Connecting two jumplike unravelings for non-Markovian open quantum systems

The development and use of Monte Carlo algorithms plays a visible role in the study of non-Markovian quantum dynamics due to the provided insight and powerful numerical methods for solving the system dynamics. In the Markovian case, the connections between the various types of methods are fairly well understood while, for the non-Markovian case, there has so far been only a few studies. We focus here on two jumplike unravelings of non-Markovian dynamics: the non-Markovian quantum jump (NMQJ) method and the property state method by Gambetta, Askerud, and Wiseman (GAW). The results for simple quantum optical systems illustrate the connections between the realizations of the two methods and also highlight how the probability currents between the system and environment, or between the property states of the total system, are associated with the decay rates of time-local master equations and, consequently, with the jump rates of the NMQJ method.

Figure 1

Probability currents $J_{1_k,0}\left(t\right)$ as a function of time $t$ for TLA in the Markovian case. Initial state is $|e\rangle |0\rangle$, $\delta =3\lambda$, ${\gamma }_0=0.8\lambda$, and we use 180 environmental modes. Units of time and mode frequency ${\omega }_c-{\nu }_k$ are $1/\lambda$ and $\lambda$, respectively. When $t\approx 1$ we see that there occurs negative probability currents.

Figure 2

Probability currents $J_{1_k,0}\left(t\right)$ as a function of time $t$ for TLA. The parameters are as in Fig. 1 except that ${\gamma }_0=4\lambda$ and $\delta =-4\lambda$. Units of time and mode frequency ${\omega }_c-{\nu }_k$ are $1/\lambda$ and $\lambda$, respectively.  We can identify the modes for which ${\nu }_k\approx {\omega }_c$, which are responsible for the non-Markovian effects. See the text for details.

Figure 3

Decay rate $\Delta \left(t\right)$ and excited state population ${\rho }_{ee}\left(t\right)$ as a function of time $t$ for TLA. The parameters are as in Fig. 2. Units of time and decay rate are $1/\lambda$ and $\lambda$, respectively.

Figure 4

Probability currents $J_{1_k,0}\left(t\right)$ as a function of time $t$ in the nonsecular case for the V system. The initial state is $\frac{1}{\sqrt{2}}\left(\right|a\rangle |0\rangle +|b\rangle |0\rangle )$, ${\gamma }_0=4\lambda$, ${\delta }_a=3\lambda$, ${\delta }_b=-3\lambda$, and we have used 240 environmental modes. Units of time and mode frequency ${\omega }_c-{\nu }_k$ are $1/\lambda$ and $\lambda$, respectively.

Figure 5

Probability currents $J_{1_k,0}^b\left(t\right)$ (left panel) and $J_{1_k,0}^a\left(t\right)$ (right panel) as a function of time $t$ in the secular case for the V system. The parameters are as in Fig. 4 but the initial state is $\frac{1}{\sqrt{2}}\left(\right|a\rangle |0\rangle |0\rangle +|b\rangle |0\rangle |0\rangle )$. Units of time and mode frequency ${\omega }_c-{\nu }_k$ are $1/\lambda$ and $\lambda$, respectively.

Figure 6

Density matrix elements as a function of time $t$ for the V system The parameters are as in Figs. 4 and 5. Units of time are $1/\lambda$. For the chosen parameter values ${\rho }_{aa}\left(t\right)={\rho }_{bb}\left(t\right)$.

Figure 7

Top: Sum of probability currents from the system to the environment as a function of time $t$ for the V system. Solid red line is the nonsecular case where we cannot distinguish different decay channels. Blue circles are the secular case where we have two different decay channels. Bottom: Decay rates as a function of time $t$ calculated from the GAW method for the secular case (solid blue line) and decay rate of TCL2 master equation (red circles) for V system. Parameters are as in Figs. 4 and 5. Units of time, probability current, and decay rate are $1/\lambda$, $\lambda$, and $\lambda$, respectively.