Exact master equation for a spin interacting with a spin bath: Non-Markovianity and negative entropy production rate

An exact canonical master equation of the Lindblad form is derived for a central spin interacting uniformly with a sea of completely unpolarized spins. The Kraus operators for the dynamical map are also derived. The non-Markovianity of the dynamics in terms of the divisibility breaking of the dynamical map and the increase of the trace distance fidelity between quantum states is shown. Moreover, it is observed that the irreversible entropy production rate is always negative (for a fixed initial state) whenever the dynamics exhibits non-Markovian behavior. In continuation with the study of witnessing non-Markovianity, it is shown that the positive rate of change of the purity of the central qubit is a faithful indicator of the non-Markovian information backflow. Given the experimental feasibility of measuring the purity of a quantum state, a possibility of experimental demonstration of non-Markovianity and the negative irreversible entropy production rate is addressed. This gives the present work considerable practical importance for detecting the non-Markovianity and the negative irreversible entropy production rate.

Figure 1

Variation of $q\left(t\right)$ with time $t$ for various interaction strengths $\alpha$. The number of bath spins is kept fixed at $N=20$. Positive $q\left(t\right)$ implies the non-Markovian nature of the dynamics according to the RHP measure.

Figure 2

Variation of $q_{\mathrm{dis}}\left(t\right)$ with time $t$ for various interaction strengths $\alpha$. The number of bath spins is kept fixed at $N=20$. To distinguish the effect on the thermal part of the quantum channel, we separately plot $q_{\mathrm{dis}}\left(t\right)$. It can be seen from the plot that the non-Markovian revival for the thermal part of the channel increases with the increase of the interaction strength $\alpha$ for fixed $N$.

Figure 3

Variation of $q\left(t\right)$ with time $t$ for different numbers of bath spins $N$. Interaction strength $\alpha =0.03$ is taken.

Figure 4

Variation of $q_{\mathrm{dis}}\left(t\right)$ with time $t$ for different numbers of bath spins $N$. Interaction strength $\alpha =0.03$ is taken. A magnified view of the rectangular region is shown in the inset. The plot depicts that the revival of $q_{\mathrm{dis}}\left(t\right)$ increases with the increase of bath spins $N$.

Figure 5

Variation of $p\left(t\right)$ with time $t$ for the two states $|\pm \rangle =\frac{1}{\sqrt{2}}|0\pm 1\rangle$, for different interaction strengths $\alpha$ (where $N=20$). Positive $p\left(t\right)$ implies non-Markovianity according to the BLP measure.

Figure 6

Variation of $p\left(t\right)$ with time $t$ for two density matrices $|\pm \rangle =\frac{1}{\sqrt{2}}|0\pm 1\rangle$, for different numbers of bath spins $N$ and interaction strength $\alpha =0.03$. A magnified view of the rectangular region is shown in the inset.

Figure 7

Variation of the rate of change of the irreversible entropy production $\sigma \left(t\right)$, the rate of change of the purity $\frac{dP}{dt}$, and ${\mathrm{\Gamma }}_{\mathrm{dis}}$ with time $t$ for the initial state $|1\rangle$ with the interaction strength $\alpha =0.03$ and the number of bath spins $N=20$. It is evident that $\sigma \left(t\right)$ and $\frac{dP}{dt}$ are negative and positive, respectively, whenever ${\mathrm{\Gamma }}_{\mathrm{dis}}$ is negative. This implies that the non-Markovian information backflow revives the purity of the state and causes a negative irreversible entropy production rate.