The quantum nature of light is one of the fundamental aspects of quantum theory.  The optical process of Parametric Down-Conversion (PDC) is an efficient source of non-classical light.  In this process, single photons from a pump beam are converted into pairs of signal and idler photons with correlated properties such as polarization, frequency and momentum. The PDC process is currently the primary photon source for quantum optics schemes that are based on non-classical photon-number correlations.

It is of great importance to ascertain that the signal and idler photons are collected from correlated optical modes. However, due to the effects of imperfect detection efficiencies and dark counts of photon detection schemes, the number correlations between the modes become less distinct. In this work, we use Silicon Photomultiplier (SiPM) [1] number-resolving detectors in order to measure the joint photon-number distributions of the two polarization modes in a collinear type-II PDC process.  The signal and idler photons are collected from modes with different degrees of correlations controlled by varying the amount of spatial and spectral overlap between the two modes. The corresponding changes in the photon-number distributions are observed and characterized.

We obtain the full joint-probability matrices up to 10 photon events. The joint distributions are compared to their corresponding uncorrelated product  distributions obtained by multiplying the individual photon-number probabilities of the two polarization modes. Despite the distortions caused by dark counts, loss and optical crosstalk in the SiPM detectors, highly correlated states show a clear distinction from their corresponding product result, whereas the zero-correlated distributions are almost identical to their product result.  Throughout the measurements, the individual photon-number probabilities in both polarization modes were held constant, thus maintaining the same product values for all measurements.  As the amount of spatial overlap is decreased, the joint probability matrix approaches that of a product state.

We present two quantitative measures for the degree of correlation of the photon-number distributions for different degrees of spatial overlap. The first is  based on the singular value decomposition of the joint probability matrix, and provides a direct measure for the deviations from a product state.  This measure shows significant increase as the spatial overlap is increased.  An additional measure for the degree of correlations is obtained from the ratio    R = P (n_A,n_B )/P(n_A)·P(n_B) between the joint probabilities and the probabilities of the product state for n_A = n_B. This value increases linearly with the degree of correlations. The linear dependence of R is explained by considering the joint probability distribution of an arbitrary degree of correlations to be a linear combination of the PDC joint distribution and the product distribution. Numerical analysis of this probability under the effects of loss and crosstalk confirms the linear dependence obtained in our measurements.

[1] G. Bondarenko, B. Dolgoshein, V. Golovin, A. Ilyin, R. Klanner, and E. Popova, Nucl. Phys. B. (Proc. Suppl) 61 , 347 (1998).