Timescales of quantum and classical chaotic spin models evolving toward equilibrium

Abstract: We investigate quench dynamics in a one-dimensional spin model, comparing both quantum and classical descriptions. Our primary focus is on the different timescales involved in the evolution of the observables as they approach statistical relaxation. Numerical simulations, supported by semi-analytical analysis, reveal that the relaxation of single-particle energies (global quantity) and on-site magnetization (local observable) occurs on a timescale independent of the system size $L$. This relaxation process is equally well-described by classical equations of motion and quantum solutions, demonstrating excellent quantum-classical correspondence, provided the system be strongly chaotic. The correspondence persists even for small quantum spin values ($S=1$), where a semi-classical approximation is not applicable. Conversely, for the participation ratio, which characterizes the initial state spread in the many-body Hilbert space and which lacks a classical analogue, the relaxation timescale is system-size dependent.