Dynamics of Fluctuations in the Open Quantum SSEP and Free Probability

The quantum symmetric simple exclusion process (Q-SSEP) is a simple model of spinless fermions hopping on a one-dimensional chain to neighbouring sites with random amplitudes. It was proposed as a toy model to study the fluctuations of coherent (i.e. purely quantum mechanical) effects in out-of-equilibrium many-body quantum systems. The aim is to describe these coherent effects at a mesoscopic scale and to establish a universal framework which might be applicable to a larger class of mesoscopic systems. Here we formulate the scaling limit of the time dynamics of fluctuating coherences in the open Q-SSEP, i.e. in the case when the one-dimensional chain is coupled to particle reservoirs on the boundaries to maintain it out-of-equilibrium. We show that the fluctuating coherences have a natural interpretation in terms of free cumulants, a concept from free probability theory which thus seems to be an appropriate framework to deal with fluctuating quantum many body systems out-of-equilibrium