## Abstract

Electron spins in silicon quantum dots are excellent qubits because they have long coherence times and high gate fidelities and are compatible with advanced semiconductor manufacturing techniques. For qubits based on single spins, electron spin resonance with real or effective time-varying magnetic fields is the standard method for universal quantum control. Here we show that spin–valley coupling in Si, which drives transitions between states with different spin and valley quantum numbers, enables coherent control of single- and multi-electron spin states without oscillating electromagnetic fields. We demonstrate Rabi oscillations between effective single-spin states in a Si/SiGe double quantum dot that are driven by spin–valley coupling. Together with the exchange coupling between neighbouring electrons, spin–valley coupling also enables universal control of effective two-spin states, driving singlet–triplet and triplet–triplet oscillations that feature coherence times on the order of microseconds. Our results establish spin–valley coupling as a promising mechanism for coherent control of qubits based on electron spins in semiconductor quantum dots.

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## Data availability

The processed data that support the plots are available as Source Data files and at https://doi.org/10.5281/zenodo.7192171. The raw data are available from the corresponding author upon request.

## Code availability

The code used to analyse the data in this work is available from the corresponding author upon request.

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## Acknowledgements

This work was sponsored by the Army Research Office under grants W911NF-17-1-0260 (X.C., E.J.C. and J.M.N.) and W911NF-19-1-0167 (J.M.N.), and by the National Science Foundation under grant OMA 1936250 (X.C.). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

## Author information

### Authors and Affiliations

### Contributions

X.C., E.J.C. and J.M.N. conceptualized the experiment, conducted the investigation and participated in writing. J.M.N. supervised the effort. L.F.E. grew the Si/SiGe wafer.

### Corresponding author

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

*Nature Physics* thanks Ryan Jock, Chih-Hwan Yang and Amin Hosseinkhani for their contribution to the peer review of this work.

## Additional information

**Publisher’s note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Spin funnel pulse sequence and data analysis.

**a**, Pulse sequence used to measure the spin funnels. The system is initialized in the (4,0) singlet ground state, pulsed to variable *ϵ* for a fixed time *t*_{e}, and then pulsed to the PSB region for measurement. The dashed line indicates the (4,0)-(3,1) transition point, where *ϵ*=0. **b**, Voltage dependence of the valley splitting in dot 1, \({E}_{1}^{\nu }\), versus *ϵ* extracted from the second spin funnel (blue dots) and fit (red line). **c**, Voltage dependence of the valley splitting in dot 2, \({E}_{2}^{\nu }\), versus *ϵ* extracted from the third spin funnel (blue dots) and fit (red line). **d**, \({E}_{1}^{\nu }+{E}_{2}^{\nu }\) versus *ϵ*, extracted from the fourth spin funnel (blue dots) and prediction (red line). The prediction is the sum of the two fits from panels **b** and **c**.

### Extended Data Fig. 2 Voltage dependence of the exchange interaction, *J*(*ϵ*).

**a**, Pulse sequence used to measure the exchange oscillations in **b** (left panel) and schematic of the relevant energy levels (right panel). After initializing the system in the (4,0) singlet ground state, we prepare the superposition state (\(\left\vert {S}^{+-}\right\rangle +i\left\vert {T}_{0}^{+-}\right\rangle\))\(\sqrt{2}\) via a 3*π*/2 pulse around Δ*E*_{Z} at *ϵ*=19 mV. The state evolves at different values of *ϵ* (therefore under different values of *J*) for a variable time *t*_{e}. After the evolution, we use a *π*/2 pulse around Δ*E*_{Z} to map this superposition state to \(\left\vert {S}^{+-}\right\rangle\) and measure the singlet return probability *P*_{S}. **b**, Measurement of exchange oscillations between superpositions of \(\left\vert {S}^{+-}\right\rangle\) and \(\left\vert {T}_{0}^{+-}\right\rangle\) at different values of *ϵ* with *B*^{z} = 350 mT. Inset: Absolute value of the fast Fourier transform of the data. **c**, Values of *J* as a function of *ϵ* extracted from the exchange oscillation measurement in **b** (blue circles) and the spin funnel measurement (red dots). The *J* values are displayed on a logarithmic scale. The solid line is an empirical fit of the data.

