Fresnel Operator for Deriving the Propagator of 2D Harmonic Oscillator with Cross Coupling

Using the identity of operator decomposition we obtain a normal ordered form of the time-evolution operator for cross coupling quantum harmonic oscillator Hamiltonian system in two dimensions, which is just a special two-mode Fresnel operator. The Feynman propagator for the Hamiltonian system is found by a direct calculation by means of the method deriving the matrix element of two-mode Fresnel operator in the entangled state representation. The technique of integration within an ordered product (IWOP) of operators is employed to derive the matrix elements of the operator in the coherent state and the entangled state representations.     

Optical Four-Wave Mixing Operator, Fresnel Operators and Three-Mode Entangled State Representation

We analyse the optical four-wave mixing operator S and relate it to the two-mode Fresnel operator. It is shown that the direct product of the two-mode Fresnel operator and the single-mode Fresnel operator has a natural representation on the basis of a three-mode entangled state, which is constructed by S and a beam splitter transform.     

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.     

Representation of the coherent state for a beam splitter operator and its applications

A beam splitter operator is a very important linear device in the field of quantum optics and quantum information. It can not only be used to prepare complete representations of quantum mechanics, entangled state representation, but it can also be used to simulate the dissipative environment of quantum systems. In this paper, by combining the transform relation of the beam splitter operator and the technique of integration within the product of the operator, we present the coherent state representation of the operator and the corresponding normal ordering form. Based on this, we consider the applications of the coherent state representation of the beam splitter operator, such as deriving some operator identities and entangled state representation preparation with continuous-discrete variables. Furthermore, we extend our investigation to two single and two-mode cascaded beam splitter operators, giving the corresponding coherent state representation and its normal ordering form. In addition, the application of a beam splitter to prepare entangled states in quantum teleportation is further investigated, and the fidelity is discussed. The above results provide good theoretical value in the fields of quantum optics and quantum information.     

Phase Properties of Two-Mode Squeezing-Rotating Entangled Representation

By virtue of the squeezing-rotating entangled representation, we mainly establish thc new two-mode phase operator and phase angle operator, which is a general form including the foregoing formalist in two-mode Fock space.In addition, the corresponding phase distribution function is given in the entangled representation. In terms of this definition, we also analyze the phase behavior of some simple two-mode states such as squeezing-rotating coherent state,squeezing-rotating vacuum state, and so on. It is found that the results exactly agree with the foregoing phase theory.     

Simple approach to deriving operator identities and mathematical formulas regarding to two-variable Hermite polynomials

By virtue of operator ordering technique and the generating function of polynomials, we provide a simple and neat approach to studying operator identities and mathematical formulas regarding to two-variable Hermite polynomials, which differs from the existing mathematical ways. We not only derive some new integration formulas and summation relations about two-variable Hermite polynomial, but also draw a conclusion that two-variable Hermite polynomial excitation of two-mode squeezed vacuum state is a squeezed two-mode number state. This may open a new route of developing mathematics by virtue of the quantum mechanical representations and operator ordering technique.     

Quantum optical ABCD theorem in two-mode case

By introducing the entangled Fresnel operator （EFO） this paper demonstrates that there exists ABCD theorem for two-mode entangled case in quantum optics. The canonical operator method as mapping of ray-transfer ABCD matrix is explicitly shown by EFO＇s normally ordered expansion through the coherent state representation and the technique of integration within an ordered product of operators.     

Evolution of a two-mode squeezed vacuum in the amplitude dissipative channel

For the first time we derive the dissipating result of an initial two-mode squeezed pure vacuum state passing through a two-mode amplitude dissipative channel described by the direct product of two independent single-mode master equations. Although these two master equations do not mix the two modes (there is no coupling between them), since the two-mode squeezed state is simultaneously an entangled state, the final state which emerges from passing this channel is a two-mode mixed density operator. The compact expression of the outcoming state is obtained, which manifestly shows that as time evolves, the squeezing effect decreases.     

Evolution of a two-mode squeezed vacuum in the amplitude dissipative channel

For the first time we derive the dissipating result of an initial two-mode squeezed pure vacuum state passing through a two-mode amplitude dissipative channel described by the direct product of two independent single-mode master equations. Although these two master equations do not mix the two modes (there is no coupling between them), since the two-mode squeezed state is simultaneously an entangled state, the final state which emerges from passing this channel is a two-mode mixed density operator. The compact expression of the outcoming state is obtained, which manifestly shows that as time evolves, the squeezing effect decreases.     

