Weyl Ordering of Two-Mode Fresnel Operator and Its Applications

By using the Weyl ordering operator formula and the Weyl transformation rule, we derive Weyl ordering of the two-mode Fresnel operator, and then obtain its matrix element in the coordinate representation, which is the integral kernel of the generalized two-mode Fresnel transformation in classical optics. By means of the optical operator method, we obtain a decomposition of the two-mode Fresnel operator’s Weyl ordering and an operator identity.     

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.     

Two-mode Integral Form Projection Operator and Its Applications in Quantizing the Mesoscopic Coupling Circuit

A ket-bra form quantum operator is constructed and its transformation properties are researched. Furthermore, we quantize two L-C circuits with coupling and investigate what happens to the two circuits when this system is quantized.     

The Generalized Eigenvector Expansions of the Liouville Operator

In this paper, we study the generalized eigenvector expansions of the Liouville operator, and construct the corresponding rigged Liouville space. As an example, we obtain the rigged Liouville space for the harmonic oscillator of one-dimensional.     

Time Evolution Caused by Hamiltonian Composed of Quadratic Combination of Canonical Operators and Time-Dependent Two-Mode Fresnel Operator

We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU（1,1） algebra. This is an approach for obtaining the time-dependent Hamiltonian from the preassigned time evolution in classical phase space, an approach which is in contrast to Lewis-Riesenfeld＇s invariant operator theory of treating timedependent harmonic oscillators.     

Time Evolution Caused by Hamiltonian Composed of Quadratic Combination of Canonical Operators and Time-Dependent Two-Mode Fresnel Operator

We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU(1,1)algebra. This is an approach for obtaining the time-dependent Hamiltonian from the preassigned time evolution in classical phase space, an approach which is in contrast to Lewis-Riesenfeld‘s invariant operator theory of treating time-dependent harmonic oscillators.     

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

We find that the mapping from classical optical transformations to the optical operator method can be realized by using the coherent state representation and the technique of integration within an ordered product of operators. The optical Fresnel operator derived in (Commun. Theor. Phys. (Beijing, China) 38 (2002) 147) can unify those frequently used optical operators. Various decompositions of Fresnel operator into the exponential canonical operators are obtained.     

Weyl Ordering Operator Formula Derived by IWOP Technique and ItsApplication for Fresnel Operator

Based on the technique of integration within an ordered product ofoperators, the Weyl ordering operator formula is derived and the Fresneloperators' Weyl ordering is also obtained, which together with the Weyltransformation can immediately lead to Fresnel transformation kernel inclassical optics.     

Power Series Expansions for Vacuum Projection Operator of Paraparticles

Some power series expansions for the vacuum projection operator of paraboson and parafermion are derived. A recursion method is essential to these derivations of the power series expansion in terms of creation and annihilation operators of the paraparticles.     

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

Expansions of Algebras and Superalgebras and Some Applications

After reviewing the three well-known methods to obtain Lie algebras and superalgebras from given ones, namely, contractions, deformations and extensions, we describe a fourth method recently introduced, the expansion of Lie (super)algebras. Expanded (super)algebras have, in general, larger dimensions than the original algebra, but also include the ?nönü–Wigner and generalized IW contractions as a particular case. As an example of a physical application of expansions, we discuss the relation between the possible underlying gauge symmetry of eleven-dimensional supergravity and the superalgebra osp(1|32).     

单模转动压缩关联态的WIGNER函数

主要是提出一种新的单模转动压缩关联态,并根据已有的wigner算符求出了它的wigner函数,然后再进一步给出相应的wigner边缘分布.基于wigner边缘分布,我们可以看出该压缩态在坐标空间或动量空间中的几率分布为高斯分布.     

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.     

Optical Fresnel Transform as a Correspondence of the SU(1,1) Squeezing Operator Composed of Quadratic Combination of Canonical Operators and the Entangled State

We study optical Fresnel transforms by finding the appropriate quantum mechanical SU(1,1) squeezing operators which are composed of quadratic combination of canonical operators. In one-mode case, the squeezing operator's matrix element in the coordinate basis is just the kernel of one-dimensional generalized Fresnel transform (GFT); while in two-mode case, the matrix element of the squeezing operator in the entangled state basis leads to the two-dimensional GFT kernel. The work links optical transforms in wave optics to generalized squeezing transforms in quantum optics.     

Two-mode excited entangled coherent states and their entanglement properties

This paper introduces two types of two-mode excited entangled coherent states (TMEECSs) |Ψ_±(α,m,n)>, studies their entanglement characteristics, and investigates the influence of photon excitations on quantum entanglement. It shows that for the state |Ψ₊(α,m,m)> the two-mode photon excitations affect seriously entanglement character while the state |Ψ_-(α,m,m)> is always a maximally entangled state, and shows how such states can be produced by using cavity quantum electrodynamics and quantum measurements. It finds that the entanglement amount of the TMEECSs is larger than that of the single-mode excited entangled coherent states with the same photon excitation number.     

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。     

Parabose Squeezed Operator and Its Applications

By virtue of the parabose squeezed operator,propagator of a parabose parametric amplifier,explicit forms of parabose squeezed number states and normalization factors of excitation states on a parabose squeezed vacuum state are calculated,which generalize the relevant results from ordinary Bose statistics to the parabose case.     

Normally Ordered and Antinormally Ordered Expansions of Some Exponential Operators in Hilbert Space

In this paper, the completeness relations of the fundamental representations in quantum mechanics, together with the "integration within orderd product" technique are exploited to derive normally ordered and antinormally ordered expansions of some exponential operators in Hilbert space. Applications of the normally ordered exponential operators to evaluating Feynman matrix element in coherent state representation are given, which seems to be a new approach.     

The converging prolate spheroidal functions and their use in Fresnel optics

The converging prolate spheroidal functions are introduced. They are shown to exhibit phase conjugacy properties under both finite and infinite Fresnel transformation. Their use in Fresnel optics is illustrated. In particular, the number of degrees of freedom of a Fresnel diffracted field is evaluated.