New Application of Weyl Correspondence in Studying Husimi Operator

In this paper, the so-cMled Husimi operator △h（q,p; κ）, which is introduced by smoothing out the Wigner operator △ω（q,p） br averaging over the ＂coarse graining＂ function exp[-κ（q＇ - q）^2- （p＇- p）^2/κ], is now regarded as a Weft correspondence connecting the Husimi operator △h（q, p; κ） with its classical correspondence, since the integration kernel is just the Wigner operator. In this way we can easily identify ｜p, q; κ ） such that △ h （ q, p; κ ） = ｜p, q;κ ） （P, q; κ｜, where ｜P, q;κ） is a new kind of squeezed coherent states. The entangled Husimi operator is also treated in this way. Thus a simple way to tnd the Husimi operator is presented.     

New Approach for Calculating Tomograms of Quantum States by Virtue of the IWOP Technique

We show that for tomographic approach there exist two mutual conjugate quantum states [p,σ, τ） and [x,μ, ν） （named the intermediate coordinate-momentum representation）, and the two Radon transforms of the Wigner operator are just the pure-state density matrices [p）σ1τσ1τ （p｜ and （p）λ,ν,λ,ν,（x｜ respectively. As a result, the tomogram of quantum states can be considered as the module-square of the states＇ wave function in these two representations. Throughout the paper we fully employ the technique of integration within an ordered product of operators. In this way we establish a new convenient formalism of quantum tomogram.     

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.     

General Wigner Transforms Studied by Virtue of Weyl Ordering of the Wigner Operator

By virtue of the property that Weyl ordering is invariant under similar transformations we show that the Weyl ordered form of the Wigner operator, a Dirac δ-operator function, brings much convenience for derivingmiscellaneous Wigner transforms. The operators which engender various transforms of the Wigner operator, can alsobe easily deduced by virtue of the Weyl ordering technique. The correspondence between the optical Wigner transformsand the squeezing transforms in quantum optics is investigated.     

General Wigner Transforms Studied by Virtue of Weyl Ordering of the Wigner Operator

By virtue of the property that Weyl ordering is invariant under similar transformations we show that the Weyl ordered form of the Wigner operator, a Dirac δ-operator function, brings much convenience for deriving miscellaneous Wigner transforms. The operators which engender various transforms of the Wigner operator, can also be easily deduced by virtue of the Weyl ordering technique. The correspondence between the optical Wigner transforms and the squeezing transforms in quantum optics is investigated.     

Fractional Fourier Transformation for Quantum Mechanical Wave Functions Studied by Virtue of IWOP Technique

Starting from the optical fractional Fourier transform (FFT) and using the technique of integration withinan ordered product of operators we establish a formalism of FFT for quantum mechanical wave functions. In doing so, theessence of FFT can be seen more clearly, and the FFT of some wave functions can be derived more directly and concisely.We also point out that different FFT integral kernels correspond to different quantum mechanical representations. Theyare generalized FFT. The relationship between the FFT and the rotated Wigner operator is studied by virtue of theWeyl ordered form of the Wigner operator.     

Applications of Weyl Ordered Two-Mode Wigner Operator for Quantum Mechanical Entangled System

Based on the technique of integral within a Weyl ordered product of operators, we present applications of the Weyl ordered two-mode Wigner operator for quantum mechanical entangled system, e.g., we derive the complex Wigner transform and its relation to the complex fractional Fourier transform, as well as the entangled Radon transform.     

Weyl-Ordered Operator Moyal Bracket by Virtue of Method of Integral Within an Weyl Ordered Product of Operators

The Moyal bracket is an exemplification of Weyl's correspondence to formulate quantum mechancis in terms of Wigner function. Here we present a formalism of Weyl-ordered operator Moyal bracket by virtue of the method of integral within a Weyl ordered product of operators and the Weyl ordering operator formula.     

Wigner function for the generalized excited pair coherent state

This paper introduces the generalized excited pair coherent state （GEPCS）. Using the entangled state 〈η〉 representation of Wigner operator, it obtains the Wigner function for the GEPCS. In the ρ-γ phase space, the variations of the Wigner function distributions with the parameters q, α, k and l are discussed. The tomogram of the GEPCS is calculated with the help of the Radon transform between the Wigner operator and the projection operator of the entangled state ｜η1, η2, τ1, τ2｜. The entangled states ｜η〉 and ｜η1, η2, τ1, τ2〉 provide two good representative space for studying the Wigner functions and tomograms of various two-mode correlated quantum states.     

