The transfer tensor method: An analytical study case

The transfer tensor method (TTM) is a versatile tool for analyzing and propagating general open quantum systems. It captures in a compact manner all memory effects in a non-Markovian system through a straightforward transformation of a set of dynamical maps. Transfer tensors (TTs) provide the exact convolutional propagator associated with a given time discretization over the past evolution of an open quantum system. Here we show that, for any finite time discretization, the memory kernel of the Nakajima-Zwanzig equation deviates from the exact TTs, although both converge in the continuous-time limit, as expected. We examine this behavior in the context of an analytically solvable model: a two-level atom resonant with a lossy cavity in the Jaynes-Cummings limit. The atomic dynamics separate into two decoupled subspaces-the coherence and the population difference. We derive exact expressions for the dynamical map, the TTs, and the memory kernel governing the coherence, and we relate them to their counterparts for the population difference. As a function of the ratio between the cavity loss rate and the atom-cavity coupling strength, we identify regions of enhanced non-Markovianity in which the dynamics can be described as fully Markovian for certain TTM time-step choices.