Coherent coupling between a quantum dot and a donor in silicon
Individual donors in silicon chips are used as quantum bits with extremely low error rates. However, physical realizations have been limited to one donor because their atomic size causes fabrication challenges. Quantum dot qubits, in contrast, are highly adjustable using electrical gate voltages. This adjustability could be leveraged to deterministically couple donors to quantum dots in arrays of qubits. In this work, we demonstrate the coherent interaction of a \ce^31P donor electron with the electron of a metal-oxide-semiconductor quantum dot. We form a logical qubit encoded in the spin singlet and triplet states of the two-electron system. We show that the donor nuclear spin drives coherent rotations between the electronic qubit states through the contact hyperfine interaction. This provides every key element for compact two-electron spin qubits requiring only a single dot and no additional magnetic field gradients, as well as a means to interact with the nuclear spin qubit.
The silicon industry’s fabrication capability promises to be a differentiating accelerator for the future development of quantum computers built with silicon quantum bits (qubits). Silicon is, furthermore, an appealing material for qubits because it provides an ultra low decoherence environment . In particular, extremely high fidelities have been demonstrated for both the electron [2, 1, 3, 4, 5] and nuclear spins  of a single dopant atom in isotopically-enriched silicon nanostructures . Assembling these exceptional solid-state qubits into a full quantum processor, as first envisioned by Kane , will require coupling donor atoms to one another in a controllable way. This has proven extremely challenging, demanding near-atomic precision in the placement of the donors [8, 9, 10, 11, 12]. In contrast, single electron spins confined in quantum dots (QDs) [13, 14, 15, 16] are routinely coupled to one another since quantum dots are highly tunable and fabricated in engineered locations, allowing for controllable and scalable two-qubit interactions [17, 18, 19, 13, 20]. For this reason, QDs have been theoretically discussed as intermediates to couple donor qubits [7, 21, 22, 23, 24, 25]. Recently, spin blockade has been observed in a silicon QD-donor device . However, the coherent spin coupling between donor- and quantum dot-based qubits has remained elusive. It is the cornerstone advance necessary for exploiting the advantages of these two complementary qubit systems.
Here, we advance silicon-based quantum information processing by coherently coupling a phosphorus donor’s electron spin to a metal-oxide-semiconductor (MOS) QD. In our system, the QD is tuned to few-electron occupancy while simultaneously keeping a nearby donor (D) tunnel-coupled to the QD. The combination of the QD and donor electron qubits gives rise to a joint singlet-triplet (ST) logical encoding analogous to those in double-QD qubits [27, 28]. Specifically, the two logical states are the singlet and unpolarized triplet . The encoding takes advantage of the contact hyperfine interaction between the donor electron spin and donor nuclear spin. This interaction makes the electron spin on the donor precess at a rate different from the QD electrons, where is the hyperfine coupling strength. The hyperfine interaction thus amounts to an effective magnetic field gradient produced by the single phosphorus nucleus and drives rotations between singlet and triplet states . By electrically controlling the donor charge configuration between ionized and neutral, the rotations can be turned off and on. The electron-electron exchange coupling and the hyperfine interaction with the donor nucleus define two orthogonal control axes for the qubit, and their relative strength is controlled using fast electrical pulses.
The electron qubit formed by the QD-D coupled system is analogous to other ST qubits, while introducing important advantages. It features full electrical control with a uniquely compact design requiring only one QD. The QD-D ST qubit avoids the integration complexities of other Si spin control schemes such as micromagnets [30, 31], microwave striplines [32, 33] or additional QDs for full electrical control [34, 14]. The hyperfine coupling to the single nuclear spin introduces a nature-defined and potentially very stable (i.e. low noise) rotation axis for the ST qubit. Furthermore, the system has a natural access to the nuclear spin, which is one of the highest performing solid state qubits . Integration of a coil for nuclear magnetic resonance could enable full control over the nuclear spin qubit. Nuclear spin readout schemes based on ST interactions with the donor have already been proposed , making complete control of these two coupled qubits foreseeable in the near future. The engineered coupling of the QD and D spins constitutes a possible path to realize over nineteen years of different theoretical proposals of donor qubit architectures [7, 35, 22, 36, 25, 24]. For example, the large lithographic quantum dot can facilitate the coupling of neighboring QD-D cells using capacitive coupling [37, 19, 20] or exchange interaction .
