The Glauber–Sudarshan P representation is a suggested way of writing down the phase space distribution of a quantum system in the phase space formulation of quantum mechanics. The P representation is the quasiprobability distribution in which observables are expressed in normal order. In quantum optics, this representation, formally equivalent to several other representations, is sometimes preferred over such alternative representations to describe light in optical phase space, because typical optical observables, such as the particle number operator, are naturally expressed in normal order. It is named after George Sudarshan and Roy J. Glauber, who worked on the topic in 1963. Despite many useful applications in laser theory and coherence theory, the Sudarshan–Glauber P representation has the peculiarity that it is not always positive, and is not a bona-fide probability function.
Definition
We wish to construct a function P ( α ) {\displaystyle P(\alpha )} with the property that the density matrix ρ ^ {\displaystyle {\hat {\rho }}} is diagonal in the basis of coherent states { | α ⟩ } {\displaystyle \{|\alpha \rangle \}} , i.e.,
ρ ^ = ∫ P ( α ) | α ⟩ ⟨ α | d 2 α , d 2 α ≡ d R e ( α ) d I m ( α ) . {\displaystyle {\hat {\rho }}=\int P(\alpha )|{\alpha }\rangle \langle {\alpha }|\,d^{2}\alpha ,\qquad d^{2}\alpha \equiv d\,{\rm {Re}}(\alpha )\,d\,{\rm {Im}}(\alpha ).}
We also wish to construct the function such that the expectation value of a normally ordered operator satisfies the optical equivalence theorem. This implies that the density matrix should be in anti-normal order so that we can express the density matrix as a power series
ρ ^ A = ∑ j , k c j , k ⋅ a ^ j a ^ † k . {\displaystyle {\hat {\rho }}_{A}=\sum _{j,k}c_{j,k}\cdot {\hat {a}}^{j}{\hat {a}}^{\dagger k}.}
Inserting the resolution of the identity
I ^ = 1 π ∫ | α ⟩ ⟨ α | d 2 α , {\displaystyle {\hat {I}}={\frac {1}{\pi }}\int |{\alpha }\rangle \langle {\alpha }|\,d^{2}\alpha ,}
we see that
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