Two-electron spin correlations in precision placed donors in silicon

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 utilising 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.

where ) * is the lever arm from gate to QD and ) * is the voltage shift of the potential of QD due to QD . This value must be the same for both QDs and . Therefore, we can write, In the same manner, " * is given by, where now, ∆ ) * is the voltage difference between two charge transitions in the space of gate . We can eliminate the (usually) unknown lever arms by combing equations Eq.(2) and Eq.(3), We now consider the possibility that " # ≠ " 2 such that multiple charging events can occur in the voltage range ∆ ) * . In this case, the measured ∆ ) * is actually the sum of the true ∆ ) * , which we denote ∆ ) * and the number of charging events, < of the other QD, Since we require ∆ ) * in Eq.(4), we must substitute Eq. (5) in such that it now reads, 4 5 6 4 5 9 = >7 8 9 (∆7 8 6 @A 9 >7 8 6 ) >7 8 6 (∆7 8 9 @A 6 >7 8 9 ) . (6) This is the general form of the equation which relates the two charging energies between QDs and . We now look at the specific case where ∆ ) # ≫ ∆ ) 2 . 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 / (SET/donor) terminology in place of = and = . If the condition ∆ ) G ≫ ∆ ) H holds then there will be multiple charging events of the SET ( H ≠ 0) and exactly zero for the donor ( G = 0) within the voltage ranges ∆ ) G and ∆ ) H , respectively. As a result, we can simplify Eq.(6) to, Supplementary Figure 1 shows the measurement of all required parameters { " H , ) H , ) G , ∆ ) H , ∆ ) G }, where we have chosen to measure along the right gate, L , i.e. = 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 (2P) and (1P), respectively. These values are consistent with theoretical [2,3] and previously measured [3,4] charging energies for 2P and 1P donor qubits respectively.

Sequential spin readout of two donor qubits
The top panels of Supp. Fig. 2a 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 spindependent tunnelling process [5], this can be seen in the data presented in the bottom panels of Supp. Fig. 2a 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 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 Supp. Fig. 2c. 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 (positions 1 and 2). 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.
As described in the main text, the mechanisms used for readout at qubit-L and -R differ. For qubit-R, at the (1,0)-(1,1) charge transition, we perform a spin-dependent unloading mechanism and in the case of qubit-L, at (1,1)-(2,1), a spin-dependent loading mechanism is used to discriminate between spin-up and -down states. In a similar vein the loading of random spins is different for each qubit. For qubit-R we plunge below the Fermi level of the SET by approximately 0.35meV (5mV in detuning along the axis ) loading from (1,0)→(1,1), whereas for qubit-L we unload from the singlet state, (2,1) →(1,1), by plunging above the Fermi level by the same detuning. shown by the circle markers. Spin readout of the electron at this donor relies on a spin-dependent unloading mechanism from the qubit to the SET reservoir at the position marked `R' in the diagram [5]. The read voltage is stepped along the SET Coulomb blockade peak, shown by the white arrow, and is equivalent to axis described in the main text. (lower) The average of 200 single-shot SET current traces, H4c , as a function of the read voltage along . The voltage at which spin readout is performed during the experiments is shown by the white dashed line. The range over which spin-up electrons can selectively tunnel off of the dot, L , is shown by the yellow arrow. All measurements were performed at e = 2.5 T. b, A similar readout method is employed for qubit L. Here, a spin-dependent loading mechanism from the 1 → 2 charge state is utilised. c, Schematic of the pulsing sequence used to sequentially readout L and R in that order, as well as initialise both qubits with random spin states. The order of spin readout is chosen to minimise the effects of spin relaxation since L has the shorter g time. To initialise spin-down deterministically on either qubit, we skip phases 3,4 for and/or 5,6 for L. d, An example of 40 single shot traces for the sequential spin readout and initialisation of random spins on L and for the sequence shown in c. The two read phases occur at the beginning of each trace. A short `blip' in the SET current indicates the tunnelling of an electron during the read phases (1 and 2) in approximately 50% of the traces. This occurs due to the presence of a spin-up electron on the dot. Example traces for the outcomes {↑↑, ↑↓ , ↓↑, ↓↓} are shown.  [6,7] is used to estimate the optimum readout time, ∆ , which optimises the successful assignment of a spin-up or -down, and respectively. From these individual fidelities the maximum spin-to-charge conversion fidelity, Hcn can be estimated for both qubits, shown in g,h.

Electrical readout fidelity
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.
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 l (see Supp. Fig. 2). From a simulation of 10,000 SET traces 50% of which contain a `blip', and with added white Gaussian noise equivalent the signal-tonoise ratio observed in the experiment (average of = 17 dB for readout of both qubits), histograms of peak voltages r are generated and shown in Supp. Fig. 3a and b for L and R respectively. Note the use of peak voltage not current due to the use of a current amplifier on the drain of the SET charge sensor.
From these histograms the fidelity of assigning either spin-up or -down ( ↑ or ↓ ) to each current trace is calculated using the following set of equations, where l is the equivalent voltage threshold for l after the current amplifier and * is the fraction of spin state . The results are shown in Supp. Fig. 3c

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; Following from the work of Buch [6] and Watson [7] we use a rate equation model to determine the optimum readout time, ∆ , based on the probability of a successful assignment of spin-up or -down, and 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, ↑{|l and ↓*A are shown in Supp. Fig. 3e and f. In addition, the spin-down tunnelling time from the qubit site to the SET, ↓{|l 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. [6] for further details on this model. The readout time, ∆ vs the fidelities and are shown in Supp. Fig. 3e-h. Similarly for the electrical readout, the visibility of spin-to-charge conversion is calculated as Hcn = + − 1. The optimum readout time is chosen where Hcn is maximised. Supplementary Table 1 gives a summary of the fidelity calculations for both qubits, where the final measurement fidelity is given by, w = ( ↓ + ↑ )/2.

