How colors influence numbers: Photon statistics of parametric down-conversion

Parametric down-conversion (PDC) is a technique of ubiquitous experimental significance in the production of nonclassical, photon-number-correlated twin beams. Standard theory of PDC as a two-mode squeezing process predicts and homodyne measurements observe a thermal photon number distribution per beam. Recent experiments have obtained conflicting distributions. In this article, we explain the observation by an a priori theoretical model solely based on directly accessible physical quantities. We compare our predictions with experimental data and find excellent agreement.

Figure 1

Photon number distribution depending on the number of effectively contributing modes as given by the Schmidt number $K$ (and thus on the angle of the SDF) of a type-II PDC process. The $x$ axes depict photon numbers, whereas the $y$ axes show probabilities.

Figure 2

Solid line and dashed line show the distance between the convoluted photon number distribution and Poissonian or thermal statistics, respectively, plotted against the Schmidt number. For a single effective mode, the distribution is exactly thermal, but the more modes contribute, the closer it gets to a Poissonian distribution. The inset fixes $\theta =\pi /4$ and varies ${\sigma }_x^2$, which is drawn on a logarithmic scale.

Figure 3

(Color online) Comparison between experimentally measured and theoretically obtained photon number distributions for a multimode PDC process at various pump strengths. The bottom inset shows the real part of the joint spectral intensity, while the top inset demonstrates the parametrization of the analytical Gaussian approximation of the SDF. Loss inversion and error estimation was performed using non-negative least-squares optimization.

Figure 4

Relation between pump power and coupling parameter. The expected square-root dependency [11] is correctly obtained for a wider range of pump intensities than can be resolved with current TMDs, which ensures the validity of our approach also for high powers. Notice that this knowledge would also allow for computing the expected mean photon number for a given pump power or a determination of ${\chi }^{\left(2\right)}$, as described in [23].