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Loschmidt echo in many-spin systems: contrasting time-scales of local and global measurements

A local excitation in a quantum many-spin system evolves deterministically. A time-reversal procedure, involving the inversion of the signs of every energy and interaction, should produce the excitation revival. This idea, experimentally coined in NMR, embodies the concept of the Loschmidt echo (LE). While such an implementation involves a single spin autocorrelation $M_{1,1}$, i.e. a local LE, theoretical efforts have focused on the study of the recovery probability of a complete many-body state, referred here as global or many-body LE $M_{MB}$. Here, we analyze the relation between these magnitudes, in what concerns to their characteristic time scales and their dependence on the number of spins $N$. We show that the global LE can be understood, to some extent, as the simultaneous occurrence of $N$ independent local LEs, i.e. $M_{MB}\sim \left( M_{1,1}\right) ^{N/4}$. This extensive hypothesis is exact for very short times and confirmed numerically beyond such a regime. Furthermore, we discuss a general picture of the decay of $M_{1,1}$ as a consequence of the interplay between the time scale that characterizes the reversible interactions ($T_{2}$) and that of the perturbation ($τ_{Σ}$). Our analysis suggests that the short time decay, characterized by the time scale $τ_{Σ}$, is greatly enhanced by the complex processes that occur beyond $T_{2}$ . This would ultimately lead to the experimentally observed $T_{3},$ which was found to be roughly independent of $τ_{Σ}$ but closely tied to $T_{2}$.