Felix Fritzsch: Solvable Quantum Circuit Models for Free Independence

Recent experimental and theoretical developments in many-body quantum systems motivate the study of their out-of-equilibrium properties through multi-time correlation functions. We consider the dynamics of higher-order out-of-time-order correlators (OTOCs) in minimal circuit models for quantum dynamics. The first model mimics the dynamics of a structured subsystem locally coupled to a maximally random environment, whereas the environment is given by a spatially local quantum circuit with a staircase design in the second model. In both settings we prove the exponential decay of higher-order OTOCs and identify all relevant time scales. Using ideas from free probability theory, we characterize the equilibration of higher-order OTOCs as the emergence of free independence between time-evolved and static observables. Our results built on capturing the effects of the environment on the local subsystem in a higher-order influence matrix, which allows for a Markovian description of the dynamics provided an auxiliary degree of freedom is introduced.