Overdamping by weakly coupled environments

A quantum system weakly interacting with a fast environment usually undergoes a relaxation with complex frequencies whose imaginary parts are damping rates quadratic in the coupling to the environment, in accord with Fermi’s “Golden Rule”. We show for various models (spin damped by harmonic-oscillator or random-matrix baths, quantum diffusion, quantum Brownian motion) that upon increasing the coupling up to a critical value still small enough to allow for weak-coupling Markovian master equations, a new relaxation regime can occur. In that regime, complex frequencies lose their real parts such that the process becomes overdamped. Our results call into question the standard belief that overdamping is exclusively a strong coupling feature.