A generalized Hadamard transformation from new entangled state

A new entangled state | eta ;theta rangle is proposed by the technique of integral within an ordered product. A generalized Hadamard transformation is derived by virtue of | eta ;theta rangle , which plays a role of Hadamard transformation for (hat a_1 sin theta - hat a_2 cos theta ) and (hat a_1 cos theta + hat a_2 sin theta ).     

From fractional Fourier transformation to quantum mechanical fractional squeezing transformation

By converting the triangular functions in the integration kernel of the fractional Fourier transformation to the hyperbolic function,i.e.,tan α→ tanh α,sin α→ sinh α,we find the quantum mechanical fractional squeezing transformation(FrST) which satisfies additivity.By virtue of the integration technique within the ordered product of operators(IWOP) we derive the unitary operator responsible for the FrST,which is composite and is made of e~(iπa~+a/2) and exp[iα/2(a~2 +a~(+2)).The FrST may be implemented in combinations of quadratic nonlinear crystals with different phase mismatches.     

On the entangled fractional squeezing transformation

We propose an entangled fractional squeezing transformation (EFrST) generated by using two mutually conjugate entangled state representations with the following operator: e-iα(a1?a2?+a1a2)eiπa2?a2; this transformation sharply contrasts the complex fractional Fourier transformation produced by using e-iα(a1?a2?+a2?a2)eiπa2?a2 (see Front. Phys. DOI 10.1007/s11467-014-0445-x). The EFrST is obtained by converting the triangular functions in the integration kernel of the usual fractional Fourier transformation into hyperbolic functions, i.e., tanα → tanhα and sinα → sinhα. The fractional property of the EFrST can be well described by virtue of the properties of the entangled state representations.     

基于纠缠态表象的复脊波变换理论

本文基于连续变量量子态构造小波变换的研究结果, 从经典信息的连续脊波变换出发, 利用有序算符内ket-bra型积分, 构造连续复脊波变换对应的量子算符和表象表示, 采用表象的内积运算与态矢投影展开, 研究量子光学态的复脊波变换理论. 关键词： 有序算符内积分技术 复脊波变换 纠缠态表象 相干态     

Squeezing entangled state of two particles with unequal masses

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 two particles' mass difference becomes large, the component of the two single-mode squeezings becomes dominant.     

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.     

利用热纠缠态表象获得Caldeira-Leggett密度算符方程的积分形式解

开放量子系统,即系统-热库模型,可以用一个关于密度算符的主方程来描述,比如,用来描述固态物理中耗散现象的Caldeira.Leggett主方程.虽然已经有人为了求解此主方程的约化密度矩阵的精确表达式而做过一些努力,但迄今还未见有解答.本文使用了一种全新的方法来求解Caldeira-Leggett方程,用这个新方法可以得到积分形式的显式表达.该方法的要点在于利用有序算符内积分技术把关于密度算符的微分方程首先转化成关于密度态矢量的微分方程,再将密度态矢量投影到热纠缠态表象中,Caldeira-Leggett方程就转变成了关于波函数的微分方程,而波函数是函数.这样就可以使用数学中求解微分方程的方法来求解出波函数.再次利用有序算符内积分技术,再将波函数转化为态矢量和算符,就得到了Caldeira-Leggett方程的积分形势解.     

基于双模相干-纠缠态表象的算符恒等式构造法

利用最近引进的相干-纠缠态|x,α〉表象和有序算符内的积分技术,给出了关于双模光场正交分量（X₁+X₂）/2^1/2和参数化的正交分量（μX₁+νX₂）/λ的函数的若干正规乘积展开式,这对于研究场的高阶压缩行为和构建广义压缩态是有用的. 关键词： 双模相干-纠缠态 算符恒等式 有序算符内的积分技术     

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 F_α[F_β[f]] = F_α+β[f], which the fractional Fourier transformation (FrFT) F_α[f] should satisfy, we reveal that the position-momentum mutualtransformation 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.     

