Stochastic representation of a class of non-Markovian completely positive evolutions

By modeling the interaction of an open quantum system with its environment through a natural generalization of the classical concept of continuous time random walk, we derive and characterize a class of non-Markovian master equations whose solution is a completely positive map. The structure of these master equations is associated with a random renewal process where each event consist in the application of a superoperator over a density matrix. Strong nonexponential decay arise by choosing different statistics of the renewal process. As examples we analyze the stochastic and averaged dynamics of simple systems that admit an analytical solution. The problem of positivity in quantum master equations induced by memory effects [S. M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001)] is clarified in this context.

Figure 1

Stochastic realizations for the CTQRW defined by the depolarizing operators, Eq. (52), and the fractional waiting time distribution, Eq. (44). The graphs correspond to the quantum average of the Pauli matrixes, $M_j\left(t\right)=\text{Tr}\left[\rho \left(t\right){\sigma }_j\right]$, $j=x,y,z$. The realization for the normalized average $M_y\left(t\right)∕M_y\left(0\right)$ is equal to that of the $x$ direction. The parameters were chosen as $p_x=p_y=0.5$ and $\alpha =0.5$, $A_{\alpha }=1∕\sqrt{2}{\sec }^{-1∕2}$, $T=A_{\alpha }^{-1∕\alpha }.$

Figure 2

Theoretical result (full line) and average over ${10}^4$ realizations (circles) for $M_x\left(t\right)$. The inset shows the short time regime together with the theoretical results for the Markovian evolution (dashed line) with $A_1=0.5{\sec }^{-1}$, and the exponential kernel (full line) with $\gamma =2{\sec }^{-1}$, $A_{ϵ}=1{\sec }^{-2}$.

Figure 3

Linear entropy for the CTQRW defined by Eq. (52) $\left(p_x=p_y=1∕2\right)$. Long dashed line, Markovian kernel with $A_1=0.5{\sec }^{-1}$. Dashed line, fractional kernel with $\alpha =0.5$, $A_{\alpha }=1∕\sqrt{2}{\sec }^{-1∕2}$. Full line, exponential kernel with $\gamma =2{\sec }^{-1}$, $A_{ϵ}=1{\sec }^{-2}$. Dotted line, exponential kernel with $\gamma =0.5{\sec }^{-1}$, $A_{ϵ}=0.25{\sec }^{-2}$.

Figure 4

Linear entropy for the CTQRW defined by Eq. (74) with $p_{\downarrow }=1$, $p_{\uparrow }=0$, and $\kappa =0.75$. Long dashed line, Markovian kernel with $A_1=1{\sec }^{-1}$. Dashed line, fractional kernel with $\alpha =0.5$, $A_{\alpha }=1{\sec }^{-1∕2}$. Full line, exponential kernel with $\gamma =4{\sec }^{-1}$, $A_{ϵ}=4{\sec }^{-2}$. Dotted line, exponential kernel with $\gamma =1{\sec }^{-1}$, $A_{ϵ}=1{\sec }^{-2}$.