Entanglement dynamics in Heisenberg spin chains coupled to a dissipative environment at finite temperature

We consider a finite one-dimensional Heisenberg $\mathit{XYZ}$ spin chain under the influence of a dissipative Lindblad environment obeying the Born-Markovian constraint in presence of an external magnetic field with closed and open boundary conditions. We present an exact numerical solution for the Lindblad master equation of the system in the Liouville space. The dynamics and asymptotic behavior of the nearest-neighbor and beyond-nearest-neighbor pairwise entanglements in the system are investigated under the effect of spatial anisotropy, temperature, system size, and different initial states. The entanglements in the free spin system exhibit nonuniform oscillatory behavior that varies significantly depending on the system size, anisotropy, and initial state. The $xy$ spatial anisotropy dictates the asymptotic behavior of the different entanglements in the system under the influence of the environment regardless of the initial state. Higher anisotropy yields higher steady-state value of the nearest-neighbor entanglement whereas a complete isotropy wipes it out. The longer range entanglements respond differently to the anisotropy variation. The anisotropy in the $z$ direction may enhance the entanglements depending on the interplay with the magnetic field applied in the same direction. As the temperature is raised, the steady state of the short-range entanglements is found to be robust within very small nonzero temperature range that depends critically on the spatial anisotropy. Moreover, the end to end entanglement transfer time and speed through the open boundary chain vary considerably based on the degree of anisotropy and temperature of the environment.

Figure 1

Time evolution of (a) ${\tau }_1,{\tau }_2$, and $R$, (b) $C_{12}$, (c) $C_{13}$ and $C_{14}$ in the free $\left(\mathrm{\Gamma }=0\right)$ Ising system starting from an initially disentangled state, where $N=7$.

Figure 2

Time evolution of (a) ${\tau }_1,{\tau }_2$, and $R$, (b) $C_{12},C_{13}$, and $C_{14}$, (c) $C_{15},C_{16}$, and $C_{17}$ in the free $\left(\mathrm{\Gamma }=0\right)$ Ising system starting from an initial maximally entangled state, where $N=7$.

Figure 3

Time evolution of the free Ising system $\left(\mathrm{\Gamma }=0\right)$ starting from an initial (a) disentangled state; (b) maximally entangled state; and the free $\mathit{XX}$ system starting from an initial (c) disentangled state; (d) maximally entangled state, where $N=5$. The legend is as shown in panel (a).

Figure 4

Time evolution of the free $XY$ system $\left(\mathrm{\Gamma }=0\right)$ starting from an initial (a) disentangled state; (b) maximally entangled state; and the free $\mathit{XYZ}$ system starting from an initial (c) disentangled state; (d) maximally entangled state, where $N=5$. The legend is as shown in panel (a).

Figure 5

Time evolution of $C_{12}$ and ${\tau }_2$ in the Ising system in presence of the environment $\left(\mathrm{\Gamma }=0.05\right)$ starting from an initial disentangled state in (a) and (b) and an entangled $W$ state in (c) and (d) respectively at different temperatures $\overline{n}=0,0.05$, and 0.1, where $N=7$. The legend is as shown in panel (a).

Figure 6

Time evolution of (a) $C_{12}$ and (b) ${\tau }_2$ in the Ising system in presence of the environment $\left(\mathrm{\Gamma }=0.05\right)$ starting from an initial maximally entangled state at different temperatures $\overline{n}=0,0.05$, and 0.1, where $N=7$. The legend is as shown in panel (a).

Figure 7

Time evolution of $C_{12}$ and ${\tau }_2$ in the $\mathit{XYZ}$ (or $XY)$ system in presence of the environment $\left(\mathrm{\Gamma }=0.05\right)$ starting from an initial disentangled state in (a) and (b) and a maximally entangled state in (c) and (d) at different temperatures $\overline{n}=0,0.05$, and 0.1, where $N=7$. The legend is as shown in panel (a).

Figure 8

Time evolution of $C_{12}$ and ${\tau }_2$ in the $\mathit{XX}$ $(\mathit{XXX}$ or $\mathit{XXZ})$ system in presence of environment $\left(\mathrm{\Gamma }=0.05\right)$ starting from an initial (a) disentangled state; (b) entangled $W$ state and maximally entangled state in (c) and (d) at different temperatures $\overline{n}=0,0.05$, and 0.1, where $N=7$. The legend is as shown in panel (a).

