New Form of Legendre Polynomials Obtained by Virtue of Excited Squeezed State and IWOP Technique in Quantum Optics

Using the technique of integration within an ordered product of operators and the intermediate coordinate momentum representation in quantum optics, as well as the excited squeezed state we derive a new form of Legendre polynomials.     

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.     

Successively Squeezed States Studied by Virtue of the IWOP Technique

Two kinds of successively squeezed states which are generated by re-squeezing two single mode squeezed states by the two-mode squeezing operator, or by re-squeezing a two-mode squeezed state by two single-mode squeezing operators, are studied in terms of the newly developed technique of integration within an ordered product (IWOP) of operators. The fluctuations in quadrature phases for the resqueezed states are analyzed.     

New Operator Ordering Formulas Related to Hermite Polynomials Derived by Virtue of IWOP Technique

By virtue of the technique of integration within an ordered product of operators and the fundamentaloperator identity Hn(X) = 2n : Xn :, where X is the coordinate operator and Hn is the n-order Hermite polynomials,:: is the normal ordering symbol, we not only simplify the derivation of the main properties of Hermite polynomials,but also directly derive some new operator identities regarding to Hn(X). Operation for transforming f(X) → :f(X) :is also discussed.     

New Operator Ordering Formulas Related to Hermite Polynomials Derived by Virtue of IWOP Technique

By virtue of the technique of integration within an ordered product of operators and the fundamental operator identity Hn(X)=2^n : X^n :, where X is the coordinate operator and Hn is the n-order Hermite polynomials,: : is the normal ordering symbol, we not only simplify the derivation of the main properties of Hermitc polynomials, but also directly derive some new operator identities regarding to Hn(X). Operation for transforming f(X) → : f(X) :is also discussed.     

New Approach for Calculating Tomograms of Quantum States by Virtue of the IWOP Technique

We show that for tomographic approach there exist two mutual conjugate quantum states [p,σ, τ） and [x,μ, ν） （named the intermediate coordinate-momentum representation）, and the two Radon transforms of the Wigner operator are just the pure-state density matrices [p）σ1τσ1τ （p｜ and （p）λ,ν,λ,ν,（x｜ respectively. As a result, the tomogram of quantum states can be considered as the module-square of the states＇ wave function in these two representations. Throughout the paper we fully employ the technique of integration within an ordered product of operators. In this way we establish a new convenient formalism of quantum tomogram.     

Quantum Hadamard Operators and Their Decomposition Derived by Virtue of IWOP Technique

We introduce the quantum Hadamard operator in continuum state vector space and find that it can be decomposed into a single-mode squeezing operator and a position-momentum mutual transform operator. The two-mode Hadamard operator in bipartite entangled state representation is also introduced, which involves the two-mode squeezing operator and [η〉 ←→｜ξ〉 mutual transformation operator, where [η〉 and ｜ξ〉 are mutual conjugate entangled states. All the discussions are proceeded by virtue of the IWOP technique.     

Completeness Relation of Squeezed States in Squeezing Parametric Space Gained by Virtue of IWOP Technique

Using the newly developed technique of integration within an ordered product of operators, we prove that single-mode (or two-mode) squeezed states span a complete space in squeezing parametric space. The proof is concise and direct, and no further knowledge of SU(1,l) general coherent state theory is needed in our discussion.     

Quantum Mechanics Version of Wavelet Transform Studied by Virtue of IWOP Technique

Using the technique of integral within an ordered product (IWOP) of operators we show that the wavelet transform can be recasted to a matrix element of squeezing-displacing operator between the mother wavelet state vector and the state vector to be transformed in the context of quantum mechanics. In this way many quantum optical states' wavelet transform can be easily derived.     

New Operator Identities and Combinatorics Studied by Virtue of IWOP Technique

Using the coherent state representation we derive some new operator identities and study some mathematical relations in combinatorics. The technique of integral within an ordered product (IWOP) of operators plays an essential role in realizing our goal.     

