Mixtures of coherent and thermal fields
When I started my PhD, there arose the need to describe mixtures of coherent and thermal states, specifically the diagonal elements $p(n)$ of the density matrix (or probability to have $n$ particles) and the postdoc there came up with something that looked hideous to me: $p_{\mathrm{coh}}(n)+p_{\mathrm{th}}(n)$ (normalized).
I was stuyding Glauber's $P$ function formalism then, where it was stated how to admix various fields from their $P$ function. Strangely enough, the statistics $p(n)$ for a mixture of coherent and thermal fields were not worked out, so I did it myself and we used this version instead for the paper.[1]
Of course, such a basic result had been derived before. Apparently the first time by G. Lachs[2]. One can also find the result in Teich and Saleh's excellent book Fundamentals of Photonics.
Possibly also related B. Picinbono and M. Rousseau, Phys. Rev. A 1:635 (1970).
