Coarse graining can beat the rotating-wave approximation in quantum Markovian master equations

We present a first-principles derivation of the Markovian semigroup master equation without invoking the rotating-wave approximation (RWA). Instead we use a time coarse-graining approach that leaves us with a free time-scale parameter, which we can optimize. Comparing this approach to the standard RWA-based Markovian master equation, we find that significantly better agreement is possible using the coarse-graining approach, for a three-level model coupled to a bath of oscillators, whose exact dynamics we can solve for at zero temperature. The model has the important feature that the RWA has a nontrivial effect on the dynamics of the populations. We show that the two different master equations can exhibit strong qualitative differences for the population of the energy eigenstates even for such a simple model. The RWA-based master equation misses an important feature which the coarse-graining-based scheme does not. By optimizing the coarse-graining time scale the latter scheme can be made to approach the exact solution much more closely than the RWA-based master equation.

Figure 1

Exact dynamics of the populations (a) and (b) and coherences (c) and (d) for $g=0.001{\omega }_c$ and ${\rho }_S\left(0\right)=|1\rangle \langle 1|$. In (a) and (c) we chose ${\omega }_1=0.095{\omega }_c$ and ${\omega }_2=0.105{\omega }_c$; in (b) and (d) ${\omega }_1=0.09975{\omega }_c$ and ${\omega }_2=0.10025{\omega }_c$. In (a) and (b) the blackdotted line is ${\rho }_{00}$, the blue solid line is ${\rho }_{11}$, and the red dashed line is ${\rho }_{22}$. In (c) and (d) we display the coherence between levels 1 and 2, i.e., ${\rho }_{12}$.

Figure 2

Dynamics of the populations ${\rho }_{ii}$ calculated with the different approximate equations [(a) CG-SME, (b) RWA-SME, and (c) exact] compared to the exact dynamics at $g=0.001{\omega }_c$, ${\omega }_1=0.095{\omega }_c$, and ${\omega }_2=0.105{\omega }_c$. The dashed red line is ${\rho }_{22}\left(t\right)$, the solid blue line is ${\rho }_{11}\left(t\right),$ and the black dotted line is ${\rho }_{00}\left(t\right)$. The optimal parameter for the averaged evolution generator is ${\omega }_c\Delta t=63.7$ for this set of system parameters, in agreement with Eq. (16). As initial condition we set ${\rho }_S\left(0\right)=|1\rangle \langle 1|$.

Figure 3

Trace-norm distance of the different approximate solutions to the exact solution, for the same paramters as in Fig. 2. The dashed red line corresponds to the RWA-SME solution and the solid blue one to the CG-SME solution.

Figure 4

Integrated trace-norm distance of the solution of the CG-SME and the exact dynamics (green solid) for different averaging times $\Delta t$ compared to the RWA-SME (black, dashed). The integration intervals are chosen according to the relaxation times that result from the different system parameters. The parameters used for each plot are (a) $\omega =0.1{\omega }_c$, $\delta \omega =0.01{\omega }_c$, $g=0.001{\omega }_c$, (b) $\omega =0.1{\omega }_c$, $\delta \omega =0.01{\omega }_c$, $g=0.002{\omega }_c$, (c) $\omega =0.1{\omega }_c$, $\delta \omega =0.01{\omega }_c$, $g=0.003{\omega }_c$, (d) $\omega =0.4{\omega }_c$, $\delta \omega =0.01{\omega }_c$, $g=0.001{\omega }_c$, and (e) $\omega =0.3{\omega }_c$, $\delta \omega =0.03{\omega }_c$, $g=0.003{\omega }_c$.

Figure 5

Trace-norm distance of the different approximate solutions to the exact solution, for the same paramters as in Fig. 2. The dotted red line corresponds to the RWA-SME solution, the solid blue one to the optimal $\Delta t_{\mathrm{opt}}$ CG-SME solution, and the dashed green and dash-dotted black lines are for $\Delta t=\Delta t_{\mathrm{opt}}/2$ and $\Delta t=2\Delta t_{\mathrm{opt}}$, respectively.

Figure 6

The rate functions of the CG-SME. The dotted red line is $b_{{\omega }_1{\omega }_2}/{\omega }_c$, the solid blue line is $b_{{\omega }_1{\omega }_1}/{\omega }_c$, the dash-dotted red line is $S_{{\omega }_1{\omega }_2}/{\omega }_c$ and the dashed blue line corresponds to $S_{{\omega }_1{\omega }_1}/{\omega }_c$. The horizontal lines are the asymptotic values of the rate functions, i.e., the values they converge to for high ${\omega }_c\Delta t$. The vertical lines are at ${\omega }_c\Delta t_{\mathrm{opt}}$. The parameters used for each plot are (a) $\omega =0.1{\omega }_c$, $\delta \omega =0.01{\omega }_c$, (b) $\omega =0.4{\omega }_c$, $\delta \omega =0.01{\omega }_c$, and (c) $\omega =0.2{\omega }_c$, $\delta \omega =0.02{\omega }_c$. Note the qualitatively similar location of the ${\omega }_c\Delta t_{\mathrm{opt}}$ values relative to the features of the rate functions. This may be a clue for future studies in terms of independent determination of $\Delta t_{\mathrm{opt}}$.