Supplemental Material for Demonstration of a coherent electronic spin cluster in diamond

Transcription

1 Supplemental Material for Demonstration of a coherent electronic spin cluster in diamond Helena S. Knowles, Dhiren M. Kara and Mete Atatüre Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK Contents: 1. Setup and Pulse protocols 2. Dipole-dipole interaction and polarization transfer under optical excitation 3. NV evolution in the magnetic field generated by dark spins 4. Determining the location of N spins 5. N spin lock about y axis 6. IDSE with optical and Hartmann-Hahn polarization 1. Setup and Pulse protocols The NV ground and excited states both consist of a spin triplet, S=1, with zero-field splittings of 2.9 GHz and 1.4 GHz, respectively. Under optical illumination (532 nm, Ventus, Laser Quantum) the system is polarized into 0. We use a confocal microscope with 0.9 numerical aperture objective (Nikon 100x) to excite, initialize and readout the NV spin state. The collected NV luminescence is sent to a single-photon counting module (PerkinElmer). We dropcast a suspension of nanodiamonds (NaBond, milled HPHT 99.95% purity) onto a quartz plate and use a 20-µm copper wire across the sample to deliver the radiowave frequency (RF) and microwave frequency (MW) excitation to the Nitrogen spins. A Tektronix 70002A arbitrary waveform generator provides the pulses used for spin control and the TTL signals for optical excitation and readout. We amplify the microwave and radiowave signals using a 20-W (Minicircuits) and 16-W amplifier (Minicircuits) and sum the

2 two signals together using a power splitter. The laser is pulsed using an acousto-optical modulator (AA Opto-Electronic). 2. Dipole-dipole interaction and polarization transfer under optical excitation We consider dipolar interaction between two spins close to the resonance condition, where E &' E ) ~ E E. We restrict ourselves to the NV-states 0 and 1 in the description of the dipolar interaction between the NV and N spins as the state 1 is far detuned from this resonance. Due to the rotational symmetry of the interaction we set the y coordinate to zero. The dipole-dipole interaction in the xz-plane between the NV spin (m / = {0, 1}, located at (0, 0) and a N spin (m / = { ', ' }, located at r = x, 0, z ) is given by 5 5 H :;< = μ )g 5 μ? 5 4π 1 1 r B h 3 r s GH r s G r 5 s GH s G where μ? is the Bohr magneton, μ ) the vacuum permeability, g = 2 the electron Landé factor, h the Planck constant, r = Norm(r) and s O,OP = ' (σ 5 R, σ S, σ T ) with the Pauli spin operators σ R, σ S and σ Y. In a magnetic field, B [<<, along the z-axis (defined as the NV crystal axis) the system is described by the following Hamiltonian (in frequency units and in the 0,, 0,, 1,, 1, basis): H \]\,^_ = H :;< + H abbc[d,^_ = Δ Q Q Ω Q Δ Δ Q Q Δ Δ Q Ω Q Q Δ + b/ b/ b/2 + D^_ b/2 + D^_ where

3 Δ = k lk m n op ' q r (1 3Cos5 (θ)), Q = k lk m n op ' q r 3 Cos(θ)Sin(θ), Ω = k lk m n op ' q r 3 Sin5 θ, b = 2 k m x B [<<, and θ is the angle between the quantization axis of both spins (z-axis) and the separation vector r from the NV to the N spin. The zero-field-splitting parameter of the NV excited state is D^_ 1.4 GHz. At the resonance condition, b = D^_ /2, we have b} H \]\,^_ = Δ + D^_ /4 Q Q Ω Q Δ D^_ /4 Δ Q Q Δ Δ + 3D^_ /4 Q Ω Q Q Δ + D^_ /4. Since D^_ is large compared to all other terms and the states 0, and 1, are degenerate we can neglect all off-diagonal terms except Ω giving, b} H \]\,^_ Δ + D^_ /4 0 0 Ω 0 Δ D^_ / Δ + 3D^_ /4 0 Ω 0 0 Δ + D^_ /4. (S1) The Ω term mixes the two degenerate states and corresponds to the coupling that leads to the optical polarization of the N spin. From our measurements we can extract Δ (see section 3) and Ω allowing the localisation of the dark spins (see section 4).

