Dissipative systems fractionally coupled to a bath

Quantum diffusion is a major topic in condensed-matter physics, and the Caldeira–Leggett model has been one of the most successful approaches to study this phenomenon. Here, we generalize this model by coupling the bath to the system through a Liouville fractional derivative. The Liouville fractional Langevin equation is then derived in the classical regime, without imposing a non-Ohmic macroscopic spectral function for the bath. By investigating the short- and long-time behavior of the mean squared displacement, we show that this model is able to describe a large variety of anomalous diffusion. Indeed, we find ballistic, sub-ballistic, and super-ballistic behavior for short times, whereas for long times, we find saturation and sub- and super-diffusion.