Recent experiments with ultracold atomic gases have triggered a great deal of interest in some fundamental aspects of the non-equilibrium dynamics of strongly correlated quantum systems. In particular, they raised an intense discussion on the general relation between system integrability and thermalization in the long-time dynamics of such systems. Instead of focusing on the asymptotic values of significant observables, we take a rather different perspective and study the dependence of the intrinsic dynamical time-scale on the initial state. We do this by considering the quantum Ising chain: we drive it out-of-equilibrium by abruptly quenching the transverse field. We focus on the onsite autocorrelation function of the order parameter, extracting the phase coherence time from its asymptotic decay. This phase coherence time is shown to be determined only by the final Hamiltonian gap, and an effective temperature which depends on the energy of the initial state after the quench. Moreover, the dependence of the coherence time on the effective temperature fairly agrees with that obtained in thermal equilibrium as a function of the equilibrium temperature. These results pertain to the Ising universality class, while the integrability of the model seems not to be crucial.
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