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.