Quantum regression theorem and non-Markovianity of quantum dynamics

We explore the connection between two recently introduced notions of non-Markovian quantum dynamics and the validity of the so-called quantum regression theorem. While non-Markovianity of a quantum dynamics has been defined looking at the behavior in time of the statistical operator, which determines the evolution of mean values, the quantum regression theorem makes statements about the behavior of system correlation functions of order two and higher. The comparison relies on an estimate of the validity of the quantum regression hypothesis, which can be obtained exactly evaluating two-point correlation functions. To this aim we consider a qubit undergoing dephasing due to interaction with a bosonic bath, comparing the exact evaluation of the non-Markovianity measures with the violation of the quantum regression theorem for a class of spectral densities. We further study a photonic dephasing model, recently exploited for the experimental measurement of non-Markovianity. It appears that while a non-Markovian dynamics according to either definition brings with itself violation of the regression hypothesis, even Markovian dynamics can lead to a failure of the regression relation.

Figure 1

(a) BLP measure of non-Markovianity ${\mathcal{N}}_s\left(\lambda \right)$, see Eq. (37), and (b) RHP measure of non-Markovianity ${\mathcal{I}}_s\left(\lambda \right)$, see Eq. (38), as a function of the coupling strength $\lambda$ for increasing values of the parameter $s$. In both panels the curves are evaluated for $s=3$ (black thick solid line), $s=3.5$ (blue solid line), $s=4$ (magenta dashed line), $s=4.5$ (green dashed thick line), $s=5$ (red dot-dashed line), and $s=5.5$ (orange dotted line).

Figure 2

(a) BLP measure of non-Markovianity ${\mathcal{N}}_s\left(\lambda \right)$, see Eq. (37), as a function of the parameter $s$, for $\lambda =1$. (b) and (c) Decoherence function ${\gamma }_s\left(t\right)$ as a function of rescaled for $\lambda =0.5$ and different values of $s$ (b), and for $s=4$ and different values of $\lambda$ (c).

Figure 3

(a) $Z_s\left(\lambda \right)$ as a function of the parameter $s$ and of the coupling strength $\lambda$, see Eq. (55), for $\Omega t_1=1$ and $\Omega t_2=2$. (b) Section of (a) for $s=2,3,4$.

Figure 4

Violation of the quantum regression theorem, as quantified by the estimator $Z$ in Eq. (60) (a) in the time-inhomogeneous Markovian case, ${\omega }_{0,1}={\omega }_{0,2}={\omega }_0$, as a function of $\Delta \delta \omega =\delta {\omega }_1-\delta {\omega }_2$ and ${\omega }_0\tau ={\omega }_0(t_2-t_1)$, for ${\omega }_0{t}_1=1$ and $r=1$; (b) in the non-Markovian case, $\delta {\omega }_1=\delta {\omega }_2=\delta \omega$, as a function of $\Delta {\omega }_0={\omega }_{0,1}-{\omega }_{0,2}$ and $\delta \omega \tau$, for $\delta \omega t_1=1$ and $r=2$. In both panels $\Delta n=1$.