### Extended Data Fig. 3 Measured Δ*E*_{Z} oscillation frequency as a function of external magnetic field *B*^{z}.

Linear fit to the data above 100 mT shows dependence of 7.21 MHz/T, corresponding to a *g*-factor gradient Δ*g* = *g*_{1} − *g*_{2} = 5.15 × 10^{−4}.

### Extended Data Fig. 4 Measurement of the sign of Δ*E*_{Z} = *g*_{1}*μ*_{B}*B*^{z} − *g*_{2}*μ*_{B}*B*^{z}.

**a**, Charge stability diagram of the (3,1) charge region illustrating the pulse positions for the relevant measurements. After adiabatically preparing either \(\left\vert {\downarrow }^{+}{\uparrow }^{-}\right\rangle\) or \(\left\vert {\uparrow }^{+}{\downarrow }^{-}\right\rangle\), we ramp *V*_{1} and *V*_{2} near either the dot-1 or dot-2 charge transitions (yellow lines) and wait for a variable time. We then reverse the pulse sequence and use PSB to distinguish between singlet and triplet states. **b**, Schematic of the relevant energy levels during relaxation near the dot-1 transition. Here, \(\left\vert {\uparrow }^{+}{\downarrow }^{-}\right\rangle\) may relax to \(\left\vert {\downarrow }^{-}{\downarrow }^{-}\right\rangle\) or \(\left\vert {\downarrow }^{+}{\downarrow }^{-}\right\rangle\) via exchange with (2,1) states. \(\left\vert {\downarrow }^{+}{\uparrow }^{-}\right\rangle\) cannot transition to any of \(\left\vert {T}_{-}^{+-}\right\rangle\), \(\left\vert {T}_{-}^{--}\right\rangle\), or \(\left\vert {T}_{-}^{++}\right\rangle\) without undergoing, at minimum, a spin flip in dot 2 and therefore does not appreciably relax during the microsecond timescales in these measurements. **c**, Schematic of the relevant energy levels during relaxation near the dot-2 transition. Here, \(\left\vert {\downarrow }^{+}{\uparrow }^{-}\right\rangle\) may relax to \(\left\vert {\downarrow }^{+}{\downarrow }^{+}\right\rangle\) or \(\left\vert {\downarrow }^{+}{\downarrow }^{-}\right\rangle\) via exchange with (3,0) states. \(\left\vert {\uparrow }^{+}{\downarrow }^{-}\right\rangle\) cannot transition to \(\left\vert {T}_{-}^{+-}\right\rangle\), \(\left\vert {T}_{-}^{--}\right\rangle\), or \(\left\vert {T}_{-}^{++}\right\rangle\) without incurring a spin flip in dot 1, and therefore does not appreciably relax. **d**, Triplet return probability near the dot-1 transition, corresponding to **b**. When preparing the higher-energy state, there is a slight enhancement in the triplet return probability inside the (3,1) region, suggesting that we have prepared the state \(\left\vert {\uparrow }^{+}{\downarrow }^{-}\right\rangle\). **e**, Triplet return probability near the dot-2 transition, corresponding to **c**. When preparing the lower-energy state, there is a strong enhancement in the triplet return probability inside the (3,1) region, indicating that we have prepared the state \(\left\vert {\downarrow }^{+}{\uparrow }^{-}\right\rangle\). The overall visibility of the higher-energy state traces is lower due to imperfections in the preparation and readout. We suspect that the stronger enhancement in the triplet return probability observed near the dot 2 transition compared to the enhancement observed near the dot 1 transition in **d** may be due to the details affecting the relaxation rates between the states involved in the relaxation processes. **f**, Plot of the triplet return probability for the high-energy-state measurement, \({P}_{T}^{e}\), minus the triplet return probability of the low-energy-state measurement, \({P}_{T}^{g}\), as a function of the wait time and wait position near the dot-1 transition. **g**, Plot of \({P}_{T}^{e}-{P}_{T}^{g}\) as a function of the wait time and wait position near the dot-2 transition. From the data shown in **d**-**g**, we conclude that the low-energy state is \(\left\vert {\downarrow }^{+}{\uparrow }^{-}\right\rangle\), and therefore Δ*E*_{Z} > 0. All data shown in this figure are acquired at *B*^{z}=600 mT.

