一种分解指数算符的简洁方法

提出一种新的分解指数算符的方法,这种方法不仅简单易懂,而且有助于找到更多的算符恒等式.     

Operators’s-parameterized ordering and its classical correspondence in quantum optics theory

In reference to the Weyl ordering xmpn→(1/2)mΣι=0m(ιm)Xm-ιPnXι , where X and P are coordinate and momentum operator, respectively, this paper examines operators’s-parameterized ordering and its classical correspondence, finds the fundamental function-operator correspondence ((1-s)/2)(n+m)/2Hm,n((2/(1-s))～(1/2)α,(2/(1-s))～(1/2)α)→a+man,and its complementary relation αnα*m→(-i)n+m((1-s)/2)(m+n)/2:Hm,n(i(2/(1-s))～(1/2)a+,i((2/(1-s))～(1/2)a):,where H m,n is the two-variable Hermite polynomial, a, a+ are bosonic annihilation and creation operators respectively, s is a complex parameter. The s’-ordered operator power-series expansion of s-ordered operator sa+mans in terms of the two-variable Hermite polynomial is also derived. Application of operators’s-ordering formula in studying displaced-squeezed chaotic field is discussed.     

New two-fold integration transformation for the Wigner operator in phase space quantum mechanics and its relation to operator ordering

Using the Weyl ordering of operators expansion formula (Hong-Yi Fan, \emph{ J. Phys.} A {\bf 25} (1992) 3443) this paper finds a kind of two-fold integration transformation about the Wigner operator $\varDelta \left( q',p'\right) $ ($\mathrm{q}$-number transform) in phase space quantum mechanics, $\iint_{-\infty}^{\infty}\frac{{\rm d}p'{\rm d}q'}{\pi }\varDelta \left( q',p'\right) \e^{-2\i\left( p-p'\right) \left( q-q'\right) }=\delta \left( p-P\right) \delta \left( q-Q\right),$ and its inverse% $ \iint_{-\infty}^{\infty}{\rm d}q{\rm d}p\delta \left( p-P\right) \delta \left( q-Q\right) \e^{2\i\left( p-p'\right) \left( q-q'\right) }=\varDelta \left( q',p'\right),$ where $Q,$ $P$ are the coordinate and momentum operators, respectively. We apply it to study mutual converting formulae among $Q$--$P$ ordering, $P$--$Q$ ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. The formula of the Weyl ordering of operators expansion and the technique of integration within the Weyl ordered product of operators are used in this discussion.     

New approach to Q-P （P-Q） ordering of quantum mechanical operators and its applications

In this paper, we introduce a new way to obtain the Q-P （P-Q） ordering of quantum mechanical operators, i.e., from the classical correspondence of Q-P （P-Q） ordered operators by replacing q and p with coordinate and momentum operators, respectively. Some operator identities are derived concisely. As for its applications, the single （two-） mode squeezed operators and Fresnel operator are examined. It is shown that the classical correspondence of Fresnel operator’s Q-P （P-Q） ordering is just the integration kernel of Fresnel transformation. In addition, a new photo-counting formula is constructed by the Q-P ordering of operators.     

Mutual transformations between the P–Q,Q–P,and generalized Weyl ordering of operators

Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator Ok （p, q） with a real k parameter and can unify the P-Q, Q-P, and Weyl ordering of operators in k = 1, - 1,0, respectively, we find the mutual transformations between 6 （p - P） （q - Q）, （q - Q） 3 （p - P）, and （p, q）, which are, respectively, the integration kernels of the P-Q, Q-P, and generalized Weyl quantization schemes. The mutual transformations provide us with a new approach to deriving the Wigner function of quantum states. The - and - ordered forms of （p, q） are also derived, which helps us to put the operators into their - and - ordering, respectively.     

New useful special function in quantum optics theory

By virtue of the operator Hermite polynomial method [Fan H Y and Zhan D H 2014 Chin. Phys. B 23 060301] we find a new special function which is useful in quantum optics theory, whose expansion involves both power-series and Hermite polynomials, i.e.,min(m,n)∑l-0n!m!(-1)l l!(n-l)!(m-l)!/Hn-l(x)ym-l≡■n,m(x,y).By virtue of the operator Hermite polynomial method and the technique of integration within ordered product of operators(IWOP) we derive its generating function. The circumstance in which this new special function appears and is applicable is considered.     

Wigner functions and quantum interference in optics

A new, approximate generalized Wigner function based on discrete coherent state superpositions is introduced. It is shown that in contrast to the exact generalized Wigner function that may not exist in some region of its parameter the proposed approximate function exists everywhere.     

