Two-Electron Spin Correlations in Precision Placed Donors in Silicon
M. A. Broome S. K. Gorman M. G. House S. J. Hile J. G. Keizer D. Keith C. D. Hill T. F. Watson W. J. Baker L. C. L. Hollenberg M. Y. Simmons





These authors contributed equally to this work
Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia
 
These authors contributed equally to this work
Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia
 
Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia
 
Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia
 
Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia
 
Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia
 
Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, The University of Melbourne, Parkville, Victoria 3010, Australia
 
Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia

Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia
 
Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, The University of Melbourne, Parkville, Victoria 3010, Australia

michelle.simmons@unsw.edu.au
Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia





§ ABSTRACT

Substitutional donor atoms in silicon are promising qubits for quantum computation with extremely long relaxation and dephasing times demonstrated. One of the critical challenges of scaling these systems is determining inter-donor distances to achieve controllable wavefunction overlap while at the same time performing high fidelity spin readout on each qubit. Here we achieve such a device by means of scanning tunnelling microscopy lithography. We measure anti-correlated spin states between two donor-based spin qubits in silicon separated by 16±1 nm. By utilizing an asymmetric system with two phosphorus donors at one qubit site and one on the other (2P-1P), we demonstrate that the exchange interaction can be turned on and off via electrical control of two in-plane phosphorus doped detuning gates. We determine the tunnel coupling between the 2P-1P system to be 200 MHz and provide a roadmap for the observation of two-electron coherent exchange oscillations.



Two-Electron Spin Correlations in Precision Placed Donors in Silicon
    M. Y. Simmons
    December 30, 2023
====================================================================





§ INTRODUCTION




Controlling the interaction strength between two quantum particles lies at the heart of quantum information processing. One must have access to classical control fields that, whilst tuning the environment of quantum particles, are sufficiently decoupled from them as to not disturb their quantum states <cit.>. Physical systems ranging from trapped ions <cit.>, single photons <cit.>, superconducting circuits <cit.> and semiconductor quantum dots <cit.> have demonstrated this exquisite level of control. In 1998 Loss and Divincenzo <cit.> proposed the use of a controllable exchange interaction in semiconductor quantum dots to perform a two-qubit logic gate. In the same year Kane <cit.> proposed how this could be achieved in donor based devices. Here, the wavefunction overlap between two electrons on neighbouring donor atoms placed ∼20 nm apart is controlled using an exchange gate between them. Harnessing this exchange interaction to perform a universal two-qubit quantum logic gate is the next step for donor based architectures. 

Three approaches exist for donor qubits: a controlled-phase (CZ) gate <cit.>; the controlled-rotation (C-ROT) gate <cit.> and a direct two-electron SWAP operation <cit.>. Whilst the first two protocols require the use of high frequency microwave fields for electron spin resonance <cit.>, a direct two-electron SWAP necessitates the ability to turn on and off the exchange interaction between the electrons over orders of magnitude for high fidelity two-qubit operations. Notably, whilst the extent of a single donor wavefunction is well understood <cit.> modelling the exchange coupling between two donor electrons is more  complex <cit.> due to multi-valley interference effects <cit.>. To this end, a critical challenge for donor based architectures is to know the distance required between the donors in order to turn the exchange interaction on and off with external gates <cit.>. 

To date, two main methods for donor placement in silicon exist; ion implantation <cit.> and atomic manipulation via scanning-tunnelling-microscopy (STM) hydrogen lithography <cit.>. Despite much success in accessing randomly placed donor spins, ion implantation has yet to demonstrate donor placement precision below ∼6 nm, whilst STM lithography has demonstrated donor placement down at the atomic scale <cit.>. 

In this paper we use STM lithography which allows both the precision placement of donor atoms for direct and independent spin-measurement of electrons near a readout structure and, most importantly, the control of the exchange interaction between them. We measure the anti-correlated spin states that arise due to the formation of two electron singlet-triplet states as a function of their wavefunction overlap, which is controlled by in-plane detuning gates. By observing the onset of these anti-correlated spin states as a function of detuning pulse voltage and time, we estimate the magnitude of tunnel coupling between the two donor qubits, and provide a roadmap toward coherent exchange gates for future devices. 



§ RESULTS



 §.§ Independent spin readout of a 2P-1P system


In the original Kane proposal an exchange gate between the donors was suggested to directly tune the exchange coupling between the qubits <cit.>. Recent tight binding simulations have shown that it is difficult to tune the exchange energy in a 1P-1P donor configuration using such a gate <cit.>. Instead, it has been proposed that the exchange energy could be tuned over five orders of magnitude <cit.> by confining electrons in an asymmetric 2P-1P configuration and by utilising `tilt' control using two opposing detuning gates rather than a central J-gate, see Fig. <ref>a. Motivated by these predictions with estimates for the required inter-donor separation, in this paper we demonstrate the ability to control exchange coupling in donor based qubits at the (1,1)-(2,0) charge region using a 2P-1P donor system.

