Spatial correlations in hexagons generated via a Kerr nonlinearity

We consider hexagonal pattern forming in the cross section of an optical beam produced by a Kerr cavity, and we study the quantum correlations characterizing this structure. By using arguments related to the symmetry broken by the pattern formation, we identify a complete scenario of six-mode entanglement. Five independent phase quadrature combinations connecting the hexagonal modes are shown to exhibit sub-shot-noise fluctuations. By means of a nonlinear quantum calculation technique, quantum correlations among the mode photon numbers are demonstrated and calculated.