Markovian and non-Markovian master equations versus an exactly solvable model of a qubit in a cavity

We investigate the dynamics of a qubit in a leaky cavity interacting with a bosonic bath, using the Jaynes–Cummings model, characterized by three different spectral densities: an impulse spectral density, an Ohmic spectral density, and a proportional spectral density with a sharp cutoff. Specifically, we focus on the behavior of the first excitation state and explore its non-Markovian features, such as oscillatory amplitudes. We derived solutions from various approximation methods to investigate their ability to approximate the exact solution and discuss their optimal performance with respect to relevant parameters. We consider the time-convolutionless (TCL) master equation up to the second order (TCL2) and the fourth order (TCL4), the coarse-graining Lindblad equation (CG-LE), and the rotating-wave approximation Lindblad equation (RWA-LE). Notably, we compare two variants of CG-LE: one based on a completely positive (CP) map that derives the semigroup master equation from first principles and another employing the Born approximation along with bath correlation functions. We obtain the optimal coarse-graining time by optimizing a metric that quantifies the deviation between the approximated and exact solutions. We demonstrate that CG-LE outperforms the Markov limit derived from RWA-LE for the cases of low coupling or high cavity frequency where the Markovian approximation is valid. In the presence of non-Markovian effects characterized by highly oscillatory and non-decaying behavior, the TCL approximation closely matches the exact solution for a short duration. Additionally, for the spectral density with a sharp cutoff, the TCL approximation accurately captures the non-zero asymptotic behavior.