### Extended Data Fig. 5 Single-spin Rabi oscillation pulse sequence and control measurements.

**a** We prepare the state \(\left\vert {\downarrow }^{+}{\uparrow }^{-}\right\rangle\) by adiabatically separating a singlet state in the presence of Δ*E*_{Z}. After an evolution period of variable time, *t*_{e}, we reverse the initialization process to map \(\left\vert {\downarrow }^{+}{\uparrow }^{-}\right\rangle\) back to \(\left\vert {S}^{+-}\right\rangle\) for PSB readout. **b** We prepare \(\left\vert {\uparrow }^{+}{\downarrow }^{-}\right\rangle\) in a similar way, except that we add a Δ*E*_{Z}*π* pulse before separating and after recombining the electrons. The initial *π* pulse rotates the state \(\left\vert {S}^{+-}\right\rangle\) to \(\left\vert {T}_{0}^{+-}\right\rangle\). The \(\left\vert {T}_{0}^{+-}\right\rangle\) state, in turn, evolves to the state \(\left\vert {\uparrow }^{+}{\downarrow }^{-}\right\rangle\) upon adiabatic separation. The readout follows the same steps in reverse, where the adiabatic pulse maps \(\left\vert {\uparrow }^{+}{\downarrow }^{-}\right\rangle\) back to \(\left\vert {T}_{0}^{+-}\right\rangle\) and then the *π* pulse maps \(\left\vert {T}_{0}^{+-}\right\rangle\) to \(\left\vert {S}^{+-}\right\rangle\). **c**,**d** Control measurements at a magnetic field away from the spin funnels, *B*^{z}=350 mT, using the pulse sequence in **a** and the pulse sequence in **b**, respectively. No oscillations are observed in either case. **e** Measured single-spin Rabi oscillation near the third spin funnel with *B*^{z}=380.8 mT, using the pulse sequence in **a**. The Rabi frequency is about 15 MHz, and the \({T}_{2}^{* }\) time is about 3 *μ**s*, resulting in a quality factor Q ~ 40 − 50, which is close to the typical values obtained using electric-dipole spin resonance in a magnetic field gradient for natural Si devices^{68}.

### Extended Data Fig. 6 Numerical simulations and comparison measurements on the impact of the residual exchange.

**a**,**b** Numerical simulations corresponding to Fig. 3c in the main text. In both cases, the state is prepared and measured along \(\left\vert \widetilde{{\downarrow }^{+}{\uparrow }^{-}}\right\rangle\). **a** Simulation assuming *J*_{r} = 0 and *η* = 0. **b** Simulation assuming *J*_{r}=0.5 MHz and *η*=0.0167. **c**,**d** Measurements corresponding to Figs. 3e,f in the main text, taken in a separate cooldown where the device was tuned differently. **e**,**f** Same measurements as in **c** and **d**, respectively, but conducted with a voltage pulse of -30 mV applied to the barrier gate between the two quantum dots. The feature visible in **c** and **d** near *ϵ*=26 mV vanishes.

