Characterization of classical and quantum Poisson systems by thinnings andsplittings

There exist several well-known characterizations of Poisson and mixed Poiss on point processes (Cox processes) by thinning and splitting procedures. So a point process is necessarily a Cox process if for arbitrary small thinni ng parameter it can be obtained by a thinning of some other point process [ 30]. Poisson processes are characterized by the independence of the two ran dom subconfigurations obtained by an independent splitting of the configura tion into two parts [11]. For quantum mechanical particle systems beam splittings which are well-know n in quantum optics provide analogous procedures. It is shown that coherent states respectively mixtures of them can be characterized in the same way as Poisson processes and Cox processes. Moreover, for the position distributions of these states which are "classic al" point processes just the above mentioned characterizations are obtained . As example of mixed coherent states we consider Gaussian states which arise as equilibrium states of ideal Bose gases.