or 2n if the n sampling grid points are chosen carefully. In fact, the approximate solutions are equivalent to the generalized Padé approximations. The resultant algorithms are therefore unconditionally stable with controllable numerical dissipation. The corresponding sampling grid points are found to be given by the roots of the modified shifted Legendre polynomials. From the numerical examples, the accuracy of the quadrature solutions obtained by using the proposed sampling grid points is found to be better than those obtained by the commonly used uniformly spaced or Chebyshev–Gauss–Lobatto sampling grid points. Copyright © 2001 John Wiley & Sons, Ltd.     

Density Matrix Elements and Moments for Generalized Gaussian State Fields

Abstract

A single-mode radiation field with Gaussian Wigner functions (Gaussian state field) is studied. The single-mode Gaussian-field generating function for both the density matrix elements and the moments of the creation and annihilation operators is calculated. It is shown that the matrix elements as well as the moments of different orders can be expressed in terms of Hermitian polynomials of two variables. Furthermore, the cumulants of the Gaussian state field are calculated. Finally, we generalize the Gaussian state field and compute the corresponding moments and density matrix elements.     

Length-scale distributions of the concentration pulsations of a passive impurity in a homogeneous turbulent flow

A relation for calculating the probability density function f _t ^λ (ϕ) of the length scales of a passive concentration field in homogeneous turbulence has been obtained by consideration of the joint statistics of the concentration field and its gradient. The closed equation derived for f _t ^λ (ϕ) has been solved numerically using the data of direct numerical modeling of homogeneous turbulence for the mean characteristics involved in the equation as the coefficients. The results obtained for different values of the Schmidt number have been compared. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 78, No. 6, pp. 131–142, November–December, 2005.     

Percolation Analyses on Transport Properties of YBa2Cu3O7−δ-Ag System

The transport properties of YBa ₂ Cu ₃ O _7– -Ag composite ceramic system have been studied from the viewpoint of the percolation theory. The percolation threshold volume fraction f _c of Ag determined from the electrical resistivity and from the thermal conductivity is f _c =0.125±0.005. This markedly smaller value than the theoretical value ( f _c =0.16) suggests the segregative distribution of Ag. The critical exponent t depends on the component conductivity ratio h = _YBCO / _Ag , which can be explained on the basis of the scaling hypothesis.     

Simple and precise measurements of fibre volume and void fractions in metal matrix composite materials

A method for determining the fibre volume fraction, V _f and the void fraction, V _g, in a metal matrix composite (MMC) material is described. These quantities are determined from specimen weight measurements in air and in a liquid using a laboratory balance. For a material without voids, V _f can be determined with an uncertainty less than 0.5% with a balance precision of 0.01%. By making the same measurements before and after etching away the matrix, using the same balance precision, V _f and V _g can be determined to an uncertainty of about 3 and 6%, respectively. It is also shown theoretically that by indenting a specimen containing no fibres and only a uniform distribution of small voids, the void fraction can also be determined from weight measurements before and after indentation.     

A Note on Moments of Gamma Order Statistics

Previously quadrature approximations were developed to determine the moments of a distribution of the response of a multivariable function when each of the variables is a random variable from a normal distribution. The error was shown to be of the order of the sixth power of the standard deviations of the random variables, but a more useful bound is desired in applied work. Only limited success has been achieved in this direction. It is shown that the best approximating distribution is a Beta distribution of the first kind with β₂ equal to 3 and mean, variance, and β₁ obtained by quadrature, where β₁ and β₂ are standard measures of skewness and kurtosis, respectively. A parametric study of the function X = C_o (a _o, ± y ₁ ± y ₂ ± … ±y_n ) ^m + b _o where the y_i all have the same standard deviation, σ, is conducted both analytically and by quadrature. The mean and variance obtained by quadrature are essentially exact in the range of interest. It is shown that for a large range of σ the above distribution is both a good approximation and a much better approximation than either a normal approximation with the same mean and variance or a linear approximation. The example also shows that the β₂ obtained by quadrature is a poor indicator of the precision of the quadrature approximation.     

