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 operatorformula is derived and the Fresnel operators' Weyl ordering is also obtained,which together with the Weyl transformationcan immediately lead to Fresnel transformation kernel in classical optics.

New decomposition of the Fresnel operator corresponding to the optical transformation in ABCD-systems

By virtue of the coherent state representation of the newly introduced Fresnel operator and its group product property we obtain new decomposition of the Fresnel operator as the product of the quadratic phase operator, the squeezing operator, and the fractional Fourier transformation operator, which in turn sheds light on the matrix optics design of ABCD-systems The new decomposition for the two-mode Fresnel operator is also obtained by the use of entangled state representation.

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

基于求婚在的纠缠的夫累Ｘ尔操作员(EFO )[Commun。Theor。Phys。46 (2006 ) 559 ] ，光操作员方法由 IWOP 技术学习了(妈等， Commun。Theor。Phys。49 (2008 ) 1295 ) 被扩大到二模式的盒子，它给纠缠的夫累Ｘ尔操作员的分解，相应于光线转移矩阵的分解[一， B， C， D ] 。EFO 能在二模式的情况中统一那些光操作员。进指数的正规操作员的 EFO 的各种各样的分解被获得。纠缠的州的表示在研究是有用的。

Optical Operator Method Studied via Fresnel Operator Decomposition and Coherent State Representation

我们发现从古典光转变印射到光操作员方法能被在操作员的一个订的产品以内使用协调州的表示和集成的技术认识到。光夫累Ｘ尔操作员发源在(Commun。Theor。Phys。(中国北京) 38 (2002 ) 147 ) 能统一经常使用的那些光操作员。进指数的正规操作员的夫累Ｘ尔操作员的各种各样的分解被获得。

Weyl Ordering Expansion of Power Product of Coordinate and Momentum Operators

By virtue of the technique of integration within Weyl ordered of operators we derive the formula of Weylordering expansion of power product of coordinate and momentum operators (2^(1/2)Q)~m (2^(1/2)iP)~r =∷H_(m:r)(2^(1/2)Q,2^(1/2)iP)∷,the introduction of two-variable Hermite polynomial H_(m,r) brings much convenience to the study of Weyl correspondence.

Generalized Fresnel operators corresponding to optical Fresnel diffraction and the squeezed states

Corresponding to optical Fresnel diffraction, we show that the exponential quadratic operator exp{ i[ P2+ Q2+ (PQ+QP)]/2} is actually a generalized single-mode Fresnel operator (GFO) in compact form, where [Q,P]=i . We also demonstrate that exp{i [(Q1+Q2)2+(P1 P2)2]+i [(Q1 Q2)2+(P1+P2)2]+i (Q1P2+Q2P1)} is a two-mode GFO. Their disentangling formula and normal ordering form are derived with the use of technique of integration within an ordered product (IWOP) of operators and the coherent state representation. The squeezed states generated by these two GFOs are obtained.

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

用 Wigner 操作符的通常订的 Gaussian 形式，我们概括量阶段空格表示，我们为寻找量的古典通讯导出一个新公式机械操作符；我们也看那是否在那里存在特徵向量 | q 坐标和动量操作员的线性联合，(Q + P ) ，实数，和 | q 是完全的，然后投影机 | qq |一定是 Wigner 操作员的氡变换。这条途径似乎简明、物理呼吁。

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。

Deriving new operator identities by alternately using normally,antinormally,and Weyl ordered integration technique

Dirac's ket-bra formalism is the language of quantum mechanics.We have reviewed how to apply Newton-Leibniz integration rules to Dirac's ket-bra projectors in previous work.In this work,by alternately using the technique of integration within normal,antinormal,and Weyl ordering of operators we not only derive some new operator ordering identities,but also deduce some new integration formulas regarding Laguerre and Hermite polynomials.This may open a new route of directly deriving some complicated mathematical integration formulas by virtue of the quantum mechanical operator ordering technique,without really performing the integrations in the ordinary way.

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.

The s-parameterized Weyl-Wigner correspondence in the entangled form and its applications

Based on the theory of integration within s-ordering of operators and the bipartite entangled state representation we introduce s-parameterized Weyl-Wigner correspondence in the entangled form. Some of its applications in quantum optics theory are presented as well.