Figure 9

Time evolution at different chain sizes $(N=3,5$, and 7) in presence of the environment $\left(\mathrm{\Gamma }=0.05\right)$ of (a) ${\tau }_2$ in the Ising system starting from an initial disentangled state at zero temperature; (b) $C_{12}$ in the $\mathit{XYZ}$ system starting from an initial disentangled state at temperature $\overline{n}=0.05$; (c) $C_{12}$ in the Ising system starting from an initial $W$-entangled state at temperature $\overline{n}=0.05$; (d) ${\tau }_2$ in the $\mathit{XX}$ system starting from an initial maximally entangled state at zero temperature. The legend is as shown in panel (a).

Figure 10

The asymptotic behavior in the $\gamma \text{−}\delta$ space of the Heisenberg $\mathit{XYZ}$ system in presence of the environment $\left(\mathrm{\Gamma }=0.05\right)$, with $J=0.05$ starting from any initial state (disentangled, entangled or maximally entangled) of (a) $C_{12}$ at $\overline{n}=0$, (b) $C_{13}$ at $\overline{n}=0$, (c) $C_{12}$ at $\overline{n}=0.05$, and (d) the contour plot of $C_{12}$ at $\overline{n}=0.05$ but for $J=0.1$.

Figure 11

The asymptotic behavior of $C_{12}$ and $C_{13}$ in the Heisenberg system in presence of the environment $\left(\mathrm{\Gamma }=0.05\right)$ vs the anisotropy parameter $\gamma$ and the temperature parameter $\overline{n}$ starting from any initial state (disentangled, entangled, or maximally entangled) and at any value of $0\le \delta \le 1$.

Figure 12

The dynamics of the entanglements $C_{12},C_{13},C_{14},C_{15},C_{16}$, and $C_{17}$ starting from a maximally entangled state in the free Ising system $\left(\mathrm{\Gamma }=0\right)$ in (a) and (b) and the free $\mathit{XX}$ system in (c) and (d) with open boundary condition, where $N=7$. The inner panels illustrate the rise up of the entanglement from zero.

Figure 13

The dynamics of the entanglements $C_{12},C_{13},C_{14},C_{15},C_{16}$, and $C_{17}$ starting from a maximally entangled state of the free $\mathit{XXZ}$ system $\left(\mathrm{\Gamma }=0\right)$ in (a) and (b) and the free $\mathit{XYZ}$ system in (c) and (d) with open boundary condition, where $N=7$. The inner panels illustrate the rise up of the entanglement from zero.

Figure 14

The dynamics of the entanglements $C_{12},C_{13},C_{14},C_{15},C_{16}$, and $C_{17}$ starting from a maximally entangled state in the Ising system in presence of the environment $\left(\mathrm{\Gamma }=0.05\right)$ at $\overline{n}=0$ in (a) and (b) and $\overline{n}=0.01$ in (c) and (d), where $N=7$. In (a) and (c), the left inner panels illustrate the rise up of entanglement while the right ones illustrate its death. In (b) the inner panel shows both of the rise up and death of entanglement.

Figure 15

The dynamics of the entanglements $C_{12},C_{13},C_{14},C_{15},C_{16}$, and $C_{17}$ starting from a maximally entangled state in the $\mathit{XX}$ system in presence of the environment $\left(\mathrm{\Gamma }=0.05\right)$ at $\overline{n}=0$ in (a) and (b) and $\overline{n}=0.01$ in (c) and (d), where $N=7$. The left inner panels illustrate the rise up of entanglement while the right ones illustrate its death.

Figure 16

The dynamics of the entanglements $C_{12},C_{13},C_{14},C_{15},C_{16}$, and $C_{17}$ starting from a maximally entangled state in the $\mathit{XYZ}$ system in presence of the environment at $\overline{n}=0$ in (a) and (b) and $\overline{n}=0.01$ in (c) and (d), where $N=7$. In (a) and (c), the left inner panels illustrate the rise up of entanglement while the right ones illustrate its death. In (b) the inner panel shows both of the rise up and death of entanglement.

Figure 17

The asymptotic behavior of (a) $C_{12}$, (b) $C_{13}$, and the value of (c) $C_{14}$, (d) $C_{15}$ (at $T=300)$ in $\gamma \text{−}\delta$ space of the Heisenberg $\mathit{XYZ}$ system in presence of the environment $\left(\mathrm{\Gamma }=0.05\right)$ starting from any initial state (disentangled, entangled, or maximally entangled) at zero temperature, where $N=5$.

Figure 18

The asymptotic behavior of (a) $C_{12}$, (b) $C_{13}$, and the value of (c) $C_{14}$, (d) $C_{15}$ (at $T=300)$ in the Heisenberg $\mathit{XYZ}$ system in presence of the environment $\left(\mathrm{\Gamma }=0.05\right)$ vs the anisotropy parameter $\gamma$ and the temperature parameter $\overline{n}$ starting from any initial state (disentangled, entangled, or maximally entangled) and for $\delta =1$.