A silicon quantum-dot-coupled nuclear spin qubit††thanks: These authors contributed equally††thanks: These authors contributed equally
Nuclear spins in the solid state have long been envisaged as a platform for quantum computingKane1998silicon-based (); Ladd2002All-Silicon (); Skinner2003Hydrogenic (), due to their long coherence timesMaurer2012Room-Temperature (); Saeedi2013Room-Temperature (); Muhonen2014Storing () and excellent controllabilityVandersypen2005NMR (). Measurements can be performed via localised electrons, for example those in single atom dopantsPla2012single-atom (); Pla2013High-fidelity (); Pla2014Coherent () or crystal defectsChildress2006Coherent (); Neumann2010Single-Shot (); Abobeih2018One-second (). However, establishing long-range interactions between multiple dopants or defects is challengingBernien2013Heralded (); Tosi2017Silicon (); Humphreys2018Deterministic (). Conversely, in lithographically-defined quantum dots, tuneable interdot electron tunnelling allows direct coupling of electron spin-based qubits in neighbouring dotsNowack2011Single-Shot (); Veldhorst2014addressable (); Veldhorst2015two-qubit (); Watson2018programmable (); Zajac2018Resonantly (); Huang2018Fidelity (). Moreover, compatibility with semiconductor fabrication techniquesMaurand2016CMOS () provides a compelling route to scaling to large numbers of qubits. Unfortunately, hyperfine interactions are typically too weak to address single nuclei. Here we show that for electrons in silicon metal–oxide–semiconductor quantum dots the hyperfine interaction is sufficient to initialise, read-out and control single silicon-29 nuclear spins, yielding a combination of the long coherence times of nuclear spins with the flexibility and scalability of quantum dot systems. We demonstrate high-fidelity projective readout and control of the nuclear spin qubit, as well as entanglement between the nuclear and electron spins. Crucially, we find that both the nuclear spin and electron spin retain their coherence while moving the electron between quantum dots, paving the way to long range nuclear-nuclear entanglement via electron shuttlingSkinner2003Hydrogenic (). Our results establish nuclear spins in quantum dots as a powerful new resource for quantum processing.
Electrons bound to single dopant atoms or localised crystal defects strongly interact with the host donor nuclear spinsPla2013High-fidelity (); Neumann2010Single-Shot (), as well as nearby lattice nuclear spinsPla2014Coherent (); Childress2006Coherent (), due to their highly confined wavefunctions. In quantum dots, on the other hand, the electron wavefunction typically overlaps with many nuclear spins, leading to undesired effects such as loss of coherence and spin relaxationJohnson2005Tripletsinglet (); Chekhovich2013Nuclear (). In silicon metal–oxide–semiconductor quantum dots, however, the strong confinement of the electrons against the interface, together with the possibility of small gate dimensions, result in a relatively small electron wavefunctionYang2013Spin-valley (); Ruskov2018Electron (), see Fig. 1d. This leads to strong hyperfine interactions with only a few nuclei when using isotopically-enriched base material. Indeed, simulating the distribution of expected hyperfine couplingsAssali2011Hyperfine () in such a quantum dot with a 10 nm diameter and 800 ppm nuclei, we may expect two to three nuclei per quantum dot that have a hyperfine coupling larger than 100 kHz, and a maximally possible hyperfine coupling of 700 kHz.
In this work we experimentally investigate the effect of individual nuclear spins on the operation of a double quantum dot device (Fig. 1), that was previously characterised in Ref. Huang2018Fidelity (). The quantum dots QD1, QD2 can be completely emptied, and single electrons can be loaded from the nearby electron reservoir. Using an external magnetic field T to split the electron spin eigenstates by 39 GHz allows spin readout via the spin-selective unloadingElzerman2004Single-shot () of an electron from QD2. Furthermore, a single electron can be transferred between QD1 and QD2, while maintaining its spin polarisationHuang2018Fidelity (); Baart2016Single-spin (). Electron spin resonance (ESR) pulses applied to an on-chip microwave antenna allow coherent manipulations of the electron spin, with intrinsic spin transition linewidths around 50 kHzVeldhorst2014addressable (). When monitoring the spin resonance frequency () for an electron loaded in QD1 (with QD2 empty) over extended periods of time, discrete jumps can be observed (Fig. 1b, left). Indeed, the histogram in Fig. 1b (right) suggests the presence of two distinct two-level systems, resulting in shifts in of approximately 120 kHz and 500 kHz.