4 3. NV evolution in the magnetic field generated by dark spins We start with a closer look at the N spin energy levels. The N spectrum shown in Supplementary Fig. 1 (b) was modelled using the following Hamiltonian, where each Jahn- Teller orientation case is described by: H = a S T I T + a S R I R + S S I S qi T 5 + gμ /h B S ˆR,S,T where a = MHz and a = 81.8 MHz are the parallel and perpendicular hyperfine constants with respect to the Jahn-Teller axis defined as the z-direction, q = 3.97 MHz is the quadrupole interaction[1,2], the electron and nuclear spin is S=1/2 and I=1, respectively, and g is the electron g-factor. We ignore the nuclear spin Zeeman term as it is negligible compared to the hyperfine term at the magnetic fields we operate. The model is covered in detail in Reference [3]. As the transitions are mostly nuclear spin conserving, we restrict ourselves to describing the N spins as spin-1/2 particles with a given probability for each Jahn-Teller and hyperfine state, as detailed below. The IDSE measurement is performed in the ground state of the NV centre, where the interaction with a N spin is described by H \]\, _ = Δ + b/2 Q Q J Q Δ b/2 Δ Q, Q Δ Δ b/2 + D _ Q J Q Q Δ 3b/2 + D _ where D _ 2.9 GHz is the ground state zero field splitting. At b = D^_ /2 there are no degeneracies and we only need to consider the diagonal elements of H \]\, _. Under perfect optical polarization the system ends up in state 0,. During the echo protocol, p 5 Œ 5 π Ž 5 p 5, this state evolves to ie 0, + 1,. The readout pulse, a π/2 rotation about an axis with angle α to the x-axis, yields a NV z-projection of Cos α + Δ τ. The phase shift

5 extracted from the IDSE measurement thus reveals directly the magnitude of the dipolar coupling term Δ. We model the IDSE measurement shown in Figure 2d of the main text by calculating the evolution of the NV spin in a magnetic field created by nearby N spins. A fit of the model to the data allows us to extract the polarisation and coupling strength Δ of individual spins within the cluster. The B-field at the NV site created by the N spin cluster is dependent on the spin state and position of each N. As discussed in the paper we define a degree of polarisation, p ;, for each spin N i as p ; = 1( 1) for ( ) and a contribution to the net B-field, B <], in units of frequency as Δ (Δ ) for ( ). During a given IDSE sequence we assume a static B <] since the sequence duration (~µμs) is shorter than a typical N spin lifetime ( 10µμs). However, we average over many repetitions of the sequence to acquire an interference signal. Therefore, we probe a distribution of B <], arising from the probabilistic distribution (in time) of the N spin states. To account for this, we take our measured IDSE fringe to be a sum of sinusoidal curves each with an amplitude given by the probability of a particular cluster state and a phase given by the B-field it produces at the NV site. Supplementary Figure 1: Illustration of N electron spin resonances for Jahn-Teller axis along an external magnetic field aligned along the NV axis (resonance group N a, shown in (a)) and

6 Jahn-Teller axes along the three other crystal axes (resonance groups N a, N b and N c shown in (b)). (c) shows the resulting spectrum including the probability that a given N spin will be driven by excitation resonant with the transition. The shaded area highlights the transitions driven by the RF field in the IDSE experiment. When performing an IDSE the radio wave excitation is only resonant with the central N spin transition, where all four Jahn-Teller subgroups overlap in energy (when the nuclear spin, I, is in the m = 0 magnetic sublevel, see Supplementary Figure 1) for a given N spin. Therefore, out of the 12 individual resonances (3 nuclear spin states and 4 Jahn-Teller orientations) the 4 overlapping resonances are driven in IDSE, which results in a probability of 4/12 = 1/3 for a given N spin to contribute to the signal detected by the NV centre. This means that two thirds of the time a given N contributes no net field: In the non-resonant cases the field produced by the N spin is cancelled in the second half of the NV spin echo sequence. Only the N spins that were resonantly flipped can contribute a non-zero phase pickup on the NV. Thus, three different spin state categories k ; for each N atom N i (where i runs from 1 to the number of N spins, n) are needed to represent the polarisation weightings (probabilities of cluster state occupancy) and contribution to B <] : 1) Category k ; = 1 ;, ( m, m _ = 0, is rotated by the RW and detected by NV): weight w ;'Ÿ = (p ; + 1)/6 and contribution to B <] u ;'Ÿ = Δ ; 2) k ; = 2 ; ( 0, is rotated by the RW and detected by NV): w ;5Ÿ = (1 p ; ) 6 and u ;5Ÿ = Δ ; 3) k ; = 3 ; ( ±1, and ±1, are not rotated by the RW driving): w ;BŸ = 2 3 and u ;BŸ = 0

