The Spin Density Matrix I: General Theory and Exact Master Equations

TL;DR

This paper develops exact master equations for the spin-density matrix of electrons in quantum dots, revealing non-unitary effects due to singlet-triplet coupling and implications for quantum computation.

Contribution

It introduces a Hamiltonian-based approach to model spin-only dynamics, deriving exact Lindblad and TCL master equations accounting for non-unitary effects without a true bath.

Findings

Non-unitary dynamics occur when singlet-triplet coupling exists.

Heisenberg interaction influences non-unitary but not unitary evolution.

Exact master equations are derived for spin-density matrices in quantum dots.

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 (one cannot unambiguously ascribe a spin to a QD) 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 non-unitary even without interaction with a true bath. Our primary focus is an investigation of non-unitary effects, based on exact master equations we derive for the spin-density matrix in the Lindblad and time-convolutionless (TCL) 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 non-unitary contributions to 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.