In order to determine whether the two-level fluctuations can be attributed to nuclear spins, we focus our attention on the 500 kHz shift. We first apply a radiofrequency (RF) tone with quantum dot QD1 unloaded, and then check whether the ESR frequency has shifted, by repeatedly probing the inversion probability around kHz or kHz, with the average ESR frequency. In Fig. 2e we show the probability that the electron spin resonance frequency has switched between and after applying an RF pulse with varying frequency and duration, and find coherent oscillations centred around MHz. This frequency corresponds to a gyromagnetic ratio of MHz/T (where the uncertainty comes from the accuracy with which we can determine the applied field ), consistent with the bulk gyromagnetic ratioPla2014Coherent () of MHz/T. We therefore conclude that the electron in quantum dot QD1 couples to a nucleus with hyperfine coupling kHz. We describe the joint spin system by a Hamiltonian of the form
where , are the electron and spin operators and GHz/T, the electron gyromagnetic ratio. Here, the interaction is dominated by the contact hyperfine term, with dipole-dipole terms expected to be several orders of magnitude smallerAssali2011Hyperfine (). The Hamiltonian results in energy eigenstates shown in Fig. 2b, with both electron and nuclear spin transitions splitting by A when the electron is loaded onto QD1. Indeed, repeating the same experiment as above, but applying the nuclear magnetic resonance (NMR) pulses after loading QD1 with a spin-down electron, we find the NMR frequency has shifted to , see Fig. 2f (right peak), yielding an accurate measurement of kHz at this control point. Finally, we repeat the experiment once more, where we load QD1 with an electron with spin up (by applying adiabatic ESR inversion, see Methods), and confirm , see Fig. 2f (left peak). In Extended Data Fig. 1 we present results for another nuclear spin coupled to the electron spin in quantum dot QD2.
Having confirmed the ability to controllably address individual nuclear spins, we proceed to characterise a qubit encoded in this new resource. As observed from the interval between jumps in Fig. 1b, the nuclear spin lifetime (while repeatedly probing the electron resonance frequency, a process which likely determines that flip rateZhao2018Coherent ()) extends to tens of minutes. By fitting an exponential decay to the intervals between the 500 kHz and 120 kHz jumps for the data in Fig. 1b, we find hours and minutes. This means we can perform multiple quantum-non-demolition measurements of the nuclear spin state to boost the nuclear spin state readout fidelityPla2013High-fidelity (); Neumann2010Single-Shot (). A simple simulation (Methods) for repeated readouts, taking into account the 8 ms measurement cycle and the electron spin readout visibility of 76% results in an optimal number of readouts , and an obtainable nuclear spin readout infidelity of . In this work we limit to 20, resulting in a measured nuclear spin readout fidelity of 99.8% for the dataset in Fig. 3, see Methods for details. We determine the nuclear spin coherence times by performing nuclear Ramsey and Hahn-echo sequences, with the electron unloaded (Fig. 2g,i) and loaded (see Fig. 2h,j). We find nuclear coherence times between two and three orders of magnitude longer than those measured for the electron in this deviceHuang2018Fidelity (), but shorter than previously measured for nuclear spins coupled to donors in enriched siliconMuhonen2014Storing (), possibly due to the closer proximity of the device surface. A full overview of the measured coherence times is presented in Extended Data Table I.
In Fig. 3, we use coherent control to prepare entangled states of the joint electron-nuclear two-qubit system. We perform all operations with the electron loaded and construct the required unconditional rotations from two consecutive conditional rotationsDehollain2016Bells (), see Fig. 3a,b. After careful calibration of the AC Stark shifts induced by the off-resonant conditional ESR pulses (Fig. 3c,d), we characterise the prepared Bell state fidelity by measuring the two-qubit expectation values , and of the joint x,y,z-Pauli operator on the nucleus and electron, respectively. We correct the two-qubit readout probabilities for final electron readout errors only, which we calibrate by interleaving the entanglement measurement with readout fidelity characterisations (Methods). Note that because the nuclear spin is initialised by measurement here, we obtain an identical dataset corresponding to initial nuclear state , which we show in Extended Data Fig. 3. We find an average Bell state preparation fidelity of .
