Dynamical role of system-environment correlations in non-Markovian dynamics

We analyze the role played by system-environment correlations in the emergence of non-Markovian dynamics. By working within the framework developed in Breuer et al. [Phys. Rev. Lett. 103, 210401 (2009)], we unveil a fundamental connection between non-Markovian behavior and dynamics of system-environment correlations. We derive an upper bound to the rate of change of the distinguishability between different states of the system that explicitly depends on the establishment of correlations between system and environment. We illustrate our results using a fully solvable spin-chain model, which allows us to gain insight into the mechanisms triggering non-Markovian evolution.

Figure 1

(a) We show $\sigma (t,{\rho }_{1,2})$ (solid blue line) and the right-hand side of Eq. (2) (which we label $\mathcal{B}$ for convenience and display as a dashed orange line) against the rescaled interaction time $Jt$ for a chain of $N=9$ particles (i.e., an environment with 8 spins) when $S$ is prepared in either $\left|+\right.〉$ or $\left|-\right.〉$ (see inset for a sketch of the physical situation at hand). (b) We compare the behavior of $\sigma (t,{\rho }_{12})$ (solid blue line) with the evolution of the environmental indistinguishability $\mathcal{E}=1-D({\rho }_1^E\left(t\right),{\rho }_2^E\left(t\right))$ (amplified by a factor of 10 for ease of visualization, dotted red line) and the quantity $\mathcal{X}$ defined in the body of the manuscript (dashed green line). In both panels, $J_0/J=1$ and $B={10}^{-2}J$, while shaded regions highlight the regions of non-Markovianity.

Figure 2

(a) We show $\sigma \left(t,{\rho }_{1,2}\right)$ (solid blue line) and ${\dot{\mathcal{S}}}_{vN}\left(t\right)$ (dashed red curve) against the interaction time $Jt$ for $J_0/J=1,B/J={10}^{-2}$. The trace distance $D({\rho }_1^S\left(t\right),{\rho }_2^S\left(t\right))$ and entanglement associated with this situation are reported for the same time window. The shaded regions highlight non-Markovianity. (b) Same analysis but for the Markovian point corresponding to $J_0/J=1$ and $B/J=1/2$.