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Predicting Imperfect Echo Dynamics in Many-Body Quantum Systems

Echo protocols provide a means to investigate the arrow of time in macroscopic processes. Starting from a nonequilibrium state, the many-body quantum system under study is evolved for a certain period of time $τ$. Thereafter, an (effective) time reversal is performed that would -- if implemented perfectly -- take the system back to the initial state after another time period $τ$. Typical examples are nuclear magnetic resonance imaging and polarization echo experiments. The presence of small, uncontrolled inaccuracies during the backward propagation results in deviations of the "echo signal" from the original evolution, and can be exploited to quantify the instability of nonequilibrium states and the irreversibility of the dynamics. We derive an analytic prediction for the typical dependence of this echo signal for macroscopic observables on the magnitude of the inaccuracies and on the duration $τ$ of the process, and verify it in numerical examples.