The QD-D device is fabricated with isotopically-enriched \ce^28Si and a foundry-compatible process (i.e. no lift-off processing). We use a poly-silicon gate stack, shown in Fig. 1a, that allows self-aligned ion implantation and subsequent activation annealing process. Phosphorus donors are implanted using the AG gate as a mask. This processing maximizes the probability of placing a D in a suitable location next to the QD. It also facilitates future multi-qubit fabrication that could take advantage of single ion implantation  and a planar QD geometry [24, 25]. Fabrication details are found in the Supplementary Sec. S1 and are similar to Ref. 38. A channel of electrons is formed at the MOS interface underneath the wire-shaped accumulation gate (AG) by applying a positive voltage, depicted as a blue overlay in Fig. 1a. Next, a QD island is isolated by applying suitable negative voltages on neighboring gates. A single-electron transistor (SET) is formed in the upper wire to monitor the electron occupation of the QD and the relevant donor, denoted . The SET charge sensor (CS) is also used for spin readout via spin-to-charge conversion. An in-plane magnetic field of is applied throughout the experiments and the electron temperature is measured to be . Detailed information about fabrication, gate biasing and electron counting is provided in the Supplementary Sec. S2.
To investigate coherent coupling dynamics between the donor and the QD, we first identify an effective QD-D charge transition with a total of four electrons, as shown in Fig. 1b-d [39, 40]. We use the spin filling structure, measured through magnetic fields, to engineer a sufficiently large energy difference between the singlet and triplet states , which we observe to be substantially larger for four electrons () than for two electrons (). Details are available in the Supplementary Sec. S3. In Si MOS, the valley splitting can be tuned to large values by increasing the electric field perpendicular to the interface, which was verified in this device . Simultaneously keeping the donor in resonance with the few electron QD states, however, constrained the available range of voltage in this design leading to the relatively small two-electron valley splitting. We note two general benefits of using the four-electron configuration: (i) filled shells might be a general approach to circumvent the obstacle of low valley splitting in any material with conduction band degeneracy [43, 14]; and (ii) increased electron numbers can extend the size of the QD due to the increased filling of the potential well, which in turn allows more range in selecting a suitable tunnel coupling to remote donor sites.
Hyperfine-driven spin rotations
Rotations between and can be driven by an effective magnetic field gradient between the QD and the donor (in the remainder of the text we will drop the ket notation). These rotations provide a signature of the single \ce^31P donor. The source of the effective is the contact hyperfine interaction between the donor electron spin and the nuclear spin . We expect the nuclear spin state to be projected onto a eigenstate by the repetitive experimental measurement. Rapidly separating a singlet state by pulling one electron onto the donor triggers coherent rotations between the and states. Reuniting the electrons onto the QD projects the state onto or . We note that spin preparation, manipulations and readout act self-consistently with respect to a fixed but unknown state of the nuclear spin (i.e. the sign of ) in sufficiently large magnetic fields such that the interaction with the polarized triplets is suppressed (which is the case in this experiment). Moreover, nuclear states are known to be long lived ( seconds) compared to the timescale of electron manipulations , therefore, errors caused by random flips while an electron is on the donor are expected to be negligible. The nuclear state could still have implications for single or multi-qubit operation. In the future, this could be addressed by deterministically setting the nuclear state through various pulsing schemes, such as a single-spin version of dynamic nuclear polarization . To demonstrate the hyperfine-driven rotations, we use the pulse sequence shown in Fig. 2. We prepare a state by first emptying the QD and loading an electron between the singlet and triplet loading lines. Then, we plunge the system at point P (see Fig. 2b). Next, we rapidly separate the electrons by pulsing the system to point A with a ramp time. After waiting for a given manipulation time, the system is pulsed back to point P in . The ramp time is such that the charge transition is adiabatic, but fast enough to prepare a . Finally, we use an enhanced latching readout developed for this experiment and described in the Supplementary Sec. S4 to measure the triplet return probability.
