Non-Markovianity and system-environment correlations in a microscopic collision model

We show that the use of a recently proposed iterative collision model with interenvironment swaps displays a signature of strongly non-Markovian dynamics that is highly dependent on the establishment of system-environment correlations. Two models are investigated: one in which such correlations are canceled iteratively and one in which they are kept all across the dynamics. The degree of non-Markovianity, quantified using a measure based on the trace distance, is found to be much greater for all coupling strengths, when system-environment correlations are maintained.

Figure 1

Pictorial sketch of the iterative process on the basis of the collision model for system-environment interaction. System $S$ interacts sequentially with the elements of an $N$-party environment consisting of subsystems $\left\{E_j\right\}(j=1,\cdots ,N)$. Following the sense of the interpanel arrows, we distinguish four stages: (i) $S$ interacts with the first element $E_1$ of the environment. It then shift by one site, while $E_1$ interacts with its rightmost nearest neighbor $E_2$ [step (ii)]. In step (iii), $S$ interacts with $E_2$ and then moves forward [step (iv)], while $E_2$ and $E_3$ interact. The specific form of the $S-E_j$ and $E_j-E_{j+1}$ interaction considered in our analysis is given in the body of the manuscript.

Figure 2

The trace distance between evolved system states [panel (a)] and the fidelity of the system state at step $n$ with the target one [panel (b)] are shown against the number of steps of the evolution for the case of the basic collision model that forbids inter-ancilla interactions and a superenvironment prepared in $\left|\right\{0\}\rangle$. We have taken the swap strength $\gamma =0.05$, while the initial states used to calculate the trace distance in panel (a) correspond to ${\theta }_1=\pi /2$ and ${\theta }_2=0$.

Figure 3

The trace distance $D$ plotted against the number of iterations of the collision model for both strategies and for different swap strengths put in place in our analysis. Panels (a) and (b) [(c) and (d)] report the results valid for strategy 2 (strategy 1 ). We have used $\delta =\pi /2$ in panels (a) and (c) and $\delta =0.95\times \pi /2$ in panels (b) and (d) to show how varying the inter-ancilla swap strength changes substantially the degree of non-Markovianity. We have taken $\gamma =0.05$ for the system-environment interaction strength, consistently with the assumption of weak $S$-$E$ coupling.

Figure 4

Measure $\mathcal{N}$ plotted against the intra-environment interaction strength $\delta$, for both strategies addressed in this work. We have used $\gamma =0.05,n=3\times {10}^4$, and the superenvironment is initialized in $\left|\right\{0\}\rangle$. By approaching the full-swap condition embodied by $\delta =\pi /2$, the non-Markovianity measure shoots up. The vertical axis of the plot has been truncated to $\mathcal{N}=10$ so as to improve the visibility of the details of the plot. The qualitative and quantitative differences inherent in the different strategies for the modeling of $S$-$E$ interactions are evident in the different thresholds in the value of $\delta$ above which $\mathcal{N}>0$. The measure is optimized over all possible pairs of initial $S$ states.

Figure 5

The derivative of the trace distance (dashed curve) and the part of the upper bound Eq. (12) dependent on SECs (solid curve) plotted against $n$ for $\delta =\pi /2$ and $\gamma =0.05$.

Figure 6

The Non-Markovianity measure, $\mathcal{N}$, plotted against the strength of the” coin.” For this graph we used $\delta =\pi /2$, $\eta =0.01$, and $n=10000$. For $\delta =\eta =\pi /2$ and a unit threshold, we have $\mathcal{N}=n$.