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 $\tau$. 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.

5 pages, 4 figures, accepted for publication in EPL

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