Figure 2c shows the triplet return probability as a function of the manipulation time. All the details on the pulse sequence can be found in the Supplementary Sec. S5 and S6. We find a ST rotation frequency . This frequency is the vector sum of the exchange energy and , such that , where is the Planck constant . We estimate a residual exchange of for this detuning from numeric fits to the frequency dependence on detuning (described below). The inset shows that this frequency is very stable over time. Such behavior differs from GaAs systems, for which dynamic nuclear polarization must be used to generate and maintain a particular of similar magnitude . The magnitude and stability of provides a strong indication that the rotations are driven by a single \ce^31P. A small and relatively constant frequency drift of around is observed over a period of which is consistent with the drift in the electrostatics of the device through the experimentally measured relation. Additionally, the observed linewidth is less than natural silicon, which has linewidths greater than for single donors  and is qualitatively consistent with an enriched \ce^28Si background. Noise in is believed to presently limit the linewidth, discussed below in terms of .
Characterization of exchange interaction
The detuning dependence of the ST rotations reveals additional information about this QD-D system. In Fig. 3a, we plot the triplet return probability against both detuning and manipulation time. As the detuning gets closer to zero, the frequency of the exchange rotations increases, as shown in Fig. 3c. This is consistent with a ST model where the exchange energy between the and states is not negligible and drives rotations around a tilted axis in the qubit Bloch sphere. To better understand the exact shape of the oscillations of Fig. 3a, we simulate the quantum dynamics of the system using a master equation approach and time-dependent controls. We describe the system using the basis states , similarly to previous treatments such as Taylor et al. . The details of the model are given in the Supplementary Sec. S7. The numerical simulation results are shown in Fig. 3b. The phase and shape of the oscillations is very well reproduced; however, the mechanisms limiting the visibility are numerous and detailed in the Supplementary Sec. S8. At the moment, we think that addressing the various causes could ultimately produce results on-a-par or even better than state-of-the-art ST qubits. The key fitting parameters of the model are the triplet tunnel coupling , singlet tunnel coupling , and hyperfine interaction . We can determine these parameters using a fit to the data of Fig. 3c, knowing that equals the energy gap between the and states. We find , and . Shifts in of this magnitude relative to the bulk value (of , Ref. 32) have been reported in single donor electron spin resonance (ESR) experiments  and have been attributed to Stark shifts of the contact hyperfine interaction due to the large electric fields in the vicinity of the neighboring QD. The measured value in this work is both consistent with a shallow phosphorus donor and is inconsistent with likely alternatives, such as arsenic. Following the fit procedure, we can extract by subtracting the contribution. The result is shown in Fig. 3c.
Decoherence of MOS QDs  and single donors  has been characterized in separate systems, but the charge noise and magnetic noise properties of strongly hybridized QD-D systems are not well established. Our system provides a unique platform to study these important properties in an effective two-electron case where entanglement is delocalized in the form of a spatially separated singlet or triplet. We measure long time traces and plot the visibility of the oscillations versus manipulation time in Fig. 4a. The data and method are presented in the Supplementary Sec. S9. We then fit the decay using a slow detuning noise model that produces a Gaussian decay of the visibility , where is an arbitrary initial visibility. We find that depends on the detuning (Fig. 4b). To understand this dependence, we use a charge noise model represented by noise with a characteristic standard deviation and producing decoherence through . Details about the model are given in the Supplementary Sec. S10. We find that is consistent with the observed . In this model, we neglect magnetic noise that could be caused by residual \ce^29Si or other sources. Our observations are consistent with being limited by charge noise, a mechanism that is expected to play an important role when varies as a function of . We note that is approximately the electronic temperature . The noise magnitude has previously been correlated with the electronic temperature . We further tabulate noise magnitudes in a variety of material systems, like GaAs/AlGaAs heterostructures , Si/SiGe heterostructures [48, 14] and MOS (this work), and show the results in Fig. 4c.
