Dimension of quantum phase space measured by photon correlations

We show that the different values 1, 2 and 3 of the normalized second-order correlation function g super((2))(0) corresponding to a coherent state, a thermal state and a highly squeezed vacuum originate from the different dimensionality of these states in phase space. In particular, we derive an exact expression for g super((2))(0) in terms of the ratio of the moments of the classical energy evaluated with the Wigner function of the quantum state of interest and corrections proportional to the reciprocal of powers of the average number of photons. In this way we establish a direct link between g super((2,)(0) and the shape of the state in phase space. Moreover, we illuminate this connection by demonstrating that in the semi-classical limit the familiar photon statistics of a thermal state arise from an area in phase space weighted by a two-dimensional Gaussian, whereas those of a highly squeezed state are governed by a line-integral of a one-dimensional Gaussian.