This work is a sequel to our preceding work [Phys. Rev. B 77, 045319 (2008)]. Here we compare pure-spin and pseudospin dynamics using as an example a system of two quantum dots, a pair of localized conduction-band electrons in an $n$-doped GaAs semiconductor. Pure-spin dynamics is obtained by tracing out the orbital degrees of freedom, whereas pseudospin dynamics retains (as is conventional) an implicit coordinate dependence. We show that magnetic field inhomogeneity and spin-orbit interaction result in a nonunitary evolution in pure-spin dynamics, whereas these interactions contribute to the effective pseudospin Hamiltonian via terms that are asymmetric in spin permutations, in particular, the Dzyaloshinskii-Moriya (DM) spin-orbit interaction. We numerically investigate the nonunitary effects in the dynamics of the triplet-state population, purity, and Lamb energy shift as a function of interdot distance and magnetic field difference $\Delta \vec{B}$. The spin-orbit interaction is found to produce effects of roughly 4 orders of magnitude smaller than those due to $\Delta \vec{B}$ in the pure-spin model. We estimate the spin-orbit interaction magnitude in the DM-interaction term. Our estimate gives a smaller value than that recently obtained by Kavokin [Phys. Rev. B 64, 075305 (2001)], who did not include double-occupancy effects. We show that a necessary and sufficient condition for obtaining a universal set of quantum logic gates, involving only two spins, in both pure-spin and pseudospin models is that the magnetic field inhomogeneity $\Delta \vec{B}$ and the Heisenberg interaction are both nonvanishing. We also briefly analyze pure-spin dynamics in the electron on liquid helium system recently proposed by Lyon [Phys. Rev. A 74, 052338 (2006)].

• Received 24 September 2007

DOI:https://doi.org/10.1103/PhysRevB.77.045320