A “square-root” method for the density matrix and its applications to Lindblad operators

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 formρnm(t)=1A∑α=1Aγnα(t)γmα*(t).The so called “square-root factors”, the γ(t)'s, are non-square matrices and are averaged over A systems (α) of the ensemble. This square-root description is exact. Evolution equations are then postulated for the γ(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”γ(t) are not unique and the equations for the γ(t)'s depend on the specific representation chosen. Two criteria can be suggested for fixing the choice of γ(t)'s one is simplicity of the resulting equations and the other has to do with the reduction of the difference between the γ(t) formalism and the Lindblad-equations.