Open system dynamics with non-Markovian quantum jumps

We discuss in detail how non-Markovian open system dynamics can be described in terms of quantum jumps [J. Piilo et al., Phys. Rev. Lett. 100, 180402 (2008)]. Our results demonstrate that it is possible to have a jump description contained in the physical Hilbert space of the reduced system. The developed non-Markovian quantum jump approach is a generalization of the Markovian Monte Carlo wave function (MCWF) method into the non-Markovian regime. The method conserves both the probabilities in the density matrix and the norms of the state vectors exactly and sheds new light on non-Markovian dynamics. The dynamics of the pure state ensemble illustrates how local-in-time master equation can describe memory effects and how the current state of the system carries information on its earlier state. Our approach solves the problem of negative jump probabilities of the Markovian MCWF method in the non-Markovian regime by defining the corresponding jump process with positive probability. The results demonstrate that in the theoretical description of non-Markovian open systems, there occurs quantum jumps which recreate seemingly lost superpositions due to the memory.

Figure 1

(Color online) Initially, all the $N$ ensemble members share the same initial state $|{\psi }_0⟩$, i.e., $N_0\left(0\right)=N$. Quantum jumps during the positive decay rate (arrows to the right) spread the ensemble members to a wider set of different states. On the contrary, the non-Markovian quantum jumps during the negative decay rate (arrows to the left) transfer ensemble members always to states which already exist in the ensemble.

Figure 2

Example cases introducing the level notations. (a) Jaynes-Cummings model, (b) $\Lambda$ system, (c) V system, and (d) ladder system. Only transitions expressed by arrows contribute since they reside close to the resonance frequency of the cavity.

Figure 3

Effective ensembles for the example cases. (a) Jaynes-Cummings model, (b) $\Lambda$ system, (c) V system, and (d) ladder system. The number in the arrow indicates the jump channel. The expressions for states $|{\psi }_{\alpha }⟩$ are given in text.

Figure 4

(Color online) Dynamics of the Jaynes-Cummings model. Initial state is $|{\psi }_0\left(0\right)⟩=\left(3|a⟩+2|b⟩\right)/\sqrt{13}$. (a) Decay rate ${\Delta }_1\left(t\right)$. (b) Populations ${\rho }_{aa}$ (initially higher line) and ${\rho }_{bb}$ (initially lower line). (c) Absolute value of the coherence ${\rho }_{ab}$.

Figure 5

(Color online) Example of the dynamics of a single realization in the Jaynes-Cummings model (parameters as in Fig. 4). In this realization, the deterministic evolution of the initial state $|{\psi }_0⟩$ (higher line) was followed until $t\approx 0.4{\Gamma }^{-1}$, when a Markovian quantum jump (arrow down) brought the state to the ground state $|{\psi }_1⟩=|b⟩$ (lower line). Then, deterministic evolution of the ground state was followed until a non-Markovian quantum jump (arrow up) recreated the $|{\psi }_0⟩$ state at $t\approx 0.8{\Gamma }^{-1}$, whereafter the evolution was deterministic.

Figure 6

(Color online) Dynamics of a $\Lambda$ system with an initial state $|{\psi }_0\left(0\right)⟩=\left(4|a⟩+2|b⟩+|c⟩\right)/\sqrt{21}$. (a) Decay rates have momentarily opposite signs. (b) A plateau emerges to the excited-state population ${\rho }_{aa}$ (initially highest line) as the decay channels counteract each other. Other populations ${\rho }_{bb}$ (initially middle line) and ${\rho }_{cc}$ (initially lowest line) behave according to the two separate decay channels. (c) Also, coherences ${\rho }_{ab}$, ${\rho }_{ac}$, and ${\rho }_{cb}$ (initially highest, middle, and lowest line, respectively) have plateaus.

Figure 7

(Color online) Dynamics of a V system with an initial state $|{\psi }_0\left(0\right)⟩=\left(|a⟩+|b⟩+|c⟩\right)/\sqrt{3}$. (a) Decay rates. (b) Populations of the upper states ${\rho }_{aa}$ and ${\rho }_{bb}$ (lowest and middle line, respectively) decay according to single decay channels and the ground-state population ${\rho }_{cc}$ (highest line) increases correspondingly. (c) The upper-state coherence ${\rho }_{ab}$ (lowest line) has a plateau as the decay channels are counteracting each other, while coherences involving the ground states ${\rho }_{ac}$ (middle line) and ${\rho }_{bc}$ (highest line) are related to individual decay channels.

Figure 8

(Color online) Dynamics of a ladder system with an initial state $|{\psi }_0\left(0\right)⟩=\left(4|a⟩+2|b⟩+|c⟩\right)/\sqrt{21}$. (a) Decay rates. (b) The ladder structure is clearly visible as the decaying population of the highest excited state ${\rho }_{aa}$ appears first as increase of the middle state population ${\rho }_{bb}$ and eventually everything ends up to the ground state ${\rho }_{cc}$ (initially highest, middle, and lowest line, respectively). (c) Only the initial state $|{\psi }_0⟩$ contributes to the coherences ${\rho }_{ab}$, ${\rho }_{ac}$, and ${\rho }_{bc}$ (initially highest, middle, and lowest line, respectively) and therefore their dynamics is as simple as in the $\text{V}$ system.

Figure 9

(Color online) Dynamics of a ladder system starting from an initial state $|{\psi }_0\left(0\right)⟩=|a⟩$; other parameters are as in Fig. 8. Populations of the middle state ${\rho }_{bb}$ (higher line) and the ground state ${\rho }_{cc}$ (lower line) are shown. The NMQJ solution follows the formal analytical solution of the master equation faithfully until the state loses its positivity at $t\approx 1.0{\Gamma }^{-1}$ as the ground-state populations tend negative (thin black line marks the zero value).

Figure 10

(Color online) (a) Sketch of a time-dependent decay rate with periods of positive and negative values (arbitrary units). (b) Examples of single realizations encountered in the ensemble. The system is assumed to be such that there is only one decay channel and two physically different states: the initial state $|{\psi }_0\left(t\right)⟩$ and the target state of a quantum jump $|{\psi }_1\left(t\right)⟩$ (deterministic evolution is given by thin horizontal lines). The state of an ensemble member at the given time is indicated by the thick line. Quantum jumps from $|{\psi }_0⟩$ to $|{\psi }_1⟩$ (arrows down) occur at random times during the positive decay rate, while non-Markovian quantum jumps from $|{\psi }_1⟩$ back to $|{\psi }_0⟩$ (arrows up) occur during the negative decay rate. The total probability to have state $|{\psi }_0⟩$ at the end of the shown evolution period is the sum of the paths 1 and 4 while the probability to have state $|{\psi }_1⟩$ is the sum of the paths 2, 3, and 5.