Dynamics of nonequilibrium thermal entanglement

The dynamics of a simple spin chain (two spins) coupled to bosonic baths at different temperatures is studied. The analytical solution for the reduced density matrix of the system is found. The dynamics and temperature dependence of spin-spin entanglement is analyzed. It is shown that the system converges to a steady state. If the energy levels of the two spins are different, the steady-state concurrence assumes its maximum at unequal bath temperatures. It is found that a difference in local energy levels can make the steady-state entanglement more stable against high temperatures.

Figure 1

Dynamics of the concurrence $C\left(t\right)$ for the initial reduced density matrix ${\widehat{\rho }}_0=\mid 1,0⟩⟨1,0\mid$. The parameters of the model are chosen to be ${\gamma }_1={\gamma }_2=0.02$, ${ϵ}_1=2$, ${ϵ}_2=1$, $K=1$ for different temperatures of baths: curve 1 corresponds to $T_1=0,5$; $T_2=0,2$; curve 2 $T_1=1$; $T_2=0,5$; curve 3 $T_1=1,5$; $T_2=1$.

Figure 2

Dynamics of the concurrence $C\left(t\right)$ for different initial states of the reduced density matrix of qubits; $T_1=1$, $T_2=0.5$, $\gamma =0.02$, ${ϵ}_1=2$, ${ϵ}_2=1$, $K=1$. The curve 1 corresponds to ${\widehat{\rho }}_0=(\mid 1,0⟩-\mid 0,1⟩)(⟨1,0\mid -⟨0,1\mid )∕2$; curve 2 corresponds to ${\widehat{\rho }}_0=\mid 1,0⟩⟨1,0\mid$; curve 3 corresponds to ${\widehat{\rho }}_0=\mid 1,1⟩⟨1,1\mid$.

Figure 3

Steady-state concurrence $C_{\infty }(T_M,\Delta T)$ as a function of the mean bath temperature $T_M=(T_1+T_2)∕2$ and temperature difference $\Delta T=T_1-T_2$ in the symmetric case ${ϵ}_1={ϵ}_2=3$ with $K=1$.

Figure 4

Steady-state concurrence $C_{\infty }(T_M,\Delta T)$ as a function of the mean bath temperature $T_M=(T_1+T_2)∕2$ and the temperature difference $\Delta T=T_1-T_2$ in the case ${ϵ}_1=3$, ${ϵ}_2=1$, $K=1$.

Figure 5

Maximal possible steady-state concurrence $C_{\infty }^{\mathrm{Max}}({ϵ}_1,{ϵ}_2)$ in the strong-coupling case ${ϵ}_1,{ϵ}_2<K$ with $K=1$.

Figure 6

Maximal possible steady-state concurrence $C_{\infty }^{\mathrm{Max}}({ϵ}_1,{ϵ}_2)$ in the weak-coupling case ${ϵ}_1,{ϵ}_2>K$ for $K=1$. In the corner: profile of the three-dimensional surface at the line ${ϵ}_1+{ϵ}_2=4$.