Trajectory tracking for non-Markovian quantum systems

We propose a systematic scheme to engineer quantum states of system governed by time-convolutionless non-Markovian master equation. According the idea reverse engineering, general algebraic equation determine control parameters, such as coherent and incoherent fields, is presented. Without artificially engineering time-dependent decay rates retaining environment-induced Lamb shifts, state can still be transferred into target in finite period time along an arbitrary designed trajectory strictly Hilbert space. As application, we apply our driven two-level realize instantaneous-steady-state tracking complete population inversion with parameters which are available experimental settings.