  We propose the method for optical visualization of Bose-Hubbard model with
two interacting bosons in the form of two-dimensional (2D) optical lattices
consisting of optical waveguides, where the waveguides at the diagonal are
characterized by different refractive index than others elsewhere, modeling the
boson-boson interaction. We study the light intensity distribution function
averaged over direction of propagation for both ordered and disordered cases,
exploring sensitivity of the averaged picture with respect to the beam
injection position. For our finite systems the resulting patterns reminiscent
the ones set in billiards and therefore we introduce a definition of discrete
quantum billiard discussing the possible relevance to its well established
continuous counterpart.