### Extended Data Fig. 7 Pulse sequences for coherent spin-valley-driven singlet-triplet oscillations.

**a** Pulse sequence for *S* − *T*_{−} Rabi oscillations. We prepare the double dot in the state \(\left\vert {S}^{+-}\right\rangle\), pulse to different values of *ϵ* for a variable evolution time *t*_{e}, and then measure the singlet return probability *P*_{S} by pulsing *ϵ* to the PSB region in (4,0). **b** Pulse sequence for *S* − *T*_{−} Ramsey oscillations, with a 3*π*/2 and *π*/2 pulse performed at the *ϵ* value of the *S* − *T*_{−} avoided crossing (denoted \({\Delta }^{S{T}_{-}}\)) before and after the evolution. Through a \({\Delta }^{S{T}_{-}}\) 3*π*/2 pulse, we prepare the double dot in a superposition of \(\left\vert {S}^{+-}\right\rangle\) and either \(\left\vert {T}_{-}^{--}\right\rangle\) or \(\left\vert {T}_{-}^{++}\right\rangle\), depending on which spin funnel we are operating near. Specifically, for the second spin funnel, we prepare \(\left\vert \psi \right\rangle =\frac{1}{\sqrt{2}}\)(\(\left\vert {S}^{+-}\right\rangle +i\left\vert {T}_{-}^{--}\right\rangle\)) and for the third spin funnel, we prepare \(\left\vert \psi \right\rangle =\frac{1}{\sqrt{2}}\)(\(\left\vert {S}^{+-}\right\rangle -i\left\vert {T}_{-}^{++}\right\rangle\)), if \({\Delta }_{1(2)}^{sv}\) are real and positive. After the evolution, we use a \({\Delta }^{S{T}_{-}}\,\)*π*/2 pulse to map \(\left\vert \psi \right\rangle\) to \(\left\vert {S}^{+-}\right\rangle\) for PSB readout. **c** Pulse sequence used to observe triplet-triplet oscillations, with a *π* pulse at the *ϵ* value of the *S* − *T*_{−} avoided crossing before and after the evolution. Through a \({\Delta }^{S{T}_{-}}\,\)*π* pulse, we prepare the excited \(\left\vert {T}_{-}\right\rangle\) state, \(\left\vert {T}_{-}^{--}\right\rangle\) for the second spin funnel or \(\left\vert {T}_{-}^{++}\right\rangle\) for the third spin funnel. After the evolution, we apply another \({\Delta }^{S{T}_{-}}\,\)*π* pulse to map the excited \(\left\vert {T}_{-}\right\rangle\) state to \(\left\vert {S}^{+-}\right\rangle\) for PSB readout. **d** Control measurement at a magnetic away from the spin funnels, *B*^{z}=350 mT, using the pulse sequence in **a**. The energy level diagram for this magnetic field is displayed in Extended Data Fig. 2a, right panel. The two vertical lines in the data near *ϵ*=0 correspond to where \(\left\vert {T}_{-}^{+-}\right\rangle\) and \(\left\vert {T}_{-}^{--}\right\rangle\) come into resonance with \(\left\vert {S}^{+-}\right\rangle\), which occur at energies below the range plotted in Extended Data Fig. 2a.

### Extended Data Fig. 8 *S* − *T*_{−} spin echo pulse sequence and data.

**a** The double dot is prepared in a superposition of \(\left\vert {S}^{+-}\right\rangle\) and \(\left\vert {T}_{-}^{++}\right\rangle\) via a *π*/2 pulse at the *S* − *T*_{−} avoided crossing on the third spin funnel. Specifically, we prepare \(\left\vert {\psi }^{{\prime} }\right\rangle =\frac{1}{\sqrt{2}}(\left\vert {S}^{+-}\right\rangle +i\left\vert {T}_{-}^{++}\right\rangle )\) for real, positive \({\Delta }_{2}^{sv}\). The state evolves near *ϵ*=9 mV, where the *S* − *T*_{−} splitting is approximately 50 MHz and varies roughly linearly with *ϵ*, for a total time *t*_{e} + *δ**t*, during which we apply a *π* pulse at the avoided crossing to refocus the dephasing. After the evolution is complete, we apply another *π*/2 pulse before PSB readout. **b** Echo amplitude decay as a function of the total qubit evolution time *t*_{e}. To determine the amplitude at each point, we fit the inhomogeneously broadened decay for fixed *t*_{e} and varying *δ**t* [27,54]. We find \({T}_{2}^{* }\approx 210\) ns and \({T}_{2}^{echo}=3.6\mu s\).