|Reference||Petersson et al.||Shi et al.||Eng et al.||This work|
|(eV)||6.9||12||6.9 – 8.6||18|
In summary, we have demonstrated coherent coupling between the electrons of two very different qubit systems: a donor atom (natural atom) and a MOS quantum dot (artificial atom ). The coherent rotations between the singlet and triplet are driven by a nuclear spin qubit through the contact hyperfine interaction, and produce rotations with a of , thus allowing over 100 rotations within the coherence time. A charge noise magnitude of fits the stationary noise model and is a characterization of the MOS interface noise properties, which are found to be of similar magnitude to other common QD material systems. Assuming this model, the could possibly be improved by a factor 10 or more by operating at larger detunings where the exchange is negligible, hence taking full advantage of isotopically pure silicon. Our experiments demonstrate the feasibility of using the QD-D system as a compact ST qubit with no additional micromagnets [30, 31] or QDs (as in all-exchange qubits [50, 51, 14]), and avoid the decoherence mechanisms associated with GaAs or Si host nuclear species [28, 52]. More sophisticated ST qubit control approaches [53, 54] and optimized preparation/readout parameters will likely increase the visibility and reduce errors of future two-axis QD-D qubit demonstrations. To further speed up the operations compared to the coherence time, it could be possible to use other donor species that have stronger contact hyperfine strengths. Beyond individual ST qubits, this work opens-up compelling possibilities. One such example is the coupling of donor-based qubits without atomic precision placement through, for example, electrostatic coupling between ST qubits [37, 19, 20]. Another example is all-electrical nuclear spin readout  and electric/nuclear magnetic resonance control without high magnetic fields or ESR, thus introducing a nuclear spin qubit as an additional resource.
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The authors would like to thank Erik Nielsen, Andrew D. Baczewski, Matthew J. Curry and Stephen Carr for valuable help and discussions regarding this work; B. Silva for support on device fabrication and room temperature measurements; and Benjamin D’Anjou and William A. Coish for discussions on readout fidelity. This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. DOE’s National Nuclear Security Administration under contract DE-NA0003525.
P.H.-C. and M.S.C. designed the experiments. P.H.-C. performed the central measurements and analysis presented in this work. M.R. performed supporting measurements on similar “control” samples that establish repeatability of many observations in this work. N.T.J., P.H.-C., M.R. and J.K.G. modelled key elements of the device structure providing critical insights. P.H.-C., M.S.C., N.T.J. and M.P.-L. analyzed and discussed central results throughout the project, including designing models for observations. J.D., T.P., G.A.T.E. and M.S.C. designed process flow, fabricated devices and designed/characterized the \ce^28Si material growth for this work. J.R.W. provided critical nanolithography steps. M.L. supplied critical laboratory set-up for the work. M.S.C. supervised combined effort including coordinating fab and identifying modelling needs for experimental path. P.H.-C., M.S.C. and M.P.-L. wrote the manuscript with input from all co-authors.
Supplementary Information is included with this paper.
Competing financial interests: The authors declare no competing financial interests.
