Lattice mapping for many-body open quantum systems and its application to atoms in photonic cystals

We present a derivation that maps the original problem of a many body open quantum system (OQS) coupled to a harmonic oscillator reservoir into that of a many body OQS coupled to a lattice of harmonic oscillators. The present method is particularly suitable to analyse the dynamics of atoms arranged in a periodic structure and coupled the EM field within a photonic crystal. It allows to solve the dynamics of a many body OQS with methods alternative to the commonly used master, stochastic Schr\"{o}dinger and Heisenberg equations, and thus to reach regimes well beyond the weak coupling and Born-Markov approximations.