Our entanglement protocol is affected by several errors. By simulating our protocol, we estimate that the dominant source of error is the electron (), followed by the uncontrolled 120 kHz coupled nucleus, observed in Fig. 1b (5%). Depending on the state of this nuclear spin, detuned ESR pulses cause an unknown phase shift. Other contributions include pulse duration calibration errors () and the reduced NMR control fidelity with QD1 loaded (3%).
A unique feature of our quantum-dot-coupled nuclear spin qubit is the large ratio of interdot tunnel coupling to hyperfine coupling , so that , with GHz in this deviceHuang2018Fidelity (). We should therefore be able to cleanly and adiabatically load and unload the electron into a neighbouring dot, without disturbing the nuclear spin state. Although this has been envisioned for donors coupled to quantum dots in siliconKane1998silicon-based (); Wolfowicz201629 (); Harvey-Collard2017Coherent (), the more tightly confining donor potential may result in insufficient control over the loading process. We can experimentally verify that the adiabatic transfer of an electron preserves the nuclear spin coherence with the pulse sequence shown in Fig. 4a. The sequence comprises a nuclear Ramsey-based experiment, where the electron is loaded onto QD1 from QD2 for a varying amount of time during free precession, while keeping the total evolution time constant. As expected, the loading of the electron causes a phase accumulation set by the hyperfine strength, see Fig. 4b, but preserves the nuclear spin coherence. To quantify the phase error induced by a load-unload cycle, we perform repeated electron shuttling, see Fig. 4c-d.
The nuclear phase preservation raises the prospect to entangle nuclei in separate quantum dots, mediated by the electron shuttlingSkinner2003Hydrogenic (); Fujita2017Coherent (). For that to work, the electron spin state itself must also remain coherent during transferFujita2017Coherent (). By performing an electron Ramsey experiment, where the first pulse is driven with the electron in QD1 and the second pulse is driven with the electron in QD2, see Fig. 4e,f, we demonstrate that this is indeed the case, presently with modest transfer fidelities. A detailed investigation of the coherent electron transfer and the potential to generate nuclear entanglement using electron shuttling will be the topic of future work.
The readout and control of nuclear spins coupled to quantum dots shown here presents us with a variety of future research possibilities. Firstly, the nuclear spin qubits could form the basis for a large-scale quantum processor, where initialisation, readout and multi-qubit interactions are mediated by electron spinsSkinner2003Hydrogenic (). Secondly, the nuclear spin qubits could be used as a quantum memoryFreer2017single-atom () in an electron-spin-based quantum processor. Implementations of quantum error correcting codes may benefit from integrated, long term quantum state storage. In particular, lossy or slow long-range interactions between quantum dots, for example mediated by photonic qubitsMi2018coherent (); Samkharadze2018Strong (), could be admissible if supplemented by local nuclear spin resources for memory or purificationWolfowicz201629 (); Bennett1996Purification (); Nickerson2013Topological (). Finally, the nuclear spin qubits can be used as a characterisation tool for electron spin-based qubits. For example, in the present experiment, the confirmed existence of a nucleus with 500 kHz hyperfine coupling bounds the electron wavefunction diameter to under 12 nm, a conclusion difficult to draw with purely electrostatic calculations or electronic measurements. Further characterisations of electron spin dynamics may be envisioned by mapping the electron spin state to the nuclear spin, and employing the nucleus as a high-fidelity readout toolDehollain2016Bells (). Limitations include the extended control times for the nuclear spin, as well as the effect of sample heating from long NMR pulses. Both limitations could be addressed by redesigning the RF delivery, for example using a global RF cavity. Presently, the are randomly distributed, resulting in a range of hyperfine couplings for each quantum dot. Although the probability of obtaining at least one addressable per quantum dot is very large, implantation of nuclei in (possibly further enrichedMazzocchi201999992 ()) silicon host material would allow designing an optimal interaction. This could be done via ion implantationDonkelaar2015Single (), requiring a relatively modest precision on the order of the size of the quantum dots, compared to the precision needed for direct donor-donor coupling. In summary, we have demonstrated coherent control, entanglement and high-fidelity readout of a single nuclear spin qubit, embedded in a lithographically-defined silicon quantum dot. We find that inter-dot electron tunnelling preserves the electron and nuclear spin coherence. The combination of controllable nuclear spin qubits with the long-range interactions afforded by electrons in silicon quantum dots provides a powerful new resource for quantum processing.
