Decoherence of an exchange qubit by hyperfine interaction

We study three-electron-spin decoherence in a semiconductor triple quantum dot with a linear geometry. The three electron spins are coupled by exchange interactions J(12) and J(23), and we clarify inhomogeneous and homogeneous dephasing dynamics for a logical qubit encoded in the (S = 1/2, S-z = 1/2) subspace. We first justify that qubit leakage via the fluctuating Overhauser field can be effectively suppressed by sufficiently large Zeeman and exchange splittings. In both J(12) = J(23) and J(12) not equal J(23) regimes, we construct an effective pure dephasing Hamiltonian with the Zeeman splitting E-Z >> J(12), J(23). Both effective Hamiltonians have the same order of magnitude as that for a single-spin qubit, and the relevant dephasing time scales (T-2*, T-FID, etc.) are of the same order as those for a single spin. We provide estimates of the dynamics of three-spin free induction decay, the decay of a Hahn spin echo, and the decay of echoes from a CPMG pulse sequence for GaAs quantum dots.