The device, shown in Fig. <ref>a, was patterned using STM hydrogen lithography. The qubits-L, and -R (left and right) are composed of 2 donors and 1 donor respectively, determined by examining the size of the lithographic patches (insets in Fig. <ref>a) as well as their measured charging energies <cit.> (see Supplementary Figure 1 and Supplementary Figure 6). Three gates, {GL, GM,  GR} control the electrostatic environment of the qubits which are  tunnel coupled to a larger readout structure made up of approximately 1000 P atoms which serves as a single-electron-transistor (SET) charge sensor. The SET quantum dot is operated with a source-drain bias of 2.5 mV, has a charging energy of ∼6 meV and is controlled predominantly via gate GS. All data in this paper was taken in a dilution refrigerator with a base temperature of ∼100 mK (electron temperature ∼200 mK). 




Figure <ref>b shows the charge stability map of the 2P-1P device. Current peaks running diagonally correspond to charge transitions of the SET island. Two sets of breaks in the SET current peaks are observed with different slopes and correspond to electron transitions from either L or R to the SET island. An avoided crossing (triple-point) between these two transitions (white dotted line) indicates the region where electrons can tunnel between L and R, in this case at the (1,1)-(2,0) charge transition. Only one more charge transition corresponding to L is observed at lower gate voltages leading to the assignment of the charge regions. 



The direct measurement of anti-correlated electrons hinges upon the ability to independently measure their spin states <cit.>. To measure the spin of R we employ an energy-selective tunnelling technique <cit.> where the electrochemical potential of the single-electron transition from the 1→0 charge state is split by the Zeeman energy in a static magnetic field B_z, see blue arrows in Fig. <ref>c. Whether the electron is able to tunnel to the SET reservoir therefore depends on its spin state, i.e. the readout is a spin-dependent unloading mechanism from the qubit to SET. This readout technique is employed for the electron at R because the (1,1) region for this qubit borders the 1→0 charge states. 

For L we use a variant of this method, first reported Watson et al. <cit.>. The charge transition for this qubit borders the 1→2 charge states, but because the chemical potential from the 1-electron spin-up and -down states to the 2-electron singlet state are also split by the Zeeman energy a similar readout method is allowed (red arrows in Fig. <ref>c <cit.>). In this case, we utilise a spin-dependent loading mechanism from the SET to L. The combination of these two distinct readout techniques avoids the need to pulse over large voltages in order to reach the (1,1)-(2,0) charge transition. Both readout methods are equivalent and give rise to a current `blip' through the SET which is used to discriminate between spin-up and -down electrons. The average readout fidelity of spin-up and -down are estimated to be 96.2±1.1% and 97.6±2.1% for qubit-L and -R respectively (See Supplementary Figures 2, 3 and Table 1 for full analysis).

Importantly, the readout of each electron must be completely independent of the spin-state of the other. That is to say, the exchange energy at the detuning-position where readout is performed must be vanishingly small, such that no spin flips-flops occur during the readout window. This is demonstrated in Fig. <ref>e and f. For these measurements we prepare one of two states, 

    ρ_↓↑ = |↓↑⟩⟨↓↑|+|↓↓⟩⟨↓↓|/2  ,  ρ_↑↓ = |↑↓⟩⟨↑↓|+|↓↓⟩⟨↓↓|/2.

where |i,j⟩ indicates the spin state i and j on qubit-L and -R respectively. Loading spin-down for one qubit is performed deterministically as a result of the spin readout protocol. Spin-up cannot be loaded deterministically, instead a random mixture of spin up and down is loaded by plunging the qubit far below the SET fermi-level. After initialisation we pulse inside the (1,1) charge region midway between the two readout positions (green square in Fig. <ref>b) and wait for up to 10 s for the randomly loaded electron spin to decay to spin down. Sequential spin-readout of L and then R is performed, in that order, to minimise the effect of the shorter T_1 of qubit-L. The spin-up fractions show relaxation of the qubit initially loaded with random spin, with T_1 times measured to be 2.9±0.5 s and 9.3±2.4 s for electrons on L and R respectively at B_z=2.5T. Importantly, the electron initially loaded as spin-down shows no significant spin-up fraction during this time, demonstrating that at these readout positions there is no significant spin-spin interaction over ∼10 s. 



