Intermediate-times dilemma for open quantum system: Filtered approximation to the refined weak-coupling limit

The famous Davies-GKSL secular Markovian master equation is tremendously successful in approximating the evolution of open quantum systems in terms of just a few parameters. However, the fully secular Davies-GKSL equation fails to accurately describe timescales short enough, i.e., comparable to the inverse of differences of frequencies present in the system of interest. A complementary approach that works well for short times but is not suitable after this short interval is known as the quasisecular master equation. Still, both approaches fail to have any faithful dynamics in the intermediate-time interval. Simultaneously, descriptions of dynamics that apply to the aforementioned "gray zone" often are computationally much more complex than master equations or are mathematically not well-structured. The filtered approximation (FA) to the refined weak-coupling limit has the simplistic spirit of the Davies-GKSL equation and allows capturing the dynamics in the intermediate-time regime. At the same time, our non-Markovian equation yields completely positive dynamics. We exemplify the performance of the FA equation in the cases of the spin-boson system and qutrit-boson system in which two distant timescales appear.