7 Here, as before, the values p ; and Δ ; are the polarization and coupling strength of each individual spin N i, and the weights w ; Ÿ and couplings u ; Ÿ for each spin category k and each spin N i can now be used to construct the overall IDSE shape. For n cluster spins this results in 3 d unique possibilities of cluster states. The probability of a given cluster state is given by the product of all weights as a n r = w ' w 5 n w B r w d, where each k ; can take values from 1 to 3 and for each cluster state there is one a value. The corresponding frequency shift on the NV due the cluster field is given by the sum of all individual couplings for each spin N i in a given k ; category, β n r = u ' + u 5 n + u B r + u d. For a given set of polarizations p ; and coupling Ω ; values, the resulting IDSE signal needs to be computed as a sum of interference curves: Each time the IDSE measurement runs it samples one of many possible cluster states with the same p ; and Ω ; values. In order to model the timeaveraged IDSE we therefore need to compute the sum of many individual IDSE measurements. We do so by using the following trigonometric identity: ; a cos(α + φ ) = A cos (α + Θ), tan Θ = ; a sin (φ ). ; a cos (φ ) In our case the phase θ of the individual sine curves is given by β τ (the total coupling strength of the cluster state acting for the length of the IDSE sequence time, τ) and the

8 amplitude a is given by the combined weights a n r. As in the main text α is the rotation angle of the last measurement pulse in the sequence and is common to all sine curves. In summary, we sum interference curves with the same periodicity but different phase shifts and amplitudes to construct the overall averaged IDSE signal detected by the NV spin. The IDSE signal then has the form, a cos α + β τ = A cos α + Cτ, [ Ÿ ]c ;d[\;]d} with tan Cτ = a sin (β τ) [ Ÿ a cos (β τ) [ Ÿ A and C both contain information about the cluster degrees of polarisation but C is more robust to signal fluctuations, NV spin dephasing and pulse errors. Therefore, we only extract the total phase shift δ <] from our IDSE data, which corresponds to 2μ? B <] τ/ħ. The fit function we use to extract polarization p ; and coupling Δ ; values is given by δ <] = 2μ? B <] τ/ħ = 2 arctan a sin β τ [ Ÿ a cos β τ [ Ÿ We use a Mathematica-based code to perform a least squares fit to the phase difference, δ <], extracted from IDSE data acquired at 2-µs and 20-µs optical initialization, keeping the coupling

9 parameters Δ ', Δ 5,, Δ common to both datasets and allowing each dataset to have distinct polarisation values for each spin. We fit the data using models including n = 1, 2, 3, 4 and 5 N spins. The resulting residual sum of squares RSS n = º (y f (x )) 5 where f (x ) is the predicted value using the n-spin model at x for each model is shown in Supplementary Figure 2. The RSS value decreases as more spins are added to the model, but the effect is most pronounced for n = 1, 2 and 3. We observe correlations between between individual residuals (y f (x )) 5 at different points x for n = 1 and 2 and therefore discard the n = 1, 2 models. Supplementary Figure 2: Residual sum of squares extracted from least square fits to the data presented in Fig. 2(d) in the main text for n=1 to 5 nitrogen electron spins. In order to quantify the likelihood of n = 3, 4 or 5 spins contributing to the signal detected at the NV we use the Akaike Information Criterion (AIC). The AIC provides a means to compare different models by assigning a number to each model based on how well it reproduces that measured data and including a penalty term for the number of parameters required. The AIC is given by the sum of the log likelihood of the fit and the penalty term[4]: AIC = n Log ½ + 2 k,

10 where k is the number of parameters used in the model. In our case k = 4n as each additional spin i has a coupling parameter, Δ, and three polarisation parameters, p, p and p. The smaller the AIC value, the more probable the model. For datasets where k ~ n the AIC value needs to be corrected by an additional term: AICc = n Log ½ + 2 k + 5 ( À') & &'. This corrected AIC value, AICc, leads to a stronger penalty per added parameter. In the limit of large n the additional term vanishes. Using the relative likelihood expression[4] we reveal that, even in the less stringent non-corrected AIC case, the 3-spin model is about 40 times more likely than the 4-spin model: P B P o = Exp Å Æ Ç&Å Æ r 5 = Phase difference(degrees) τ (ns) Supplementary Figure 3: IDSE measurements for τ = 2, 6 and 20 µs in black, blue and red, respectively, with 3-spin model fits shown as solid lines. We therefore conclude that with over 98% likelihood three N spins are coupled to the NV center and present the polarization and coupling parameters of the 3-spin fit, including standard