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Supplementary Information for:
Coherent coupling between a quantum dot and a donor in silicon
Appendix S1 Device Fabrication and Measurement
The device used for these experiments is fabricated identically to the one of Fig. 1a. Electrons are confined in a 2D electron gas at the interface between an epitaxial enriched \ce^28Si layer with residual \ce^29Si and a gate oxide. Highly n-doped poly-silicon gates ( thick) are patterned on top of the gate oxide using low pressure chemical vapor deposition and plasma etching . These are used to accumulate electrons by applying a positive voltage (in an enhancement mode) or deplete electrons (with negative voltages). Phosphorus donors are implanted in a PMMA resist window that overlaps with the AG gate on both sides of both wires, and the poly-Si gate used as a self-aligned implantation mask. The approximate relevant donor location indicated by the red dot in Fig. 1a of the main text is inferred from various donor-gate capacitance ratios. The source and drain reservoir electrons are connected by n+ regions and ohmic contacts to the instruments. The device is biased to form a SET in the upper wire that is used as a charge sensor (CS), while simultaneously forming a few-electron QD under the lower wire. The CS current is measured using an AC lock-in technique at with DC source-drain bias and (rms) AC bias. The derivative with respect to gate voltage is taken numerically to show the QD charge occupancy steps in charge stability diagrams.
Appendix S2 Few electron regime
One can form a clean single QD with this device geometry through biasing that pushes the QD towards one lead, shown in Fig. S1a. A representative set of gate voltages used for this experiment is shown in Fig. S1c. We use gates AG and CP to discriminate between QD and D states, respectively. The region where donors interact resonantly with the QD is shown in Fig. S1b. In this regime, the single QD turns into two strongly coupled QDs in series along the wire axis. This is indicated in the charge stability diagram of Fig. S1b by two sets of nearly parallel lines. This behavior is systematically reproduced in the devices we measured with such a geometry, which indicates that it is a feature produced by the electrostatics of the device. We can assign occupation numbers to the two QDs, counting from zero. We establish that the QD is in the few electron regime (i.e. emptied) by opening the tunnel barriers to the point where the QD charge-sensed lines become lifetime-broadened without detecting other states (data not shown). Donor and/or defect transitions can be seen cutting through the QD lines and are identified by red lines. These objects anti-cross with the QD lines in a way analogue to double QDs. The main difference is that they can only accommodate a limited number of charge states, like 0 or 1. We additionally performed magnetospectroscopy [55, 56, 57] to verify that the first electron fills as a spin-down electron. For this work, we treat the QD closest to the reservoir as being part of the reservoir itself and neglect its impact on the other QD.
Appendix S3 Effective (2,0)-(1,1) system
To investigate singlet-triplet dynamics, we first identify an effective QD-D charge transition with a total of four electrons, as shown in Fig. 2b. Singlet-triplet states with more than two electrons have been studied theoretically  and experimentally  in double-QD systems. Using magnetospectroscopy [55, 56, 57], we verify that the QD spin filling is indeed consistent with having a four-electron singlet ground state (see Fig. S1f). A requirement for efficient spin initialization and readout is that the energy difference between the singlet and triplet be much larger than the electron temperature of the experiment , which is in this case. Hence charge transitions have a full width at half maximum of approximately . In silicon, the valley splitting is generally the factor limiting . In our device and for the values of used, we have measured the valley splitting to be approximately . Consistent with this observation, the two-electron QD states had similarly small values for . The four-electron QD state of Fig. 2b, however, has an appreciably larger ST splitting of (as measured from both magnetospectroscopy and excited state spectroscopy). This might be understood as a shell filling effect with QD orbitals, where the pairing of spins allows to circumvent the small valley splitting, as illustrated in the schematic of Fig. 1c-d .
