Fast and slow dynamics for classical and quantum walks on mean-field small world networks

This work investigates the dynamical properties of classical and quantum random walks on mean-field small-world (MFSW) networks in the continuous time version. The adopted formalism profits from the large number of exact mathematical properties of their adjacency and Laplacian matrices. Exact expressions for both transition probabilities in terms of Bessel functions are derived. Results are compared to numerical results obtained by working directly the Hamiltonian of the model. For the classical evolution, any infinitesimal amount of disorder causes an exponential decay to the asymptotic equilibrium state, in contrast to the polynomial behavior for the homogeneous case. The typical quantum oscillatory evolution has been characterized by local maxima. It indicates polynomial decay to equilibrium for any degree of disorder. The main finding of the work is the identification of a faster classical spreading as compared to the quantum counterpart. It stays in opposition to the well known diffusive and ballistic for, respectively, the classical and quantum spreading in the linear chain.