Non-Markovian master equations from piecewise dynamics

We construct a large class of completely positive and trace-preserving non-Markovian dynamical maps for an open quantum system. These maps arise from a piecewise dynamics for statistical operators characterized by a continuous time evolution interrupted by jumps, randomly distributed in time and described by a quantum channel. The state of the open system is shown to obey a closed evolution equation, given by a master equation with a memory kernel and an inhomogeneous term. The non-Markovianity of the obtained dynamics is explicitly assessed studying the behavior of the distinguishability of two different initial system's states with elapsing time.

Figure 1

(a) Modulus of the functions $d_{-}\left(t\right)$ and $q\left(t\right)$ for a dephasing dynamics described by $D\left(t\right)=\cos \left(\lambda t\right)$, and waiting time given by the convolution of three equal exponentials $\Gamma e^{-\Gamma t}$. The growth of any of these quantities, as discussed in the Supplemental Material [21], provides a direct signature of NM of the time evolution map ${\Lambda }_d\left(t\right)$. The quantities are plotted as a function of $\lambda t$ and $\Gamma /\lambda$. The semitransparent surface corresponds to $d_{-}\left(t\right)$, while the meshed surface represents $q\left(t\right)$. It immediately appears that for growing ratio $\Gamma /\lambda$, determining the relation between the time scales inherent in ${\mathcal{F}}_d\left(t\right)$ and $f\left(t\right)$, the oscillations in $d_{-}\left(t\right)$ are suppressed. The NM is then only detected by $q\left(t\right)$, arising due to the action of the map ${\mathcal{E}}_x$ which describes the events, in this case spin flips, in between the continuous time evolution ${\mathcal{F}}_d\left(t\right)$. (b) Modulus of $g_{-}\left(t\right)$ and $h_{+}\left(t\right)$, here for a continuous dynamics ${\mathcal{F}}_{+}\left(t\right)$ involving both populations and coherences, and waiting time corresponding to the convolution of two equal exponential distributions. The ratio $\gamma /\lambda$ appearing in the function $G\left(t\right)$ given by Eq. (16) is set equal to 3, corresponding to NM of ${\mathcal{F}}_{+}\left(t\right)$ alone. Again the growth of the modulus of any of these functions warrants NM. The function $g_{-}\left(t\right)$ given by the semitransparent surface only detects NM for small $\Gamma$, while the mashed surface $h_{+}\left(t\right)$ shows that NM also takes place for frequent events, that is, large $\Gamma /\lambda$, even though it is confined to shorter and shorter time intervals.