用Weyl对应与Wigner算符的途径构造表象

由Wigner算符的完备性和Weyl对应,我们推导出一个能获得纯态密度算符的新等式。借助此公式,可以方便快捷的构造出量子力学中一些有用的新表象。     

Effect of Dissipative Environments on Atomic-State Quantum Teleportation

In the paper, taking the atomic EPR entanglement states as quantum channel, we investigate the fidelity of quantum teleportation of atomic state in thermal environment and vacuum reservoir by means of quantum theory of damping-density operator approach, and the average fidelities are calculated. the resultsshow that the atomic quantum channel state |ø> = (1/2^1/2)(| 00>+ |11>) is more robust than | Φ> = (1/2^1/2)(|01> + |10>) in teleportation process when they are subject to the dissipative environments.     

S-ordered Expansion of the Intermediate Representation Projector and Its Applications

We first deduce the s-ordered expansion of the Wigner operator. Since Radon transformation of Wigner operator is just the intermediate representation |x〉 _λ,ν projector, we naturally obtain the s-ordered product of |x〉 _λ,νλ,ν〈x|. Accordingly, the completeness relation is still preserved under the s-ordering. Finally, based on it, we obtain the s-ordered expansion of some useful operator in quantum optics, and some new operator identities are revealed accordingly.     

Algebraic aspects of the wigner distribution in quantum mechanics

The algebraic structure underlying the method of the Wigner distribution in quantum mechanics and the Weyl correspondence between classical and quantum dynamical variables is analysed. The basic idea is to treat the operators acting on a Hilbert space as forming a second Hilbert space, and to make use of certain linear operators on them. The Wigner distribution is also related to the diagonal coherent state representation of quantum optics by this method.     

SORE APPLICATIONS OF THE COHERENT STATE FORMULATION OF THE WIGNER OPERATOR

On the basis of the preceding paper^[1]we present some new applications of both the normal rroduct form and the coherent state form of the Wigner operator,which involve deriving some new quantum operator formulas, giving the coherent state generalization of the Moyal theorem, evaluating some quantum operators which correspond to the given classical functions in the weyl manner and vice versa. Were it not for the Wigner operator's coherent state formulation given by us the above-mentioned calculations would be hard to perform.     

Optical complex integration-transform for deriving complex fractional squeezing operator

We find a new complex integration-transform which can establish a new relationship between a two-mode operator's matrix element in the entangled state representation and its Wigner function. This integration keeps modulus invariant and therefore invertible. Based on this and the Weyl–Wigner correspondence theory, we find a two-mode operator which is responsible for complex fractional squeezing transformation. The entangled state representation and the Weyl ordering form of the two-mode Wigner operator are fully used in our derivation which brings convenience.     

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.     

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.     

Exact integral operator form of the Wigner distribution-function equation in many-body quantum transport theory

A formal derivation of a generalized equation of a Wigner distribution function including all many-body effects and all scattering mechanisms is given. The result is given in integral operator form suitable for application to the numerical modeling of quantum tunneling and quantum interference solid state devices. In the absence of scattering and many-body effects, the result reduces to the noninteracting-particle Wigner distribution function equation, often used to simulate resonant tunneling devices. The derivation uses a Weyl transform technique which can easily incorporate Bloch electrons. Weyl transforms of self-energies are derived. Various simplifications of a general quantum transport equation for semiconductor device analysis and self-consistent numerical simulation of a quantum distribution function in the phase-space/frequency-time domain are discussed. Recent attempts to include collisions in the Wigner distribution-function approach to the numerical simulation of tunneling devices are clearly shown to be non-self-consistent and inaccurate; more accurate numerical simulation is needed for a deeper understanding of the effects of collision and scattering.     

引入Wigner函数,Weyl对应和Wigner算符相干态表象的新途径

用坐标、动量完备性的正规乘积内积分形式直接地引入了Wigner函数和Wigner算符的相干态表象,简洁地阐述了它与Weyl对应的关系。     

Evolution of quantum states via Weyl expansion in dissipative channel

Based on the Weyl expansion representation of Wigner operator and its invariant property under similar transformation, we derived the relationship between input state and output state after a unitary transformation including Wigner function and density operator. It is shown that they can be related by a transformation matrix corresponding to the unitary evolution. In addition, for any density operator going through a dissipative channel, the evolution formula of the Wigner function is also derived. As applications, we considered further the two-mode squeezed vacuum as inputs, and obtained the resulted Wigner function and density operator within normal ordering form. Our method is clear and concise, and can be easily extended to deal with other problems involved in quantum metrology, steering, and quantum information with continuous variable.