Boson Inverse Operators and a New Family of Two-photon Annihilation Operators

Abstract

We introduce the inverse of the Hermitian operator (ââ ^?) and express the Boson inverse operators â ^?1 and â ^??1 in terms of the operators â, â ^? and (ââ ^?)^?1. We show that these Boson inverse operators may be realized by Susskind-Glogower phase operators. In this way, we find a new two-photon annihilation operator and denote it as â ²(ââ ^?)^?1. We show that the eigenstates of this operator have interesting non-classical properties. We find that the eigenstates of the operators (ââ ^?)^?1 â ², â(ââ ^?)^?1 â and â ²(ââ ^?)^?1 have many similar properties and thus they constitute a family of two-photon annihilation operators.     

Four-photon Coherent States

Abstract

We present a detailed discussion of a type of four-photon coherent state defined as right eigenstates of the operator â ⁴ where â is the usual annihilation operator. There are actually four sets of states that need to be considered, namely those containing as the lowest number states ¦0>, ¦1>, ¦2>, or ¦3>. These correspond to the possible unique superpositions of the ordinary coherent states ¦±α> and ¦±iα>. We discuss the nonclassical properties of these states such as photon antibunching and squeezing. The usual second order squeezing does not exist for these states but higher order squeezing and square field amplitude squeezing do exist. Also discussed are the quasiprobability distributions, namely the P-function, the Q-function and the Wigner function. Finally, a method of generating these states based on the competition between a four-photon parametric process and incoherent losses from four-photon absorption is presented.     

Genesis of Bimodal Distributions

Conditions for which the density function of a mixture of two normal distributions is bimodal are investigated. For fixed values of the variances σ₁ ² and σ₂ ² of the normal distributions if the difference between the means is sufficiently small, the distribution of the mixture will be unimodal, independent of the proportions p and 1 – p, 0 < p < 1. If the difference exceeds a critical value which depends on σ₁ ² and σ₂ ² the bimodality property then depends on p. Values of p sufficiently close to zero and one always exist for which the distribution is unimodal.     

Variances of locally Minimum Variance Quadratic Unbiased Estimators (“MIVQUE's”) of Variance Components

Minimum variance quadratic unbiased estimators (MIVQUE's) of variance components from unbalanced data are functions of the components they are to estimate. To use the MIVQUE expressions for estimation under the one-way classification random model, for example, the unknown between- and within-treatments components, σ² _a and σ² _e , must be replaced by a priori estimates σ² _ao and σ² _eo the resulting estimators are called “MIVQUE's.” For the one-way classification, expressions for the variances and covariance of the “MIVQUE's” are obtained under normality. Numerical comparisons indicate that when σ² _a /σ² _e > 1 (approximately) and unless σ² _ao /σ² _eo Q σ² _a /σ² _e , (a) the “MIVQUE's” have variances near their lower bounds, and (b) the “MIVQUE” of σ² _a is more efftcient than the ANOVA estimator. When σ² _a /σ² _e < 1, the “MIVQUE's” are more dependent on accurate specification of σ² _ao /σ² _eo . The “MIVQUE” and ANOVA estimator of σ² _e have nearly equal variances unless σ² _eo /σ² _eo σ² _a /σ² _e , when the ANOVA estimator has smaller variance.     

Sampling Theory

A setup involving p + 1 normal populations Π i with unknown population means μ p and known population variances σ2 i (0 ≤ i ≤ p) is considered; Π0 is the “control” population and Π i (1 ≤ i ≤ p) are the “test” populations. Based on independent observations zii (j = 1, 2, …, Ni ) from Π i (0 ≤ i ≤ p) it is desired to make an exact joint confidence statement of the form {x0 – x i – d < μO – μ i < x 0 x i + rd (1 ≤ i ≤ p)). Here x i = Σ N i i = 1 x ij /N i (0 ≤ i ≤ p) is the observed sample mean from II i , and the constants N = Σ p i , r > ?1, and d > 0 are specified prior to experimentation.

The problem of choosing the N_i (0 ≤ i ≤ p) to maximize the confidence coefficient is considered. A procedure is described which is globally optimal if σ² ₁ = σ² ₂ = … = σ² _p and is optimal in a more restricted sense if not all of the σ² _i (1 ≤ i ≤ p) are equal.

