Dynamics of two central spins immersed in spin baths

In this article we derive the exact dynamics of a two-qubit (spin $1/2$) system interacting centrally with separate spin baths composed of qubits in a thermal state. Furthermore, each spin of the bath is coupled to every other spin of the same bath. The corresponding dynamical map is constructed. It is used to analyze the non-Markovian nature of the two qubit central spin dynamics. We further observe the evolution of quantum correlations like entanglement and discord under the influence of the environmental interaction. Moreover, we demonstrate the comparison between this exact two-qubit dynamics and the locally acting central spin model in a spin bath. This work is a stepping stone towards the realization of non-Markovian heat engines and other quantum thermal devices.

Figure 1

Schematic diagram of coupled central spin model where each central spin interacts with individual spin baths.

Figure 2

Variation of trace distance, between time evolved state and initial state, with time for the reduced state of the system. In panel (a), the initial state is taken to be $|11\rangle$, and in panel (b), the initial state is taken to be $|10\rangle$. The parameters have following values: ${\omega }_1=2.0, {\omega }_2=1.9, \delta =2.5, {\omega }_a=1.1, {\omega }_b=1.2, M=N=100, T=1, {\epsilon }_1=2.6, {\epsilon }_2=2.5$.

Figure 3

Variation of trace distance for difference between local and global maps with time for the reduced state of the system. In panel (a), the initial state is taken to be $|11\rangle$, and in panel (b), the initial state is taken to be $|10\rangle$. The parameters have following values: ${\omega }_1=2.0, {\omega }_2=1.9, \delta =5, {\omega }_a=1.1, {\omega }_b=1.2, M=N=100, T=1, {\epsilon }_1=2.6, {\epsilon }_2=2.5$.

Figure 4

Variation of concurrence ${\mathcal{Q}}_{\mathcal{C}}$ and quantum discord ${\mathcal{Q}}_{\mathcal{D}}$ with time for the reduced state of the system. The parameters have following values: ${\omega }_1=2.0, {\omega }_2=1.9, \delta =3, {\omega }_a=1.1, {\omega }_b=1.2, M=N=100, T=1, {\epsilon }_1=1.3, {\epsilon }_2=1.25$.