Wigner Function of Density Operator for Negative Binomial Distribution

By using the technique of integration within an ordered product （IWOP） of operator we derive Wigner function of density operator for negative binomial distribution of radiation field in the mixed state case, then we derive the Wigner function of squeezed number state, which yields negative binomial distribution by virtue of the entangled state representation and the entangled Wigner operator.

Relations between chirp transform and Fresnel diffraction,Wigner distribution function and a fast algorithm for chirp transform

Two physical interpretations of chirp transform related to Fresnel diffraction and Wigner distribution function are given. The chirp transform can be regarded as a Fresnel diffraction observed on a spherical tangent to the diffraction plane, or a rotation and stretching transformation of the Wigner distribution function space. A general fast algorithm for the numerical calculation of chirp transform is developed by employing two fast Fourier transform algorithms. The algorithm, by which a good evaluation can be achieved, unifies the calculations of Fresnel diffraction, arbitrary fractional- order Fourier transforms and other scalar diffraction systems. The algorithm is used to calculate the Fourier transform of a Gaussian function and the Fourier transform, the Fresnel transform, the Fractional-order Fourier transforms of a rectangle function to evaluate the performance of this algorithm. The calculated results are in good agreement with the analytical results, both in the amplitude and phase.

The Wigner distribution function of a super Lorentz–Gauss SLG_(11) beam through a paraxial ABCD optical system

An orthonormal beam family of super Lorentz-Gauss （SLG） beam model is proposed to describe the higher-order mode beams with high divergence, which are generated by a high power diode laser. Here we consider the simplest case of the SLG beams, where there are four mutually orthogonal SLG beams, namely SLG00, SLG01, SLG10, and SLGll beams. The SLG00 beam is just the Lorentz-Gauss beam. Based on the Collins integral formula and the Hermite-Gaussian expansion of a Lorentz function, an analytical expression for the Wigner distribution function （WDF） of an SLG11 beam through a paraxial ABCD optical system is derived. The properties of the WDF of an SLG11 beam propagating in free space are demonstrated. The normalized WDFs of an SLG11 beam at the different spatial points are depicted in several observation planes. The influence of the beam parameter on the WDF of an SLGI 1 beam in free space is analyzed at different propagation distances. The second-order moments of the WDF of an SLG11 beam in free space are also examined. This research reveals the propagation properties of an SLGll beam from another perspective. The WDFs of SLG01 and SLG10 beams can be easily obtained by using the WDFs of Lorentz-Gauss beam and the SLG11 beam.

A new finite-dimensional thermal coherent state and its Wigner distribution function

This paper constructs a new type of finite-dimensional thermal coherent states （FDTCS）, which differs from the proceeding thermal coherent state in construction, as realisations of SU（2） Lie algebra. Using the technique of integration within an ordered product of operator, it investigates the orthonormality and completeness relation of the FDTCS. Based on the thermal Wigner operator in the thermal entangled state representation, the Wigner function of the FDTCS is obtained. The nonclassical properties of the FDTCS are discussed in terms of the negativity of its Wigner function.

Wigner Function and Phase Probability Distribution of q-Analogue of Squeezed One-Photon State

In this paper, in terms of the technique of integration within an ordered product （IWOP） of operators and the properties of the inverses of q-deformed annihilation and creation operators, normalizable q-analogue of the squeezed one-photon state, which is quite different from one introduced by Song and Fan [Int. 3. Theor. Phys. 41 （2002） 695], is constructed. Moreover, the Wigner function and phase probability distribution of q-analogue of the squeezed one-photon state are examined.

Momentum Distribution Functions in Quark-Gluon Plasma

Based on the constituent quasiparticle model of the quark-gluon plasma (QGP), the Wigner function is presented in the form of a color path integral. The Monte Carlo calculations of the quark and gluon densities, pair correlation functions and the momentum distribution functions for strongly coupled QGP plasma in thermal equilibrium at barion chemical potential equal to zero have been carried out. Analysis of the pair correlation functions points out on arising glueballs and related gluon bound states. Comparison results between the momentum distribution functions and Maxwell-Boltzmann distributions show the significant influence of the interparticle interaction on the high energy asymptotics of the momentum distribution functions resulting in the appearance of quantum “tails”.

