Loschmidt echo and dynamical fidelity in periodically driven quantum systems

We study the dynamical fidelity F(t) and the Loschmidt echo L(t), following a periodic driving of the transverse magnetic field of a quantum Ising chain (back and forth across the quantum critical point) by calculating the overlap between the initial ground state and the state reached after n periods tau. We show that [log F(n tau)]/L (the logarithm of the fidelity per site) reaches a steady value in the asymptotic limit n -> infinity, and we derive an exact analytical expression for this quantity. Remarkably, the steady-state value of [log F(n tau -> infinity)]/L shows memory of non-trivial phase information which is instead hidden in the case of thermodynamic quantities; this conclusion, moreover, is not restricted to 1-dimensional models.