Effects of the rotating-wave and secular approximations on non-Markovianity

We study the effect of the rotating-wave approximation (RWA) and the secular approximation (SA) on the non-Markovian behavior in the spin-boson model at zero temperature. We find that both the RWA and SA lead to a dramatic reduction in the observed non-Markovianity. In general, non-Markovian dynamics is observed for the whole relaxation time of the system, whereas the RWA and SA lead to such dynamics only on the short time scale of the environmental correlation time. Thus, the RWA and SA are not necessarily justified in the studies of non-Markovianity, although they can estimate the state of the system precisely. Furthermore, we derive an accurate analytical expression for the non-Markovianity measure without the RWA or SA. This expression yields important insight into the physics of the problem.

Figure 1

Numerically calculated non-Markovianity corresponding to the (a) Lorentzian and (b) Ohmic spectral densities. In (a), the top solid line is obtained using the full master equation [see Eq. (3)], while the bottom solid and dashed lines show the non-Markovianity obtained under the RWA and SA, respectively [see Eqs. (25) and (26)]. In (b), the dynamics is Markovian if the RWA or SA master equation is used. The reason for the large difference between $\mathcal{N}\left(\Phi \right)$, and $\mathcal{N}\left({\Phi }_{\mathrm{SA}}\right)$ and $\mathcal{N}\left({\Phi }_{\mathrm{RWA}}\right)$ is explained in Sec. 5. Here we have chosen $\alpha /{\omega }_A=0.01$, and in (a) $\lambda /{\omega }_A={\tau }_s/{\tau }_c=0.1$.

Figure 2

(a) The positive values of $\sigma$ obtained using the SA master equation and the Lorentzian spectral density. Here $\alpha /{\omega }_A=0.01$, $\lambda /{\omega }_A=0.10$, and $\Delta /{\omega }_A=-0.90$. The initial-state pair corresponds to the one maximizing the non-Markovianity. (b) As in (a), but for the full master equation. The initial-state pair is that of Eq. (27) with $\xi \left(0\right)=\pi$. The solid blue line gives the exact numerical solution ${\sigma }_{\perp }$, and the dashed orange line corresponds to the analytical approximation ${\sigma }_{\perp }^{\mathrm{ana}}$ given in Eq. (28). The bottom panel shows a magnification of part of the top panel. The inset gives the maximum value of ${\sigma }_{\perp }$ (calculated over one oscillation cycle) as a function of time. It decays on a time scale given by the relaxation time ${\tau }_r$. For the chosen parameters ${\tau }_r/{\tau }_c\approx 1494$.

Figure 3

Non-Markovianity calculated using (a) Lorentzian and (b) Ohmic spectral densities. The orange dotted line shows the analytical result ${\mathcal{N}}^{\text{ana}}\left(\Phi \right)$. The solid blue line shows the numerically obtained value of $\mathcal{N}\left(\Phi \right)$ corresponding to $\alpha /{\omega }_A=0.01$. In (a), we have chosen $\lambda /{\omega }_A={\tau }_s/{\tau }_r=0.1$.

Figure 4

The behavior of ${\mathfrak{g}}_{\text{i}}$ [see Eq. (32)] as a function of $\omega$ for two values of $t$ (solid and dashed lines). For comparison, we also show the behavior of the Ohmic (dotted line) and Lorentzian (dash-dotted line) spectral densities. We have chosen the parameters of $J_{\text{O}}$ and $J_{\text{L}}$ such that the maximum of these functions is at $\omega ={\omega }_A$. For $J_{\text{L}}$, we have set ${\tau }_s/{\tau }_c=\lambda /{\omega }_A=0.1$. In the case of $J_{\text{L}}$ and $J_{\text{O}}$, the scale on the vertical axis is arbitrary.