Loschmidt echo and dynamical fidelity in periodically driven quantum systems

Abstract

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 τ. 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.