Volume 150, number 8,9 PHYSICS LETTERS A 19 November 1990 Induced correlations and partition statistics in degenerated light beams D. Costantini Istituto di Statistica, Universit,~ di Genova, Corso Paganini 3, 16125 Genoa, Italy and U. Garibaldi Centro CNR Fisica delle Superfici e delle Basse Temperature, c/o Dipartimento di Fisica, Universitiz di Genova, Via Dodecaneso 33, 16146 Genoa, Italy Received 15 June 1990; revised manuscript received 10 September 1990; accepted for publication 17 September 1990 Communicated by J.P. Vigier Experimental results about partition statistics of optical photons are discUssed, introducing the notion of "exchangeable split- ting process". A very perspicuous connection between the noise distribution and the induced correlation among particles in the splitting process will be shown. A very simple description of the "interstatics regime" will be given, and an alternative suggestion for the treatment of the noise will be proposed. In a recent paper De Martini and Di Fonzo [ 1 ], starting from a work of Tersoff and Bayer [2 ], have shown experimentally that the statistical behaviour of optical photons changes according to whether the scattering probability over two channels is given by a stationary cross-section or by a stochastic one. De Martini and Di Fonzo have illustrated the the- oretical part of their experiment as follows. A mono- chromatic light beam, belonging to a field mode k, in a chaotic state and with an average photon num- ber per mode equal to (n), excites a beam splitter which can be either in a stationary or in a stochastic state, with a corresponding scattering cross-section IV,. over the output modes i= 1, 2. The field density operator is := ~ P(n) ~. p,mP,hlm, n-m)(n-h, hl, n=O m,h~O m and n- m being the photon numbers scattered over the i modes. P(n) describes the statistics of the num- ber operator of the beam, and P(m, n-m) = Ipnm[ 2 is the two-channel partition probability. In the case of a stationary cross-section P(m, n- m) is given by P(m, n-m)=(n)w7 ' W~ -m , (1) called by the authors "Maxwell-Boltzmann parti- tion statistics". In the case of a stochastic cross-section the param- eters W~ are subjected to random fluctuations. As a consequence P(m, n- m) is the average over all pos- sible values of W~. In the ease of symmetry between the two channels, and no loss, the splitting distribution becomes 1 P(m,n-m)= xr"(1-x)"-"p(x)dx, (2) 0 where p (x) is a density function describing the noise introduced at the splitter. The family of noise dis- tributions is that of uniform symmetric distributions p(x)=(x"-x') -1, x' ~x<~x", = O, elsewhere, 0375-9601/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland) 337