 §.§ Controllable exchange of precision placed donors


The realisation of a two-qubit logic gate hinges on the ability to controllably turn on and off an interaction between quantum particles. We show this here by pulsing toward the (1,1)-(2,0) charge transition where an exchange interaction between the two electrons arises as a consequence of the Pauli-exclusion principle <cit.>. The Hamiltonian is given by H_ex=JS_L·S_R, where S_L and S_R are the left and right electron spin vectors and J is the strength of the exchange interaction <cit.>. The magnitude of J is given by the energy difference between the symmetric and anti-symmetric two-electron states |T^0⟩=(|↑↓⟩+|↓↑⟩)/√(2) and |S⟩=(|↑↓⟩-|↓↑⟩)/√(2) respectively. Similar to gate defined quantum dots, it has been shown that the exchange between donors can also be parameterised in terms of the tunnel coupling and detuning between the (1,1) and (2,0) charge states <cit.>,

    J(ϵ)=ϵ/2 + √(t_c^2 + (ϵ/2)^2),
 
where ϵ is the detuning and t_c is the tunnel coupling (such that J(0)=t_c). The detuning axis ϵ is applied along V_GL=-0.9V_GR (along the SET Coulomb blockade) and is shown by the white arrow in Fig. <ref>a. It effects a tilting from the (1,1) toward the (2,0) charge state, shown schematically in Fig. 1d. The detuning energy, ϵ, is related to the applied gate voltage V_GL via the lever arm α_ϵ=0.071 eV/V, such that ϵ=α_ϵV_GL.




We start by initialising either state from Eq. <ref> by loading one qubit randomly and deterministically down on the other, and subsequently apply a 50 ms pulse along the axis ϵ to control the strength of the exchange interaction <cit.>, shown by the open black circles in Fig. <ref>a. This time is long enough to allow for a significant exchange interaction, but much shorter than any electron spin relaxation such that readout is not hindered. Upon pulsing back into the (1,1) charge region we perform independent spin readout of L and then R. In addition to the single spin outcomes for each qubit we also determine the joint probabilities P_ij for ij∈{↑↑,↑↓,↓↑,↑↑}, as shown in Fig. <ref>b-d. 



In the case where ρ_↑↓ is initialised, after pulsing to ϵ≪0 we observe P_↑↓∼0.5 and P_↓↑∼0, indicating no spin flip-flops have occurred during the 50 ms pulse duration. However, anti-correlated spins can be seen in Fig. <ref>(b,c) as we pulse closer to the (1,1)-(2,0) charge transition, at ϵ=0 where both P_↑↓ and P_↓↑→0.25. Furthermore, we see that both P_↑↑ and P_↓↓ remain constant at approximately 0 and 0.5 respectively as they represent populations of the triplet states |↑↑⟩ and |↓↓⟩ and are not subject to the exchange interaction. Statistical analysis (see Supplementary Figure 4) of these results indicates a correlation coefficient of ϕ=-0.243±0.028 with a p-value ≪0.01 for 0<ϵ<2.4 meV, demonstrating the presence of statistically significant spin anti-correlations in this region. 



 §.§ Estimate of inter-donor exchange coupling


To ascertain the value of the inter-dot tunnel coupling, t_c, we repeat the same pulsing scheme as above whilst modifying the detuning pulse duration from 0.1-2 ms and compare our results to a spin-level theoretical model, see Fig. <ref>. The quantum mechanical behaviour of a donor based two-qubit system is described by the following terms in the Hamiltonian:

    H_ze   =γ_eB·(S_L + S_R), 
    
    				H_zn   =γ_nB·(I_n_L1 + I_n_L2 + I_n_R), 
    
    				H_hf   =A_LS_L·(I_n_L1 + I_n_L2) + A_RS_R·I_n_R, 
    
    				H_ex   =JS_L·S_R,

where H_ze and H_zn are the electron and nuclear Zeeman energies, with γ_e=28.024 GHz/T and γ_n=17.235 MHz/T gyromagnetic ratios respectively <cit.>. The hyperfine term, H_hf is separated into two components as it has been predicted that the hyperfine constants will be different for varying donor cluster configurations <cit.>. Here for simplicity we assume the bulk-like value of A_L=A_R=A=117.53 MHz <cit.> and define the static field to be B=(0,0,|B_z|). We numerically calculate the time evolution of the density matrix via a 4th order Runge-Kutta method with the inclusion of relevant decoherence channels (see Supplementary Figure 5). 

