Non-Markovian dynamics in a spin star system: Exact solution and approximation techniques

The reduced dynamics of a central spin coupled to a bath of $N$ spin-$\frac{1}{2}$ particles arranged in a spin star configuration is investigated. The exact time evolution of the reduced density operator is derived, and an analytical solution is obtained in the limit $N\to \infty$ of an infinite number of bath spins, where the model shows complete relaxation and partial decoherence. It is demonstrated that the dynamics of the central spin cannot be treated within the Born-Markov approximation. The Nakajima-Zwanzig and the time-convolutionless projection operator technique are applied to the spin star system. The performance of the corresponding perturbation expansions of the non-Markovian equations of motion is examined through a comparison with the exact solution.

Figure 1

Comparison of the limit $N\to \infty$ [see Eqs. (36, 37)] with the exact functions for $N=20$ and $N=200.$ The plot shows the $v_{\pm }$ component of the Bloch vector.

Figure 2

Comparison of the limit $N\to \infty$ [see Eqs. (36, 37)] with the exact functions for $N=20$ and $N=200.$ The figure shows the $v_3$-component of the Bloch vector.

Figure 3

von Neumann entropy $S\left(t\right)$ of the reduced system for different initial conditions in the limit $N\to \infty$. $S_{\max }\equiv \ln 2$ is the maximal entropy for a qubit, representing a completely mixed state.

Figure 4

Comparison of the TCL and the NZ technique with the exact solution. The plot shows the approximations to second and fourth order in $\alpha$ and the exact solution of $v_{\pm }\left(t\right)$ [see Eqs. (23, 28)] for a bath of $100$ spins.

Figure 5

Comparison of the TCL and the NZ technique with the exact solution. The plot shows the second- and the fourth-order approximations as well as the exact solution of $v_3\left(t\right)$ [see Eqs. (24, 29)] for a bath of $100$ spins.

Figure 6

Error of TCL2 and NZ2: The plot shows the deviation $ϵ\left(t\right)\equiv \mid v_{\pm }\left(t\right)-v_{\pm }^{\text{approx}}\left(t\right)\mid$ of the exact solution $v_{\pm }\left(t\right)$ from the approximate solution $v_{\pm }^{\text{approx}}\left(t\right)$ for small $\alpha t$.

Figure 7

The TCL and the NZ approximation of the components of the Bloch vector in tenth order for a bath of $100$ spins.