Singlet-triplet decoherence due to nuclear spins in a double quantum dot

We have evaluated hyperfine-induced electron spin dynamics for two electrons confined to a double quantum dot. Our quantum solution accounts for decay of a singlet-triplet correlator even in the presence of a fully static nuclear spin system, with no ensemble averaging over initial conditions. In contrast to an earlier semiclassical calculation, which neglects the exchange interaction, we find that the singlet-triplet correlator shows a long-time saturation value that differs from $\frac{1}{2}$, even in the presence of a strong magnetic field. Furthermore, we find that the form of the long-time decay undergoes a transition from a rapid Gaussian to a slow power law $(\sim 1∕t^{3∕2})$ when the exchange interaction becomes nonzero and the singlet-triplet correlator acquires a phase shift given by a universal (parameter independent) value of $3\pi ∕4$ at long times. The oscillation frequency and time-dependent phase shift of the singlet-triplet correlator can be used to perform a precision measurement of the exchange interaction and Overhauser field fluctuations in an experimentally accessible system. We also address the effect of orbital dephasing on singlet-triplet decoherence and find that there is an optimal operating point where orbital dephasing becomes negligible.

Figure 1

(Color online) Saturation value of the triplet occupation probability $C_{T_0}\left(\infty \right)=\overline{C_n}$ vs. ${⟨\delta h^z⟩}_{\mathrm{rms}}∕J$ when the nuclear spin system has been squeezed into an eigenstate of $\delta h^z$ (dashed line) and when the state of the nuclear spin system describes a Gaussian distribution of eigenvalues $\delta h_n^z$, with mean $\overline{\delta h_n^z}=0$ and variance ${\sigma }_0^2=\overline{{\left(\delta h_n^z\right)}^2}$ (solid line). We also show the analytical asymptotics for ${\sigma }_0⪢J$, given by Eq. (15) (dotted line) and the semiclassical value $[C_{T_0}\left(\infty \right)=1∕2]$ (dash-dotted line).

Figure 2

(Color online) Decay of the correlator $C_{T_0}\left(t\right)$ evaluated by numerical integration of Eq. (13) for three ratios of ${\sigma }_0∕J$: ${\sigma }_0∕J=2$ (dashed-dotted line), ${\sigma }_0∕J=1$ (dashed line), and ${\sigma }_0∕J=1∕2$ (solid line). The analytical long-time asymptotic expressions from Eq. (17) are shown as dotted lines.

Figure 3

(Color online) The correlator $C_{T_0}\left(t\right)∕C_{T_0}\left(\infty \right)$ shows a rapid Gaussian decay when $J=0$ [solid line, from Eq. (16)], but has a much slower power-law decay $\sim 1∕t^{3∕2}$ for large exchange $J=10{\sigma }_0⪢{\sigma }_0$ [dotted line, from Eq. (21)].

Figure 4

(Color online) Decay of the correlator $C_{T_{+}}\left(t\right)$ in two cases. A typical unprepared initial state, where the nuclear spin system is in a superposition of $h^z$ eigenstates, results in a Gaussian decay [solid line, from Eq. (39)]. If the nuclear spin state is squeezed into an $h^z$ eigenstate, there is no decay, only coherent oscillations [dotted line, from Eq. (38)]. For this plot we have used $\Delta =5\alpha$.