For the theoretical data shown in the lower panel of Fig. <ref>a we prepare the initial state ρ_↑↓ from Eq. <ref> and simulate non-adiabatic pulses to detuning positions for varying pulse durations, τ_w. For this simulation we use a tunnel coupling, t_c=200 MHz, assume dynamic P nuclear spins as well as a single-spin dephasing time of T_2^*=55 ns due to the constantly fluctuating Overhauser field of the ^29Si nuclear spins <cit.>. The equivalent measured data set is shown in the upper panel of Fig. <ref>a with cuts at 0.3 and 1 ms shown in Fig. <ref>b and compared with the theoretical predictions for t_c=50, 200 and 500 MHz. From these results we can estimate the tunnel coupling within an order of magnitude accuracy to be t_c∼200 MHz for this device. Following Eq. <ref>, this result places an equivalent bound on the achievable exchange energy J<200 MHz inside the (1,1) charge region (see Fig. <ref>a).

In Ref. <cit.> the authors investigated multiple different 2P intradot configurations, and found that disorder at the lattice-site level had little effect on the final exchange energy. They showed that the exchange energy for a 2P-1P system with a 15nm separation was tunable over five orders of magnitude for electric field strengths -2<|E|<2 MV/m at the donor sites. The voltage applied to the in-plane gates in our device amount to a potential difference of ∼100 mV between V_GL and V_GR at the (1,1)-(2,0) inter-dot transition. From an electrostatic model of our device we estimate |E|=0.49±0.10 MV/m at the donor sites. Our estimate of t_c=0.2 GHz (which is equal to J at ϵ=0) is within an order of magnitude of the theoretical prediction for J in a 2P-1P system given this |E| <cit.>.



 §.§ Requirements for coherent control of exchange


In this final section we investigate the potential for achieving coherent exchange between two electrons confined to donors in natural silicon. Based on Eq. <ref> the plot in Fig. <ref>a shows the obtainable exchange energies, J, as a function of detuning and tunnel coupling, where the vertical  blue dashed line shows t_c=0.2 GHz for our device. Importantly, the boundary where the difference in magnetic field at the two qubit sites Δ B_z is equal to the exchange J, separates the two-electron product eigenstates and singlet-triplet eigenstates. For donor qubits Δ B_z is dominated by the phosphorus nuclear-spin hyperfine, A. The exact value of Δ B_z varies depending on the number of donors at each qubit site and their nuclear spin orientations: For a 2P-1P device with random nuclear spin configurations Δ B_z fluctuates between 3A/2 or A/2 with a 1:3 ratio.

It can be seen from Fig. <ref>a that for the device studied here there exists only a small range in detuning (approximately 10 μV in gate voltage) over which one could implement coherent exchange oscillations inside the (1,1) charge region (negative ϵ). When one takes into account any voltage noise on gates (which influences ϵ and ultimately J) this makes the operation of coherent oscillations challenging <cit.>. Indeed, in this particular device we measured gate RMS voltage noise of 50 μV from shot to shot, equivalent to detuning noise of δϵ=850 MHz, indicating that pulsing repeatedly to the same exchange energy would not be possible for ϵ<0. For the same reasons, charge noise also destroys coherence when adopting the approach to pulse ϵ>0. We carried out experiments with pulses down to 10 ns for both ϵ<0 and ϵ>0 but were unable to observe coherent exchange phenomenon. 

Figure <ref>b shows the predicted number of exchange oscillations (∼15) that would be observed in a device with t_c=2.5 GHz after pulsing to a detuning ϵ=-25 GHz (circle marker in Fig. <ref>a). Conversely, using the same model we estimate that a noise floor of <6V (∼100 MHz in detuning or ∼50 mK, much lower than the electron temperature) would be required to observe the signature of coherent oscillations in the present device. Note that in addition to the realistic detuning noise we have also included the effect of a constantly fluctuating ^29Si Overhauser field expected in natural silicon <cit.> as well as randomized P donor nuclear spins of the donor atoms themselves.

From these simulations we can extract the frequency of oscillations, ν as well as the dephasing time τ_d, allowing us to determine the figure of merit ντ_d as a function of tunnel coupling and detuning pulse position, see Fig. <ref>c. Interestingly, the product ντ_d only becomes significant beyond the boundary Δ B_z=3A/2 for values of t_c>2 GHz, providing a lower bound on the required tunnel coupling for coherent control. Figure <ref>d gives ντ_d as a function of tunnel coupling for a detuning pulse to the boundary Δ B_z=3A/2. These results indicate that at high tunnel coupling, the observation of exchange oscillations will ultimately be hindered by the presence of the fluctuating ^29Si Overhauser field. In the case where qubits exist in a spin vacuum, as in ^28Si, only charge noise is relevant and ντ_d can be seen to increase monotonically as a function of tunnel coupling (dashed line in Fig. <ref>d). 



