Time evolution and decoherence of a spin-$\frac{1}{2}$ particle coupled to a spin bath in thermal equilibrium

The time evolution of a spin-$\frac{1}{2}$ particle under the influence of a locally applied external magnetic field, and interacting with anisotropic spin environment in thermal equilibrium at temperature $T$ is studied. The exact analytical form of the reduced density matrix of the central spin is calculated explicitly for a finite number of bath spins. The case of an infinite number of environmental spins is investigated using the convergence of the rescaled bath operators to normal Gaussian random variables. In this limit, we derive the analytical form of the components of the Bloch vector for antiferromagnetic interactions within the bath, and we investigate the short-time and long-time behavior of reduced dynamics. The effect of the external magnetic field, the anisotropy, and the temperature of the bath on the decoherence of the central spin are discussed.

Figure 1

Time evolution of the components ${\lambda }_3\left(t\right)$ and ${\lambda }_1\left(t\right)$ for different values of the number of spins in the environment: $N=100$ (dotted lines), $N=200$ (dashed lines), and $N=400$ (solid lines). The other parameters are $\gamma =0$, $g=1$, $\beta =0.5$, $\Delta =0$, and $\mu =\alpha$. The initial conditions are ${\lambda }_3\left(0\right)=\frac{1}{2}$ and ${\lambda }_{1,2}\left(0\right)=\frac{3}{8}$.

Figure 2

The Gaussian decay of the Bloch vector components ${\lambda }_3\left(t\right)$ and ${\lambda }_1\left(t\right)$ in the case of ferromagnetic interactions: (top) $N=100$ and (bottom) $N=200$. The plot on the left of each subfigure corresponds to ${\lambda }_3\left(t\right)$, whereas the one on the right corresponds to ${\lambda }_1\left(t\right)$. The other parameters are $\gamma =2\alpha$, $\beta =1$, $g=-5$, $\Delta =0.5$, $\mu =0$, ${\lambda }_3\left(0\right)=\frac{1}{2}$, and ${\lambda }_{1,2}\left(0\right)=\frac{3}{8}$.

Figure 3

Evolution in time of ${\lambda }_3\left(t\right)$ and ${\lambda }_1\left(t\right)$ for $N=400$ (dotted lines) and $N\to \infty$ (solid lines); the dashed lines correspond to the asymptotic values. Other parameters are $\gamma =0$, $g=1$, $\beta =5$, $\Delta =0.5$, $\mu =0.4\alpha$, ${\lambda }_3\left(0\right)=\frac{1}{2}$, and ${\lambda }_{1,2}\left(0\right)=\frac{3}{8}$.

Figure 4

The decay of the components ${\lambda }_3\left(t\right)$ and ${\lambda }_1\left(t\right)$ for $g=0$ (dotted lines), $g=5$ (dashed lines), and $g=10$ (solid lines). Other parameters are $\beta =1$, $\gamma =\Delta =0$, $\mu =0.1\alpha$, ${\lambda }_3\left(0\right)=\frac{1}{2}$, and ${\lambda }_{1,2}\left(0\right)=\frac{3}{8}$.

Figure 5

Dependence of ${\lambda }_3\left(t\right)$ and ${\lambda }_1\left(t\right)$ on the strength of the magnetic field in the case $N\to \infty$: $\mu =0$ (dotted lines), $\mu =0.5\alpha$ (dashed lines), and $\mu =2\alpha$ (solid lines). The parameters are $\gamma =0$, $g\beta =2$, ${\lambda }_3\left(0\right)=\frac{1}{2}$, and ${\lambda }_{1,2}\left(0\right)=\frac{3}{8}$.

Figure 6

Dependence of ${\lambda }_3\left(t\right)$ and ${\lambda }_1\left(t\right)$ on the bath temperature in the case $N\to \infty$: $\beta =0$ (dotted lines), $\beta =1$ (dashed lines), and $\beta =10$ (solid lines). The parameters are $\gamma =0$, $g=2$, $\mu =0.5\alpha$, ${\lambda }_3\left(0\right)=\frac{1}{2}$, and ${\lambda }_{1,2}\left(0\right)=\frac{3}{8}$.

Figure 7

The short-time behavior of ${\lambda }_3\left(t\right)$ and ${\lambda }_1\left(t\right)$ in the case $N\to \infty$. The solid lines correspond to the exact solutions, and the dotted lines denote the approximations (93, 94). Here, $\gamma =0$, $g\beta =1$, $\mu =0.5\alpha$, ${\lambda }_3\left(0\right)=\frac{1}{2}$, and ${\lambda }_{1,2}\left(0\right)=\frac{3}{8}$.

Figure 8

(Color online) Purity evolution for different values of the bath temperature in the case $N\to \infty$ with $\gamma =0$, $g=1$, and $\mu =0$. The initial conditions are ${\lambda }_3\left(0\right)=\sqrt{\frac{7}{8}}$ and ${\lambda }_{1,2}\left(0\right)=\frac{1}{4}$.

Figure 9

(Color online) Evolution in time of the purity for different values of the bath temperature in the case $N\to \infty$ with $\gamma =0$, $g=1$, and $\mu =\alpha$. Same initial conditions as in Fig. 8.

Figure 10

Evolution of ${\lambda }_3\left(t\right)$ and ${\lambda }_1\left(t\right)$ for $N=600$ (dotted lines) and $N\to \infty$ (solid lines). The dashed lines correspond to the asymptotic values. Here, $\gamma =2\alpha$, $g=1$, $\beta =1.5$, $\Delta =1$, $\mu =0.3\alpha$, ${\lambda }_3\left(0\right)=\frac{1}{2}$, and ${\lambda }_{1,2}\left(0\right)=\frac{3}{8}$. The plots corresponding to ${\lambda }_3\left(t\right)$ are almost identical; the latter component saturates with respect to $N$ faster than ${\lambda }_1\left(t\right)$.

Figure 11

Dependence of ${\lambda }_3\left(t\right)$ and ${\lambda }_1\left(t\right)$ on the anisotropy constant in the case $N\to \infty$: $\Delta =0$ (solid lines), $\Delta =5$ (dashed lines), and $\Delta =10$ (dotted lines). Here, $\gamma =2\alpha$, $g\beta =1$, $\mu =0.1\alpha$, ${\lambda }_3\left(0\right)=\frac{1}{2}$, and ${\lambda }_{1,2}\left(0\right)=\frac{3}{8}$.

Figure 12

The short-time behavior of ${\lambda }_3\left(t\right)$ and ${\lambda }_1\left(t\right)$ in the case $N\to \infty$. The solid lines correspond to the exact solutions, and the dotted lines denote the approximations (98, 99). The parameters are $\gamma =1\alpha$, $\Delta =0$, $g\beta =1$, and $\mu =5\alpha$. The initial conditions are the same as in Fig. 11.

Figure 13

(Color online) Purity evolution for different values of the bath temperature in the case $N\to \infty$ with $\gamma =2\alpha$, $\Delta =1$, $g=1$, and $\mu =\alpha$. The initial conditions are ${\lambda }_3\left(0\right)=\sqrt{\frac{7}{8}}$ and ${\lambda }_{1,2}\left(0\right)=\frac{1}{4}$.