Analysis and optimal control of discrete time‐varying systems Via Hahn polynomials

Abstract

Recently, the problem of analysis and optimal control of discrete time‐invariant systems has been extensively studied using finite series expansion of discrete orthogonal polynomials. This paper is to extend the applicable scope of discrete orthogonal polynomials to discrete time‐varying systems. The finite set of Hahn polynomials {q_ik)], i=0, 1, …, N} is chosen as the finite series expansion basis due to its general form and useful properties. First, for treating the product of two discrete‐time functions by Hahn series expansion, a new algorithm is derived to compute the Hahn series expansion coefficients of products qi(k)q_j (k), i, j=0, 1, …, N. These Hahn coefficients are then used to establish a product operational matrix for relating the Hahn coefficient vector of a product function to those of its component functions. This product operational matrix, along with the relations for connecting the Hahn coefficient vectors of a discrete function x(k) and its time‐shifted x(k+1), is finally applied to derive computational algorithms for solving the problems of analysis and optimal control of discrete time‐varying systems via finite Hahn series. Computed results are provided to illustrate the applicability of the proposed algorithms.     

Maximum entropy and discretization of probability distributions

Using gaussian quadrature we can find m concentrations of probability that replace the density function of a random variable X and match 2m - 1 of its moments. This reduces a probabilistic analysis to m deterministic ones. Even small values of m provide excellent accuracy in many practical circumstances. When fewer than 2m - 1 moments are known there is arbitrariness in the choice of the concentrations, which is overcome by resorting to the maximum entropy formalism. Its use is here systematized for the case in which αXb and we know N moments of the density of X, so that calculation of N - 1 integrals suffices for finding the density function and any number of its moments. The approach is illustrated for m = 2 and 3, N = 2, 3, α = 0, B = ∞ and graphs are provided for finding the equivalent concentrations.     

Multivariate L-estimation

In one dimension, order statistics and ranks are widely used because they form a basis for distribution free tests and some robust estimation procedures. In more than one dimension, the concept of order statistics and ranks is not clear and several definitions have been proposed in the last years. The proposed definitions are based on different concepts of depth. In this paper, we define a new notion of order statistics and ranks for multivariate data based on density estimation. The resulting ranks are invariant under affinc transformations and asymptotically distribution free. We use the corresponding order statistics to define a class of multivariate estimators of location that can be regarded as multivariate L-estimators. Under mild assumptions on the underlying distribution, we show the asymptotic normality of the estimators. A modification of the proposed estimates results in a high breakdown point procedure that can deal with patches of outliers. The main idea is to order the observations according to their likelihoodf(X ₁),...,f(X _n ). If the densityf happens to be cllipsoidal, the above ranking is similar to the rankings that are derived from the various notions of depth. We propose to define a ranking based on a kernel estimate of the densityf. One advantage of estimating the likelihoods is that the underlying distribution does not need to have a density. In addition, because the approximate likelihoods are only used to rank the observations, they can be derived from a density estimate using a fixed bandwidth. This fixed bandwidth overcomes the curse of dimensionality that typically plagues density estimation in high dimension. The research was partially supported by grant #37 from the CONICYT and by grant 5-81089 from NSERC.     

Simulation and quantitative assessment of finite­size particle distributions in metal matrix composites

Abstract

A series of finite­size particle distributions were simulated to investigate the effects of particle size, shape, orientation, and area fraction on the quantification of homogeneity in structural particulate metal matrix composites (MMCs). It is found that, for nominally random distributions, the values of conventional centre-to-centre nearest-neighbour spacing parameters are influenced by particle morphology, and, as such, are unsuitable for characterising distributions of finite-size particles. However, the coefficient of variation of the mean near-neighbour distance COV(d _mean), derived from particle interfaces using finite-body tessellation, appears independent of particle shape, size distribution, orientation, and area fraction, while showing great sensitivity to particle clustering. In the range of particle morphological characteristics studied, the random distributions were found to exhibit a consistent value of COV(d _mean) equal to 0.36±0.02. The degree of inhomogeneity of any given distribution may then be evaluated by simply comparing the measured COV(d _mean) with this value.