It is shown that the globally optimal proportion of observations _i to be taken from II; (0 ≤ i ≤ p) depends only on the θ _i = σ² _i /σ² ₀ (1 ≤ i ≤ p) and λ = d √N/σ₀. The behavior of the _i as functions of r and λ is studied analytically; formulae are derived for determining the _i . Detailed results are given for p = 2 including tables of the _i (0 ≤ i ≤ 2) as a function of λ for selected r when σ² ₀ = σ² ₁ = σ² ₂. Examples of the use of the tables are provided. These results generalize earlier results for one-sided comparisons (r. = + ∞) given by Bechhofer.     

Generating a Variable from the Tail of the Normal Distribution

The performance of three rules for dealing with outliers in small samples of size n from the normal distribution N(μ, σ²) is investigated when the primary objective of sampling is to obtain an accurate estimate of σ². It is assumed that at most one observation in the sample may be biased, arising from either N(μ + aσ, σ²) or N(μ, (1 + b) σ²). Performance of each rule is measured in terms of “Protection”, the fractional decrease in MSE obtained by using the rule when a biased observation actually is present in the sample. Numerical results have been obtained for n 5 ≤ 11 when μ is known and n = 3 when μ is unknown.     

Ground state cooling,quantum state engineering and study of decoherence of ions in Paul traps

Abstract

We investigate single ions of ⁴⁰Ca⁺ in Paul traps for quantum information processing. Superpositions of the S_½ electronic ground state and the metastable D_5/2; state are used to implement a qubit. Laser light on the S_½ ? D_5/2 transition is used for the manipulation of the ion's quantum state. We apply sideband cooling to the ion and reach the ground state of vibration with up to 99.9% probability. Starting from this Fock state (n = 0), we demonstrate coherent quantum state manipulation. A large number of Rabi oscillations and a ms-coherence time is observed. Motional heating is measured to be as low as one vibrational quantum in 190ms. We also report on ground state cooling of two ions.     

A new ordered sold solution with a bixbyite structure: The compounds Cu2+xTi2−xFe2−xSbxO9 (0 ⩽ x ⩽ 1.4)

A new solid solution with the formula Cu_2+xTi_2−xFe_2−xSb_xO₉ was prepared by solid state reaction in air at 900°C. A rather large homogeneity range is observed: 0 ⩽ x ⩽ 1.4. The X-ray powder diffraction shows a (Fe, Mn)₂O₃ bixbyite type structure with a cubic elementary cell. The a parameter increases with the Cu⁺² amount. Two types of polyhedra are present with on ordered cationic distribution over them. Cu⁺² exhibits a highly distorted octahedral coordination: PII sites and Sb⁺⁵ an almost regular one: PI sites. The octahedral distortion is discussed in terms of two parameters: the bond length deviation Δoct and the bond angle variance σ² oct. The stabilization of the bixbyite structure is well understood in the context of a strong correlation between Δoct of the PII polyhedra and σ²oct of the PI polyhedra.     

Investigation of Rules for Dealing With Outliers in Small Samples from the Normal Distribution: I: Estimation of the Mean

The performance of three rules for dealing with outliers in small samples of size n from the normal distribution N(μ, σ²) are investigated when the primary objective of sampling is to obtain an accurate estimate of μ. It is assumed that at most one observation in the sample may be biased, arising from either N(μ + aσ, σ²) or N(r, (1 + b)σ²). Performance of each rule is measured in terms of “Protection”, the fractional decrease in the Mean Square Error (MSE) obtained by using the rule when a biased observation actually is present in the sample. Although numerical results have been obtained for n ≤ 10 when σ² is known, computational difficulties have prevented evaluation of protections when σ² is unknown except when n = 3.     

