以双变量厄米多项式表达的量子光学基本恒等式  

Two Basic Operator Identities in Quantum Optics Obtained by Virtue of the Two-Variable Hermite Polynomials

作　　者：展德会 范洪义[2] ZHAN De-hui;FAN Hong-yi(College of Mechanic and Electronic Engineering,Wuyi University,Wuyishan 354300,China;Department of Material Science and Engineering,University of Science and Technology of China,Hefei 230026,China)

机构地区：[1]武夷学院机电工程学院,福建武夷山354300 [2]中国科学技术大学材料科学与工程系,安徽合肥230026

出　　处：《量子光学学报》2024年第1期14-17,共4页Journal of Quantum Optics

基　　金：武夷学院引进人才科研启动经费项目(YJ201808)。

摘　　要：在量子光学理论计算中,经常遇到算符的正规排序和反正规排序问题,我们从双变量厄米多项式Hm,n(x,y)的母函数出发,导出两个简洁的重要的基本算符恒等式并由此可以给出一些推论公式。Quantum optics theory needs an advanced method to tackle density operator’various physical quantities,such as expectation value,variance,cumulant,etc.To be specific,since photon creation and annihilation operators do not commute,we need to deal with the problems of how to convert normally ordered operators into anti-normally ordered operators,and how to convert anti-normally ordered operators into normally ordered operators.In short,the operator re-ordering problem is often encountered in quantum optics theory.In this paper we employ the generating function of two-variable Hermite polynomials to derive two basic operator identities.The first basic operator identity is a~na~(+m)=(-i)~(m+n):H_(m,n)(ia~+,ia):,which converts anti-normally ordered operators into normally ordered operators.As an application of the basic operator identity we compute and get■,meanwhile,we give the commutation relation of[a~m,a~(+n)].The second basic operator identity is■,which converts normallyyordered operators into anti-normally ordered operators.When m=n,in virtue of laguerre's polynomials we get the equality H_(n,n)(x,y)=(-1)~nn!L_n(xy).We derive a formula for the transformation between normal product and the anti-normal product in the end.The two basic operator identities are easily remembered and useful in quantum optics.The application of two-variable Hermite polynomials,such as for studying quantum entangled state representation,is greatly developed by Fan Hong-yi in recent years.One can also apply the new basic operator identities to develop binomial and negative-binomial theory which involves twovariable Hermite polynomials.

关 键 词：双变量厄米多项式 算符恒等式 正规排序 反正规排序 

分 类 号：O431[机械工程—光学工程]