Charge Noise Spectroscopy Using Coherent Exchange Oscillations in a Singlet-Triplet Qubit

Two level systems that can be reliably controlled and measured hold promise as qubits both for metrology and for quantum information science. Since a fluctuating environment limits the performance of qubits in both capacities, understanding environmental coupling and dynamics is key to improving qubit performance. We show measurements of the level splitting and dephasing due to the voltage noise of a GaAs singlet-triplet qubit during exchange oscillations. Unexpectedly, the voltage fluctuations are non-Markovian even at high frequencies and exhibit a strong temperature dependence. This finding has impacts beyond singlet-triplet qubits since nearly all solid state qubits suffer from some kind of charge noise. The magnitude of the fluctuations allows the qubit to be used as a charge sensor with a sensitivity of $2\times {10}^{-8}e/\sqrt{\mathrm{Hz}}$, 2 orders of magnitude better than a quantum-limited rf single electron transistor. Based on these measurements, we provide recommendations for improving qubit coherence, allowing for higher fidelity operations and improved charge sensitivity.

Figure 1

(a) The device used in these measurements is a gate-defined $S\mathrm{\text{−}}{T}_0$ qubit with an integrated rf sensing dot. (b) The Bloch sphere that describes the logical subspace features two rotation axes ($J$ and $\Delta B_Z$) controlled with dc voltage pulses. (c) An energy diagram of the relevant states as a function of $ϵ$. States outside of the logical subspace of the qubit are grayed out. (d) $J(ϵ)$ and $dJ/dϵ$ in three regions: the (1, 1) region where $J$ and $dJ/dϵ$ are both small and $S\mathrm{\text{−}}{T}_0$ qubits are typically operated, the transitional region where $J$ and $dJ/dϵ$ are both large where the qubit is loaded and measured, and the (0, 2) region where $J$ is large but $dJ/dϵ$ is small and large quality oscillations are possible.

Figure 2

Ramsey oscillations reveal low-frequency environmental dynamics. (a) The pulse sequence used to measure exchange oscillations uses a stabilized nuclear gradient to prepare and read out the qubit and has good contrast over a wide range of $J$. (b) Exchange oscillations consistently show larger $T_2^{*}$ as $J$, and hence $dJ/dϵ$, shrinks until dephasing due to nuclear fluctuations sets in at very negative $ϵ$. (c) Extracted values of $J$ and $dJ/dϵ$ as a function of $ϵ$. (d) The decay curve of FID exchange oscillations shows Gaussian decay. (e) Extracted values of $T_2^{*}$ and $dJ/dϵ$. $T_2^{*}$ is proportional to $(dJ/dϵ{)}^{-1}$, indicating that voltage noise causes the dephasing. (f) Exchange oscillations measured in (0, 2), showing slow dephasing at very positive $ϵ$. Line cuts at different values of $ϵ$ show beats between the frequency of the oscillations and the sampling frequency.

Figure 3

Spin-echo measurements reveal high-frequency bath dynamics. (a) The pulse sequence used to measure exchange echo rotations. (b) A typical echo signal. The overall shape of the envelope reflects $T_2^{*}$, while the amplitude of the envelope as a function of $\tau$ (not pictured) reflects $T_2^{\mathrm{echo}}$. (c) $T_2^{\mathrm{echo}}$ and $Q\equiv J{T}_2^{\mathrm{echo}}/2\pi$ as a function of $J$. A comparison of the two noise models: Noise with a power-law spectrum fits over a wide range of $ϵ$ (constant $\beta$), but the apparent relative contributions of white and $1/f$ noise change. (d) A typical echo decay is nonexponential but is well fit by $\exp [-(\tau /T_2^{\mathrm{echo}}{)}^{\beta +1}]$. (e) $T_2^{\mathrm{echo}}$ varies with $dJ/dϵ$ in a fashion consistent with dephasing due to power-law voltage fluctuations.

Figure 4

Higher temperatures speed dephasing. (a) $T_{2,\mathrm{nuclear}}^{*}$ does not depend significantly on $\mathcal{T}$. (b) $T_2^{*}$ in (1, 1) (red squares) and (0, 2) (green crosses) has the same weak, scaled dependence on $\mathcal{T}$. (c) $T_2^{\mathrm{echo}}$ shows a strong temperature dependence near ${\mathcal{T}}^{-2}$ (red line) across over an order of magnitude in coherence times. (d) As $\mathcal{T}$ is increased, $\beta$ approaches zero, indicating that the noise leading to decoherence becomes nearly white. Uncertainties in $\beta$ are larger at higher temperatures due to the fast decay of the echo.