Geometrical characterization of non-Markovianity

We introduce a tool for the quantitative characterization of the departure from Markovianity of a given dynamical process. Our tool can be applied to a generic $N$-level system and extended straightforwardly to Gaussian continuous-variable systems. It is linked to the change of the volume of physical states that are dynamically accessible to a system and provides qualitative expectations in agreement with some of the analogous tools proposed so far. We illustrate its predictive power by tackling a few canonical examples.

Figure 1

Time evolution of the determinant of the map for a generic Markovian (a) and non-Markovian dynamics (b). Quantitatively, the curves displayed in the figure correspond to the spontaneous emission of a two-level system in a resonant leaky cavity (cf. Example 1) with the Lorentzian spectral density given in Eq. (13). (a) shows the (Markovian) case with ${\gamma }_0/\lambda =0.1$ (bad cavity limit), while (b) shows a (non-Markovian) evolution in the good cavity limit with ${\gamma }_0/\lambda =10$. We also picture the set of accessible states, whose volume changes in time according to the behavior of the determinant.

Figure 2

(a) Example of an ellipsoid representing the image of a random map for a qubit. The canonical basis of ${\mathbb{R}}^3$ is mapped onto the corresponding colored circles highlighted on the ellipsoid, from which the volume can be obtained. (b) ${\mathcal{N}}_V$ against $\Delta$ for the spontaneous-emission dynamics into a reservoir with Lorentzian spectral density (cf. Example 1).

Figure 3

Non-Markovianity measure ${\mathcal{N}}_V$ as a function of $J/\Delta$ (with the coefficient $J$ defined through $\kappa =\sqrt{{\Delta }^2+J^2/N_m}$), and at various values of $N_m$. The inset shows ${\mathcal{N}}_V$ against the detuning $\Delta =\omega -{\omega }_0$. The measure is calculated integrating over the time window $[0,4000/\kappa ]$.