Coherent Operations and Screening in Multielectron Spin Qubits

Multielectron spin qubits are demonstrated, and performance examined by comparing coherent exchange oscillations in coupled single-electron and multielectron quantum dots, measured in the same device. Fast ($>1\mathrm{GHz}$) exchange oscillations with a quality factor $Q\sim 15$ are found for the multielectron case, compared to $Q\sim 2$ for the single-electron case, the latter consistent with experiments in the literature. A model of dephasing that includes voltage and hyperfine noise is developed that is in good agreement with both single- and multielectron data, though in both cases additional exchange-independent dephasing is needed to obtain quantitative agreement across a broad parameter range.

Figure 1

(a) Scanning electron micrograph of a lithographically identical device, indicating double dot (blue) and charge sensor (yellow). Scale bar indicates 500 nm, arrow indicates magnetic field direction [200 mT for (1,1), 50 mT for (7,5)] and the [110] crystal axis. (b) Schematic exchange pulse sequence. An adiabatic ramp to $J=0$ initializes the system in the lowest energy $m_{\mathrm{S}}=0$ eigenstate before an exchange pulse of duration $\tau$ to detuning $ϵ$ is applied. Detuning noise $\delta ϵ$ induces exchange fluctuations $\delta J$, which limits the number of visible exchange oscillations at high detuning. (c) Probability of detecting a singlet $P(S)$ as a function of detuning and exchange time for single-electron dots. (d) Same as (c) but for multielectron dots. Coherence is significantly improved for the multielectron exchange case. Insets: Simulated exchange oscillations (see text). Color scale shared for all 2D plots.

Figure 2

(a) Exchange coupling $J(ϵ)$ as a function of detuning $ϵ$ for the single- and multielectron exchange gate. Data extracted from exchange oscillations (represented in insets) below (square, circle) and above (triangle) the range of the waveform generator (Tektronix AWG5014), measured by externally stepping the clock, and from the location of the $S\text{−}{T}_{+}$ anticrossing (diamond), where Zeeman and exchange energies are equal. Fits are to biexponential models (see text). (b) $(dJ/dϵ)/J$, obtained numerically from the fits in (a), reflects the dephasing per exchange pulse due to $ϵ$ noise. As $ϵ$ approaches zero, the multielectron exchange gate should display improved coherence for equal amounts of $ϵ$ noise. This is consistent with the observation of improved coherence in the multielectron case.

Figure 3

Plotting the left side integral $L({ϵ}_i)$ versus the right side integral $R({ϵ}_i)$ of Eq. (4) allows us to extract a rms $ϵ$ noise of (a) $\delta ϵ=2.0\mu \mathrm{eV}$ for (1,1) and (b) $\delta ϵ=1.2\mu \mathrm{eV}$ for (7,5) dot occupations (dashed lines). Deviations from linear behavior indicate the presence of non-$ϵ$ noise. Using the extracted values for $dJ/dϵ$ and $\delta ϵ$, the dephasing rate due to $ϵ$ noise ${\mathrm{\Gamma }}_{ϵ}$ can be predicted without any free parameters. For both (c) (1,1) and (d) (7,5), the system is dominated by $ϵ$ noise at large $J$, but decoheres due to an unknown source at small $J$. The excess dephasing rate is not explained by nuclei (${\mathrm{\Gamma }}_n$), but is well captured by a model (${\mathrm{\Gamma }}_{\mathrm{\Sigma }}$) that includes a constant dephasing rate ${\mathrm{\Gamma }}_0$ as its only free parameter (see text).

Figure 4

(a) Quality of exchange rotations $Q$ as a function of $J$ for (7,5) and (1,1). For large $J$, the multielectron exchange gate significantly outperforms the single-electron gate. The model (solid lines) includes dephasing due to $ϵ$ noise, nuclei, and a constant ${\mathrm{\Gamma }}_0$ to account for the unknown noise source discussed in Fig. 3. Also shown (dotted lines) are the same models without the contribution from ${\mathrm{\Gamma }}_0$. (b) The observed reduction of $Q$ when applying external voltage fluctuations of fixed amplitude $\delta V_{\mathrm{ext}}$ to the detuning axis is in good agreement with the model including ${\mathrm{\Gamma }}_0$ (solid line), and is in disagreement with the model excluding ${\mathrm{\Gamma }}_0$ (dotted line). All model parameters determined from other measurements. This serves as an independent check of our model parameters.