Spectral narrowing and spin echo for localized carriers with heavy-tailed Levy distribution of hopping times

We study analytically the free induction decay and the spin echo decay originating from the localized carriers moving between the sites which host random magnetic fields. Due to disorder in the site positions and energies, the on-site residence times, \tau, are widely spread according to the Levy distribution. The power-law tail \propto \tau^{-1-\alpha} in the distribution of waiting times does not affect the conventional spectral narrowing for \alpha >2, but leads to a dramatic acceleration of the free induction decay in the domain 2>\alpha >1. The next abrupt acceleration of the decay takes place as the tail parameter, \alpha, becomes smaller than 1. In the latter domain the decay does not follow a simple-exponent law. To capture the behavior of the average spin in this domain, we solve the evolution equation for the average spin using the approach different from the conventional approach based on the Laplace transform. Unlike the free induction decay, the tail in the distribution of the residence times leads to the slow decay of the spin echo. The echo is dominated by realizations of the carrier motion for which the number of sites, visited by the carrier, is minimal.