New light fields based on integration theory within the Weyl ordering product of operators

The development of quantum optics theory based on the method of integration within an ordered product of operators(IWOP)has greatly stimulated the study of quantum states in the light field,especially non-Gaussian states with various non-classical properties.In this paper,the two-mode squeezing operator is derived with integral theory within the Weyl ordering product of operators using a combinatorial field in which one mode is a chaotic field and the other mode is a vacuum field.The density operator of the new light field,its entanglement property and photon number distribution are analyzed.We also note that tracing a three-mode pure state can yield this new light field.These methods represent a theoretical approach to investigating new density operators of light fields.

Unifying the theory of integration within normal-,Weyl- and antinormal-ordering of operators and the s-ordered operator expansion formula of density operators

By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators （which considers normally ordered, antinormally ordered and Weyl ordered product of operators as its special cases）. The s-ordered operator expansion （denoted by s…s ） formula of density operators is derived, which isρ=2/1-s∫d^2β/π〈-β｜ρ｜β〉sexp{2/s-1（s｜β｜^2-β＊α＋βa-αα）}s The s-parameterized quantization scheme is thus completely established.

Wigner function of optical cumulant operator and its dissipation in thermo-entangled state representation

To conveniently calculate the Wigner function of the optical cumulant operator and its dissipation evolution in a thermal environment, in this paper, the thermo-entangled state representation is introduced to derive the general evolution formula of the Wigner function, and its relation to Weyl correspondence is also discussed. The method of integration within the ordered product of operators is essential to our discussion.

Quantum entangled fractional Fourier transform based on the IWOP technique

In our previous papers,the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics,and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform.The core function of the coordinate-momentum exchange operators in the addition law of fractional Fourier transform was analyzed too.In this paper,the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators(IWOP)are used to establish the entanglement fractional Fourier transform theory to the extent of quantum.A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.

Some new generating function formulae of the two-variable Hermite polynomials and their application in quantum optics

We derive some new generating function formulae of the two-variable Hermite polynomials, such as ∞∑n=0tm/m!Hn,2m（x）,∞∑n=0sntm/n!m!H2n,2m（x,y）,and ∞∑n=0sntm/n!m!H2n＋l,2m＋k（x,y）.We employ the operator Hermite polynomial method and the technique of integration within an ordered product of operators to solve these problems, which will be useful in constructing new optical field states.

New decomposition of the Fresnel operator corresponding to the optical transformation in ABCD-systems

By virtue of the coherent state representation of the newly introduced Fresnel operator and its group product property we obtain new decomposition of the Fresnel operator as the product of the quadratic phase operator, the squeezing operator, and the fractional Fourier transformation operator, which in turn sheds light on the matrix optics design of ABCD-systems The new decomposition for the two-mode Fresnel operator is also obtained by the use of entangled state representation.

Wigner function and density operator of the photon-subtracted squeezed thermal state

By using the technique of integration within an ordered product of operators, the normal ordered density operator of the photon-subtracted squeezed thermal state （PSSTS） is derived. Then the corresponding Wigner function is presented by using the coherent state representation of the Wigner operator. The nonclassical properties of the PSSTS are discussed based on the negativity of the Wigner function.

The squeezing entangled state of two particles with unequal mass

For two unequal-mass particles,we construct the entangled state representation and then derive the corresponding squeezing operator.This squeezing operator has a natural realization in the entangled state representation,which exhibits the intrinsic relation between squeezing and quantum entanglement.This squeezing operator involves both two-mode squeezing and the direct product of two single-mode squeezings.The maximum squeezing occurs when the two particles possess equal mass.When the two particles＇ mass difference becomes large,the component of the two single-mode squeezings becomes dominant.

New generating function formulae of even- and odd-Hermite polynomials obtained and applied in the context of quantum optics

By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials which will be useful in constructing new optical field states. We then show that the squeezed state and photon-added squeezed state can be expressed by even- and odd-Hermite polynomials.

Wigner functions and tomograms of the even and odd binomial states

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product （IWOP） of operators, the Wigner functions of the even and odd binomial states （EOBSs） are obtained. The physical meaning of the Wigner functions for the EOBSs is given by means of their marginal distributions. Moreover, the tomograms of the EOBSs are calculated by virtue of intermediate coordinate-momentum representation in quantum optics.

