Stochastic wave-function unravelling of the generalized Lindblad equation

We investigate generalized non-Markovian stochastic Schrödinger equations (SSEs), driven by a multidimensional counting process and multidimensional Brownian motion introduced by A. Barchielli and C. Pellegrini [J. Math. Phys. 51, 112104 (2010)]. We show that these SSEs can be translated in a nonlinear form, which can be efficiently simulated. The simulation is illustrated by the model of a two-level system in a structured bath, and the results of the simulations are compared with the exact solution of the generalized master equation.

Figure 1

A two-state system, with level distance $\omega$, couples to an environment consisting of two energy bands, each with finite number of evenly spaced levels $N_1$ and $N_2$. $\delta \epsilon$ is the width of the bands and $V$ is the system-environment interaction potential.

Figure 2

The ground-state population for different initial values: red dashed line, ${\psi }_1\left(0\right)={\psi }_2\left(0\right)=\{\frac{1}{\sqrt{2}},0\}$; blue dotted line, ${\psi }_1\left(0\right)=\{\frac{1}{\sqrt{2}},0\}$ and ${\psi }_2\left(0\right)=\{0,\frac{1}{\sqrt{2}}\}$; green solid line, ${\psi }_1\left(0\right)=\{0,\frac{1}{\sqrt{2}}\}$ and ${\psi }_2\left(0\right)=\{0,\frac{1}{\sqrt{2}}\}$. The parameters are: ${\gamma }_0=1,{\gamma }_1=0.5,{\gamma }_2=0.3,{\omega }_1={\omega }_2=\sqrt{37}/2,\varkappa =2,\mathrm{\Delta }t=0.001$. The error bars have the same size as the dots on the figure.

Figure 3

Plot of the mean occupation number of the excited state for different values of ${\gamma }_1$: red dashed line, ${\gamma }_1=0.05$; blue dotted line, $\gamma =0.5$; green solid line, ${\gamma }_1=2.5$. Other parameters are: ${\psi }_1\left(0\right)=\{\frac{1}{\sqrt{2}},0\},{\psi }_2\left(0\right)=\{0,\frac{1}{\sqrt{2}}\},{\gamma }_0=1,{\gamma }_2=0.3,{\omega }_1={\omega }_2=\sqrt{37}/2,\varkappa =1,\mathrm{\Delta }t=0.001$. The error bars have the same size as the dots on the figure.

Figure 4

Plot of the mean occupation number of the excited state for different values of ${\gamma }_2$: red dashed line, ${\gamma }_2=0.05$; blue dotted line, ${\gamma }_2=0.3$; green solid line, ${\gamma }_2=0.01$. Other parameters are: ${\psi }_1\left(0\right)=\{\frac{1}{\sqrt{2}},0\},{\psi }_2\left(0\right)=\{0,\frac{1}{\sqrt{2}}\},{\gamma }_0=1,{\gamma }_1=0.5,{\omega }_1={\omega }_2=\sqrt{37}/2,\varkappa =2,\mathrm{\Delta }t=0.001$. The error bars have the same size as the dots on the figure.

Figure 5

Plot of the mean occupation number of the excited state for different values of $\varkappa$: red dashed line, $\varkappa =0.05$; blue dotted line, $\varkappa =2$; green solid line, $\varkappa =0.1$. Other parameters are: ${\psi }_1\left(0\right)=\{\frac{1}{\sqrt{2}},0\},{\psi }_2\left(0\right)=\{0,\frac{1}{\sqrt{2}}\},{\gamma }_0=1,{\gamma }_1=0.5,{\gamma }_2=0.3,{\omega }_1={\omega }_2=\sqrt{37}/2,\mathrm{\Delta }t=0.001$. The error bars have the same as the dots on the figure.

Figure 6

A single realization of the process for the parameters: ${\psi }_1\left(0\right)=\{\frac{1}{\sqrt{2}},0\},{\psi }_2\left(0\right)=\{0,\frac{1}{\sqrt{2}}\},{\gamma }_0=1,{\gamma }_1=5/2,{\gamma }_2=3/2,{\omega }_1={\omega }_2=\sqrt{37}/2,\varkappa =1,\mathrm{\Delta }t=0.005$.

Figure 7

The conditional probability of having the next count at time $t$, while the previous count was at time $t_0$. Here $P\left(t\right)=$ exp $\{-{\int }_0^{t}I\left(s\right)ds\}$ after one realization for the parameters: ${\psi }_1\left(0\right)=\{\frac{1}{\sqrt{2}},0\},{\psi }_2\left(0\right)=\{0,\frac{1}{\sqrt{2}}\},{\gamma }_0=1,{\gamma }_1=5/2,{\gamma }_2=3/2,{\omega }_1={\omega }_2=\sqrt{37}/2,\varkappa =1,\mathrm{\Delta }t=0.005$.

Figure 8

Plot of the mean occupation number of the excited state for the parameters ${\gamma }_1=5/2,{\gamma }_2=3/2,{\omega }_1={\omega }_2=\sqrt{37}/2,\varkappa =1,\mathrm{\Delta }t=0.005$ and ${10}^5$ realizations. The red (light gray) line corresponds to the case with the diffusion in Eq. (21), while the blue (dark gray) line shows the evolution without diffusion part for the state-dependent intensities. The error bars have the same size as the dots on the figure.