SOME APPLICATIONS OF COHERENT STATE FORMULATION OF THE WIGNER OPERATOR

In this paper both the normal product form and the coherent state form of the Wigner operator are derived. Furthermore, the applications of the new forms of the Wigner operator are also presented, which are involved in deriving some new quantum operator formulas, in the coherent state generalization of the Moyal theorem, and in evaluating some quantum operators which corresponds to the given classical functions in the Weyl manner and vice versa.     

SOME NEW QUANTUM OPERATOR FORMULAS --NEW APPLICATIONS OF THE COHERENT STATE

We present a novel method for deriving some new quantum operator formulas by carrying out,the integrations within a normal product symbol. The idea underlying this method is com- bining the properties of normal product and of coherent state together. For example, we can deduce operator ordering theorems, the Fujiwara formula, and some disentangling theorems simply and directly. Bymeans of this method we also get the explicit operator form and coherent state form for the Wigner function with these new forms, the wigner function's eigen-state is found and its zero point value property is discussed. Furthermore some famous quantum unitary transformations and the transformed matrix element for functional obtained conveniently. integral may be also     

COHERENT STATE FORMULATION OF THE WEYL CORRESPONDENCE AND THE WIGNER FUNCTION

We point out that there is a close connection between the Weyl correspondence and coherent state correspondence. Bu virtue of differential and integral method within normal ordered products we demonstrate that the essence of the Gleyl correspondence is just coherent state correspondence in both boson and fermion cases. Furthermore, coherent state form and explicit operator form for the Wigner function are given. With these forms we can derive some important operator formulas from the Wey1 correspondence. The Wigner function's zero-point value is also discussed.     

量子扩散通道中Wigner算符的演化规律

众所周知,量子态的演化可用与其相应的Wigner函数演化来代替.因为量子态的Wigner函数和量子态的密度矩阵一样,都包含了概率分布和相位等信息,因此对量子态的Wigner函数进行研究,可以更加快速有效地获取量子态在演化过程的重要信息.本文从经典扩散方程出发,利用密度算符的P表示,导出了量子态密度算符的扩散方程.进一步通过引入量子算符的Weyl编序记号,给出了其对应的Weyl量子化方案.另外,借助于密度算符的另一相空间表示-Wigner函数,建立了Wigner算符在扩散通道中演化方程,并给出了其Wigner算符解的形式.本文推导出了Wigner算符在量子扩散通道中的演化规律,即演化过程中任意时刻Wigner算符的形式.在此结论的基础上,讨论了相干态经过量子扩散通道的演化情况.     

Quantum statistical properties of photon-added spin coherent states

The photon-added spin coherent state as a new kind of coherent state has been defined by iterated actions of the proper raising operator on the ordinary spin coherent state. In this paper, the quantum statistical properties of photon-added spin coherent states such as photon number distribution, second-order correlation function and Wigner function are studied. It is found that the Wigner function shows the negativity in some regions and the second-order correlation function is less than unity. Therefore, the photon-added spin coherent state is a nonclassical state.     

Normally Ordered Form of Smoothed Wigner Operator and its Applications

Using the IWOP (integration within ordered product) technique, we derive a normally ordered form of the smoothed Wigner operator with which the smoothed Wigner function, that is specified only in the integral form in the literature, can be simplified. The application of the new form of the smoothed Wigner operator to the coherent state is also presented.     

相干态表象在量子相空间分布函数中的应用

利用相干态表象和IWOP技术导出了自由热态密度矩阵的正规乘积形式,进而根据相干态表象下的Wigner函数定义重构了自由热态和热相干态的Wigner函数.结果表明利用相干态表象下的Wigner函数定义和算符的正规乘积形式可以方便简捷重构一些量子态的Wigner函数.     

New approach for solving the Wigner function from the Husimi function in the two-mode entangled state representation

We introduce a new method to calculate the Wigner function when its corresponding Husimi function is given. A new formula is derived for calculating conveniently the Wigner function in two-mode entangled state representation. As application, we derive Wigner functions of some quantum states, such as two-mode entangled state, the electron's two-mode squeezed canonical coherent state, and the electron's coordinate eigenstate.     

量子相空间分布函数与压缩相干态表示间的变换关系

基于Husimi算符具有压缩相干态投影子形式, 首先介绍了一个新的量子算符表示, 即压缩相干态表示.当高斯展宽参数κ = 1时, 该函数约化为通常的P函数. 作为例子, 研究了热态的压缩相干态表示, 通过图示说明了压缩相干态表示与P函数的区别. 为更好地在量子光学问题中使用该表示, 我们揭示了压缩相干态表示与Wigner函数、Q函数以及Husimi函数间的积分变换关系.     

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.     