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We thank T.D. Ladd and C. Escott for valuable discussions and feedback on the manuscript, and J.J. Pla for technical assistance. We acknowledge support from the US Army Research Office (W911NF-17-1-0198), the Australian Research Council (CE170100012), and the NSW Node of the Australian National Fabrication Facility. 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 authorised to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. B.H. acknowledges support from the Netherlands Organisation for Scientific Research (NWO) through a Rubicon Grant. K.M.I. acknowledges support from a Grant-in-Aid for Scientific Research by MEXT.
.2 Author contributions
B.H. and W.H. performed the experiments. K.W.C. and F.E.H. fabricated the devices. K.M.I. prepared and supplied the wafer. B.H., W.H., C.H.Y., J.Y., T.T. and A.L. designed the experiments. B.H., W.H. and J.Y. analysed the data. B.H. wrote the manuscript with input from all co-authors. A.M., A.L., and A.S.D. supervised the project.
.3.1 Experimental methods
Details of sample fabrication and experimental setup can be found in Huang et al.Huang2018Fidelity (). NMR pulses were generated by a secondary Agilent E8267D microwave vector signal generator, combined with the ESR pulses using a resistive combiner. Conditional ESR pulses for nuclear readout are performed via adiabatic inversions with a frequency span of kHz to kHz and kHz to kHz for reading nuclear spin , respectively. The adiabatic pulse has a duration of and a power corresponding to a 100 kHz Rabi frequency. Electron spin transfers between QD1 and QD2 are performed with a linear ramp, except where indicated otherwise.
.3.2 Nuclear spin readout fidelity
We model the repetitive nuclear readout as a stochastic process, where, as a function of the number of shots , the fidelity is limited by the nuclear decay on the one hand,
with ms the measurement time per shot, hour, and the factor 2 comes from the fact that for each shot we read out the electron twice; once for an inversion around and once for . On the other hand the fidelity is limited by the cumulative binomial distribution representing the majority voting of M single shots:
A first order estimate for the nuclear readout is then obtained by
resulting in a minimum infidelity for , for , where we have taken the average electron spin readout fidelity recorded for the dataset in Fig. 3e.
.3.3 Nuclear-electron entanglement experiment: experimental details
For the datasets presented in Fig. 3e and Extended Data Fig. 3c, we interleave the following measurement sequences:
Bell state preparation + ZZ projection
Bell state preparation + XX projection
Bell state preparation + YY projection
Electron spin- readout characterisation for ZZ projection
Electron spin- readout characterisation for ZZ projection
Electron spin- readout characterisation for XX and YY projections
Electron spin- readout characterisation for XX and YY projection
Prior to running each sequence, we initialise the electron spin state to , using a spin relaxation hotspot at the (0,1)-(1,0) charge transition (details in et al.Huang2018Fidelity ().). After running a sequence once, we read the state of both electron and nuclear spin (corresponding to a total of electron spin readouts). We then perform an ESR frequency check and, if necessary, calibration. The frequency check proceeds by applying a weak, resonant ESR pulse (60 kHz Rabi frequency), and fails if the spin inversion probability drops below 0.3. If the check fails for both and , the ESR frequency is recalibrated using a series of Ramsey sequences to estimate the detuning. Details of this ESR frequency calibration are described in the Supplementary Information of Huang et al.Huang2018Fidelity (). After recording 10 datapoints of sequence 1 in this manner, we switch to sequence 2, and so forth. After sequence 7 we loop back to sequence 1, until the end of the measurement. The nuclear spin initialisation is given by the readout result in the previous sequence. Total measurement time for the presented dataset was 9.5 hours, resulting in 4320 Bell state preparations.
