Long-time coherence in echo spectroscopy with π/2-π-π/2 pulse sequence

Motivated by atomic optics experiments, we investigate a class of fidelity functions describing the reconstruction of quantum states by time-reversal operations as MDa(t)=∣∣⟨ψ∣∣eiH2t/2eiH1t/2e−iH2t/2e−iH1t/2∣∣ψ⟩∣∣2. We show that the decay of MDa is quartic in time at short times and that it freezes well above the ergodic value at long times, when H2−H1 is not too large. The long-time saturation value of MDa contains easily extractable information on the strength of decoherence in these systems.