Crack layer analysis of fatigue crack propagation in rubber compounds

A methodology to characterize the resistance of rubber compounds to crack propagation (fracture toughness) is presented. A constitutive model based on the crack layer theory is utilized to extract the specific energy of damage ^*, a material parameter characteristic of the material's resistance to crack propagation and the dissipative characteristic, . The model expresses the rate of crack propagation as% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaamaalaaabaGaam% izaGqaciaa-fgaaeaacaWGKbGaa8Ntaaaaaaa!3AFA!\[\frac{{da}}{{dN}}\]= % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaeq% OSdiMaamOsamaaDaaaleaacaaIXaaabaGaaGOmaaaaaOqaaiaadMha% caGGQaGaamOuamaaBaaaleaacaaIXaaabeaakiabgkHiTiaadQeada% WgaaWcbaGaaGymaaqabaaaaaaa!41A5!\[\frac{{\beta J_1^2 }}{{y*R_1 - J_1 }}\]where da/dN is the cyclic rate of fatigue crack propagation (FCP), J ₁ is the energy release rate (tearing energy) and R ₁ is the resistance moment which accounts for the amount of damage associated with the crack advance. Microscopic examination revealed that crack tip microcracking is the dominant damage mechanism. Hence, R ₁ was evaluated as the area (m²) of microcracking surfaces per unit crack advance.Fatigue crack propagation data for a particular rubber compound have been analyzed using the present model. The proposed equation describes the entire FCP history in the compound. According to this model, ^* and for the compound investigated, are found to be 9.3 kJ m^-2 and 9.7×10^-9 m⁴/J-cycle, respectively.

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Résumé On présente une méthodologie pour caractériser la résistance de composés de caoutchouc à la propagation des fissures du point de vue de la ténacité à la rupture. Un modèle constitutif basé sur la théorie de la couche de fissuration est utilisé pour obtenir l'énergie spécifique d'endommagement ^*, un paramètre du matériau représentatif de sa résistance à la propagation d'une fissure, et une caractéristique de dissipation . Le modèle exprime la vitesse de propagation d'une fissure de fatigue par cycle da/dN en fonction de ces deux paramètres, de la vitesse de relaxation de l'énergie de cisaillement J ₁, et du moment résistif R ₁ qui tient compte de état de l'endommagement associé à la progression de la fissure. Un examen microscopique révèle que la microfissuration à l'extrémité de la fissure est le mécanisme déterminant de l'endommagement. Dès lors, on évalue R ₁ en fonction de l'aire de microfissuration (en m²) par unité de progression de la fissure.Des données de propagation de fissure de fatigue sont analysées à l'aide du présent modèle pour un composé de caoutchouc particulier. L'équation proposée décrit l'entièreté de la propagation de la fissure dans le composé. Des valeurs numériques pour ^* et pour de respectivement 9,3 kJ m^-2 et 9,7×10^-9 m^-4/J-cycle sont trouvées.

     

High pressure and time resolved luminescence spectra of Gd3Ga5O12:Pr crystal

Luminescence spectra and time resolved luminescence spectra of GGG crystal doped with Pr³⁺ were measured at high hydrostatic pressure from ambient to 220 kbar. Effect of pressure results in the red shift of all luminescence lines related to Pr³⁺ ion emission equals from −0.32 to −1.02 cm⁻¹/kbar and in the diminishing of the luminescence lifetimes. The luminescence decay related to emission from ³P₀ state was single-exponential and diminished with pressure from 23 μs at ambient pressure to 6.5 μs at 165 kbar. Luminescence decay related to transition form ¹D₂ state was two-exponential with longer decay equal to 162 μs at ambient pressure and 120 μs at 165 kbar. We discussed effect of pressure on the ¹D₂ → ³H₄ luminescence and emission from ³P₀ state in the context of non-radiative processes that depopulate the ³P₀ and populate the ¹D₂ state, considering mainly multiphonon relaxation processes and depopulation via the praseodymium trapped exciton state.     

Characterization of La1 −x Sr1 +x Cu5 −x Fe x O12 + δ (0 ≤x ≤ 1·0) by dc electrical conductivity,magnetic susceptibility and EPR measurements(0 ≤x ≤ 1·0) by dc electrical conductivity,magnetic susceptibility and EPR measurements