The aim of sequence 4-7 is to record the actual average electron spin readout fidelity while recording the dataset. To estimate the spin-down readout fidelity (sequence 4,6), we apply no ESR pulses and measure the spin-down readout probability. To estimate the spin-up readout fidelity we apply an adiabatic inversion of the electron spin, consisting of a long 2.8 MHz wide frequency sweep centred around , with a power corresponding to a 100 kHz Rabi frequency, and measure the spin-up readout probability. Sequence 4,5 each have the same NMR pulses applied as sequence 1, but applied far detuned, in order to mimic the effect of the ZZ-projection NMR pulses on the electron spin readout fidelity, while unchanging the nuclear spin state itself. Similarly, sequence 6,7 have the same NMR pulses as sequence 2, but far detuned, to mimic the effect of XX, YY-projection pulses.
Finally, data from sequence 4-7 is also used to obtain an estimate for the nuclear spin readout fidelity: since all NMR pulses are applied off-resonant, the nuclear spin should remain unchanged. If the nuclear spin is read out differently after running sequence 4-7, this indicates a readout error has occurred. We find 5 readout errors in 3200 nuclear spin readouts, identified as such by a single outcome being different in a sequence of 10. The readout fidelity is estimated as the fraction of readout errors.
Bar plots shown in Fig. 3e and Extended Data Fig. 3c are corrected using their respective electron spin readout fidelity characterisation, using direct inversion. We estimate the Bell state fidelity using , where , , for nuclear spin- initialised data (Fig. 3e), and , for nuclear spin- initialised data (Extended Data Fig. 3c).
.3.4 Nuclear-electron entanglement: error analysis
Using the two-spin Hamiltonian, eq. (1), with two control fields and , and taking the secular and rotating wave approximation, we perform a time evolution simulation to estimate the effects of various noise sources on the nuclear-electron Bell state fidelity. We simulate the exact control sequences used in the experiment. The simulation calculates the operator at any specific time resulting in a final operator . We incorporate quasi-static noise along , and directions following a Gaussian distribution with standard deviation of , , respectively, and repeat the simulation for 1000 times to obtain the average final measurement probabilities. We use values ms, ms, . The value for has a large uncertainty, ranging from 8 to depending on the exact ESR frequency feedback settings and intervalHuang2018Fidelity (). To simulate the effect of the uncontrolled 120 kHz coupled spin, we estimate the probability that the nuclear spin flips within the time between ESR frequency checks, resulting in an unnoticed frequency shift. Using min, and an average time between ESR frequency checks of 40 seconds, we find a probability of 7% of running the entanglement sequence with 120 kHz detuned ESR pulses. Finally, to simulate the effect of pulse calibration errors, we estimate our pulse-length calibration is accurate within 5%. Error percentages quoted in the Main text are the reduction in final Bell state fidelity resulting from incorporating the corresponding error mechanism only, with all other error mechanisms turned off in the simulation.
.3.5 Coherent loading dephasing analysis
We model the effect of transferring the electron between QD1 and QD2 on the nuclear spin state as a dephasing channel
This model yields an exponentially decaying off-diagonal matrix element magnitude as a function of channel transfers , , which is the measured coherence defined in the caption of Fig. 4d,e.
.4 Extended Data
|QD1||(0,1) - unloaded||450 kHz||ms||3.2 hrs||Fig. 2g||ms||Fig. 2i|
|QD1||(1,0) - loaded||450 kHz||ms||1.1 hrs||Fig. 2h||ms||Fig. 2j|
|QD1||(0,0) - unloaded||450 kHz||ms||7 hrs||Ext. 1a||ms||Ext. 1b|
|QD2||(0,1) - loaded||90 kHz||ms||6.3 hrs||Ext. 2b||ms||Ext. 2c|
|QD2||(0,0) - unloaded||90 kHz||Not measured||ms||Ext. 2d|