EPR entangled states and complex fractional Fourier transformation

Starting from the Einstein-Podolsky-Rosen entangled state representations of continuous variables we derive a new formulation of complex fractional Fourier transformation (CFFT). We find that two-variable Hermite polynomials are just the eigenmodes of the CFFT. In this way the CFFT is linked to the appropriate operator transformation between two kinds of entangled states in the context of quantum mechanics. In so doing, the CFFT of quantum mechanical wave functions can be derived more directly and concisely. Received 21 February 2002 / Received in final form 1st June 2002 Published online 24 September 2002 RID="*" ID="*"Work supported by National Natural Science Foundation of China under grant 10175057 and the President Foundation of Chinese Academy of Science. RID="a" ID="a"e-mail: fhym@ustc.edu.cn     

Some new generating function formulae of the two-variable Hermite polynomials and their application in quantum optics

We explore the theoretical possibility of extending the usual squeezed state to those produced by nonlinear singlemode squeezing operators. We derive the wave functions of exp[-（ig/2）（（1-X2）1/2P ＋ P（1-X2）1/2）]｜0 in the coordinate representation. A new operator＇s disentangling formula is derived as a by-product.     

Four-mode continuous-variables entangled state: generation from beam splitter, a parametric down-conversion and a polarizer

We introduce a new kind of four-mode continuous variable entangled state in Fock space. The completeness relation and the partly nonorthonormal property of such a state are proven. The scheme to generate this state is presented by combining a symmetrical beamsplitter, a parametric down-conversion and a polarizer. After making a single-mode quadrature amplitude measurement, the remaining three modes are kept in entanglement. And its applications are also discussed.     

New decomposition of the Fresnel operator corresponding to the optical transformation in AB CD-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.     

量子阻尼振子产生的压缩

用有序算符内的积分技术,求解一维量子阻尼振子的时间演化态.结果表明系统的时间演化态是压缩态.     

奇偶二项式光场态的小波变换

利用有序算符内积分方法,小波变换可以表示为被转换态矢|f〉在压缩平移算符U(μ,s)作用下向母小波态矢〈ψ|转换的矩阵元〈ψ|U(μ,s)|f〉. 在此基础上,计算了奇、偶二项式态的小波变换,得到小波变换谱. 结果表明,小波变换谱可以起到识别这些量子力学态的作用,具有直观易辨的优点. 关键词： 小波变换 奇偶二项式态 有序算符内积分     

EPR型连续变量纠缠态的正规乘积方法求解

提出了一种求解EPR型连续变量纠缠态在Fock表象中的具体形式的方法.该方法利用量子场论中的正规乘积的性质,通过对正规乘积形式的玻色子算符函数的运算,导出了Fock表象中两粒子EPR型连续变量纠缠态(即两粒子算符X1-X2和P1 P2的共同本征态)的具体形式.该方法可以进一步推广至多粒子EPR型纠缠态相应具体形式的求解.因而,这是一种求解此类纠缠态在Fock表象中具体形式的普遍技术.     

A new finite-dimensional pair coherent state studied by virtue of the entangled state representation and its statistical behavior

In this paper we construct a new type of finite-dimensional pair coherent states |ξ, q〉 as realizations of SU(2) Lie algebra. Using the technique of integration within an ordered product of operator, the nonorthogonality and completeness properties of the state |ξ, q〉 are investigated. Based on the Wigner operator in the entangled state |τ〉 representation, the Wigner function of |ξ, q〉 is obtained. The properties of |ξ, q〉 are discussed in terms of the negativity of its Wigner function. The tomogram of |ξ, q〉 is calculated with the aid of the Radon transform between the Wigner operator and the projection operator of the entangled state |η, κ₁, κ₂〉. In addition, using the entangled state |τ〉 representation of |ξ, q〉 to show that the states |ξ, q〉 are just a set of energy eigenstates of time-independent two coupled oscillators.     

Coherent state evolution in a Raman dispersion process

By using the technique of integration within an ordered product of operators, we derive the evolution of an initial coherent state in a Raman dispersion process.     

New bipartite coherent-entangled state in two-mode Fock space and its applications

A new bipartite coherent-entangled state (CES) is found which exhibits both coherent state and entangled state properties. The set of CESs constitute a complete and partly nonorthogonal representation. Using an asymmetric beamsplitter a experimental scheme to produce the CES is proposed. Applications of CESs in quantum optics are also presented.