§ DISCUSSION


In summary, we have demonstrated a controllable exchange interaction resulting in two-electron spin anti-correlations on precision placed 2P-1P donors qubits in Si using in-plane `detuning' gates. The results are consistent with the exchange interaction behaviour expected at the (1,1)-(2,0) charge transition and represent the first direct measurement of correlated electron spins in donor based devices. Whilst the small tunnel coupling (0.2 GHz) in the present device prohibited measurement of coherent oscillations, we show our results agree with recent studies <cit.> in which much smaller distances than previously predicted are required to achieve a sufficiently large exchange coupling for coherent control. Furthermore, while detuning noise presents a problem for devices with a small tunnel coupling, we theoretically predict that for larger tunnel couplings of t_c>2 GHz it can be overcome. Theoretical work on coupled donor systems suggest a separation of 13-14nm between a 1P-2P system will be required to achieve this magnitude of exchange coupling <cit.>. Importantly, there is no reason to believe that this small change in donor site separation will lead to a significant reduction in electrical controllability based on previous experimental works <cit.>. This benchmark for a larger interaction strength between neighbouring donor-based qubits provides the focus for future experiments. 

With the atomic precision placement of donors using STM lithography it will be possible to further optimise the inter-donor distance to control the coherent coupling between two donors qubits with order-of-magnitude accuracy <cit.>. Whilst extensive studies have been conducted for deterministic single P donor incorporation <cit.> similar studies will need to be developed to determine the optimal lithographic patch for deterministic 2P incorporation. Crucially, recent theory predicts that the 1P-2P configuration we present in this paper both increases the tunability of the tunnel coupling and at the same time suppresses the `exchange fluctuations' known for two single donors, and may therefore be less sensitive to the exact atomistic donor positions than two coupled single donors <cit.>. Furthermore, our ability to directly place the donor with <1 nm accuracy along with the reproducible demonstration of high fidelity single-shot spin-readout in multiple devices <cit.>, bode well for the future scalability of donor qubit quantum computers. 



§ METHODS





 §.§ Device fabrication



The device, shown in Fig. <ref> was fabricated using scanning-tunnelling-microscopy hydrogen lithography to selectively remove hydrogen from a passivated Si(100) 2×1 reconstructed surface. The lithographic mask is subsequently dosed with PH_3 and annealed (320^∘C) to incorporate P atoms into the silicon substrate <cit.> with ∼1/4 ML density ( 2× 10 ^14cm^-2) allowing for quasi-metallic conduction in all electrodes <cit.>. 




 §.§ Measurement setup



For all electrical measurements, the device was mounted on a high-frequency printed circuit board within a copper enclosure, thermally anchored to the cold finger of a dilution refrigerator with a base temperature of 50 mK. Voltage pulses were applied to gates GL and GR by an arbitrary waveform generator (Agilent 81180A), connected via a bias tee to the gate along with a constant-voltage source. The SET current, I_SET, was amplified and converted into a voltage signal at room temperature, low-pass filtered to 1 kHz bandwidth, and acquired with a fast digitizing oscilloscope.



 §.§ Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.



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§ AUTHOR CONTRIBUTIONS

M.A.B., S.K.G., J.G.K. and W.B. fabricated the device. M.A.B., S.K.G., M.H. and S.J.H. obtained all measurements. Calculation of donor charging energies was performed by S.J.H. and S.K.G. The data analysed by M.A.B., S.K.G. and M.H. and was discussed critically with all authors. Theoretical modelling of coherent exchange coupling was carried out by M.A.B. and D.K. with input from S.K.G. and C.D.H. Spin-readout analysis was performed by M.A.B., D.K. and T.F.W. The manuscript was written by M.A.B., S.K.G and M.Y.S. with input from all other authors. M.Y.S. supervised the project. 





§ COMPETING FINANCIAL INTERESTS

All authors declare no competing financial interests.



§ ACKNOWLEDGEMENTS

We thank Sven Rogge for enlightening discussions. This research was conducted by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (project no. CE110001027) and the US National Security Agency and US Army Research Office (contract no. W911NF-08-1-0527). M.Y.S. acknowledges an ARC Laureate Fellowship.













§ SUPPLEMETARY INFORMATION






§ CHARGING ENERGY CALCULATION


The charging energy, E_c^a of a quantum dot (QD) a can be calculated by knowing the charging energy of another QD, b which is capacitively coupled to a <cit.>. They are related through their mutual charging energy, E_m, which is given by,

    E_m = α_g^i δ V_g^i,

where α_g^i is the lever arm from gate g to QD i and δ V_g^i is the voltage shift of the potential of QD i due to QD j. This value must be the same for both QDs a and b. Therefore, we can write,

    α_g^a δ V_g^a = α_g^b δ V_g^b.

In the same manner, E_c^i is given by,

    E_c^i = α_g^i Δ V_g^i,

where now, Δ V_g^i is the voltage difference between two charge transitions in the space of gate g. We can eliminate the (usually) unknown lever arms by combing equations Eq.(2) and Eq.(3),

    E_c^a/Δ V_g^aδ V_g^a = E_c^b/Δ V_g^bδ V_g^b.


