Optical equivalence theorem

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The optical equivalence theorem says that the expectation value of any normally ordered function of Creation and annihilation operators:

$g^{(N)}({\hat {a}}^{\dagger },{\hat {a}})=\sum _{n,m}c_{nm}{\hat {a}}^{\dagger n}{\hat {a}}^{m}$

can be found by replacing the operators by their eigenvalues and averaging over the resulting complex number, $g^{(N)}(\alpha ,\alpha ^{*})\,$ using the weighting function from the Glauber-Sudarshan P-representation, $\phi (\alpha )\,$.

This theorem implies the formal equivalence between expectation values of normally ordered operators in quantum optics and the corresponding complex numbers in classical optics and was derived by Sudarshan in 1963.