Effect of nuclear polarization on spin dynamics in a double quantum dot

We study spin dynamics and singlet-triplet decoherence due to the hyperfine interaction in a parabolic double quantum dot, focusing on the effect of nuclear-spin polarization on the time evolution of the singlet probability. The probabilities for the singlet state exhibit damped oscillations, which do not change considerably when the nuclear-spin polarization is small. We derive expressions for the mean and variance of the saturation value of the singlet probability in cases where the hyperfine field has a nonzero mean. We demonstrate that the polarization could be deduced from experiments by measuring both the mean and variance of the asymptotic singlet probability.

Figure 1

(Color online) Singlet probability $⟨P_S⟩$ as a function of time in the cases (a) $⟨P_S\left(\infty \right)⟩=0.82$, (b) 0.91, and (c) 0.95, correspondingly. The values of the mean and standard deviation used in each curve are given in the Table . The amplitude of the oscillation increases with the polarization of the hyperfine field $h_0$.

Figure 2

(Color online) Asymptotic triplet probability $⟨{\overline{P}}_T⟩$ as a function of the mean of the hyperfine field $\frac{h_0}{J}$ for different values of the standard deviation of the hyperfine field. From bottom to top at $h_0=0:\frac{{\sigma }_0}{J}=0.1$, 0.2, 0.5, 1.0, and 2.0.

Figure 3

(Color online) Asymptotic triplet probability $⟨{\overline{P}}_T⟩$ as a function of the of the standard deviation of the hyperfine field $\frac{{\sigma }_0}{J}$ for different values of the mean of the hyperfine field. From bottom to top: $\frac{h_0}{J}=0$, 0.3, 0.5, 1.0, 2.0, and 5.0.

Figure 4

(Color online) Asymptotic variance of the triplet probability ${\sigma }^2\left(⟨{\overline{P}}_T⟩\right)$ as a function of the asymptotic triplet probability $⟨{\overline{P}}_T⟩$ for different values of ratio of the mean and variance of the hyperfine field. From top to bottom: $\frac{h_0}{{\sigma }_0}=0$, 1.0, 2.0, 3.0, and 4.0.