ABCD rule for Gaussian beam propagation in the context of quantum optics derived by the IWOP technique

The development of technique of integration within an ordered product (IWOP) of operators extends the Newton-Leibniz integration rule, originally applying to permutable functions, to the non-commutative quantum mechanical operators composed of Dirac’s ket-bra, which enables us to obtain the images of directly mapping symplectic transformation in classical phase space parameterized by [A, B; C, D] into quantum mechanical operator through the coherent state representation, we call them the generalized Fresnel operators (GFO) since they correspond to Fresnel transforms in Fourier optics. Based on GFO we find the ABCD rule for Gaussian beam propagation in the context of quantum optics (both in one-mode and two-mode cases) whose classical correspondence is just the ABCD rule in matrix optics. The entangled state representation is used in discussing the two-mode case.     

Newton-Leibniz integration for ket-bra operators in quantum mechanics (V)—Deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism

We show that Newton-Leibniz integration over Dirac’s ket-bra projection operators with continuum variables, which can be performed by the technique of integration within ordered product (IWOP) of operators [Hong-yi Fan, Hai-liang Lu, Yue Fan, Ann. Phys. 321 (2006) 480], can directly recast density operators and generalized Wigner operators into normally ordered bivariate-normal-distribution form, which has resemblance in statistics. In this way the phase space formalism of quantum mechanics can be developed. The Husimi operator, entangled Husimi operator and entangled Wigner operator for entangled particles with different masses are naturally introduced by virtue of the IWOP technique, and their physical meanings are explained.     

Newton-Leibniz integration for ket-bra operators (III)—Application in fermionic quantum statistics

The Newton-Leibniz integration over Dirac’s ket-bra operators introduced in Ref. [Hong-yi Fan, Hai-liang Lu, Yue Fan, Ann. Phys. 321 (2006) 480-494] is generalized to Newton-Leibniz-Berezin integration over fermionic ket-bra projection operators, the corresponding technique of integration within an ordered product (IWOP) of Fermi operators is proposed which is then used to develop fermionic quantum statistics. The generalized partition function formula of multi-mode quadratic interacting fermion is derived via the fermionic coherent state representation and the IWOP technique. The two-mode fermionic squeezing operators and their group property studied by their fermionic coherent state representation as well as fermionic permutation operator are also deduced in this way. Thus Dirac’s symbolic method for Fermi system can also be developed, which exhibits Bose-Fermi supersymmetry in this aspect.     

s-Parameterized phase space distribution for describing no counts registered on a photonic detector

By introducing the generalized Wigner operator for s-parameterized quasiprobability distribution and employing the technique of integration within ordered product (IWOP) of operators (normally ordered, Weyl ordered or antinormally ordered), we derive two new quantum-mechanical formulas for describing no counts registered on a photonic detector when a light field’s density operator ρ is known, one involves ρ’s s-parameterized distribution function, and the other involves ρ’s coherent state mean value, when these information is known then using the new formulas to calculate no-photocount would be convenient.     

量子光学态的Ridgelet变换

利用有序算符内积分技术推导出一个有用的双模算符正规乘积公式. 然后在量子力学框架下, 计算出相干态、特殊压缩相干态、中介纠缠态表象的Radon变换. 在此基础上, 通过选取“墨西哥帽”母小波函数, 分别分析了以上三种量子光学态的Ridgelet变换. 关键词： 有序算符内积分技术 Radon变换 Ridgelet变换     

基于坐标表象的脊波变换研究

在小波变换量子力学机制的启发下, 通过采用Fock空间里双模坐标本征态改写经典Ridgelet变换, 定义了量子光学态的Ridgelet变换. 然后利用IWOP技术给出不对称积分算符的显式, 并推导出了两个有用的双模算符正规乘积公式. 在此基础上, 通过选取双模“墨西哥帽”母小波函数, 分析了相干态、特殊压缩相干态、中介纠缠态表象的Ridgelet变换. 关键词： IWOP技术 Ridgelet变换 相干态     

Normally Ordered Form of Smoothed Wigner Operator and its Applications

Using the IWOP (integration within ordered product) technique, we derive a normally ordered form of the smoothed Wigner operator with which the smoothed Wigner function, that is specified only in the integral form in the literature, can be simplified. The application of the new form of the smoothed Wigner operator to the coherent state is also presented.     

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.     

Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations

Newton-Leibniz integration rule only applies to commuting functions of continuum variables, while operators made of Dirac’s symbols (ket versus bra, e.g., |q〉〈q| of continuous parameter q) in quantum mechanics are usually not commutative. Therefore, integrations over the operators of type |〉〈| cannot be directly performed by Newton-Leibniz rule. We invented an innovative technique of integration within an ordered product (IWOP) of operators that made the integration of non-commutative operators possible. The IWOP technique thus bridges this mathematical gap between classical mechanics and quantum mechanics, and further reveals the beauty and elegance of Dirac’s symbolic method and transformation theory. Various applications of the IWOP technique, including constructing the entangled state representations and their applications, are 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.     

Wigner function and tomogram of the Hermite polynomial state

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 function for the Hermite polynomial state (HPS). The tomogram of the HPS is calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.     

Mapping of Householder Transformation in Euclidean Space to Rotation-Reflection Operator

By virtue of the technique of integration within an ordered product (IWOP) of operators we maoifestly show that Householder transformation in 3-dimensional Euclidean space directly maps to the normally-ordered rotation-reflection operator. A new "dyad" form of it in three-mode Fock space is also deduced.     

New operator identities with regard to the two-variable Hermite polynomial by virtue of entangled state representation

By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial （TVHP）. By them and the technique of integration within an ordered product （IWOP） of operators we further derive new generating function formulas of the TVHP. They are useful in quantum optical theoretical calculations. It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.     

Normally Ordered Operator Realization of U(1/1) Supergroup and Super-Jaynes-Cummings Model

By making use of the technique ofintegration within an ordered product (IWOP) of operators and the bosonic and fermionic coherent states we derive the normally ordered operator realization of U(1/1) supergroup. Its application in calculating the superpartition function of the supersymmetric Jaynes-Cummings model is demonstrated. The antinormally ordered realization of U(1/1) is also obtained.     

Wigner Functions and Tomograms of?the?Klauder-Perelomov Coherent States for?the?Pseudoharmonic Oscillator

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, the Wigner functions of the Klauder-Perelomov coherent states (KP-CSs) for the pseudoharmonic oscillator (PHO) are obtained and the variations of the Wigner functions with the parameters k and z are discussed. Moreover, the tomograms of the KP-CSs for the PHO are calculated by virtue of intermediate coordinate-momentum representation in quantum optics. Project 10574060 supported by the National Natural Science Foundation of China and project X071049 supported by Science Foundation of Liaocheng University.     

Normal ordering and antinormal ordering of the operator （fQ＋gP）n and some of their applications

In this paper by virtue of the technique of integration within an ordered product (IWOP) of operators and the intermediate coordinate-momentum representation in quantum optics, we derive the normal ordering and antinormal ordering products of the operator (f Q + gP )n when n is an arbitrary integer. These products are very useful in calculating their matrix elements and expectation values and obtaining some useful mathematical formulae. Finally, the applications of some new identities are given.     

单双模连续压缩真空态及其量子统计性质

利用有序算符内的积分技术研究了通过双模压缩算符作用于两个单模压缩 态上得到的单双模 连续压缩态. 导出了单双模连续压缩算 符的正规乘积形式, 并在此基础上研究了单双模连续压缩真空态的量子统计性质. 特别是利用Weyl编 序算符在相似变换 下的不变性, 简洁地导出了单双模连续 压缩真空态的Wigner函数. 最后, 还简单地提出 了单双模连续压缩 真空态的实验产生方案.     

Husimi Functions of Excited Squeezed Vacuum States

Based on the Husimi operator in pure state form introduced by Fan et al., which is a squeezed coherent state projector, and the technique ofintegration within an ordered product (IWOP) of operators, as well as theentangled state representations, we obtain the Husimi functions of theexcited squeezed vacuum states (ESVS) and two marginal distributions of theHusimi functions of the ESVS.     

Some Expressions for Legendre Polynomials and Jacobian Polynomials Gained by Virtue of the IWOP Technique

By virtue of the coherent state and the technique of integration within an ordered product (IWOP) of operators we derive some expressions, which are in differential forms, for Legendre polynomials and Jacobian polynomials.     

Operator Fredholm Integration Equation in Intermediate Coordinate-Momentum Representation and Its Analogue to Thermo Conduction Equation

Using the technique of integration within an ordered product （IWOP） of operators we construct intermediate coordinate-momentum representation, with which we build a type of operator Fredholm integration equation that is an operator generalization of the solution of thermo conduction equation. Then we seach for the solution of operator Fredholm integration equations, which provides us with a new approach for deriving some operator identities.