The Wigner function and phase properties of superposition of two coherent states with the vacuum state

This paper discusses some statistical properties of the superposition of two coherent states with a vacuum state, such as sub-Poissonian photon statistics and negativity of the Wigner function. Phase probability distribution and phase variance are calculated. Special cases of the constructed superposition states are presented. The results show that depending on the vacuum state coefficient γ and the coherent state coefficient a, it can generate a variety of nonclassical states.

Wigner functions and tomograms of the even and odd binomial states

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product （IWOP） of operators, the Wigner functions of the even and odd binomial states （EOBSs） are obtained. The physical meaning of the Wigner functions for the EOBSs is given by means of their marginal distributions. Moreover, the tomograms of the EOBSs are calculated by virtue of intermediate coordinate-momentum representation in quantum optics.

Wigner function and the entanglement of a quantized Bessel-Gaussian vortex state of a quantized radiation field

A new kind of quantum non-Gaussian state with a vortex structure, termed a Bessel-Gaussian vortex state, is constructed, which is an eigenstate of the sum of squared annihilation operators a2 ＋ b2. The Wigner function of the quantum vortex state is derived and exhibits negativity which is an indication of nonclassicality. It is also found that a quantized vortex state is always in entanglement. And a scheme for generating such quantized vortex states is proposed.

Phase Space Path Integral Representation for Wigner Function

Quantum interference and exchange statistical effects can affect the momentum distribution functions making them non-Maxwellian. Such effects may be important in studies of kinetic properties of matter at low temperatures and under extreme conditions. In this work we have generalized the path integral representation for Wigner function to strongly coupled three-dimensional quantum system of particles with Boltzmann and Fermi statistics. In suggested approach the explicit expression for Wigner function was obtained in harmonic approximation and Monte Carlo method allowing numerical calculation of Wigner function, distribution functions and average quantum values has been developed. As alternative more accurate single-momentum approach and related Monte Carlo method have been developed to calculation of the distribution functions of degenerate system of interacting fermions. It allows partially overcoming the well-known sign problem for degenerate Fermi systems.

基于Wigner分布函数的光学元件评价方法

为了有效地表征和评价光学元件局部小尺度波前畸变,提出了能同时反映空间频率和空间位置的Wigner分布函数评价方法。通过理论分析Wigner分布函数与功率谱密度之间的关系,得到了局部波前畸变的评价方法和指标。模拟计算和实验结果验证了该方法的有效性,并表明:该方法不但能判断光学元件的中频误差是否满足惯性约束聚变系统要求,而且还能确定不满足要求的特定区域,从而有效地指导光学元件的返修。

Wigner分布函数及非整数傅里叶变换

本文研究了Wigner分布函数的特性，利用Wigner分布函数的旋转，将整数域傅里叶变换推广到了非整数域，描述了自由空间光场的演变过程．

基于经验模态分解-Wigner分布的光学元件中频误差识别

对于大尺寸高精密光学元件,不仅要对光学元件表面低频面形精度和高频粗糙度进行控制,还需要严格限制中频误差,以保证其使用性能和稳定性.为了确定光学元件的不合格区域并指导其返修,引入经验模态分解（EMD）和Wigner分布（WVD）函数方法,通过理论分析确定该方法与功率谱密度函数间的关系,实现对光学元件表面中频误差的辨识与定位.实验结果表明：EMDWVD方法不仅可以识别分布在实验光学元件表面15～27 mm空间频率为0.1mm-1的中频误差,还可以减小多分量信号所引起的空间频率为1.0～1.5 mm-1的交叉项干扰,提高中频误差辨识的准确率.

短时傅里叶变换和拟Wigner分布最佳窗函数

短时傅里叶变换和拟Wigner分布是应用最广泛的两种时频分析工具,其窗函数的选择是其应用的前提,对此仍有待深入研究。本文详细研究了其窗函数的选取准则,给出了在最佳频率分辨率意义上的最佳窗函数。

任意两个相干态的叠加态的相位分布和Wigner函数(英文)

从相位分布和Wigner函数两个方面研究了任意两个相干态|β〉and |mβeiδ〉的叠加态的量子统计性质.结果表明这种叠加态的非经典特性与β2,振幅系数m,相干态间的位相差δ以及叠加系数间的位相差都有关.当参量选择合适,这种叠加态存在着量子效应.计算了两个相干态等几率混合系综的相位分布和Wigner函数,经过与前者比较,结果表明由于相干项的存在,使得叠加态具有很好的量子力学行为.

