Statistical properties of four-mode quantum states in coupled optical parametric down-conversions

We study the quantum properties of four coupled parametric down-conversion processes which can be realized in an aperiodical nonlinear photonic crystal. In the interaction under consideration, each mode is created by two parametric processes. The photon statistics of particular modes and correlation functions for photon numbers and their fluctuations are studied in detail. It is shown that photon number correlations are nonclassical at small interaction lengths. We calculate the covariance matrix for continuous quantum variables of the generated modes and make use of the criterion of separability based on the partial scaling transform to reveal the entanglement between the variables. It is established that cluster entangled states are created in these coupled parametric processes.