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 Bosonic Operator Ordering Identities Gained by the Entangled State Representation and Two-Variable Hermite Polynomials

Based on the technique of integration within an ordered product of operators, we derive new bosonicoperators' ordering identities by using entangled state representation and the properties of two-variable Hermite poly-nomials H and vice versa. In doing so, some concise normally (antinormally) ordering operator identities, such asa+man =:Hm,n(a+,a):, ana+m = (-i)m+n:Hm,n(ia+,ia): are obtained.

New operator identities with regard to the two-variable Hermite polynomial by virtue of entangled state representation

By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial （TVHP）. By them and the technique of integration within an ordered product （IWOP） of operators we further derive new generating function formulas of the TVHP. They are useful in quantum optical theoretical calculations. It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.

New Bosonic Operator Ordering Identities Gained by the Entangled State Representation and Two－Variable Hermite Polynomials

Based on the technique of integration within an ordered product of operators, we derive new bosonic operators, ordering identities by using entangled state representation and the properties of two-variable Hermite polynomials , and vice versa. In doing so, some concise normally (antinormally) ordering operator identities, such as : are obtained.

导出双粒子纠缠态表象的新途径(英文)

对于量子光学的双粒子纠缠态表象,我们给出一个新途径以分析其在福克空间中的表达式。此分析自然导致双变数厄密多项式的引入,它截然不同于单变数厄密多项式。

Weyl Ordering Expansion of Power Product of Coordinate and Momentum Operators

By virtue of the technique of integration within Weyl ordered of operators we derive the formula of Weyl ordering expansion of power product of coordinate and momentum operators （√2Q）^m（√2iP） ^τ=：： Hm,r （√2Q, √2iP）：：, the introduction of two-variable Hermite polynomial Hm,r brings much convenience to the study of Weyl correspondence.

Operators's-parameterized ordering and its classical correspondence in quantum optics theory

In reference to the Weyl ordering xmpn→ （1/2）m ∑l=0m （ml）Xm-lPnXl , where X and P are coordinate and momentum operator, respectively, this paper examines operators＇ s-parameterized ordering and its classical correspondence, finds the fundamental function-operator correspondence （1-s/2）（n＋m）/2Hm,n（/2/1-sα,/2/1-sα）→αman and its complementary relation anam→（-i）n＋m（1-s/2）（m＋n）/2：Hm,n（i√2/1-sa,i√2/1-sa）,where Hrn,n is the two-variable Hermite polynomial, a, at are bosonic annihilation and creation operators respectively, s is a complex parameter. The s＇-ordered operator power-series expansion of s-ordered operator atraan in terms of the two-variable Hermite polynomial is also derived. Application of operators＇ s-ordering formula in studying displaced- squeezed chaotic field is discussed.