Appendix S4 Spin preparation and readout
We show that we can initialize and read out ST spin states. To do so, we use the pulse sequence of Fig. S2a. The system is initialized into a or state (where stands for any triplet) by first ejecting the fourth electron at point R (as defined in Fig. 2b), and then loading either a singlet (S) or triplet (T) state by carefully tuning the load level of point L. A deeper load tends to prepare T states due to their times faster loading rate. After passing through an intermediate point P, which will be important for spin manipulations, the gate voltages are pulsed to point M for spin readout. The readout mechanism is shown in Fig. S2b. Through Pauli-blockade, the spin state is converted to either a or charge state depending on whether the initial spin state was a singlet or a triplet, respectively. The mechanism relies on a charge hysteresis effect caused by the absence of direct access to a charge reservoir for the donor (Fig. S2c). Hence, the donor lead transitions are very slow because they have to go through a co-tunneling process to equilibrate . Placing point M between the S and T charge preserving transitions then allows a fast relaxation path to the charge ground state only if the initial state was . If the state was , the system is locked in a metastable charge configuration. The resulting CS signal is enhanced because the final charge configuration differs by one electron and lasts longer than the relaxation time of the state. A charge enhancement effect like this has been previously highlighted by Studenikin et al. [47, 61]. This readout mechanism allows us to use averaged measurements instead of single-shot. Since the measurement step is the longest in the pulse sequence, the current at point M in Fig. S2d-e is proportional to the triplet probability. All state measurements throughout this work are averaged over many (150 to 200) cycles. Details about the pulse sequence, loading rates, relaxation rates and probability calibration are given in the next section.
Appendix S5 Pulse sequence, loading and relaxation rates
The AC component of pulses in the experiment is applied using an Agilent 33500B arbitrary waveform generator using two synchronized channels for the AG and CP gates. The waveform is composed of DC and AC components and applied to the gates through a room temperature bias tee. The waveforms are applied such that all target points are fixed in the charge stability diagram, except the ones explicitly varied for a particular measurement (e.g. manipulation time or position of point M). The loading rate is approximately , and the loading rate approximately . The relaxation time is approximately , determined by preparing mostly and measuring the triplet probability decay versus time. The metastable state lifetime is roughly 2 to . We define zero detuning (the energy difference between the QD and D) at the QD-D charge transition, and positive detunings along in the direction with a lever arm. Then, we plunge the system to at point P (see Fig. 2b). Next, we rapidly pulse the system to (point A, Fig. 2) or a variable detuning (Fig. 3) with a ramp time. After waiting for a given manipulation time, the system is pulsed back to point P in .
|Point||Ramp time (s)||Wait time (s)|
Appendix S6 Probability calibration
To calibrate the triplet probability, the following procedure is used. First, the CP gate voltage of the measurement point M and the loading point L are swept to tune the readout and initialization, respectively, using the same waveform as for state manipulation except for point A (such that no manipulations are done). The resulting CS current is mapped in Fig. S3a. Given a certain load level, the CS current is then plotted versus CP measurement level, Fig. S3b. The current has a downward linear trend because of the CS Coulomb peak flank and a step that is similar in origin to a normal charge sensing signal. To the left of the measurement window the current always corresponds to a singlet signal, and to the right it always corresponds to a triplet signal. By extrapolating what this current would be assuming a linear background, one can determine what the pure singlet and triplet signals should be in the measurement window. The actual triplet probability is determined using a linear transformation that maps to triplet probability. When manipulations are performed, the duty cycle of the waveform is changed by at most , so the calibration is largely unaffected. Any systematic error introduced by this method (e.g. broadening of transitions due to temperature) would tend to underestimate the visibility of oscillations.
Appendix S7 Singlet-triplet dynamics model
We model the ST system with a Hamiltonian in the basis given by
\mat\eps(t) & 0 & -t_S & 0
0 & 2J_(4,0)+\eps(t) & 0 & -t_T
-t_S & 0 & -\eps(t) & -A/2
0 & -t_T & -A/2 & 2J_(3,1)-\eps(t) , where and are the exchange between singlet and triplet states in the and charge sectors, respectively, () is the QD-D tunnel coupling for the singlet (triplet) states, is the effective magnetic field gradient due to the contact hyperfine interaction in the configuration, and is the detuning. We fix and . For a given control schedule , we numerically integrate to solve for the time evolution of the density matrix generated by the von Neumann equation \ma \derivρ(t)t = -iℏ \commH(t)ρ . To model the effect of finite control bandwidth, the pulse sequence we consider in our numerical simulations is given by the ideal pulse sequence after having been filtered through a (low pass) RC filter, \ma ~\eps(t) = ∫_-∞^∞ \Dτh_RC(τ) \eps(t-τ) , where is the impulse response function and is the Heaviside step function. Considering various filtered control schedules , we find that a time constant is consistent with the experiment.
