The dynamics of a spin-
1∕2 particle coupled to a nuclear spin bath through an isotropic Heisenberg interaction is studied as a model for the spin decoherence in quantum dots. The time-dependent polarization of the central spin is calculated as a function of the bath-spin distribution and the polarizations of the initial bath state. For short times, the polarization of the central spin shows a Gaussian decay, and at later times it is revived displaying nonmonotonic time dependence. The decoherence time scale depends on moments of the bath-spin distribuition, and also on the polarization strengths in various bath-spin channels. The bath polarizations have a tendency to increase the decoherence time scale. The effective dynamics of the central spin polarization is shown to be described by a master equation with non-Markovian features.
Decoherence of a central spin coupled to an interacting spin bath via inhomogeneous Heisenberg coupling is studied by two different approaches, namely an exact equations of motion (EOMs) method and a Chebyshev expansion technique (CET). By assuming a wheel topology of the bath spins with uniform nearest-neighbor
XX-type intrabath coupling, we examine the central spin dynamics with the bath prepared in two different types of bath initial conditions. For fully polarized baths in strong magnetic fields, the polarization dynamics of the central spin exhibits a collapse-revival behavior in the intermediate-time regime. Under an antiferromagnetic bath initial condition, the two methods give excellently consistent central spin decoherence dynamics for finite-size baths of $N\ensuremath{\le}14$ bath spins. The decoherence factor is found to drop off abruptly on a short time scale and approach a finite plateau value which depends on the intrabath coupling strength nonmonotonically. In the ultrastrong intrabath coupling regime, the plateau values show an oscillatory behavior depending on whether
N/2 is even or odd. The observed results are interpreted qualitatively within the framework of the EOM and perturbation analysis. The effects of anisotropic spin-bath coupling and inhomogeneous intrabath bath couplings are briefly discussed. Possible experimental realization of the model in a modified quantum corral setup is suggested.
Decoherence of a center spin or qubit in a spin bath is essentially determined by the many-body bath evolution. We develop a cluster-correlation expansion (CCE) theory for the spin-bath dynamics relevant to the qubit decoherence problem. A cluster-correlation term is recursively defined as the evolution of a group of bath spins divided by the cluster correlations of all the subgroups. This correlation accounts for the authentic (nonfactorizable) collective excitations within a given group. The bath propagator is the product of all possible cluster correlation terms. For a finite-time evolution as in the qubit decoherence problem, a convergent result can be obtained by truncating the expansion up to a certain cluster size. The two-spin cluster truncation of the CCE corresponds to the pair-correlation approximation developed previously [W. Yao et al., Phys. Rev. B 74, 195301 (2006)]. In terms of the standard linked cluster expansion, a cluster-correlation term is the infinite summation of all the connected diagrams with all and only the spins in the group flip-flopped, and thus the expansion is exact whenever convergence occurs. When the individual contribution of each higher-order correlation term to the decoherence is small (while all the terms combined in product could still contribute substantially), as the usual case for relatively large baths, where the decoherence could complete well within the bath spin flip-flop time, the CCE coincides with the cluster expansion [W. M. Witzel and S. Das Sarma, Phys. Rev. B 74, 035322 (2006)]. For small baths, however, the qubit decoherence may not complete within the bath spin flip-flop time scale and thus individual higher-order cluster correlations could become significant. In such cases, only the CCE converges to the exact coherent dynamics of multispin clusters. We check the accuracy of the CCE in an exactly solvable spin-chain model.