Two Basic Operator Identities in Quantum Optics Obtained by Virtue of the Two-Variable Hermite Polynomials
Quantum optics theory needs an advanced method to tackle density operator'various physical quantities,such as expec-tation 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 ana+m=(-i)m+n:Hm,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 am|n〉=√n!/(n-m)!|n-m〉,meanwhile,we give the commutation relation of[am,a+n].The second basic operator identity is a+man =…Hn,m(a+,a)…,which converts nor-mally ordered operators into anti-normally ordered operators.When m=n,in virtue of laguerre's polynomials we get the equal-ity Hn,n(x,y)=(-1)nn!Ln(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 two-variable Hermite polynomials.
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