Spin density matrix of a two-electron system. I. General theory and exact master equations

Abstract

We consider a scenario where interacting electrons confined in quantum dots (QDs) are either too close to be resolved, or we do not wish to apply measurements that resolve them. Then, the physical observable is an electron spin only and the system state is fully described by the spin density matrix. Accounting for the spatial degrees of freedom, we examine to what extent a Hamiltonian description of the spin-only degrees of freedom is valid. We show that as long as there is no coupling between singlet and triplet states, this is indeed the case, but when there is such a coupling, there are open systems effects, i.e., the dynamics is nonunitary even without interaction with a true bath. Our primary focus is an investigation of nonunitary effects, based on exact master equations we derive for the spin density matrix in the Lindblad and time-convolutionless forms, and the implications for quantum computation. In particular, we demonstrate that the Heisenberg interaction does not affect the unitary part (apart from a Lamb shift) but does affect the nonunitary contributions to the time evolution of the spin density matrix. In a sequel paper, we present a detailed analysis of an example system of two quantum dots, including spin-orbit effects.