Coarse graining a non-Markovian collisional model

The dynamics of systems subjected to noise is called Markovian in the absence of memory effects, i.e., when its immediate future depends only on its present. Time correlations in the noise source may generate non-Markovian effects that, sometimes, can be erased by appropriately coarse graining the time evolution of the system. In general, the coarse-graining time $t_{\text{CG}}$ is taken to be much larger than the correlation time $\tau$ but no direct relation between them is established. Here we analytically obtain a relation between $t_{\text{CG}}$ and $\tau$ for the dynamics of a qubit subjected to a time-correlated environment. Our results can be applied in principle to any distribution of the environmental correlations and can be tested through a collisional model where the qubit sequentially interacts with correlated qutrits.

Figure 1

Plot of the average correlation function $〈\mathrm{\Gamma }\left(h\right)〉$ for ${10}^7$ trajectories and the total number of collisions of $n=100$ for (a) the step function and (b) the exponential decay. The red line represents a fit, where the estimated correlation length is ${\overline{n}}_{\text{cor}}=10.009$ in panel (a) and ${\overline{n}}_{\text{cor}}=10.28$ in panel (b), in agreement with the expected value of $n_{\text{cor}}=10$.

Figure 2

Plot of the smallest eigenvalue ${\lambda }_{2n,n}$ of the dynamical matrix of the map ${\mathrm{\Lambda }}_{2n,n}$ vs the number of collisions $n$, for $\epsilon =0.001, {10}^8$ trajectories and for (a) the step function and (b) the exponential decay function. Different lines represent different correlation lengths as is shown in the plot. The lines are just for illustrative reasons, since the number of collisions is a discretized quantity.

Figure 3

Plot of the coarse-grain $n_{\text{CG}}$ vs the correlation length $n_{\text{cor}}$ for the exponential function (dots) and the step function (squares) for $\epsilon =0.001$ and ${10}^8$ trajectories.