Sudarshan-Glauber representation

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 championed over 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 were working on the topic in 1963. It was the subject of a controversy when Glauber was awarded a share of the 2005 Nobel Prize in Physics for his work in this field and George Sudarshan's contribution was not recognized. Despite many useful applications in laser theory and coherence theory, the Glauber–Sudarshan P representation has the drawback that it is not always positive, and therefore is not a true probability function.

We wish to construct a function ${\displaystyle P(\alpha )}$ with the property that the density matrix ${\displaystyle {\hat {\rho }}}$ is diagonal in the basis of coherent states ${\displaystyle \{|\alpha \rangle \}}$, i.e.

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