Coherent Fluctuations in Noisy Mesoscopic Systems, the Open Quantum SSEP, and Free Probability

Quantum coherences characterize the ability of particles to quantum mechanically interfere within some given distances. In the context of noisy many-body quantum systems, these coherences can fluctuate. A simple toy model to study such fluctuations in an out-of-equilibrium setting is the open quantum symmetric simple exclusion process (Q-SSEP), which describes spinless fermions in one dimension hopping to neighboring sites with random amplitudes coupled between two reservoirs. Here, we show that the dynamics of fluctuations of coherences in Q-SSEP have a natural interpretation as free cumulants, a concept from free probability theory. Based on this insight, we provide heuristic arguments as to why we expect free probability theory to be an appropriate framework to describe coherent fluctuations in generic mesoscopic systems where the noise emerges from a coarse-grained description. In the case of Q-SSEP, we show how the link to free probability theory can be used to derive the time evolution of connected fluctuations of coherences as well as a simple steady-state solution.