Non-Markovian waiting-time distribution for quantum jumps in open systems

Simulation methods based on stochastic realizations of state vector evolutions are commonly used tools to solve open quantum system dynamics, both in the Markovian and non-Markovian regimes. Here, we address the question of the waiting time distribution (WTD) of quantum jumps for non-Markovian systems. We generalize Markovian quantum trajectory methods in the sense of deriving an exact analytical WTD for non-Markovian quantum dynamics and show explicitly how to construct this distribution for certain commonly used quantum optical systems.

Figure 1

Schematic figures of the different systems. (a) Two level system, (b) $\Lambda$ system, and (c) ladder system.

Figure 2

Initial state is $|{\psi }^0\left(0\right)\rangle =|0\rangle$, parameters are ${\gamma }_0=5\lambda$ and $\delta =8\lambda$, and sample size was ${10}^5$. In panel (a) we have the decay rate, and in (b)–(d) we have three different realizations. In panel (e) we have WTDs for the realizations. Line styles and color coding match with the sample realizations. Lines are for exact numerical solutions and markers for sample estimates.

Figure 3

Initial state is $|{\psi }^0\left(0\right)\rangle =|0\rangle$, parameters are ${\gamma }_0^{(1,2)}=5\lambda$, ${\delta }^{\left(1\right)}=4\lambda$, and ${\delta }^{\left(2\right)}=8\lambda$, and sample size was ${10}^5$. In panel (a) we have the decay rate, and in (b)–(d) we have three different realizations. In panel (e) we have WTDs for the realizations. Line styles and color coding match with the sample realizations. Lines are for exact numerical solutions and markers for sample estimates.

Figure 4

Initial state is $|{\psi }^0\left(0\right)\rangle =|0\rangle$, parameters are ${\gamma }_0^{(1,2)}=5\lambda$, ${\delta }^{\left(1\right)}=8\lambda$, and ${\delta }^{\left(2\right)}=4\lambda$, and sample size was ${10}^6$. In panel (a) we have the decay rate, and in (b)–(d) we have three different realizations. In panel (e) we have WTDs for the realizations. Line styles and color coding match with the sample realizations. Lines are for exact numerical solutions and markers for sample estimates.