Diffusive behavior from a quantum master equation

We study a general class of translation invariant quantum Markov evolutions for a particle on Z. We assume locality of the spatial jumps and exponentially fast relaxation in momentum space. It is shown that the particle position diffuses in the long time limit. We employ a fiber decomposition in momentum space of the evolution made possible by the translation invariance of the dynamics. A central limit theorem follows from perturbation theory around a single fiber whose evolution in the momentum representation is described by a Markov jump process on the d-dimensional torus.