Expansion of time-convolutionless non-Markovian quantum master equations: A case study using the Fano-Anderson model

We explore the performance of the time-convolutionless (TCL) projection operator technique using the Fano-Anderson model as a test case. Comparing the exact TCL master equation with an expansion in powers of the strength of the system-environment coupling, we analyze the transient dynamics as well as the steady-state behavior. For a Lorentzian spectral density we demonstrate that the dimensionless expansion parameter corresponds to the ratio of the environmental correlation time to the relaxation time of the system, and we derive the convergence radius for the TCL expansion, which is seen to depend on the ratio of detuning and width of the spectral density. We further study the quantum non-Markovianity of the model based on the evolution of the Bures distance between quantum states and how it is represented by the second and fourth order of the expansion. Our results highlight both the strengths and the limitations of the TCL formalism in capturing key features of open quantum systems and, in particular, the challenges of accurately describing strongly coupled systems and non-Markovian dynamics.