Evolution of the reduced density matrix: a generalized projection-operator approach
A convolutionless equation of motion for the reduced density matrix of a system coupled to its environment, where the system+environment is closed, may be obtained using a projection-operator technique. We show that, when both the system and the environment Hilbert spaces are finite-dimensional, it is possible to eliminate the need for the partial trace over the environment states by constructing a simple and transparent basis-induced isomorphism between the system Liouville space and the unit-eigenspace of a special projection operator. Consequently, an equation of motion for the reduced density matrix is derived by a mere basis transformation within the system+environment Hilbert space and the explicit dependence of the reduced density matrix on the matrix elements of the Hamiltonian is uncovered, in a form well suited for numerical calculation.
density matrix time-convolutionless equation memory effects quantum liouville equation