Spin density matrix of a two-electron system. II. Application to a system of two quantum dots

This work is a sequel to our work "The Spin Density Matrix I: General Theory and Exact Master Equations" (eprint arXiv:0708.0644 [cond-mat]). Here we compare pure- and pseudo-spin 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 pseudo-spin dynamics retains (as is conventional) an implicit coordinate dependence. We show that magnetic field inhomogeneity and spin-orbit interaction result in a non-unitary evolution in pure-spin dynamics, whereas these interactions contribute to the effective pseudo-spin Hamiltonian via terms that are asymmetric in spin permutations, in particular, the Dzyaloshinskii-Moriya (DM) spin-orbit interaction. We numerically investigate the non-unitary effects in the dynamics of the triplet states population, purity, and Lamb energy shift, as a function of interdot distance and magnetic field difference. The spin-orbit interaction is found to produce effects of roughly four orders of magnitude smaller than those due to magnetic field difference 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- and pseudo-spin models is that the magnetic field inhomogeneity and the Heisenberg interaction are both non-vanishing. We also briefly analyze pure-spin dynamics in the electron on liquid helium system recently proposed by Lyon [Phys. Rev. A 74, 052338 (2006)].