Partial-trace-free time-convolutionless equation of motion for the reduced density matrix

Evolution of a system, coupled to its environment and influenced by external driving fields, is an old problem that remains of interest. In this paper, we derive an equation of motion for the reduced system density matrix, which is time convolutionless and free of the partial trace with respect to the environment states. This new approach uses an extension of the projection-operator technique, which incorporates an isomorphism between the system’s Liouville space and the unit eigenspace of the projection operator induced by the uniform environment density matrix. Numerical application of the present approach is particularly useful in large externally driven systems, as the partial-trace-free equation is given in terms of submatrices significantly smaller than the matrices in the conventional time-convolutionless approaches, which alleviates the computational burden. We also show that all time-convolutionless approaches, conventional or partial-trace-free, are based upon a hidden underlying assumption of time reversibility of the system’s evolution. This feature puts significant constraints on applicability of time-convolutionless approaches when employing approximations that yield time irreversibility. Also, we investigate the application of the approach in the description of far-from-equilibrium systems.