Quantification of memory effects in the spin-boson model

Employing a recently proposed measure for quantum non-Markovianity, we carry out a systematic study of the size of memory effects in the spin-boson model for a large region of temperature and frequency cutoff parameters. The dynamics of the open system is described utilizing a second-order time-convolutionless master equation without the Markov or rotating wave approximations. While the dynamics is found to be strongly non-Markovian for low temperatures and cutoffs, in general, we observe a special regime favoring Markovian behavior. This effect is explained as resulting from a resonance between the system's transition frequency and the frequencies of the dominant environmental modes. We further demonstrate that the corresponding Redfield equation is capable of reproducing the characteristic features of the non-Markovian quantum behavior of the model.

Figure 1

The trace distance as a function of time for various temperatures (top) and cutoffs (bottom). The parameter values of the middle curve in both plots are $T={\omega }_0$, $\Omega ={\omega }_0$, $\gamma =0.1{\omega }_0$, and ${\vec{v}}_{1,2}\left(0\right)=\pm {(1,0,0)}^T$. The other curves show the effect of a variation of one of these parameters.

Figure 2

Non-Markovianity $\mathcal{N}\left(\Phi \right)$ as a function of $\Omega$ and $T$ up to $\Omega ,T=50{\omega }_0$, indicated by the logarithmic colorbar.

Figure 3

Non-Markovianity $\mathcal{N}\left(\Phi \right)$ as a function of $\Omega$ and $T$ up to $\Omega ,T=10{\omega }_0$. For this range, the maximization has been refined with additional initial state pairs to yield a higher resolution. The white curve shows the parametrization of the resonance effect given by Eq. (23).

Figure 4

The effective spectral density $J_{\text{eff}}\left(\omega \right)$ for a fixed cutoff $\Omega =1.55{\omega }_0$ and three different values of temperature $T$. The middle curve ($T=0.4{\omega }_0$) shows the spectral density obtained from the resonance condition (23); the other two curves correspond to detuned temperature values. The vertical line indicates the system's transition frequency ${\omega }_0$. The bottom plot shows a magnification of the top plot.

Figure 5

Non-Markovianity $\mathcal{N}\left(\Phi \right)$ as a function of $\Omega$ and $T$ up to $\Omega ,T=10{\omega }_0$ for the master equation with stationary rates (Redfield equation).