Appendix S8 Factors limiting the visibility
In the main text Fig. 2c, the visibility of the coherent rotations is low. This discussion identifies the different contributions to the visibility. It should be noted that the rotations are approximately 100 times faster than the coherence time. Therefore, the fidelity of the rotation itself should be quite high. Factors contributing to the reduced visibility are state preparation and measurement (SPAM) errors, additional incoherent or leakage processes during the fast ramp in/out of the region, and the control protocol itself. It should be noted that the control protocol used is not expected to produce full visibility according to our simulations. This is in part due to the limited bandwidth of the pulse in this setup (i.e. part of the wavefunction remains in the ground state because of partial adiabatic transfer of the spin state in the strong gradient field). Through various measurements we estimate that preparation errors alone are responsible for the triplet probability background in the region and limit the visibility to . Singlet preparation was limited by the slow QD-lead tunnel rate which required long loading steps that were competing with the bias tee time constant. The readout process could also yield additional errors at the and levels for singlets and triplets respectively due to triplet relaxation and various technical compromises. The dynamics model in the main text takes preparation errors into account and predicts a visibility of approximately , which is the simulated data shown in the main text. Adding measurement errors further reduces the expected visibility to . In the main text data, the visibility is around . This additional loss of visibility is dominated by an error process that occurs when the zero-detuning line is crossed. The exact mechanism is unknown. We speculate that it could be due to incoherent charge excitation/relaxation near the zero detuning point itself. If this is the case, a faster pulsing rise time and a bigger tunnel coupling would be expected to reduce errors.
Appendix S9 Coherence time analysis
To extract the visibility of the ST oscillations of Fig. 4a, the following method is employed. The source data is shown in Fig. S4. Because of CS drift over the long periods of time required to acquire these longer time traces (2 hours each in this case), the CS current (proportional to triplet probability) has a general downward trend and some residual fluctuations. To remove these fluctuations and smooth the data, the time trace is divided into time bins of approximately . The oscillations in each time bin are fitted with a sine function of fixed frequency. The amplitude for each time bin is then reported as visibility in Fig. 4a. The visibility decay is then fitted using a Gaussian decay, as detailed in the “Detuning noise model” section. We have verified that this time binning method agrees well with other methods such as maximum likelihood analysis. We now look at the apparent modulations of the oscillations in Fig. 2c of the main text and Fig. S4. These are believed to arise from the averaging of a limited ensemble of traces with slightly different frequencies. This is expected because of the slow charge noise and light drift, and leads to beating-like features. We also calculate the Fourier transform of the data to verify the spectral content of the signal and find a single large peak at the expected frequency.
Appendix S10 Detuning noise model
Since our device is fabricated with enriched \ce^28Si, the fluctuations in the “magnetic” control axis are expected to be small. Other work in ST qubits has shown that a dominant mechanism limiting the coherence is noise in exchange induced by quasi-static noise on the detuning [46, 62], i.e. “charge” noise. Given a quasi-static noise on the detuning having zero mean and standard deviation , an ensemble average leads to a Gaussian decay of the coherence of the form
&= ∫_-∞^∞ \DηP(η) cos\offractΔ(\eps+η)ℏ
&= \Exp-\offractT_2^*^2 cos\offractΔ(\eps)ℏ , where , is the energy gap , and \ma T_2^* = 2ℏσ\eps\abs∂Δ/∂\eps . Since the values we report for pertain to an ensemble average of measurements over a timescale of hours, our estimated detuning noise strength includes the effects of a secular drift component as well. While sufficiently large variations of the detuning can lead to Stark shifting of the contact hyperfine strength , this Stark shifting effect should be small compared to the that we observe.