The Transfer Tensor Method: an Analytical Study Case

Abstract

The transfer tensor method 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 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 transfer tensors, although both converge in the continuous-time limit, as expected. We examine this behaviour 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 degrees of freedom -- the coherence and the population inversion. We derive exact expressions for the dynamical map, the transfer tensors and the memory kernel governing the coherence, and we relate them to their counterparts for the population inversion. 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 system can be described as fully Markovian for certain time-step choices.

Figures (6)

• Figure 1: a) Diagram of the considered atom-cavity model. The atom is represented by two levels $\ket e$ and $\ket g$. The cavity loses photons at a rate $\kappa$. Cavity and atom are coupled by a Jaynes-Cummings term of strength $g$. b) Level diagram representing the lowest states of the full Hilbert space, the strength of the couplings among themselves and the decay rates.
• Figure 2: Temporal evolution of the coherence dynamical map $\mathcal{E}_c$ for different values of $\kappa/4g$, illustrating the underdamped (solid line), overdamped (dot-dashed line) and critically damped (dashed line) regimes.
• Figure 3: Absolute value of the difference between the memory kernel $\mathcal{K}_c$ and the transfer tensor $\mathcal{T}_{k,c}$ for different timestep values and $\kappa/4g = 0.2$, showing that $\mathcal{T}_{k,c}$ converges to $\mathcal{K}_c$ as the timestep is reduced.
• Figure 4: $\mathcal{T}_{2,c}$ as a function of $gt$ and $\kappa/4g$. The dashed line corresponds to the frontier between the underdamped and the overdamped regimes. The solid lines show the zeros of $\mathcal{T}_{2,c}$ where the evolution of the system is guaranteed to be fully Markovian.
• Figure 5: Temporal evolution of the coherence dynamical map $\mathcal{E}_c$ in the underdamped regime. The dots on each line correspond to the three first zeros of $\mathcal{T}_{2,c}(t)$, where the evolution is fully Markovian for that given timestep.