Calculation of lever arms along the detuning axis ε
The lever arm of the gates along the detuning axis 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 * in the lower panels of Supp. Fig. 2a and b, is proportional to the Zeeman splitting via the lever arms, } and } k for qubit-L and -R respectively, by where € = 28.024 GHz/T and e is the magnetic field. From Supp. Fig. 2 the lever arms were calculated to be } = 0.041 ± 0.004 and } = 0.030 ± 0.003. The sum of these two lever arms represents scaling of the detuning, between the qubit-L and -R to the gate voltage •~ along this axis.
For perfectly anti-correlated spins = −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 = −0.25. We measure an average of ƒ = −0.243 ± 0.028 in the detuning range 0 < < 2.4 meV for both ↓↑ and ↑↓ , see Supp. Fig. 4a. The statistical significance of these anti-correlations was deduced from the p-value of • = • / , where = 1000 is the number of measurement repetitions. A p-value ≪ 0.01 is shown in Supp. Fig. 4b over the same range of , demonstrating statistically significant two-electron anti-correlated spins.

Modelling the 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 > ∆ e . The spins are allowed to evolve for some time ' 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 [8] defined by a standard deviation of 850MHz in detuning energy, based on gate noise measurements of approximately = 50 V 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 • * = 55 ns measured in previous electron spin resonance experiments in natural silicon [9]. 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 ' times, giving rise to the coherent oscillations that can be seen by the green markers in Fig. 3b of the main text.
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, , depends on the relative magnitude of the exchange energy and the difference in magnetic field between the two qubits ∆ e . For donor systems, ∆ e is dominated by the donor nuclear spin orientation resulting in a difference in hyperfine strength between the two qubits, . For a particular nuclear spin configuration the frequency is given by, similarly, the undamped amplitude of these oscillations Λ is given by, 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 = /2.  Fig.4a of main text) for a time ' . The two electron state is initialised as |↑↓ at a point where the exchange energy is negligible, and subsequently a non-adiabatic detuning pulse is applied to = −25 GHz. We have assumed voltage noise equivalent to 850~MHz along the detuning axis, (obtained from measurements) as well as a single electron dephasing time of • * = 55 ns measured in previous works [9]. The results for a numerically simulated full quantum model are shown by the green markers, while the blue line gives the predicted curve based on an analytical expression in Eq.(17). b, Theoretical prediction of • along the line ∆ e = as a function of tunnel coupling for a 2P-1P donor qubit system. Solid (dashed) lines show analytical results including (excluding) the 29 Si Overhauser field, whilst the green markers are results from a numerical simulation.
As in the numerical case, the dephasing is a combination of detuning noise and an Overhauser field. The standard deviation of the detuning noise = 50 V can be transformed into an exchange frequency noise by considering the minimum and maximum oscillation frequencies given the , Note here we have also averaged over all possible nuclear spin orientations which give rise to different values of . For a 2P-1P donor dot system, = 3 /2 or /2 with a 1 to 3 ratio. Based on the width, , in the frequency domain the resulting dephasing timeis calculated by converting to the time domain and is given by, Note that the decay induced byis expected to be Gaussian based on the nature of the noise [9]. The total dephasing time • which includes dephasing from the constantly fluctuating Overhauser field is approximated using the formula, Finally, the form of the analytical coherent exchange oscillations is given by, Both the characteristic decay time • and oscillation frequency change as a function of the pulse detuning position and tunnel coupling › , and their product • 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 Supp. Fig. 5 we shown a comparison of the analytical expressions derived above against a numerical simulation with equivalent parameters as described in previous section.

Supplementary Discussion
B-field dependence of single-electron g relaxation Supplementary Figure 6a 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 e « dependence of the 1/ g relaxation rate as shown in Supp. Fig. 6c, indicating that the relaxation processes are driven by phonon coupling to the electron spins as previously observed [6,9,10]. However, one thing we do have to be careful of is the magnetic field orientation during the measurement. Note that the results for independent readout presented in the main text are shown by square markers in Supp. Fig. 6c at e = 2.5 T and were obtained with a field orientation of e (g) || [110]. The magnetic field dependence of g are shown by the red and blue circular markers and were taken during a different cool down with a field orientation e (•) ||[100]. Since g relaxation times are known to be highly sensitive to the magnetic field orientation [11--13], the difference in the observed g values between the two cool downs can be explained by this.
There are two interesting aspects of the results presented in Supp. Fig. 6c. Firstly, we measure the spin relaxation rate of a 2P donor qubit in this device to be greater than that for the 1P donor case. This is in contrast to recent experimental [14] and theoretical results [15], which show a slower relaxation rate for multi-donor qubits in the one-electron case due to its tighter confining potential [15]. At this time, we do not have a theoretical model to describe this discrepancy, but speculate that it could be due to the specific spatial configuration of the donors inside the 2P cluster. Furthermore, we note that an unexpected anisotropy in g in a magnetic field has recently been observed in donor based systems by the present group and further work is needed to determine whether this can explain the effect observed herein.