11 errors on fit parameters, in Supplementary Table 1. The corresponding fits are plotted in Supplementary Figure 3. Fit values for N=3 Standard error Δ ' (khz) Δ 5 (khz) Δ B (khz) p ' (%) 2 1 p 5 (%) 7 1 p B (%) 31 1 p ' (%) 2 1 p 5 (%) 8 1 p B (%) 52 1 p ' (%) 5 1 p 5 (%) 7 1 p B (%) 72 2 Supplementary Table 1: Best fit parameters of 3-spin model with standard errors. 4. Determining the location of N spins The unique dependence of and Ω on r and θ means that N spins can be localized relative to the NV. In the previous section we showed that ; can be extracted from the IDSE data. In this section we use the dependence of p ; on the laser excitation time to find Ω ;. To quantify Ω ; we model the optical polarization transfer by solving the master equation of an NV coupled to a single N spin. NV-N master equation steady state: Since Ns have low decay rates Γ G = 1/T ',O and low coupling rates Ω in comparison to the NV re-initialisation rate (~ few MHz), we can assume the NV dynamics under optical excitation are not strongly perturbed by the presence of a coupled N spin. Importantly, this means that in the case of multiple N spins being pumped we can treat each NV-N coupling independently. This is similar to the familiar problem of a classical optical field driving a small ensemble of

12 two-level dipoles, where the driven system does not affect the driving field. The standard solution (for the excited state population) of the optical Bloch equation is: P bb t = 5 Ω ]<\ 2 γ Γ + δ 5 5 ]<\ (Γ γ ) + Ω ]<\ where Ω ]<\ is the driving field strength, δ ]<\ is the detuning of the driving field w.r.t. the dipole transition, γ is the dephasing rate between the field and the dipole, and Γ is the system decay rate. We can re-write this for the case of a NV spin (acting now as the driving field) and a nearby N spin (the two-level system) as p G t = p GH Ð Ñ Ò n Ó ÔÕ,Ô Ö Ô À n (Ö Ô Ó ÔÕ,Ô )ÀÐ Ñ Ò n, (S2) 5 where p G, Ω and Γ G are defined as in the main text. We modify Ω ]<\ by the fraction of time, f Ù, that the NV spends in the states 0 ^_ and 1 ^_ which drive the N spins. The dephasing rate is now γ GH,G, the dephasing between the NV spin and the N spin. And the overall scaling factor, p GH, is the NV state polarization given by Ú l ÛÜ &Ú Ý ÛÜ Ú l ÛÜ ÀÚ Ý ÛÜ, where P denotes populations. At the resonance condition of B [<< = 24 mt we can set δ = 0. These three modifying parameters will enable us to develop a time-dependent model of our coupled NV-N system, required to extract Ω ; values. Their values are most easily extracted from steady state solutions of the system. To evaluate these terms, we begin by defining the radiative and non-radiative ES to GS decay channels of the NV as shown in Supplementary Figure 4. The state labels follow the notation of reference [5] such that states 1 and 3 correspond to 0 _ and 0 ^_, 2 and 4 to ±1 _ and ±1 ^_, and 5 is the metastable singlet state. The m } = ±1 spin states can be treated

13 equivalently since their decay channels are identical (this condition holds at low B fields aligned with the NV axis such as the B [<< = 24 mt in our experiments). 4 γ o5 γ oà 3 γ Bà γ B' 5 γ à5 2 γ à' 1 Supplementary Figure 4: Decay rates of the NV center We use the known values of the NV decay rates which are displayed in Supplementary Table 2[5].

14 Decay parameter Rate (MHz) γ B' 32.2 γ o γ Bà 12.6 γ oà 80.7 γ à' 3.1 γ à5 2.5 Supplementary Table 2: Decay rates of the NV in MHz. The radiative decays in Supplementary Table 2, i.e. γ B'. and γ o5, include a factor to account for the different refractive index environment of an NV in a ND compared to bulk. This factor modifies the bulk radiative decay such that γ ã<\ ã<\ Gâ = γ?ä ' d å + d æ d å, where n : = and n } = 1.46 are the diamond and quartz refractive indices respectively [6]. Laser excitation is mostly spin conserving, with a small fraction ε = 0.01, that results in spin flips [5]. Explicitly, we have the spin conserving η 'B = η 5o = 1 ε r é, and non-conserving η 'o = 2 η 5B = 2 ε r é excitation rates, where r é is the net laser excitation rate and subscripts denote the initial and final states, respectively. We now consider the rate equations of the NV: c ' = 1 + ε r é c ' + γ B' c B + γ à' c à c 5 = r é c 5 + γ o5 c o + γ à5 c à c B = 1 ε r é c ' + ε r é c 5 (γ B' + γ Bà )c B c o = 2ε r é c ' + 1 ε r é c 5 (γ o5 + γ oà )c o c à = γ Bà c B + γ oà c o (γ à' + γ à5 )c à,

