Photon-number correlations in waveguide lattices with second order coupling

We introduce second-order coupling into the transport and coupling of photons in waveguide lattices. Second-order coupling, as well as high-order couplings, can be significant in long, compact or some special lattices, and is important for large-scale quantum circuits. Photon-number correlations with various amounts of second-order couplings are calculated. Some distinctive and interesting effects arise, and these are helpful for our comprehension of quantum walks in waveguide lattices.