Exotic Stochastic Processes from Complex Quantum Environments

Stochastic processes are shown to emerge from the time evolution of complex quantum systems. Using parametric, banded random matrix ensembles to describe a quantum chaotic environment, we show that the dynamical evolution of a particle coupled to such environments displays a variety of stochastic behaviors, ranging from turbulent diffusion to L\'evy processes and Brownian motion. Dissipation and diffusion emerge naturally in the stochastic interpretation of the dynamics. This approach provides a derivation of a fractional kinetic theory in the classical limit and leads to classical L\'evy dynamics.