Electron spin-flip correlations due to nuclear dynamics in driven GaAs double dots

We present experimental data and associated theory for correlations in a series of experiments involving repeated Landau-Zener sweeps through the crossing point of a singlet state and a spin-aligned triplet state in a GaAs double quantum dot containing two conduction electrons, which are loaded in the singlet state before each sweep, and the final spin is recorded after each sweep. The experiments reported here measure correlations on time scales from 4 $\mu \mathrm{s}$ to 2 ms. When the magnetic field is aligned in a direction such that spin-orbit coupling cannot cause spin flips, the correlation spectrum has prominent peaks centered at zero frequency and at the differences of the Larmor frequencies of the nuclei, on top of a frequency-independent background. When the spin-orbit field is relevant, there are additional peaks, centered at the frequencies of the individual species. A theoretical model which neglects the effects of high-frequency charge noise correctly predicts the positions of the observed peaks, and gives a reasonably accurate prediction of the size of the frequency-independent background, but gives peak areas that are larger than the observed areas by a factor of 2 or more. The observed peak widths are roughly consistent with predictions based on nuclear dephasing times of the order of 60 $\mu \mathrm{s}$. However, there is extra weight at the lowest observed frequencies, which suggests the existence of residual correlations on the scale of 2 ms. We speculate on the source of these discrepancies.

Figure 1

Energy-level diagram of two electrons in a DQD as a function of detuning $\epsilon$.

Figure 2

Figures (a) and (b) are the plots of the power spectrum $S$ of the Landau-Zener probability ($\langle P_{\text{LZ}}\rangle =0.4$ in both cases), at $B=0.19\text{and}0.4\text{T}$, respectively. Here, $\varphi =0$, so spin-orbit effects are absent. The black curves are the experimental data while the magenta curves are an empirical fit to a sum of Gaussian peaks sitting on a frequency-independent background. Vertical lines are at the difference frequencies between two different species: ${\nu }_{{}^{69}\mathrm{Ga}}-{\nu }_{{}^{71}\mathrm{Ga}}$ (blue), ${\nu }_{{}^{71}\mathrm{Ga}}-{\nu }_{{}^{75}\mathrm{As}}$ (green), and ${\nu }_{{}^{75}\mathrm{As}}-{\nu }_{{}^{69}\mathrm{Ga}}$ (red). All frequencies are folded back to the interval $-125$ to $+125$ KHz. As a result, peaks appear in different order in the two plots.

Figure 3

(a) Power spectrum of $P_{\text{LZ}}$ for $B=0.1\text{T}, \langle P_{\text{LZ}}\rangle =0.6$, and the direction of the field $\varphi =5^{\circ }$. (b) Power spectrum of $P_{\text{LZ}}$ for $B=0.19\text{T}, \langle P_{\text{LZ}}\rangle =0.42$, and the direction of the field $\varphi ={10}^{\circ }$. The black curves are the experimental data while the magenta curves are an empirical fit to a sum of Gaussian peaks on a frequency-independent background. In both the figures solid vertical lines are at the difference frequencies between two different species: ${\nu }_{{}^{69}\mathrm{Ga}}-{\nu }_{{}^{71}\mathrm{Ga}}$ (blue), ${\nu }_{{}^{71}\mathrm{Ga}}-{\nu }_{{}^{75}\mathrm{As}}$ (green), and ${\nu }_{{}^{75}\mathrm{As}}-{\nu }_{{}^{69}\mathrm{Ga}}$ (red), while the dashed vertical lines are at the bare frequencies: ${\nu }_{{}^{69}\mathrm{Ga}}$ (blue), ${\nu }_{{}^{71}\mathrm{Ga}}$ (green), and ${\nu }_{{}^{75}\mathrm{As}}$ (red). All frequencies are folded back to the interval $-125$ to $+125$ KHz.

Figure 4

The weight at low frequencies in the power spectrum $F\left({\nu }_n\right)$. Panel (a) is at field orientation $\varphi =0$, where spin-orbit effects are absent, with magnetic field strengths $B=0.19$ and 0.40 T. Panel (b) is at field strengths $B=0.10$ and 0.19 T, at orientations $\varphi =5^{\circ }$ and ${10}^{\circ }$, respectively, where SO is effective. In both cases, the excess weight at low frequencies is prominent as non-Gaussianity in the experimental data.

Figure 5

Distribution of shifts in the Larmor frequency of a ${}^{69}\mathrm{Ga}$ or ${}^{71}\mathrm{Ga}$ nucleus due to interactions with the four nearest-neighbor ${}^{75}\mathrm{As}$ nuclei, leading to an approximately Gaussian decay of the autocorrelation function. The red vertical lines are the normalized probability of a frequency shift $n{J}_{69}$ or $n{J}_{71}$, while the blue curve is a Gaussian fit.