Decoherence of an exchange qubit by hyperfine interaction

Abstract

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 ${J}_{12}\ensuremath{\ne}{J}_{23}$ regimes, we construct an effective pure dephasing Hamiltonian with the Zeeman splitting ${E}_{Z}\ensuremath{\gg}{J}_{12},{J}_{23}$. Both effective Hamiltonians have the same order of magnitude as that for a single-spin qubit, and the relevant dephasing time scales (${T}_{2}^{*},{T}^{\mathrm{FID}}$, 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.