We now consider the possibility that E_c^a ≠ E_c^b such that multiple charging events can occur in the voltage range Δ V_g^i. In this case, the measured Δ V_g^i is actually the sum of the true Δ V_g^i, which we denote ΔV̂_g^i and the number of charging events, n^j of the other QD,

    Δ V_g^i = ΔV̂_g^i + n^j δ V_g^i,

Since we require ΔV̂_g^i in Eq.(4), we must substitute Eq.(5) in such that it now reads,

    E_c^a/E_c^b  = δ V_g^b(Δ V_g^a - n^b δ V_g^a)/δ V_g^a(Δ V_g^b - n^a δ V_g^b).




This is the general form of the equation which relates the two charging energies between QDs a and b. We now look at the specific case where Δ V_g^a ≫Δ V_g^b. This is the case most commonly seen when we have a large QD used as a charge sensor to measure smaller QDs or single donors (which will necessarily have larger charging energies than the charge sensor QD). For clarity, we switch to the S/D (SET/donor) terminology in place of a (= D) and b (= S). If the condition Δ V_g^D ≫Δ V_g^S holds then there will be multiple charging events of the SET (n^S ≠ 0) and exactly zero for the donor (n^D = 0) within the voltage ranges Δ V_g^D and Δ V_g^S, respectively. As a result, we can simplify Eq.(6) to,

    E_c^D = E_c^Sδ V_g^S/Δ V_g^S(Δ V_g^D/δ V_g^D - n^S).

Figure <ref> shows the measurement of all required parameters {E_c^S,δ V_g^S, δ V_g^D, Δ V_g^S, Δ V_g^D}, where we have chosen to measure along the right gate, G_R, i.e. g→R. Using Eq.(7) for the 1→2 electron transitions for both qubits, we find charging energies of 65±8 and 43±5 meV and for L (2P) and R (1P), respectively. These values are consistent with theoretical <cit.> and previously measured <cit.> charging energies for 2P and 1P donor qubits respectively.





§ SEQUENTIAL SPIN READOUT OF TWO DONOR QUBITS

The top panels of Fig. <ref>a and b show close-ups of the current through the SET charge sensor in the region where spin readout is performed on R and L, respectively. The position of the three readout phases, load (L), read (R) and empty (E) are shown by the white circles in these diagrams. As discussed in the main text both readout techniques rely on a spin-dependent tunnelling process <cit.>, this can be seen in the data presented in the bottom panels of Fig. <ref>a and b. Here, the average SET current in time is shown as the read voltage is stepped along the axis shown by the white arrow in the upper panels (equivalent to the detuning axis ϵ in the main text). At certain read voltages a short current blip can be seen to occur at the start of the read phase indicating the spin-up of the electron qubit. 

Sequential spin readout of electrons on L and R is carried out using the sequence of pulses shown in Fig. <ref>c. The first phase of this sequence is the read phase of qubit-L, after which a pulse is applied to the read out position of qubit-R. Since the readout is independent at these detuning positions i.e. exchange is negligible (as shown in Fig. 1 of the main text) the electron remaining on qubit-L during the read phase of qubit-R has no effect on spin readout fidelity. The following four pulses from 3-6 serve to empty and reload electrons from and to the two qubit sites, these may or may not occur depending on the exact experimental protocol, i.e. whether or not a qubit is being prepared with random spins or deterministically with spin-down. Finally, we pulse to position 7 from which we carry out the exchange pulse as described in Fig. 2a of the main text. 



§ CALCULATION OF LEVER ARMS ALONG THE DETUNING AXIS Ε

The lever arm of the gates along the detuning axis ϵ can be ascertained from the range in read voltage over which tunnelling due to spin-up electrons on the qubits is observed (the so called spintail). This range, labelled δ_i in the lower panels of Fig. <ref>a and b, is proportional to the Zeeman splitting via the lever arms, α_ϵ^L and α_ϵ^R for qubit-L and -R respectively, by

    α_ϵ^iδ_i = γ_e B_z,

where γ_e=28.024 GHz/T and B_z is the magnetic field. From Fig. <ref> the lever arms were calculated to be α_ϵ^L=0.041±0.004 and α_ϵ^R=0.030±0.003. The sum of these two lever arms represents scaling of the detuning, ϵ, between the qubit-L and -R to the gate voltage V_GL along this axis. 



§ SPIN READOUT FIDELITIES

The assignment of a spin-up or -down electron from each SET current trace comprises of two separate parts, (i) electrical readout and (ii) spin-to-charge conversion. 