New fundamental quantum mechanical operator-ordering identities for the coordinate and momentum operators

In quantum mechanics theory one of the basic operator orderings is Q-P and P-Q ordering,where Q and P are the coordinate operator and the momentum operator,respectively.We derive some new fundamental operator identities about their mutual reordering.The technique of integration within Q-P ordering and P-Q ordering is introduced.The Q-P ordered and P-Q ordered formulas of the Wigner operator are also deduced which makes arranging the operators in either Q-P or P-Q ordering much more convenient.

A generalized Collins formula derived by virtue of the displacement-squeezing related squeezed coherent state representation

Based on the displacement-squeezing related squeezed coherent state representation ｜z〉g and using the technique of integration within an ordered product of operators, this paper finds a generalized Fresnel operator, whose matrix element in the coordinate representation leads to a generalized Collins formula （Huygens-Fresnel integration transformation describing optical diffraction）. The generalized Fresnel operator is derived by a quantum mechanical mapping from z to sz -- rz^＊ in the ｜Z〉g representation, while ｜z〉g in phase space is graphically denoted by an ellipse.

再谈相干态在算符编序中的应用

利用广义Weyl对应导出了相干态密度算符的S-编序表示.通过此表示并结合S-编序内的算符积分技术,导出了若干量子玻色算符的S-编序形式(包含了正规编序、Weyl编序和反正规编序).此方法进一步推广了相干态在算符编序中的应用.

The squeezing entangled state of two particles with unequal mass

For two unequal-mass particles,we construct the entangled state representation and then derive the corresponding squeezing operator.This squeezing operator has a natural realization in the entangled state representation,which exhibits the intrinsic relation between squeezing and quantum entanglement.This squeezing operator involves both two-mode squeezing and the direct product of two single-mode squeezings.The maximum squeezing occurs when the two particles possess equal mass.When the two particles＇ mass difference becomes large,the component of the two single-mode squeezings becomes dominant.

Four-mode coherent-entangled state and its application

A new kind of four-mode continuous variable coherent-entangled state is proposed in the Fock space by using the technique of integration within an ordered product, which exhibits both the properties of a coherent state and an entangled state, and spans a complete and orthonormal representation. The conjugate state of the four-mode continuous variable coherent-entangled state is derived by using the Fourier transformation. Moreover, a simple experimental protocol of generating a four-mode continuous variable coherent-entangled state is proposed by using beam splitters. As applications of this four-mode continuous variable coherent-entangled state, a four-mode entangled state and a four-mode squeezing-Fresnel operator are constructed.

On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations

By a quantum mechanical analysis of the additive rule Fa [Fβ[f]] = Fα＋β[f], which the fractional Fourier transformation （FrFT） Fα [f] should satisfy, we reveal that the position-momentum mutual- transformation operator is the core element for constructing the integration kernel of FrFT. Based on this observation and the two mutually conjugate entangled-state representations, we then derive a core operator for enabling a complex fractional Fourier transformation （CFrFT）, which also obeys the additive rule. In a similar manner, we also reveal the fractional transformation property for a type of Fresnel operator.

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 ex- plore 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, de- riving 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 opera- tor （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 op- tics 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＂.

四体纠缠态表象的构建及其应用

利用有序算符乘积内的积分(IWOP)技术,构建了一种四体相容算符1+2+3+4、p1-p2、p1-p3和p1-p4的共同本征态。研究表明,这种量子态具有正交性和完备性,完全可以作为一种新的表象。在动量表象和坐标表象中,对这种量子态进行了Schmidt分解,表明它是一种四体纠缠态。利用这种纠缠态构造了一个新的四模压缩算符,并分析其压缩特性,指出了纠缠与压缩的内在联系。

量子光学中的双模幺正相位算符及其经典对应

众所周知,传统定义的单模相位算符不是幺正算符。我们选择另一种定义相算符的途径,鉴于相位的测量总是相对的,就像势能的测量是相对于零势能位置一样,所以我们在双模空间中引入相位算符。具体做法是在纠缠态表象的基础上提出幺正的双模相位算符,它的本征值就是经典位相。在此理论框架内,相角对应的正则共轭算符是光子数差算符。并且给出了数差-相测不准关系以及此相位算符的Weyl-Wigner经典对应,发现它是双变数相空间(q1,p1;q2,p2)中的相exp[iargtan(p1+p1)/(q1-q1)]。