Golden ratio entanglement in hexagonally poled nonlinear crystals

This work analyses the quantum state of twin photons and twin beams generated by parametric-down conversion in a hexagonally poled photonic crystal,characterized by the simultaneous presence of two nonlinear processes sustained by two vectors of the reciprocal lattice. In those special points of the fluorescence spectrum where the two processes coexist,we show that a tripartite entangled state is realized, equivalent to a single parametric process followed by a beam-splitter. By proper angle tuning a peculiar resonance condition is reached, with a transition to a 4-mode entanglement, dominated by the Golden Ratio of the segment $\phi= (1+\sqrt 5)/2$. A maximal coherence between the two nonlinear processes is here estabilished, as the overall process is shown to be equivalent to two independent parametric processes followed by a beam-splitter. We offer an interpretation of the occurrence of the golden ratio in this system based on an analogy between the evolution of the light modes and the Fibonacci sequence.