Measurement of Temporal Correlations of the Overhauser Field in a Double Quantum Dot

In quantum dots made from materials with nonzero nuclear spins, hyperfine coupling creates a fluctuating effective Zeeman field (Overhauser field) felt by electrons, which can be a dominant source of spin qubit decoherence. We characterize the spectral properties of the fluctuating Overhauser field in a GaAs double quantum dot by measuring correlation functions and power spectra of the rate of singlet-triplet mixing of two separated electrons. Away from zero field, spectral weight is concentrated below 10 Hz, with $\sim 1/f^2$ dependence on frequency $f$. This is consistent with a model of nuclear spin diffusion, and indicates that decoherence can be largely suppressed by echo techniques.

Figure 1

(a) Schematic energy diagram of the two-electron system. Inset: false-color scanning electron microscopy image of a double dot with integrated rf QPC charge sensor similar to the one measured (scale bar is 500 nm). (b) Gate-pulse cycle that is used to prepare ($P$) the (2,0) singlet, separate ($S$) into (1,1), either to the $S\mathrm{\text{−}}{T}_0$ degeneracy [left (green) dashed line] or the $S\mathrm{\text{−}}{T}_{+}$ degeneracy [right (red) dashed line], and return to (2,0) for measurement ($M$). (c) rf QPC readout $V_{\mathrm{rf}}$ around the (1,1)–(2,0) transition during application of the cyclic gate-pulse sequence, showing the readout triangle indicated with white lines ($B=0\mathrm{mT}$, ${\tau }_S=50\mathrm{ns}$). A background plane has been subtracted. (d) $V_{\mathrm{rf}}$ as in (c) but for $S$ at the $S\mathrm{\text{−}}{T}_{+}$ degeneracy ($B=10\mathrm{mT}$). (e) Average value of $P_S({\tau }_S)$ at $B=0$, ${\tau }_M=2\mu \mathrm{s}$. Red line is a fit to the theoretical Gaussian form. (f) Average value of $P_S({\tau }_M)$ showing contrast dependence, ${\tau }_S=50\mathrm{ns}$. Red line is a fit to the exponential form (see main text).

Figure 2

(a) rf QPC sensor output $V_{\mathrm{rf}}$ as a function of $V_L$ and $V_R$ with gate-pulse cycle applied (${\tau }_S=25\mathrm{ns}$, ${\tau }_M=1.6\mu \mathrm{s}$, $B=100\mathrm{mT}$). Color scale as in Fig. 1. (b) Repeated slices of $V_L$ with $V_R=-709\mathrm{mV}$ as a function of time. Markers on left axis correspond to markers in (a). (c) Sensor output calibrated to $P_S$ (lower, blue) along with a measurement of the background QPC noise (upper, pink) from (b) at arrow positions. Bandwidth limited to $\sim 3\mathrm{Hz}$. (d) Similar to (b) but for $B=0$, color scale same as in Fig. 1. (e) Similar to (b) but with $S$ point at $S\mathrm{\text{−}}{T}_{+}$ degeneracy, $B=100\mathrm{mT}$; color scale same as in Fig. 1.

Figure 3

(a) Power spectra of $P_S$ at various magnetic fields, ${\tau }_S=25\mathrm{ns}$. Spectra obtained by fast Fourier transform (with Hamming window) of average of 8 traces sampled at 10 kHz. Background measurement noise (BG) found by setting ${\tau }_S=1\mathrm{ns}$ at $B=100\mathrm{mT}$. Inset: numerical simulation results for corresponding magnetic fields: $B=0$ (pink), $B=5\mathrm{mT}$ (blue), $B=10\mathrm{mT}$ (green), $B=100\mathrm{mT}$ (red). (b) Autocorrelation $P_S$ for ${\tau }_S=25\mathrm{ns}$ and $B=100\mathrm{mT}$ (red curve). Model function [Eq. (1)] (brown curve) and Monte Carlo result (black curve).

Figure 4

Power spectra of $P_S$ at $B=100\mathrm{mT}$ for separation times ${\tau }_S=25\mathrm{ns}$ (red line) and ${\tau }_S=100\mathrm{ns}$ (blue line). Setting ${\tau }_S=1\mathrm{ns}$ (black line) yields background noise. Inset shows simulation results for $B=100\mathrm{mT}$, ${\tau }_S=25\mathrm{ns}$ (red) and ${\tau }_S=100\mathrm{ns}$ (blue). Note the suppression of low-frequency content and enhancement of midfrequency content for long ${\tau }_S$ in the experiment and simulation.