Dynamics and decoherence in the central spin model using exact methods

The dynamics and decoherence of an electronic spin-1/2 qubit coupled to a bath of nuclear spins via hyperfine interactions in a quantum dot is studied. We show how exact results from the integrable solution can be used to understand the dynamic behavior of the qubit. It is possible to predict the main frequency contributions and their broadening for relatively general initial states analytically, leading to an estimate of the corresponding decay times, which are related to $T_1$ of the electron. Furthermore, for a small bath polarization, a distinct low-frequency time scale is observed.

Figure 1

(Color online) Time evolution $⟨S_0^z⟩\left(t\right)$ [(a) and (c)] and the corresponding Fourier transform [(b) and (d)] for $N_b=15$, $x_1=2$, and an initial uniform distribution of bath states with $M_b=1$ [in (a) and (b)] and $M_b=2$ [in (c) and (d)], respectively. The effect of two different homogeneity parameters $B=0.4$ (circles, black, lower time series) and $B=2$ (crosses, red, upper time series) for the couplings in Eq. (2) is shown. Dashed lines (green) show the estimates for the decoherence times. The dotted lines (blue) give the position of the peaks in the homogeneous limit $A_j\equiv 2$.

Figure 2

(Color online) The time evolution $⟨S_0^z⟩\left(t\right)$ (left) and its Fourier transform $⟨S_0^z⟩\left(\omega \right)$ for $S_{\text{tot}}^z=0$ from complete diagonalization, for $N_b=11$, $x_1=2$, and two different choices of the homogeneity parameter $B$ in Eq. (2). The dotted lines (blue) give the position of the peaks in the homogeneous limit $A=2$. The smaller amplitude oscillations in the left panel correspond to the inhomogeneous case $B=2$