Coherence loss and recovery of an electron spin coupled inhomogeneously to a one-dimensional interacting spin bath: an adaptive t-DMRG study

Coherence evolution and echo effect of an electron spin, which is coupled inhomogeneously to an interacting one-dimensional finite spin bath via hyperfine-type interaction, is studied using the adaptive time dependent density matrix renormalization group (t-DMRG) method. It is found that the interplay of the coupling inhomogeneity and the transverse intra-bath interactions results in two qualitatively different coherence evolutions, namely, a coherence preserving evolution characterized by periodic oscillation and a complete decoherence evolution. Correspondingly, the echo effects induced by an electron spin flip at time $tau$ exhibit stable recoherence pulse sequence for the periodic evolution and a single peak at $sqrt 2 tau$ for the decoherence evolution, respectively. With the diagonal intra-bath interaction included, the specific feature of the periodic regime is kept, while the $sqrt 2tau$-type echo effect in the decoherence regime is significantly affected. To render the experimental verifications possible, the Hahn echo envelope as a function of $tau$ is calculated, which eliminates the inhomogeneous broadening effect and serves for the identification of the different status of the dynamic coherence evolution, periodic versus decoherence.