Optical equivalence theorem

The optical equivalence theorem in quantum optics asserts an equivalence between the expectation value of an operator in Hilbert space and the expectation value of its associated function in the phase space formulation with respect to a quasiprobability distribution. The theorem was first reported by George Sudarshan in 1963 for normally ordered operators and generalized later that decade to any ordering.

Let Ω be an ordering of the non-commutative creation and annihilation operators, and let ${\displaystyle g_{\Omega }({\hat {a}},{\hat {a}}^{\dagger })}$ be an operator that is expressible as a power series in the creation and annihilation operators that satisfies the ordering Ω. Then the optical equivalence theorem is succinctly expressed as

${\displaystyle \langle g_{\Omega }({\hat {a}},{\hat {a}}^{\dagger })\rangle =\langle g_{\Omega }(\alpha ,\alpha ^{*})\rangle .}$

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