The DC electrical resistivity results of La_{4 −x} Sr_{1 +x} Cu_{5 −x} Fe _x O_{12 + δ} (0 ≤x ≤ 1·0) showed that for S1 (x = 0) and S2 (x = 0·25) the temperature coefficient of resistivity (TCR), dρ/dT, is positive and slightly increases with increasing temperature in the range 20–270 K. This shows the metallic nature of S1 and S2. For the samples S3(x = 0·5) and S4 (x = 0·75), TCR slightly increases in the range 20–270 K, with change in sign from negative to positive at ∼ 80 K and ∼ 130 K, respectively. These results show the metal-insulator type transition in S3 and S4. For the sample S5 (x = 1·0), the TCR is negative and gradually increases in the range 20–270 K, which shows its semiconductor-like behaviour. The activation energy for S5 is found to be 0·21 × 10⁻² eV. Furthermore, the DC resistivity results of S1–S5 in the range 350–660 K are in conformity with the low temperature results. The very weak temperature dependence of magnetic susceptibility results of S1–S3 show Pauli-paramagnetic behaviour in the range 77K–400 K, while S4 and S5 exhibit Pauli-paramagnetic behaviour in the range 77–850 K. Long-range antiferromagnetic interaction is observed in S5 (x = 1·0) belowT _c ∼ 100 K. The room temperature EPR lineshapes gradually improve from metallic S1 (x = 0) to semiconductor-like S5(x = 1·0). Negativeg-shift is observed in the samples S2–S5 with increasing trend ing _iso-values of 1·880 in S2 to 1·961 in S5. However, theg _iso-value for S1 could not be observed due to very poor lineshape.     

Pressure-induced band cross-over in Pb1−x Sn x Te

Resistivity and thermoelectric power studies have been carried out on two semiconductor alloy systems viz Pb_0·8Sn_0·2Te and Pb_0·6Sn_0·4Te up to 35 kbar pressure. Thermoelectric power and resistivity data on Pb_0·8Sn_0·2Te indicate that the energy gapE _g=E _{L −} ⁶ −E _{L 6} ⁺ decreases with pressure resulting in a zero gap state near 35 kbar pressure. TEP studies on the alloy system Pb_0·6Sn_0·4Te provide direct evidence for a pressure induced L ₆ ⁻ →L ₆ ⁺ cross over transition.     

Thin film CdZnS/CuInSe2 solar cells by spray pyrolysis

Cd_1−xZn_xS/CuInSe₂ solar cells having efficiencies in the range of 2·3% were fabricated by spray pyrolysis. The best cell had the following parameters:V _oc = 305 mV,J _sc = 32 mA/cm², FF = 0·32 area = 0·4 cm² and efficiency = 3·149%.V _oc versus temperature measurements showed that the electron affinity difference was 0·22 eV. Forward dark current versus voltage curves were plotted and a possible current mechanism occurring in these cells has been proposed.     

Scattering of Scalar Waves by an Impenetrable Rough Sphere

Scattering of scalar waves by an impenetrable, slightly rough sphere is considered. The incoherent differential cross-section is calculated to the lowest order in hk (where k is the wavenumber and h ² the mean-square deviation from a smooth sphere), but for any value of the size parameter α = kR₀ (where R ₀ is the radius of the sphere). Particular attention is paid to the back-scattering cross-section σ_B for large α, under the assumption that the correlation length of the roughness structure is not larger than the incident wavelength. The resulting limit of σ_B is well known: the same cross-section is obtained by approximating the sphere locally by a tangential plane and using results for slightly rough planar areas.     

Deep levels in GaN studied by deep level transient spectroscopy and Laplace transform deep-level spectroscopy

In this study we present the results of investigations on Schottky Au-GaN diodes by means of conventional DLTS and Laplace DLTS methods within the temperature range of 77–350 K. Si-doped GaN layers were grown by Molecular Beam Epitaxy technique (MBE) on sapphire substrates. DLTS signal spectra revealed the presence of four majority traps: two hightemperature and two low-temperature peaks. Using LDLTS method and Arrhenius plots the activation energy and capture cross sections were obtained. For two high-temperature majority traps they are equal to E1 = 0.65 eV, σ₁ = 8.2 × 10^?16cm² and E2 = 0.58 eV, σ₂ = 2.6 × 10^?15 cm² whereas for the two low-temperature majority traps E3 = 0.18 eV, σ₃ = 9.72 × 10^?18 cm² and E4 = 0.13 eV, σ₄ = 9.17 × 10^?18 cm². It was also found that the traps are related to point defects. Possible origin of the traps was discussed and the results were compared with the data found elsewhere [1–5].     

