Timescales of quantum and classical chaotic spin models evolving toward equilibrium

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 semianalytical 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 is strongly chaotic. The correspondence persists even for small quantum spin values (S= 1), where a semiclassical 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 analog, the relaxation timescale is system-size dependent