Ensemble of Lindblad's trajectories for non-Markovian dynamics

Although Lindblad developed a general Markovian theory for open-system dynamics while maintaining the positivity of the density matrix, a practical non-Markovian analog remains a significant problem. Here, we present an extension of Lindblad's theory through an ensemble of Lindbladian trajectories originating from different times in the system's history. This approach provides an account of the system's memory while preserving the positivity of the density matrix. We apply the theory to the Jaynes-Cummings model to capture non-Markovian dynamics in the weak and strong coupling regimes.

Figure 1

An ensemble of Lindbladian trajectories whose weighted ensemble produces the density matrix at time $t$.

Figure 2

The exact (black line), Markovian (green triangles), GME2 (green-blue inverted triangles), TCL2 and TCL4 (teal diamonds and blue squares respectively), and ELT (red circles) (a) populations of the excited level and (b) errors relative to the exact solution are shown for the weak coupling limit ($\lambda =5{\gamma }_0, {\gamma }_0=1.091, \mathrm{\Delta }=0$) in the Jaynes-Cummings model. The ELT method shows closest agreement to the exact solution.

Figure 3

The population of the Jaynes-Cummings excited level in the strong coupling limit ($\lambda =0.2{\gamma }_0, {\gamma }_0=1.091, \mathrm{\Delta }=0$) is shown as a function of time for the exact (black line), Markovian (green triangles), GME2 (green-blue inverted triangles), TCL2 and TCL4 (teal diamonds and blue squares respectively), and ELT (red circles) solutions. The ELT method agrees with the exact solution for all times.

Figure 4

The population of the excited level of the Jaynes-Cummings model in the strong coupling, detuning limit ($\lambda =0.3{\gamma }_0, {\gamma }_0=1.091, \mathrm{\Delta }=2.4{\gamma }_0$) is shown as a function of time for the Markovian (green triangles), TCL4 (blue squares), and ELT (red circles) solutions. The ELT agrees with the TCL4 solution.