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