Direct observation of the degree of correlations using photon-number-resolving detectors

Optical parametric down-conversion is a common source for the generation of nonclassical correlated photonic states. Using a parametric down-conversion source and photon-number-resolving detectors, we measure the two-mode photon-number distribution of up to 10 photons for different degrees of correlation. The degree of correlation is controlled by collecting different spatial and spectral single modes for each polarization and varying the amount of spectral and spatial overlap between them. Clear evidence for photon-number correlations is presented despite detector imperfections such as low detection efficiency and other distorting effects. Two criteria, derived directly from the raw data, are shown to be good measures for the degree of correlation. Additionally, using a fitting technique, we find a connection between the measured photon-number distribution and the degree of correlation of the reconstructed original two-mode state. These observations are only possible as a result of the detection of high photon number events.

Figure 1

(a) Photon-number resolution of a single detector, demonstrated by the good peak separation of the pulse intensity histogram. (b) The joint photon-number distribution of two polarization modes (H and V) of a type-II collinear PDC process for a maximally correlated state with $\gamma =0.18$.

Figure 2

Joint distribution measurements between (a) highly correlated modes $(\gamma =0.18$, blue circles) and (b) noncorrelated modes ($\gamma =0.06$, green up-triangles). The corresponding products of the two individual distributions are shown as black down-triangles. Poissonian errors are assumed and presented when larger than the symbol size.

Figure 3

Quantitative measures for the degree of correlation of the joint probability distributions. (a) The ratio $R$ of the joint probabilities to their respective product values. (b) The Euclidean distance $\parallel \Delta M\parallel$ between the measured distribution and its closest product state. (c) The reconstructed degree of nonclassicality $g$.

Figure 4

Measurements of distributions with fixed product statistics and different heralded efficiencies of $\gamma =0.17$ (blue circles), $\gamma =0.11$ (green diamonds), $\gamma =0.02$ (red triangles), and the calculated product state (black triangles). The solid lines are fits to Eq. (2) with the following parameters: average number of photons $\langle n\rangle =4.1\pm 0.1$; detection efficiencies, ${\eta }_A=0.012\pm 0.005$ and ${\eta }_B=0.010\pm 0.0005$; crosstalk probabilities, ${\epsilon }_A=0.12\pm 0.01$ and ${\epsilon }_B=0.11\pm 0.01$; average dark count rate, ${\Delta }_A=0.11\pm 0.01$ and ${\Delta }_B=0.14\pm 0.01$.

Figure 5

The reconstructed photon-number probability distributions for the data presented in Fig. 4. (a) $\gamma =0.17$, $g=0.47$; (b) $\gamma =0.11$, $g=0.23$; (c) $\gamma =0.02$, $g=0.06$; and (d) the product state.