A "Square-root" Method for the Density Matrix and its Applications to Lindblad Operators

TL;DR

This paper introduces a novel 'square-root' method for representing density matrices in open quantum systems, providing exact decomposition and new evolution equations that extend Lindblad dynamics to off-diagonal elements.

Contribution

The paper proposes an exact square-root factorization of density matrices and develops new evolution equations for these factors, extending Lindblad dynamics to off-diagonal terms.

Findings

Exact square-root representation of density matrices.

New evolution equations for off-diagonal elements.

Reduction to Lindblad equations for diagonal terms.

Abstract

The evolution of open systems, subject to both Hamiltonian and dissipative forces, is studied by writing the $nm$ element of the time ($t$) dependent density matrix in the form \ber \rho_{nm}(t)&=& \frac {1}{A} \sum_{\alpha=1}^A \gamma ^{\alpha}_n (t)\gamma^{\alpha *}_m (t) \enr The so called "square root factors", the $\gamma(t)$'s, are non-square matrices and are averaged over $A$ systems ($\alpha$) of the ensemble. This square-root description is exact. Evolution equations are then postulated for the $\gamma(t)$ factors, such as to reduce to the Lindblad-type evolution equations for the diagonal terms in the density matrix. For the off-diagonal terms they differ from the Lindblad-equations. The "square root factors" $\gamma(t)$ are not unique and the equations for the $\gamma(t)$'s depend on the specific representation chosen. Two criteria can be suggested for fixing the choice of $\gamma(t)$'s one is simplicity of the resulting equations and the other has to do with the reduction of the difference between the $\gamma(t)$ formalism and the Lindblad-equations.