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Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical operators; we also show that if there exists the eigenvector |qλ,ν of linear combination of the coordinate and momentum operator, (λQ + νP), where λ, ν are real numbers, and |qλν is complete, then the projector |qλνq| must be the Radon transform of Wigner operator. This approach seems concise and physical appealing.

The coherent-intermediate-entangled state |α, xλ,ν is proposed in the two-mode Fock Space, which exhibits both the properties of the coherent and entangled states. The |α,xλ,ν makes up a new quantum mechanical representation, and the completeness relation of |α,xλ,ν is proved by virtue of the technique of integral within an ordered product of operators. The corresponding squeezing operators are derived. Furthermore, Generalized P-representation is constructed in the coherent-intermediate-entangled state |α,xλ,ν and the Schmidt decomposition of |α,xλ,ν is investigated.

This paper constructs the new common eigenvectors of n intermediate coordinate-momentum operators which are complete and orthonormal. The intermediate coordinate-momentum representation of a multi-particles system is proposed and applied to a generaln-mode quantum harmonic oscillators system with coordinate-momentum coupling.

The q-analogues of two-mode squeezed states are introduced by virtue of deformation quantization methods and the technique of integration within an ordered product (IWOP) of operators. Some new completeness relations about these squeezed states composed of the bra and ket which are not mutually Hermitian conjugates are obtained. Furthermore, the antibunching effects of the two-mode squeezed vacuum state S'2(r) |00〉 are investigated. It is found that, in different ranges of the squeezed parameter r, both modes of the state exhibit the antibunching effects and the two modes of the state are always nonclassical correlation.

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.

This paper introduces the generalized excited pair coherent state (GEPCS). Using the entangled state |η〉 representation of Wigner operator, it obtains the Wigner function for the GEPCS. In the ρ-γ phase space, the variations of the Wigner function distributions with the parameters q, α, k and l are discussed. The tomogram of the GEPCS is calculated with the help of the Radon transform between the Wigner operator and the projection operator of the entangled state |η1, η2, τ1, τ2〉. The entangled states |η〉 and η1, η2, τ1, τ2〉 provide two good representative space for studying the Wigner functions and tomograms of various two-mode correlated quantum states.

Based on the scheme of damped harmonic oscillator quantization and thermo-field dynamics (TFD), the quantization of mesoscopic damped double resonance RLC circuit with mutual capacitance–inductance coupling is proposed. The quantum fluctuations of charge and current of each loop in a squeezed vacuum state are studied in the thermal excitation case. It is shown that the fluctuations not only depend on circuit inherent parameters, but also rely on excitation quantum number and squeezing parameter. Moreover, due to the finite environmental temperature and damped resistance, the fluctuations increase with the temperature rising, and decay with time.

The mesoscopic quartz piezoelectric crystal equivalent circuit is quantized by the method of damped harmonic oscillator quantization. It is shown that the quantum fluctuations of voltage and current of each loop are related to not only the equivalent circuit inherent parameter and squeezing parameter, but also the temperature, and decay according to exponent along with time in the thermal vacuum state, the thermal coherent state and the thermal squeezed state.

In the preceding paper (Commun. Theor. Phys. 51 (2009) 321) we have recommended a convenient method for disentangling exponential operators. In this work we use this method for disentangling exponential operators composed of angular momentum operators. We mainly disentangle the form of exp[2hJz + gJ+ + kJ−] as the ordering exp(...J+)exp(...Jz)exp(...J−), we employ the Schwinger Bose realization J− = b†a, J+ = a†b, Jz = (a†a − b†b)/2 to fulfil this task, without appealing to Lie algebra method. Note that this operator's disentangling is different from its decomposition in normal ordering.

In this paper, we reconstruct the explicit representation of the radiation field eigenstates in Fock space by decomposing the normally ordered Gaussian operator. Then with the help of the technique of integration within an ordered product of operators, the phase-shifting operator in quantum optics has been expressed through the Dirac's representation theory. In addition, the unitarily equivalent relation between the radiation field eigenstates and the coordinate eigenstates has been naturally established by the phase-shifting operator in quantum optics. These results deepen people's understanding to the radiation field eigenstates and phase-shifting operator 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.

Evaluation and measurement of surface profiles are very important especially to short wavelength optical research. A linear system treatment of short-wavelength surface scattering theory is introduced, and based on this, a new inverse scattering mathematical model of soft X-ray grazing incidence optics is established. By using these scattered light distributions of super-smooth surfaces measured by a soft X-ray reflectometer, the surface profiles of super-smooth surfaces are computed by means of inverse scattering mathematical model of soft X-ray grazing incidence optics. The calculating results are in accordance with those measured by WYKO. It can be concluded that the soft X-ray grazing incidence optical scattering method can calculate micro-roughness and surface auto correlation function of smooth surface accurately, and can give optical surface profiles intuitively.

We investigate how an optical squeezed chaotic field (SCF) evolves in an amplitude dissipation channel. We have used the integration within ordered product of operators technique to derive its evolution law. We also show that the density operator of SCF can be viewed as a generating field of the squeezed number state.

Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator Ωk (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 δ(p — P)δ(q — Q), δ(q — Q)δ(p — P), and Ωk(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 Ωk (p, q) are also derived, which helps us to put the operators into their — and — ordering, respectively.

A new kind of four-mode continuous variable coherent-entangled state is proposed in the Fock space by using the technique of integration within an ordered product, which exhibits both the properties of a coherent state and an entangled state, and spans a complete and orthonormal representation. The conjugate state of the four-mode continuous variable coherent-entangled state is derived by using the Fourier transformation. Moreover, a simple experimental protocol of generating a four-mode continuous variable coherent-entangled state is proposed by using beam splitters. As applications of this four-mode continuous variable coherent-entangled state, a four-mode entangled state and a four-mode squeezing-Fresnel operator are constructed.

Based on the rotation transformation in phase space and the technique of integration within an ordered product of operators, the coherent state representation of the multimode phase shifting operator and one of its new applications in quantum mechanics are given. It is proved that the coherent state is a natural language for describing the phase shifting operator or multimode phase shifting operator. The multimode phase shifting operator is also a useful tool to solve the dynamic problems of the multimode coordinate{momentum coupled harmonic oscillators. The exact energy spectra and eigenstates of such multimode coupled harmonic oscillators can be easily obtained by using the multimode phase shifting operator.

The new two mode coherent entangled state representation 1 p , β 〉 is proposed by the technique of integration within an ordered product of operators. Its characteristics and mechanism are analysed. The results show that, the common eigenvector | p , β 〉 of p 1 - p 2 and a 1 + a 2 exhibits both the characteristics of the coherent state and entangled state. The 1| p , β 〉 makes up a new quantum mechanical representation and possesses the completeness relation. The ideal single-mode squeezed state | p =0〉 2 =exp［1/2 a +2 2 |0〉 2 and the vacuum state |0〉 1 overlapping on a beam splitter may produce the coherent entangled state | p , β 〉at the output by the modulation of the laser field.

By using the idea of damped harmonic oscillator operation,a mesoscopic quartz piezoelectric crystal equivalent circuit is quantized. By this basic theory, we studied the quantum fluctuation of the voltage and current in each branch of the mesoscopic quartz piezoelectric crystal equivalent circuit under the squeezed vacuum state and the vacuum state.