Weyl Ordering Operator Formula Derived by IWOP Technique and ItsApplication for Fresnel Operator

Based on the technique of integration within an ordered product ofoperators, the Weyl ordering operator formula is derived and the Fresneloperators' Weyl ordering is also obtained, which together with the Weyltransformation can immediately lead to Fresnel transformation kernel inclassical optics.     

Weyl Ordering Operator Formula Derived by IWOP Technique and Its Application for Fresnel Operator

Based on the technique of integration within an ordered product of operators, the Weyl ordering operator formula is derived and the Fresnel operators＇ Weyl ordering is also obtained, which together with the Weyl transformation can immediately lead to Eresnel transformation kernel in classical optics.     

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.     

New Operator Ordering Formulas Related to Hermite Polynomials Derived by Virtue of IWOP Technique

By virtue of the technique of integration within an ordered product of operators and the fundamentaloperator identity Hn(X) = 2n : Xn :, where X is the coordinate operator and Hn is the n-order Hermite polynomials,:: is the normal ordering symbol, we not only simplify the derivation of the main properties of Hermite polynomials,but also directly derive some new operator identities regarding to Hn(X). Operation for transforming f(X) → :f(X) :is also discussed.     

Effective Way of Finding Common Eigenvectors of Center-of-Mass Coordinate and Mass-Weighted Relative Momenta of Multi-Partite via IWOP Technique

We present a convenient approach to finding multi-partite entangled state with continuum variables, which is the common eigenvectors of center-of-mass coordinate and mass-weighted relative momenta, by decomposing the normally ordered Gaussian-form operator expressing the completeness relation which is constructed by analyzing the eigenvector equations. The whole derivation is based on the technique of integration within an ordered product of operators.     

New Operator Ordering Formulas Related to Hermite Polynomials Derived by Virtue of IWOP Technique

By virtue of the technique of integration within an ordered product of operators and the fundamental operator identity Hn(X)=2^n : X^n :, where X is the coordinate operator and Hn is the n-order Hermite polynomials,: : is the normal ordering symbol, we not only simplify the derivation of the main properties of Hermitc polynomials, but also directly derive some new operator identities regarding to Hn(X). Operation for transforming f(X) → : f(X) :is also discussed.     

New Three-Mode Squeezing Operators Gained via Tripartite Entangled State Representation

We show that the Agarwal-Simon representation of single-mode squeezed states can be generalized to find new form of three-mode squeezed states. We use the tripartite entangled state representations ｜p, y, z） and ｜x, u, v） to realize this goal.     

Some Unitary Operators Derived Using the Quantum State and Its Application

Some unitary operators are derived using quantum state by depending on the technique of integration within an ordered product of operators, for example parity operator, displacement operator, squeezed operator, etc. The characteristics of these operators are analyzed. Their unitary transformations play an essential role in some transformations. As applications, the dynamic problems of the double momentum coupling harmonic oscillators are solved exactly.     

New Operator Identity Derived by IWOP Technique and Its Application in Claculating Photocount Distribution

Using the technique of integration within an antinormally ordered product of operators we present a convenient approach for deriving some new operator identities in quantum optics theory. Based on P-representation we also derive a new formula for evaluating photocount distribution.     

Quantum Hadamard Operators and Their Decomposition Derived by Virtue of IWOP Technique

We introduce the quantum Hadamard operator in continuum state vector space and find that it can be decomposed into a single-mode squeezing operator and a position-momentum mutual transform operator. The two-mode Hadamard operator in bipartite entangled state representation is also introduced, which involves the two-mode squeezing operator and [η〉 ←→｜ξ〉 mutual transformation operator, where [η〉 and ｜ξ〉 are mutual conjugate entangled states. All the discussions are proceeded by virtue of the IWOP technique.     

A Kind of Four-Mode Nonlinear Entangled State Representation in Quantum Optics

Based on the technique of integration within an ordered product of nonlinear bosonic operators, we construct a new four-mode nonlinear entangled state | α,β,γ〉 _λ,μ in 4-mode Fock space, which can make up a complete set. Its properties and applications are discussed. A possible scheme to generate this state is also presented.     

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。     

Recapitulating Quantum Phase Space Representation by Virtue of Normally Ordered Gaussian Form of Wigner Operator

Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical operators; we also show that if there exists the eigenvector ｜q〉λ,v of linear combination of the coordinate and momentum operator, （λQ ＋ vP）, where λ,v are real numbers, and ｜q〉λv is complete, then the projector ｜q〉λ,vλ,v〈q｜ must be the Radon transform of Wigner operator. This approach seems concise and physical appealing.     

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.     

Optical Operator Method in Two-Mode Case and Entangled Fresnel Operator＇s Decomposition

Based on the entangled Fresnel operator （EFO） proposed in [Commun. Theor. Phys. 46 （2006） 559], the optical operator method studied by the IWOP technique （Ma et al., Commun. Theor. Phys. 49 （2008） 1295） is extended to the two-mode case, which gives the decomposition of the entangled Fresnel operator, corresponding to the decomposition of ray transfer matrix [A, B, C, D]. The EFO can unify those optical operators in two-mode case. Various decompositions of EFO into the exponential canonical operators are obtained. The entangled state representation is useful in the research.     

New Operator Identities and Combinatorics Studied by Virtue of IWOP Technique

Using the coherent state representation we derive some new operator identities and study some mathematical relations in combinatorics. The technique of integral within an ordered product (IWOP) of operators plays an essential role in realizing our goal.     

Deformation of Superoperator's Kraus Representation Derived by Weyl Correspondence and Entangled State Representation

By virtue of the Weyl correspondence and based on the the technique of integration within an ordered product of operators, we show under what condition the superoperator's Kraus representation ρ^ˊ=∑_μ A_μρA_μ^† can be deformed as ρ^ˊ=(1/π)∫d²αB(α)D(α)ρD^†(α), where D(α) is the displacement operator, B(α) is a probability density related to the classical Weyl correspondence of A_μ. An alternate discussion by using the entangled state representation and through a quantum teleportation process is also presented.     

Relationship Between Wave Function and Corresponding Wigner Function Studied in Entangled State Representation

By using the explicit form of the entangled Wigner operator and the entangled state representation we derive the relationship between wave function and corresponding Wigner function for bipartite entangled systems. The technique of integration within an ordered product （IWOP） of operators is employed in our discussions.     

由光分束器和起偏器混合产生的三模纠缠态表象

在三模Fock空间中,本文构建了一种新的三模连续变量纠缠态,它构成了一个新的量子力学表象.该态可以利用非对称光束分离器和起偏器实现.态的纠缠性通过获得其相应的施密特分解得以说明.作为该表象的一个重要应用,利用它实现了单粒子态的量子隐态传输,给出了相应的传输方案. 关键词： 三模连续纠缠态表象 有序算符内的积分技术 量子纠缠 量子隐态传输