Squeezing in Representation by Quadratic Combinations of Canonical Operators

The technique of integration within an ordered product of operators and the coherent-state representationare used to convert exponential operators of basis operators (p2, Q2, pQ + Qp) to those of the basis operators (a2, a+2,ara). The coherent state representation of unitary squeezing operators in the factorized form and their normal productform are thus derived. The squeezing engendered by operators of the general form exp{-i[αP2 + βQ2 + γ/2 (QP + PQ)] }is also obtained.

A new kind of special function and its application

By virtue of the operator Hermite polynomial method and the technique of integration within the ordered product of operators we derive a new kind of special function, which is closely related to one- and two-variable Hermite polynomials.Its application in deriving the normalization for some quantum optical states is presented.

Evolution of quantum states via Weyl expansion in dissipative channel

Based on the Weyl expansion representation of Wigner operator and its invariant property under similar transformation,we derived the relationship between input state and output state after a unitary transformation including Wigner function and density operator.It is shown that they can be related by a transformation matrix corresponding to the unitary evolution.In addition,for any density operator going through a dissipative channel,the evolution formula of the Wigner function is also derived.As applications,we considered further the two-mode squeezed vacuum as inputs,and obtained the resulted Wigner function and density operator within normal ordering form.Our method is clear and concise,and can be easily extended to deal with other problems involved in quantum metrology,steering,and quantum information with continuous variable.

Some applications of a new coherent-entangled state in two-mode Fock space

Using the technique of integration within an ordered product of operators, we find a new kind of coherent-entangled state （CES）, which exhibits both coherent and entangled state properties. The set of CESs makes up a complete and partly nonorthogonal representation. Using a beam splitter, we propose a simple experimental scheme to produce the CES. Finally~ we present some applications of CESs in quantum optics.

Time evolution of negative binomial optical field in a diffusion channel

We find the time evolution law of a negative binomial optical field in a diffusion channel. We reveal that by adjusting the diffusion parameter, the photon number can be controlled. Therefore, the diffusion process can be considered a quantum controlling scheme through photon addition.

The N-mode squeezed state with enhanced squeezing

It is known that exp [iA （Q] P1 - i/2）] is a unitary single-mode squeezing operator, where Q1, P1 are the coordinate and momentum operators, respectively. In this paper we employ Dirac＇s coordinate representation to prove that the exponential operator Sn ≡exp[iλi=1∑n（QiPi＋1＋Qi＋1Pi））],（Qn＋1=Q1,Pn＋1=P1）,is an n-mode squeezing operator which enhances the standard squeezing. By virtue of the technique of integration within an ordered product of operators we derive Sn＇s normally ordered expansion and obtain new n-mode squeezed vacuum states, its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.

Two mutually conjugated tripartite entangled states and their fractional Fourier transformation kernel

We newly construct two mutually-conjugate tripartite entangled state representations, based on which we propose the formulation of three-mode entangled fractional Fourier transformation （EFFT） and derive the transformation kernel. The EFFT＇s additivity property is proved and the eigenmode of EFFT is derived. As an application, we calculate the EFFT of the three-mode squeezed vacuum state.

The s-parameterized Weyl-Wigner correspondence in the entangled form and its applications

Based on the theory of integration within s-ordering of operators and the bipartite entangled state representation we introduce s-parameterized Weyl-Wigner correspondence in the entangled form. Some of its applications in quantum optics theory are presented as well.

Thermal vacuum state for the two-coupled-oscillator model at finite temperature: Derivation and application

Following the spirit of thermo field dynamics initiated by Takahashi and Umezawa, we employ the technique of integration within an ordered product of operators to derive the thermal vacuum state （TVS） for the Hamiltonian H of the two-coupled-oscillator model. The ensemble averages of the system are derived conveniently by using the TVS. In addition, the entropy for this system is discussed based on the relation between the generalized Hellmann-Feynman theorem and the entroy variation in the context of the TVS.

New formulas for normalizing photon-added(-subtracted) two-mode squeezed thermal states

For the first time,we derive the compact forms of normalization factors for photon-added（-subtracted） two-mode squeezed thermal states by using the P-representation and the integration within an ordered product of operators（IWOP） technique.It is found that these two factors are related to the Jacobi polynomials.In addition,some new relationships for Jacobi polynomials are presented.