  §.§.§ (i) Electrical readout

The electrical readout involves determining whether a given SET current trace can be assigned as having a `blip' during the read phase i.e. whether during this time the current surpasses a threshold value I_t (see Fig. <ref>). From a simulation of 10,000 SET traces 50% of which contain a `blip', and with added white Gaussian noise equivalent the signal-to-noise ratio observed in the experiment (average of SNR=17 dB for readout of both qubits), histograms of peak voltages V_p are generated and shown in Fig. <ref>a and b for L and R respectively. Note the use of peak voltage not current due to the use of a current amplifiera on the drain of the SET charge sensor. 


From these histograms the fidelity of assigning either spin-up or -down (F_↑ or F_↓) to each current trace is calculated using the following set of equations,

    F_↑   = 1-∫_-∞^V_tN_↑(V_P)dV_p 
    
    F_↓   = 1-∫_V_t^∞N_↓(V_P)dV_p,

where V_t is the equivalent voltage threshold for I_t after the current amplifier and N_i is the fraction of spin state i. The results are shown in Fig. <ref>c and d with the addition of the calculated electrical readout visibility V_ER=F_↑+F_↓-1. From this we can determine the optimum voltage threshold, V_t, where V_ER is maximised. 





  §.§.§ (ii) Spin-to-charge conversion

Next we determine the optimum length of time for the read phase of the three level readout sequence. During spin-to-charge conversion errors are introduced from three main sources; T_1 relaxation of spin-up electrons; spin-up electrons failing to tunnel to the SET during the designated read time; and spin-down electrons tunnelling to the SET due to thermal excitation [Note that the second and third points apply specifically to the spin dependent unloading mechanism used at R. However, equivalent arguments apply for L in the case of the spin dependent loading mechanism.].

Following from the work of Buch <cit.> and Watson <cit.> we use a rate equation model to determine the optimum readout time, Δ t, based on the probability of a successful assignment of spin-up or -down, α and β respectively. As an input to the model the tunnelling times of spin-up out of the qubit site to the SET and spin-down into the qubit site from the SET, τ_↑out and τ_↓in are shown in Fig. <ref>e and f. In addition, the spin-down tunnelling time from the qubit site to the SET, τ_↓out was also measured experimentally to be and 0.61±0.06 s and 25±5 s for qubit-L and -R respectively. We refer the reader to Ref. <cit.> for further details on this model. The readout time, Δ t vs the fidelities α and β are shown in Fig. <ref>e-h. Similarly for the electrical readout, the visibility of spin-to-charge conversion is calculated as V_STC=α+β-1. The optimum readout time is chosen where V_STC is maximised. Table 1 gives a summary of the fidelity calculations for both qubits, where the final measurement fidelity is given by, F_M=(α F_↑+β F_↓)/2. 



§ STATISTICAL ANALYSIS OF DETUNING DEPENDENT SPIN CORRELATIONS

The statistical significance of the anti-correlated spin measurements presented in Fig. 2 of the main text can be ascertained using the ϕ-correlation coefficient (the Pearson correlation coefficient for two binary variables) which is given by,

    ϕ_m = P_↑↑P_↓↓ - P_↓↑P_↑↓/√(P_L↑P_L↓P_R↑P_R↓).

For perfectly anti-correlated spins ϕ=-1, however, given our choice of initial states from Eq. 1 in the main text, where one spin is randomly loaded up or down and the other is deterministically loaded with spin-down, the maximum expected value is ϕ=-0.25. We measure an average of ϕ_m=-0.243±0.028 in the detuning range 0<ϵ<2.4 meV for both ρ_↓↑ and ρ_↑↓, see Fig. <ref>a. The statistical significance of these anti-correlations was deduced from the p-value of χ^2=ϕ_m^2/n, where n=1000 is the number of measurement repetitions. A p-value ≪0.01 is shown in Fig. <ref>b over the same range of ϵ, demonstrating statistically significant two-electron anti-correlated spins. 





§ B-FIELD DEPENDENCE OF SINGLE-ELECTRON T_1 RELAXATION

Figure <ref>a and b shows the dependence of spin-up fraction of each qubit as a function of wait time for different magnetic field strengths. Both qubits show a B_z^5 dependence of the 1/T_1 relaxation rate as shown in Fig. <ref>c, indicating that the relaxation processes are driven by phonon coupling to the electron spins as previously observed <cit.>. However, one thing we do have to be careful of is the magnetic field orientation during the measurment. Note that the results for independent readout presented in the main text are shown by square markers in Fig. <ref>c at B_z=2.5 T and were obtained with a field orientation of B_z^(1)||[11̅0]. The magnetic field dependence of  T_1 are shown by the red and blue circular markers and were taken during a different cool down with a field orientation B_z^(2)||[100]. Since T_1 measurements are known to be highly sensitive to the magnetic field orientation <cit.> the difference in the T_1 values observed can be explained by this. 