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.     

Phase Properties of Two-Mode Squeezing-Rotating Entangled Representation

By virtue of the squeezing-rotating entangled representation, we mainly establish the new two-mode phase operator and phase angle operat, or, which is a general form including the foregoing formalist in two-mode Fock space. In addition, the corresponding phase distribution function is given in the entangled representation. In terms of this definition, we also analyze the phase behavior of some simple two-mode states such as squeezing-rotatlng coherent state, squeezing-rotating vacuum state, and so on. It is found that the results exactly agree with the foregoing phase theory.     

量子光学理论中双模厄米多项式的新应用

强调双模厄米多项式在量子光学理论中的地位,认为它是研究连续变量纠缠态和压缩态的必要函数,具有明确的物理意义。利用双模厄米多项式,结合有序算符内的积分技术,给出了若干新的算符恒等式和互逆的积分变换公式,证明了压缩双模粒子数态恰好是双变量厄米多项式激发压缩真空态。     

A generalized two-mode entangled state: Its generation,properties, and applications

Using the technique of integration within an ordered product of operators we construct a generalized two-mode entangled state, which can be generated by an asymmetrical beam splitter （BS）. Some important properties of this state, such as orthogonality and Schmidt decomposition, are also dis- cussed by deriving the expression of BS operator in coordinate representation. As its applications, to conjugate state, obtain operator identities, generate new squeezing operators （squeezed state） are also presented. It is shown that the fidelity of quantum teleportation can be enhanced under certain case by using the asymmetrical new squeezed state as entangled resource.     

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.     

Dissipation of Two-Mode Squeezed State in Two Independent Amplitude Dissipative Channels

We examine the dissipative result of a pure two-mode squeezed state passing through two independent amplitude dissipative channels described by the direct product of two independent single-mode master equations. Although these two master equations do not mix the two modes (no coupling between them), but since the two-mode squeezed state is simultaneously an entangled state, the output state emerging from the two channels is a two-mode density operator (mixed state). The exact and compact expression of the outcoming state is obtained, which manifently shows that as time evolves the squeezing effect and photon number decreases.     

纠缠微波信号的特性及表示方法

纠缠微波信号是电磁场微波频段量子特性的体现.在总结了现有纠缠微波信号产生及验证实验的基础上,针对目前没有统一的表达式来描述纠缠微波信号格式的问题,通过深入分析纠缠微波信号的特性,提出了两种纠缠微波信号的表示方法.一种是在量子框架下,利用双模压缩真空态表示,并分别在光子数表象下和Wigner分布下分析了其信号特征,刻画了正交分量之间的正反关联特性;另一种是在经典框架下,利用关联随机信号表示,刻画了测量后纠缠微波信号场幅度正交分量随时间变化的波形图.两种表示恰当合理地反映了纠缠微波信号连续变量纠缠的特性.     

Establishing path integral in the entangled state representation for Hamiltonians in quantum optics

Based on two mutually conjugate entangled state representations, we establish the path integral formalism for some Hamiltonians of quantum optics in entangled state representations. The Wigner operator in the entangled state representation is presented. Its advantages are explained.     

On Common Eigenvector of Parametric Interaction Hamiltonian and Number-Difference Operator Derived by Virtue of Entangled State Representation

By virtue of the properties of bipartite entangled state representation we derive the common eigenvector of the parametric Hamiltonian and the two-mode number-difference operator. This eigenvector is superposition of some definite two-mode Foek states with the coefficients being proportional to hypergeometric functions. The Gauss contiguous relation of hypergeometrie functions is used to confirm the formal solution.     

基于四波混频过程的纠缠光放大

双模纠缠态是量子信息领域一种重要的量子资源,本文基于四波混频过程从理论上提出了对双模纠缠态的单个模式(单模放大方案)和对双模纠缠态的两个模式(双模放大方案)的放大.利用光学分束器模型来模拟在光学传输过程中损耗引入的真空场噪声,利用部分转置正定判据分析了两种不同的放大方案中四波混频过程的增益对初始双模纠缠态的纠缠程度的影响.结果表明,在特定的损耗情况下,两个方案中初始双模纠缠态的纠缠度都随增益的增大而减小,直至消失,且双模放大方案中初始双模纠缠态纠缠消失得比单模放大方案中更快.本文的理论结果为实验上实现基于四波混频过程的双模纠缠态的放大奠定了理论基础.