New Approach for Solving Master Equations in Quantum Optics and Quantum Statistics by Virtue of Thermo-Entangled State Representation

By introducing a fictitious mode to be a counterpart mode of the system mode under review we introduce the entangled state representation <η |, which can arrange master equations of density operators ρ(t) in quantum statisticsas state-vector evolution equations due to the elegant properties of <η |. In this way many master equations (respectively describing damping oscillator, laser, phase sensitive, and phase diffusion processes with different initial density operators) can be concisely solved. Specially, for a damping process characteristic of thedecay constant κ we find that the matrix element of ρ (t) at time t in <η| representation is proportional to that of the initial ρ₀ in the decayed entangled state <ηe^-κt| representation, accompanying with a Gaussian damping factor. Thus we have a new insight about the nature of the dissipative process. We also set up the so-called thermo-entangled state representation of density operators, ρ=∫(d²η /π)< η | ρ> D(η), which is different from all the previous known epresentations.     

Fractional Fourier Transformation for Quantum Mechanical Wave Functions Studied by Virtue of IWOP Technique

Starting from the optical fractional Fourier transform (FFT) and using the technique of integration withinan ordered product of operators we establish a formalism of FFT for quantum mechanical wave functions. In doing so, theessence of FFT can be seen more clearly, and the FFT of some wave functions can be derived more directly and concisely.We also point out that different FFT integral kernels correspond to different quantum mechanical representations. Theyare generalized FFT. The relationship between the FFT and the rotated Wigner operator is studied by virtue of theWeyl ordered form of the Wigner operator.     

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。     

The Development of Dirac's Symbolic Method by Virtue the IWOP Technique

Dirac expected that his symbolic method (q-number theory) which could express the physical law in neat and concise way would probably get developed. In this paper, we show that the technique of integration within an ordered product of operators can fashion Dirac's symbolic method and develop representation theory and transformation theory of quantum mechanics.     

正规和反正规排列互换算符公式的推导

用文献[1]中提出的有序算符内的积分技术,简洁地导出了[量子光学学报,2002,8(1):8-12]一文中给出的有用的光场算符公式,并引入了双模厄米特多项式来表达这些公式.     

Nonlinear Squeezed States and Multiplication Rule of Nonlinear Squeezing Operators Gained via Nonlinear Coherent State Representation and IWOP Technique

Using the nonlinear coherent state representation we derive nonlinear squeezed states and the multiplication rule of nonlinear squeezing operators. We find that the symplectic matrices multiplication rule in nonlinear coherent state projection operator representation maps into the multiplication rule of successive nonlinear squeezing operators.The technique of integral within an ordered product of operators plays an essential role in deriving the multiplication rule.     

Quantum State Sharing by Using Two-Mode and Single-Mode Squeezed State Lights

We propose a quantum state sharing scheme for continuous variables using bright two-mode squeezed state and single-mode squeezed state light. The squeezing of a single-mode state is applied to enhance the security of information in quantum teleportation network. The signal-to-noise ratio of communication and the fidelity between the secret and reconstruction state are analysed. It is shown that both the receivers of Bob and Charlie cannot extract information with a high signal-to-noise ratio because of the large noise come from the other quadrature component of single mode squeezed state. Anyone of Bob and Charlie can retrieve the quantum state with a high signal-to-noise ratio if and only if the other one cooperates with the measurement.     

Nonlinear Squeezed States and Multiplication Rule of Nonlinear Squeezing Operators Gained via Nonlinear Coherent State Representation and IWOP Technique

Using the nonlinear coherent state representation we derive nonlinear squeezed states and the multiplication rule of nonlinear squeezing operators. We find that the symplectic matrices multiplication rule in nonlinear coherent state projection operator representation maps into the multiplication rule of successive nonlinear squeezing operators.The technique of integral within an ordered product of operators plays an essential role in deriving the multiplication rule.     

Invariants and Quantum Fluctuations of a Mesoscopic RLC Circuit in a Squeezed State and Squeezed Number State

The invariants for a mesoscopic RLC circuit with a power source are studied and used to construct the squeezed states and squeezed number states for the system. The quantum fluctuations of the mesoscopic RLC circuit in the squeezed states and squeezed number states are also investigated.