  Photonic lattices are usually considered to be limited by their lack of
methods to include interactions. We address this issue by introducing
mean-field interactions through optical components which are external to the
photonic lattice. The proposed technique to realise mean-field interacting
photonic lattices relies on a Suzuki-Trotter decomposition of the unitary
evolution for the full Hamiltonian. The technique realises the dynamics in an
analogous way to that of a step-wise numerical implementation of quantum
dynamics, in the spirit of digital quantum simulation. It is a very versatile
technique which allows for the emulation of interactions that do not only
depend on inter-particle separations or do not decay with particle separation.
We detail the proposed experimental scheme and consider two examples of
interacting phenomena, self-trapping and the decay of Bloch oscillations, that
are observable with the proposed technique.
