Fig. 4

Z-rotations and noise. a Energy Diagram and gate pulse schematic for controlling exchange rotations. We initialize the qubit into the S(2, 0) ground state and ramp adiabatically, such that it transfers to the ground in the (1, 1) charge sector. A fast pulse to and from a detuning, \(\epsilon\), where J is substantial drives coherent rotations around an axis depending on both J and Δ_SO. Returning to the (2, 0) charge sector adiabatically projects the states onto the S(2, 0) and T₀(1, 1) basis for measurement. b Measured charge sensor current as a function of the time spent rotating for various detuning points. Here, high current corresponds to a higher probability of measuring a singlet. c The extracted rotation frequency vs. detuning. The blue line is a fit to the form \(\sqrt {J(\epsilon )^2 + {\mathit{\Delta }}_{{\mathrm{SO}}}^2}\), where J(\(\epsilon\)) ∝ \(\epsilon\)⁻¹. d Extracted \(T_{\mathrm{2}}^ \star\) as a function of detuning. We also plot the long integration time values from Fig. 3a. The blue lines are fits to the form \(T_{\mathrm{2}}^ \star\) = \(\frac{1}{{\sqrt 2 \pi \sigma _e}} \cdot \left| {\frac{{{\mathrm{d}}f}}{{{\mathrm{d}}\epsilon }}} \right|^{ - 1}\), where the extracted charge noise, \(\sigma _\epsilon\), is 1 (dashed), 2 (solid) and 3 μeV (dotted). e Gate pulse schematic for a Hahn-echo sequence. We initialize the qubit into the S(2, 0) ground state and transfer the system to the (1, 1) charge sector with a rapid adiabatic pulse such that it remains in a singlet state. Combinations of Δ_SO-rotations about the X-axis and J-rotations about a second axis provide access to entire Bloch sphere. This echo sequence counteracts low frequency noise, prolonging qubit coherence. f Hahn-echo amplitude as a function of total time, τ′ + τ, exposed to charge noise at detuning \(\epsilon\). The error bars represent 95% confidence interval. A fit to an exponential decay gives qubit coherence time of \(T_{2e}^{{\mathrm{echo}}}\) = 8.4 μs. (inset) Measured echo signal for τ = 1 μs with B = 0.141 T along the [110] direction. The change in charge sensor current (ΔCS) due to the echo signal has an oscillation frequency corresponding to Δ_SO and a Gaussian envelope around τ = τ′ with a decay due to the inhomogeneous dephasing time of \(T_{2e}^ \ast\) = 1 μs