Ising Hamiltonian for Coupled Fibre-Based Optical Parametric Oscillators

Here, I derive an Ising Hamiltonian for $N$ coupled fibre-based Optical Parametric Oscillators (OPOs). For this, I use the Lugiato-Lefever equation. The derivation of the Hamiltonian does not pretend to be mathematically correct, although it works under the assumption of plane waves propagating through the OPOs. Its objective is rather to be as close as possible to the Hamilton formalism than to be mathematically rigorous. The approach is based on the assumptions that are justified from the physical point of view and are implemented experimentally. The resulting Ising Hamiltonian contains all necessary terms that are connected to the appearance and evolution of phase solitons.