Normal and antinormal ordering of field operators in quantum optics

The relations between the probability distributionp n _vt of the number of photonsn _vt in a volumeV at timet and the probability distributions P_N(W) and P_A(W) of the integrated intensityW related to the normal and antinormal ordering of field operators and the relations between corresponding moments are studied for a field containingM modes. As the normally ordered correlations are measured with photodetectors and the antinormally ordered correlations are measured with quantum counters the relations betweenn _vt , P_N(W) and P_A(W) permit the statistical behaviour of light to be determined from measurements with photodetectors and quantum counters. The results obtained here are used for the superposition of coherent and thermal fields. It is also shown that the antinormal correlations depend explicitly on the number of modes and that in the classical limit, when the average photon occupation number per mode becomes large, the distributionsn _vt , P_N(W) and P_A(W) become equal.     

Coherent State Representation for a Kind of Operator Disentangling Formulae

We derive a kind of operator disentangling formulae and their coherent state represen tation by virtue of the correspondence be tween symplectic transformations and their quantum mechanical images. The application of the formulae in calculating Feynman transformation matrix elements is discussed.     

New transformation of Wigner operator in phase space quantum mechanics for the two-mode entangled case

As a natural and important extension of Fan's paper (Fan Hong-Yi 2010 Chin. Phys. B 19 040305) by employing the formula of operators' Weyl ordering expansion and the bipartite entangled state representation this paper finds a new two-fold complex integration transformation about the Wigner operator Δ ( μ,v ) (in its entangled form) in phase space quantum mechanics, and its inverse transformation. In this way, some operator ordering problems regarding to ( a₁⁺-a₂) and (a₁+a₂⁺) can be solved and the contents of phase space quantum mechanics can be enriched, where a_i,a_i⁺ are bosonic creation and annihilation operators, respectively.     

双模坐标-动量积分型投影算符及其在量子光学中的应用

引进了幺正的双模坐标-动量积分型投影算符,利用有序算符内的积分（IWOP）技术分析了其变换特性,并导出了其正规乘积展开式.然后利用该积分型投影算符对角化了双模耦合量子谐振子体系的哈密顿量,从而求出了体系的本征能级与本征波函数.最后讨论了特例情形. 关键词： 积分型投影算符 有序算符内的积分技术 坐标-动量耦合     

New approach to the quantum tunneling process: Wave functions for transmission and reflection

A new approach in which tunneling of a quantum particle through a one-dimensional potential barrier is treated as a combined process involving two alternative elementary transmission and reflection processes is presented. It is demonstrated that the wave function describing the state of the entire quantum ensemble of particles can unambiguously be expressed as a sum of two other solutions of the one-dimensional Schrödinger equation (ODSE) that describe the states of sub-ensembles of transmitted and reflected particles in all stages of scattering.     

A new bipartite entangled state describing the parametric down-conversion process and its applications in quantum optics

We construct a new bipartite entangled state(NBES),which describes both the squeezing and the entanglement involved in the parametric down-conversion process and can be produced using a symmetric beam splitter.Constructing asymmetric ket-bra integrations based on the NBES leads to some new squeezing operators,which clearly exhibit the relationships between squeezing and entangled state transformations.Moreover,an entangled Wigner operator with a definite physical meaning is also presented.     

经典菲涅耳变换的量子力学对应

介绍经典菲涅耳变换的量子力学对应,它是相干态在相空间中的代表点做辛运动所对应的量子菲涅耳算符.     

厄米多项式算符的新恒等式及其在量子压缩中的应用

通过发现有关厄米多项式算符H_n(X)的恒等式, 并结合有序算符内的积分技术, 得到了一些关于量子压缩的算符新恒等式, 这对于研究压缩粒子数态波函数十分有用.     

On the quantum group and quantum algebra approach toq-special functions

Two interpretations ofq-special functions based on quantum groups and algebras have been presented in the literature. The connection between these approaches is explained using as an example the case whereU _q (sl(2)) is the basic structure.Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.     

The two-mode quantum Fresnel operator and the multiplication rule of 2D Collins diffraction formula

By using the two-mode Fresnel operator we derive a multiplication rule of two-dimensional (2D) Collins diffraction formula, the inverse of 2D Collins diffraction integration can also be conveniently derived in this way in the context of quantum optics theory.     

New Lie-algebraic structures in nonlinear problems of quantum optics and laser physics

New Lie-algebraic structures (polynomial deformations of Lie algebras) are revealed in some problems of quantum optics and laser physics. Specifically, deformations of oscillator algebras due to extensions of unitary algebras by their symmetric and skew-symmetric tensors are shown to be algebras of dynamic symmetry (ADS) in models of n-photon processes with internal symmetries. Similarly, deformed algebras su_d(2) are found as ADS in the context of generalized Dicke models and frequency conversion models. We also briefly discuss some possible schemes of employing the results to solving physical problems.Based on materials of the Second International Wigner Symposium (Goslar, Germany, July 16–20, 1991) and the International Workshop Squeezing, Groups, and Quantum Mechanics (Baku, Azerbaijan, September 16–20, 1991).Lebedev Physics Institute, Leninsky prospect 53, Moscow 117924, Russia. Published as Preprint No. 138 (1991) of the Lebedev Physics Institute (in English).