Emergence of diffusion in finite quantum systems

We study the emergence of diffusion for a quantum particle moving in a finite and translationally invariant one-dimensional subsystem described by a tight-binding Hamiltonian with a single energy band and interacting with its environment by an interaction energy proportional to some coupling parameter. We show that there exists a crossover between a nondiffusive relaxation regime for small sizes or low values of the coupling parameter and a diffusive regime above a critical size or for higher values of the coupling parameter. In the nondiffusive regime, the relaxation is characterized by oscillations decaying at rates independent of the size and proportional to the square of the coupling parameter and the temperature of the environment. In the diffusive regime, the damped oscillations have disappeared and the relaxation rate is inversely proportional to the square of the size. The diffusion coefficient is proportional to the square of the energy bandwidth of the subsystem and inversely proportional to the temperature of the environment and the square of the coupling parameter. The critical size where the crossover happens is obtained analytically.