A NEW FINITE ELEMENT FORMULATION FOR PLANAR ELASTIC DEFORMATION

For the stress analysis of planar deformable bodies, we usually refer to either plane stress or plane strain hypothesis. Three-dimensional analysis is required when neither hypothesis is applicable, e.g. bodies with finite thickness. In this paper, we derive an ‘exact’ solution for the plane stress problem based on a less restrictive hypothesis than σ_z=0. By requiring the out-plane stress σ_z to be a harmonic function, the three-dimensional solution is obtained. In addition, we present a two-dimensional finite element for planar analysis of problems where the thickness of the body 2h is comparable to other characteristic dimensions. This element is presented as a substitute for classical plane stress and plane strain finite elements. The typical plane stress and plane strain state are recovered in the case where h→0 and the case h→∞, respectively. As an example for the application of such formulation, the behaviour of a concrete gravity dam is investigated. It is shown that this structure, typically analysed by using plane strain hypothesis, has its out-plane stress underestimated. © 1997 John Wiley & Sons, Ltd.     

Comparisons of Designs and Estimation Procedures for Estimating Parameters in a Two-Stage Nested Process

For the usual two-stage nested random model, N observations are to be used to estimate some or all of these parameters: μ (the general mean), σ _a ² (the between classes variance component), σ _b ² (the within classes variance component) and ρ = σ _a ²/σ _b ² Using the traditional analysis of variance estimator, with fixed number of classes, (k), an allocation procedure to minimize the variance of the estimator is derived to estimate σ _a ² or ρ: p + 1 observations in each of r classes and p observations in each of k ? r classes, where N = pk + r, 0 ≤ r < k. The optimal number of classes (k) is conjectured to be the closest integer to

For large N, this implies that the optimal number of classes to estimate σ _a ² would be approximately k ₁ = N _ρ/(1 + ρ) with an average of 1 + ρ^?1 observations per class; the optimal number of classes to estimate ρ would be approximately Nρ/(1 + 2ρ) with an average of 2 + ρ^?1 observations per class. Hence if ρ > 1, p = 1 for σ _a ² and p = 2 for .

Investigations on the effect of using an incorrect value of ρ in k ₁ and k ₂ show that if 0.5 < ρ₁/ρ < 2.0, where ρ₁ is the value of ρ used, the loss in efficiency is generally less than 10%. Comparisons with the variances of hypothetical estimators in which k need not be an integer show that the integral requirement results in only a small loss of efficiency. Percent efficiencies of the estimates of the other parameters when a sampling plan is designed to minimize the variance of one estimator are presented for selected values of N and p.

Alternative estimators are considered for μ and σ _a ² and are found to be useful when the sampling is decidedly unbalanced and ρ is quite large. A computing procedure is developed for restricted maximum likelihood estimation of the variance components.     

Das Verhalten des Wassers im kritischen Gebiet

Zusammenfassung Callendars Versuche mit Wasser im kritischen Gebiet zeigen ein von den bisherigen Anschauungen abweichendes Bild, insofern als der kritische Punkt nicht bei 226 kg/cm² und 374°C, sondern erst bei 256,6 kg/cm² und 380,6°C liege. Die hier mitgeteilten eigenen, bei 240 kg/cm² angestellten Versuche best?tigen die Ergebnisse vonCallendar nicht. Aus dem W?rmeinhalt des Wassers bei 240 kg/cm² ergibt sich die spezifische W?rme, die bei 379,6°C ein Maximum von 36 kcal/kg °C aufweist. (Mitteilung aus dem Laboratorium für technische Physik der Technischen Hochschule München)     

The Elements of Probability

Four estimates of the current process average are compared in the situation when measurements are made at equal intervals of time and two types of variance components, σ² ₂, the variance of a random-shift, cumulative component, and σ² ₁, a transitory measurement component, are present. The four estimators are the current measurement, the m.v.u.e. (minimum variance unbiased estimate), the geometric moving average and the arithmetic moving average. The coefficients of the m.v.u.e. depend on c = σ² ₂/σ² ₁ and n, the number of periods of time at which measurements have been made. The inefficiencies caused by using the geometric moving average when c is small and n is small and by using the arithmetic moving average when c is large and n is large are studied and emphasized because of their practical importance.