There are two interesting aspects of the results to note as a result of this different field orientation in Figure <ref>c. Firstly, we measure the spin relaxation rate of a 2P donor dot in this device to be greater than that for the 1P donor case. This is in contrast to recent theoretical results, which predict a slower relaxation rate for multi-donor qubits in the one-electron case due to its tighter confining potential <cit.>. Secondly, we see that for the two field orientations B_z^(1) and B_z^(2) (see inset of Fig. <ref>c) the 1P relaxation rates are significantly slower than those measured previously  <cit.>, giving a value of  22s and  52s for qubits L and R respectively at B=1.5T. Further work is underway to examine the interplay of the magnetic field and electric field orientation in these Coulomb confined devices.  






§ NUMERICAL MODEL OF COHERENT EXCHANGE OSCILLATIONS


Using the same Hamiltonian given in Eq. 3 of the main text we model coherent exchange oscillations as described in Fig. 3 and 4 of the main text. For this proposed experiment, both electron spins are initialised at a large negative detuning position where the exchange is negligible and subsequently pulsed non-adiabatically into a region where the exchange dominates over hyperfine, that is where J>Δ B_z. The spins are allowed to evolve for some time τ_w before being pulsed back to the initial preparation positions, where spin readout can be carried out. In our simulation we replicate this pulse sequence with the inclusion of detuning noise with a Gaussian distribution <cit.> defined by a standard deviation of 850MHz in detuning energy, based on gate noise measurements of approximately δϵ=50μV measured in our device. We average over 100 repetitions of the simulation to obtain the final density matrix. In addition to detuning noise, we include a constantly fluctuating Overhauser field which equates to a single electron T^*_2=55 ns measured in previous electron spin resonance experiments in natural silicon <cit.>. Finally, we average over all eight possible donor nuclear spin configurations for the 2P-1P system, giving us an average representation of the nuclear hyperfine interaction. The detuning pulse sequence is simulated for varying τ_w times, giving rise to the coherent oscillations that can be seen by the green markers in Fig. 3b of the main text. 



§ ANALYTICAL MODEL OF COHERENT EXCHANGE OSCILLATIONS

The form of coherent exchange oscillations, as shown in Fig. 4b of the main text, can also be approximated analytically in the following way. Firstly, the oscillation frequency resulting from the exchange interaction, ν, depends on the relative magnitude of the exchange energy J and the difference in magnetic field between the two qubits Δ B_z. For donor systems, Δ B_z is dominated by the donor nuclear spin orientation resulting in a difference in hyperfine strength between the two qubits, δ A. For a particular nuclear spin configuration the frequency ν is given by,

    ν = √(J^2+δ A^2),

similarly the undamped amplitude of these oscillations 𝒜 is given by,

    𝒜=J^2/J^2+δ A^2.

The amplitude is averaged over all nuclear spin configurations but it is assumed that the frequency is dominated by the most common configuration, which in the 2P-1P case is δ A=A/2. 




As in the numerical case, the dephasing is a combination of detuning noise and an Overhauser field. The standard deviation of the detuning noise δϵ=50μV can be transformed into an exchange frequency noise δν by considering the minimum and maximum oscillation frequencies given the δϵ,

    δν = |ν(ϵ+δϵ,δ A) - ν(ϵ-δϵ,δ A) |.

Note here we have also averaged ν over all possible nuclear spin orientations which give rise to different values of δ A. For a 2P-1P donor dot system δ A=3A/2 or A/2 with a 1 to 3 ratio. Based on the width, δν, in the frequency domain the resulting dephasing time τ_J is calculated by converting to the time domain and is given by,

    τ_J=1/πδν.
 	
Note that the decay induced by τ_J is expected to be Gaussian based on the nature of the noise <cit.>. The total dephasing time τ_d which includes dephasing from the constantly fluctuating Overhauser field is approximated using the formula,

    1/τ_d≈1/τ_J + 1/T_2^*.

Finally, the form of the analytical coherent exchange oscillations is given by, 

    1/2+ [𝒜e^-(t/τ_J)^2cos(2πν t) + (1-𝒜)] e^-t/T_2^*/2.


Both the characteristic decay time τ_d and oscillation frequency ν change as a function of the pulse detuning position ϵ and tunnel coupling t_c, and their product τ_d·ν gives an indication of the number of observable oscillations. This is plotted in Fig. 3c of the main text as a function of the tunnel coupling and pulse detuning position. In Fig. <ref> we shown a comparison of the analytical expressions derived above against a numerical simulation with equivalent parameters as described in previous section. 





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