奇偶对相干态的维格纳函数和层析图函数

利用纠缠态η〉表象下的维格纳算符,重构了奇偶对相干态的维格纳函数.根据维格纳函数在相空间中随变量ρ和γ的变化规律,讨论了奇偶对相干态的非经典性质和量子干涉效应.研究发现,奇偶对相干态总呈现非经典性质,并且当q取奇数时,奇偶对相干态更容易出现非经典性质.奇偶对相干态的量子干涉效应的显著程度与q取值有关,但对于q的同一取值,奇对相干态的量子干涉效应更为显著.利用纠缠态η〉表象下的维格纳算符Δ1,2(ρ,γ)和纠缠态η,τ1,τ2〉的投影算符之间满足的拉东变换,获得了奇偶对相干态的量子层析图函数.     

Statistical Properties of Photon-Added Two-Mode Squeezed Coherent States

The nonclassical and non-Gaussian quantum states—photon-added two-mode squeezed coherent states have been theoretically introduced by adding multiple photons to each mode of the two-mode squeezed coherent states. Starting from the new expression of two-mode squeezing operator in entangled states representation, the normalization factor is obtained, which is directly related to bivariate Hermite polynomials. The sub-Poissonian photon statistics, cross-correlation between two modes, partial negative Wigner function are observed, which fully reflect the nonclassicality of the target states. The negative Wigner function often display non-Gaussian distribution meanwhile. The investigations may provide experimentalists with some better references in quantum engineering.     

复合函数算符的微商法则及其在量子物理中的应用

给出了在量子物理学、量子统计学、算符排序理论、矩阵论以及控制理论中有着重要用途的复合函数算符的一般微分法则,利用这一法则研究了Wigner算符和Weyl对应规则中的积分问题,证明了两类典型的算符恒等公式.给出了Wigner算符的有序算符内的微分形式,并得到了一些重要函数的新的微分式.最后,引入了一个参数型的Wigner算符来统一正规序、Weyl编序以及反正规序三种算符排序.     

Some Evolution Formulas on the Optical Fields Propagation in Realistic Environments

Evolution formulas of the density operator, the photon number distribution, and the Wigner function are derived for the problem on the optical fields propagation in realistic environments. Using the idea “reservoir modeled by beam splitter (BS)” and the Weyl expansion of the density operator, we obtain these formulas cleverly, which are very useful for quantum optics and quantum statistics. As an application, we study the time evolution of the photon number distribution and the Wigner function for single-photon-added coherent state in thermal environment.     

Wigner function and tomogram of the Hermite polynomial state

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, this paper derives the Wigner function for the Hermite polynomial state (HPS). The tomogram of the HPS is calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.     

Wigner functions and tomograms of the photon-depleted even and odd coherent states

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product （IWOP） of operators, this paper derives the Wigner functions for the photon-depleted even and odd coherent states （PDEOCSs）. Moreover, in terms of the Wigner functions with respect to the complex parameter a the nonclassical properties of the PDEOCSs are discussed. The results show that the nonclassicality for the state ｜β, m〉o （or ｜β,m〉e） is more pronounced when m is even （or odd）. According to the marginal distributions of the Wigner functions, the physical meaning of the Wigner functions is given. Further, the tomograms of the PDEOCSs are calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.     

New Properties of the Wigner Function of the Tripartite Entangled State

For entangled three particles one should treat their wave function as a whole, there is no physical meaning talking about the wave function (or Wigner function) for any one of the tripartite, therefore thinking of the entangled Wigner function (Wigner operator) is of necessity, we introduce the entangled Wigner operator related to a pair of mutually conjugate tripartite entangled state representations and discuss some of its new properties, such as the trace product rule, the size of an entangled quantum state and the upper bound of the three-mode Wigner function. Deriving wave function from its corresponding tripartite entangled Wigner function is also presented. Those new properties of the tripartite entangled Wigner function play significant role in quantum physics because they provide us deeper insight into the shape of quantum states.     

增光子奇偶相干态的Wigner函数

利用相干态表象下的Wigner算符, 重构了增光子奇偶相干态的Wigner函数.根据此Wigner函数在相空间中随复变量α的变化关系, 讨论了增光子奇偶相干态的非经典性质. 结果表明, 增光子奇偶相干态总可呈现非经典性质, 且在m取奇(或偶)数时, 增光子偶(或奇)相干态更容易出现非经典性质. 根据增光子奇偶相干态的Wigner函数的边缘分布, 阐明了此Wigner函数的物理意义. 同时, 利用中介表象理论获得了增光子奇偶相干态的量子tomogram函数. 关键词： 增光子奇偶相干态 Wigner函数 中介表象 tomogram函数     

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.     

Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method

By virtue of the new technique of performing integration over Dirac’s ket–bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel–Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”.