Density Operators for Coherent Fields

We study the forms which the conditions for coherence impose upon the density operators for electromagnetic fields. All fields possessing first-order coherence, we show, may be regarded as ones in which only a single mode is excited; this mode need not, however, be a monochromatic one. The higher order coherence conditions may be regarded as specifying certain moments of the distribution of the number of photons in the single excited mode. Full coherence, in particular, is shown to require that the number of photons present follow a Poisson distribution. The class of fields which possess full coherence is shown to be larger than the class of eigenstates of the annihilation operators. The unique character of the eigenstates, on the other hand, is demonstrated by means of a number of simple theorems.