基于谱相关函数-Wigner-Ville谱的轴承分布故障信号分析

针对包络解调技术不能有效提取滚动轴承中分布故障特征的问题,利用轴承分布故障振动信号的二阶循环平稳特性,研究了基于谱相关函数的Wigner-Ville谱的时频特征提取方法。与直接Wigner-Ville分布方法和匹配追踪时频分布方法不具备降噪功能不同,对含噪声的循环平稳信号,先用长数据序列计算得到谱相关函数,可通过计算中的多次平均显著削弱信号中噪声成分,再对谱相关函数作关于循环频率的逆傅里叶变换,得到的Wigner-Ville时频分布计算结果可有效降低噪声的干扰。对有轴承分布故障的振动信号,用滤波器去除一阶循环平稳成分,用获得的基于谱相关函数的Wigner-Ville时频谱消除平稳随机噪声的影响,能有效提取轴承分布故障的二阶循环平稳特征。仿真分析表明:所提方法能有效直观提取轴承分布故障,而普通Winger-Ville时频分布易受噪声影响,包络谱分析结果无法明确轴承分布故障特征。实验结果验证了所提时频算法提取齿轮箱轴承分布故障特征的有效性,而信号包络谱、平滑伪Wigner-Ville时频分布均无法有效提取该特征。

用热场动力学理论研究介观电路的Wigner函数

利用热场动力学及相干热态表象理论,重构了有限温度下介观RLC电路的Wigner函数,研究了有限温度下介观RLC电路的量子涨落.借助于Weyl-Wigner理论讨论了有限温度下介观RLC电路Wigner函数的边缘分布,并进一步阐明了Wigner函数边缘分布统计平均的物理意义.结果表明:有限温度下介观RLC电路中电荷和电流的量子涨落随着温度和电阻值的增加而增加,回路中的电荷和电流之间存在着压缩效应,这种量子效应是由于系统零点振动的涨落而引起的;有限温度下介观RLC电路Wigner函数边缘分布的统计平均正好是储存在介观RLC电路中电容和电感上的能量.

激光光路追踪Wigner分布函数方法

提出了一种考虑衍射效应的激光几何光路追踪方法。引入由激光标量场定义的Wigner激光能量相空间分布函数,并给出该函数满足的刘维尔运动方程。Wigner分布函数用来描述经过空间任一点沿任一方向传输的激光光线上的能量分配。激光能量沿由波包色散关系定义的光线轨迹保持不变(真空中)或者衰减(等离子体中)。与传统几何光路追踪方法相比,该方法从理论上给出了激光光线初始携带能量份额的计算方法,并且将激光标量场的相位信息自然地包含在Wigner分布函数的定义里。算例表明,该方法与解析模型及广泛使用的菲涅耳衍射积分方法结果一致。

基于经验模式分解的Wigner-Ville分布交叉项抑制方法

时频分析作为时变非平稳信号分析的有力工具,成为现代信号处理研究的一个热点。这种分析方法提供了时间域与频率域的联合分布信息,为我们清楚地描述了信号随时间变化的关系。Wigner-Ville分布由于其良好时频集聚性,在非平稳信号分析中得到广泛应用,本文针对Wigner-Ville分布中的交叉项问题,提出了基于经验模式分解的Wigner-Ville分布,即对多分量信号运用经验模式分解,将其分解为单分量信号,再对每个单分量信号求Wigner-Ville分布进行线性叠加。提出运用相关系数法对经验模式分解伪分量进行剔除,提高了该方法的精度,并将该方法与Cohen类方法进行比较,阐述了该方法的优点。

基于经验模态分解的Wigner-Ville分布交叉项抑制方法

针对Wigner-Ville分布(Wigner-Ville distribution,简称WVD)在多分量信号时频表示中固有的交叉项问题,提出了一种结合经验模态分解(empirical mode decomposition,简称EMD)的WVD交叉项抑制新方法。对待分析信号进行经验模态分解,得到一系列本征模态函数分量,为了进一步保证本征模态函数分量之间的正交性,对本征模态分量进行独立分量分析(independent component analysis,简称ICA),根据WVD定义,正交信号之和的WVD等于正交信号WVD的和,通过分别求分解分量WVD并求和的方式,得到最终的WVD时频表示,可以有效抑制多分量信号WVD的交叉项,并保持WVD的时频聚集性。