Loschmidt echo and dynamical fidelity in periodically driven quantum systems

We study the dynamical fidelity $\mathcal{F}(t)$ and the Loschmidt echo $\mathcal{L}(t)$, 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 $\log \mathcal{F}(n\tau )\mathrm{/}L$ (the logarithm of the fidelity per-site) reaches a steady value in the asymptotic limit $n\to \mathrm{\infty }$, and we derive an exact analytical expression for this quantity. Remarkably, the steady state value of $[\log \mathcal{F}(n\tau \to \mathrm{\infty })]\mathrm{/}L$ 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.