Dobiński relations and ordering of boson operators

We introduce a generalization of the Dobiński relation, through which we define a family of Bell-type numbers and polynomials. Such generalized Dobiński relations are coherent state matrix elements of expressions involving boson ladder operators. This may be used in order to obtain normally ordered forms of polynomials in creation and annihilation operators, both if the latter satisfy canonical and deformed commutation relations.     

Newton-Leibniz integration for ket-bra operators (II)—application in deriving density operator and generalized partition function formula

Via the route of applying Newton-Leibniz integration rule to Dirac’s symbolic operators, we show that the density operator e^−βH, where H is multi-mode quadratic interacting boson operators, is a mapping of symplectic transformation in the coherent state representation appearing in the form of non-symmetric ket-bra operator integration. By virtue of the technique of integration within an ordered product (IWOP) of operators, we deduce its normally ordered form which directly leads to the generalized partition function formula and the Wigner function. Some new representations, such as displacement-squeezing correlated squeezed coherent states, constructed by the IWOP technique, also bring convenience in deriving partition functions.     

再谈相干态在算符编序中的应用

利用广义Weyl对应导出了相干态密度算符的S-编序表示.通过此表示并结合S-编序内的算符积分技术,导出了若干量子玻色算符的S-编序形式(包含了正规编序、Weyl编序和反正规编序).此方法进一步推广了相干态在算符编序中的应用.     

Evolution of microwave quantum states in terms of measurable ordered moments of creation and annihilation operators

The detection of microwave states is complicated by strong thermal noise, which is inevitably introduced by linear amplifiers. We show how to extract from measured data normally or anti-normally ordered moments of photon creation and annihilation operators, the set of which contains complete information on the quantum state of an electromagnetic field. Equations for the evolution of the quantum state are derived in terms of moments. Using this approach, we consider in detail issues of decoherence and thermalization of microwave quantum states. Results are illustrated using the examples of Fock, coherent, squeezed, thermal, and even and odd coherent states (Schrödinger cat states).     

Normally-Ordered Time Evolution Operator for Mass-Varying Harmonic Oscillator and Wigner Function of Squeezed Number State

For investigating dynamic evolution of a mass-varying harmonic oscillator we constitute a ket-bra integration operator in coherent state representation and then perform this integral by virtue of the technique ofintegration within an ordered product of operators. The normally orderedtime evolution operator is thus obtained. We then derive the Wigner functionof$u(t)| n>, where | n> is a Fock state, which exhibits a generalized squeezing, the squeezing effect is related to the varying mass with time.     

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     

Physics of a Kind of Normally Ordered Gaussian Operators in Quantum Optics

Based on the technique of integration within an ordered product of operators we investigate a completeness relation of pure states （such as the coordinate eigenstate, the momentum eigenstate and the coherent state） into normally ordered Gaussian forms. The Weyl ordering invariance under similarity transformations is employed to reveal physical meaning of a kind of normally ordered Gaussian operators, which have the similar forms to the bivariate normal distributions in statistics, i.e., the thermo mixed state density matrix.     

Invariants and coherent states for a nonstationary fermionic forced oscillator

Invariant creation and annihilation operators and related Fock states and coherent states are built up for the system of nonstationary fermionic forced oscillator.     

The P-representation on Nonlinear Coherent State Basis

Using the technique of integration within an ordered product of nonlinear Bose operators and by composing the nonlinear coherent state's over-completeness relation, we construct the corresponding P-representation theory. The generalized P-representation for some nonlinear Bose operators can be established.     

Fresnel-Transform's Quantum Correspondence and Quantum Optical ABCD Law

Corresponding to the Fresnel transform there exists a unitary operator in quantum optics theory, which could be known the Fresnel operator (FO). We show that the multiplication rule of the FO naturally leads to the quantum optical ABCD law. The canonical operator methods as mapping of ray-transferABCD matrix is explicitly shown by the normally ordered expansion of the FO through the coherent state representation and the technique of integration within an ordered product of operators. We show that time evolution of the damping oscillator embodies the quantum optical ABCD law.     

Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach

This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron–phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.     

The nonlinear squeezed one-photon states and their nonclassical properties

By virtue of the technique of integration within an ordered product (IWOP) of operators and the properties of the inverses of annihilation and creation operators of f-oscillator, this paper obtains two new types of squeezed operators and f-analogues of squeezed one-photon states, which are quite different from ones constructed by Song and Fan ({Phys. Lett.} A {294} (2002) 66). Subsequently, some nonclassical properties of the states are investigated in detail.     