### Extended Data Fig. 9 Measured magnitudes of \({\Delta }_{1}^{sv}\) and \({\Delta }_{2}^{sv}\) versus the in-plane magnetic field angle *ϕ* (blue squares).

The amplitude of the magnetic field is fixed at \(| {B}_{ext}| =\sqrt{{({B}^{x})}^{2}+{({B}^{z})}^{2}}\) =300 mT in **a** and 375 mT in **b**, where the *B*^{x} component is parallel with the axis connecting the two quantum dots and the *B*^{z} component is perpendicular to the axis. We fit both sets of data to an equation of the form *g*_{1} (*ϕ*)=\(| A\sin \phi +B\cos \phi |\), where *A* and *B* are fit parameters (dark blue lines), and another equation of the form \({g}_{2}(\phi )=| A\sin \phi +B{e}^{i\theta }\cos \phi |\), where *A*, *B* and *θ* are fit parameters and constrained to be real numbers (orange lines). **c**, Examples of the oscillations used to extract the values of \(| {\Delta }_{1}^{sv}|\) in **a**, for three different angles as indicated. *ϕ* = − 0.79 and *ϕ* = 0.79 correspond to the maximum and minimum values of \(| {\Delta }_{1}^{sv}|\) in **a**. Data are offset vertically for clarity. The *ϵ* values of the avoided crossing are also indicated. The small variation in the *ϵ* value may indicate a shift of ∣*B*_{ext}∣ away from the setpoint due to hysteresis in the superconducting magnet in our dilution refrigerator. **d**, Examples of the oscillations used to extract the values of \(| {\Delta }_{2}^{sv}|\) in **b**. *ϕ* = − 1.36 and *ϕ* = 0.21 correspond to the maximum and minimum values of \(| {\Delta }_{2}^{sv}|\) in **b**.

### Extended Data Fig. 10 Measured and simulated *S* − *T*_{−} oscillations.

**a**-**d** Measured and simulated \({S}^{+-}-{T}_{-}^{--}\) oscillations near the second spin funnel. **a** Measured Rabi oscillations at the avoided crossing and envelope of the corresponding fit. The fitted values of \({T}_{2}^{* }\) and *β* with the 95% confidence intervals are indicated. **b** Measured Ramsey oscillations near *ϵ* = 6.5 mV and envelope of the corresponding fit. **c** Simulation and fit corresponding to panel **a**. **d** Simulation and fit corresponding to panel **b**. **e**-**h** Measured and simulated oscillations near the third spin funnel, where the two \({S}^{+-}-{T}_{-}^{++}\) avoided crossings have merged into one. **e** Measured Rabi oscillations at the avoided crossing and envelope of the corresponding fit. **f** Measured Ramsey oscillations near *ϵ* = 10 mV and envelope of the corresponding fit. **g** Simulation and fit corresponding to panel **e**. **h** Simulation and fit corresponding to panel **f**.

## Source data

### Source Data Fig. 1

Excel table with data shown in the plots.

### Source Data Fig. 2

Excel table with data shown in the plots.

### Source Data Fig. 3

Excel table with data shown in the plots.

### Source Data Fig. 4

Excel table with data shown in the plots.

### Source Data Fig. 5

Excel table with data shown in the plots.

### Source Data Fig. 6

Excel table with data shown in the plots.

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Excel table with data shown in the plots.

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Excel table with data shown in the plots.

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Excel table with data shown in the plots.

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Excel table with data shown in the plots.

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Excel table with data shown in the plots.

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Excel table with data shown in the plots.

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Excel table with data shown in the plots.

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Excel table with data shown in the plots.

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Excel table with data shown in the plots.

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Excel table with data shown in the plots.

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### Cite this article

Cai, X., Connors, E.J., Edge, L.F. *et al.* Coherent spin–valley oscillations in silicon.
*Nat. Phys.* **19**, 386–393 (2023). https://doi.org/10.1038/s41567-022-01870-y

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DOI: https://doi.org/10.1038/s41567-022-01870-y