The optical equivalence theorem in

quantum optics Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have ...

asserts an equivalence between the

expectation value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...

of an operator in

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...

and the expectation value of its associated function in the

phase space formulation The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mome ...

with respect to a

quasiprobability distribution A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Quasiprobabilities share several of general features with ordinary probabilities, ...

. The theorem was first reported by

George Sudarshan Ennackal Chandy George Sudarshan (also known as E. C. G. Sudarshan; 16 September 1931 – 13 May 2018) was an Indian American theoretical physicist and a professor at the University of Texas. Sudarshan has been credited with numerous contri ...

in 1963 for normally ordered operators and generalized later that decade to any ordering.G. S. Agarwal and E. Wolf "Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. II. Quantum Mechanics in Phase Space", ''Phys. Rev. D'',2 (1970) pp. 2187–2205. Let Ω be an ordering of the non-commutative

creation and annihilation operators Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually ...

, and let

g_(\hat,\hat^)

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 Here, is understood to be the

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...

of the annihilation operator on a coherent state and is replaced formally in the power series expansion of . The left side of the above equation is an expectation value in the Hilbert space whereas the right hand side is an expectation value with respect to the quasiprobability distribution. We may write each of these explicitly for better clarity. Let

\hat

be the density operator and

be the ordering ''reciprocal'' to Ω. The quasiprobability distribution associated with Ω is given, then, at least formally, by :

\hat = \frac \int f_(\alpha,\alpha^*) , \alpha\rangle\langle\alpha,  \, d^2\alpha.

The above framed equation becomes :

\operatorname( \hat \cdot g_\Omega(\hat,\hat^\dagger)) = \int f_(\alpha,\alpha^*) g_\Omega(\alpha,\alpha^*) \, d^2\alpha.

For example, let Ω be the

normal order In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operator ...

. This means that can be written in a power series of the following form: :

g_N(\hat^\dagger, \hat) = \sum_ c_ \hat^ \hat^m.

The quasiprobability distribution associated with the normal order is the Glauber-Sudarshan P representation. In these terms, we arrive at :

\operatorname( \hat \cdot g_N(\hat,\hat^\dagger)) = \int P(\alpha) g(\alpha,\alpha^*) \, d^2\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.

References

{{DEFAULTSORT:Optical Equivalence Theorem Quantum optics Theorems in quantum mechanics