General Formalism for Mapping of Two-Mode Classical Canonical Transformations to Quantum Unitary operators

Two types of canonical transformations in two-mode classical phase space are mapped into the quantum mechanical Hilbert space to produce some new normally ordered unitary operators. These operators are evaluated in the coordinate (momentum) representations using the "integration within ordered product technique, and the mapping is maniferrtly apparent in the derivation. New generalixed coherent states are constructed in terms of these operators, and the uncertainty relations for these states are analysed.     

Construction of coherent states for Morse potential: A su(2)-like approach

We propose a generalized su(2) algebra that perfectly describes the discrete energy part of the Morse potential. Then, we examine particular examples and the approach can be applied to any Morse oscillator and to practically any physical system whose spectrum is finite. Further, we construct the Klauder coherent state for Morse potential satisfying the resolution of identity with a positive measure, obtained through the solution of truncated Stieltjes moment problem. The time evolution of the uncertainty relation of the constructed coherent states is analyzed. The uncertainty relation is more localized for small values of radius of convergence.     

New Formulation of Weyl Quantization Scheme in Coherent State Representation

By virtue of the technique of integration within an ordered product of operators we present a new formulation of the Weyl quantization scheme in the coherent state representation, which not only brings convenience for calculating the Weyl correspondence of normally ordered operators, but also directly leads us to find both the coherent state representation and the Weyl ordering representation of the Wigner operator.     

Permutation invariant algebras, a Fock space realization and the Calogero model

We study permutation invariant oscillator algebras and their Fock space representations using three equivalent techniques, i.e. (i) a normally ordered expansion in creation and annihilation operators, (ii) the action of annihilation operators on monomial states in Fock space and (iii) Gram matrices of inner products in Fock space. We separately discuss permutation invariant algebras which possess hermitean number operators and permutation invariant algebras which possess non-hermitean number operators. The results of a general analysis are applied to the -extended Heisenberg algebra, underlying the M-body Calogero model. Particular attention is devoted to the analysis of Gram matrices for the Calogero model. We discuss their structure, eigenvalues and eigenstates. We obtain a general condition for positivity of eigenvalues, meaning that all norms of states in Fock space are positive if this condition is satisfied. We find a universal critical point at which the reduction of the physical degrees of freedom occurs. We construct dual operators, leading to the ordinary Heisenberg algebra of free Bose oscillators. From the Fock-space point of view, we briefly discuss the existence of a mapping from the Calogero oscillators to the free Bose oscillators and vice versa. Received: 26 July 2001 / Revised version: 9 January 2002 / Published online: 12 April 2002     

Unified Method for Dynamical Groups of Some Anharmonic Potentials

Realizations of the creation and annihilation operators for some important anharmonic potentials, such as the Morse potential, the modified Pöschl–Teller potential (MPT), the pseudoharmonic oscillator, and infinitely deep square-well potential, are presented by a factorization method. It is shown that the operators for the Morse potential and the MPT potential satisfy the commutation relations of an SU(2) algebra, but those of the pseudoharmonic oscillator and the infinitely deep square-well potential constitute an SU(1, 1) algebra. The matrix elements of some related operators are analytically obtained. The harmonic limits of the SU(2) operators for the Morse and MPT potentials are studied as the Weyl algebra.     

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.     

Differential-Form Relations Between Density Operators and Their P-Representations in Coherent State Basis

By virtue of the technique of integration within an ordered product of operators, we derive differential-form relations between density operators and their P-representations in the coherent state basis. Some applications of these new relations in normally ordering operators are given.     

Painlevé IV coherent states

A simple way to find solutions of the Painlevé IV equation is by identifying Hamiltonian systems with third-order differential ladder operators. Some of these systems can be obtained by applying supersymmetric quantum mechanics (SUSY QM) to the harmonic oscillator. In this work, we will construct families of coherent states for such subset of SUSY partner Hamiltonians which are connected with the Painlevé IV equation. First, these coherent states are built up as eigenstates of the annihilation operator, then as displaced versions of the extremal states, both involving the related third-order ladder operators, and finally as extremal states which are also displaced but now using the so called linearized ladder operators. To each SUSY partner Hamiltonian corresponds two families of coherent states: one inside the infinite subspace associated with the isospectral part of the spectrum and another one in the finite subspace generated by the states created through the SUSY technique.