15 where c ',5 à are the populations of the NV states. The steady state solution to these equations provides the optical saturation rate r é }[\ of the NV: r é }[\ = a b with a = γ oà γ à' γ B' + γ Bà + γ Bà γ à5 γ B' + γ oà + ε 3γ B' γ à' γ B' + γ Bà + γ à5 3γ B' + γ Bà γ B' + γ oà and b = γ oà γ à' + γ Bà γ oà + γ à5 + ε γ Bà γ oà + γ à' (2γ Bà γ oà ) + γ oà γ à5 + γ B' γ Bà + 2γ oà + 3 γ à' + γ à5 + ε 5 γ oà γ Bà γ à5 + γ B' γ Bà + γ à' + γ à5 where we have used that γ B' = γ o5. The NV fluorescence intensity, I, scales with the excitation rate as I = I ) q ë q ë æìí Àq ë, allowing r é to be extracted from a saturation curve measurement. At low laser powers (r é r é }[\ ) the NV excited state population and hence the polarization of nearby Ns is strongly dependent on r é. This dependence reduces with increasing optical power, but other detrimental effects such as thermal drifts of the confocal microscope and NV blinking become an issue. Accordingly, we run with the laser stabilised at r é = 1.53 r é }[\ = 17.0 MHz and periodically optimise the laser-nv alignment during our measurements.

16 The pumping efficiency of Ns by the NV depends on the fraction, f Ù, in Eq. S2. Since we have assumed that N spins have a negligible impact on the NV dynamics we take P ) = c B and P &' = c o /2. In the steady-state the fraction f Ù is given by: f Ù = c d = with c = r é (2γ oà γ à' + γ Bà γ à5 ) + εr é (4γ B' γ à' + 2γ Bà γ à' 2γ oà γ à' + 4γ B' γ à5 + 2γ oà γ à5 ) + ε 5 r é (2γ B' γ à' + 2γ B' γ à5 γ Bà γ à5 + 2γ oà γ à5 ) and d = 2 γ Bà γ oà γ à' + γ à5 + γ B' γ oà γ à' + γ Bà γ à5 + r é γ oà γ à' + γ Bà γ oà + γ à5 + ε r é γ oà γ à' + γ à5 + 3γ 5 B' γ à' + γ à5 + γ Bà 2r é γ à' + γ oà r é + γ à5 + γ B' 3γ Bà γ à' + γ Bà γ à5 + 3γ oà γ à5 + r é γ Bà + 2γ oà + 3 γ à' + γ à5 + ε 5 r é γ Bà + γ oà γ à5 + γ B' γ Bà + γ à' + γ à5. The degree of N spin polarization achievable is ultimately bound by p GH, p GH = 2 ô Ð ' 5 = (S3) with e = 2γ oà γ à' + ε 2γ B' γ à' 2γ oà γ à' + 2 γ B' + γ oà γ à5 + ε 5 2γ B' γ à' + 2 γ B' + γ oà γ à5 and f = 2γ oà γ à' + γ Bà γ à5 + ε(4γ B' γ à' + 2γ Bà γ à' 2γ oà γ à' + 4γ B' γ à5 + 2γ oà γ à5 ) + ε 5 (2γ B' γ à' + 2γ B' γ à5 γ Bà γ à5 + 2γ oà γ à5 ).

17 This value is independent of the laser power. The f Ù and p GH values indicate that approximately 10% of the time the NV is in a state that can pump nearby Ns. The remaining term to consider in Eq. S2 is the NV-N dephasing. The NV GS and N spin dephasing is governed by magnetic noise along the z axis, hence their dephasing rates are similar (on the order of 1 µμs). The dephasing between the NV ES and the N is dominated by the ES to GS decay rate and a spin dephasing term γ^_ (=1/10.9 ns) due to spin conserving orbital jumps within the NV excited state orbital doublet [7]: γ GH,G = 1 2 γ B' + γ Bà + γ o5 + γ oà 2 + γ^_ = 85.3 MHz. As γ GH,G is larger than any other rates in Eq. S2, small detunings 10MHz between the NV ES and N spin transitions do not have a significant impact on the N polarization. 1 Polarisation of spin Coupling Ω(MHz) Supplementary Figure 5: Blue and orange circles show numerical solution of N and NV spin polarization, respectively, with increasing coupling Ω for fixed Γ O = 10 khz. The gray lines show the analytical solution result. To test the validity of our assumptions, we numerically calculate the steady-state solution of the full NV-N coupled system. This involves a 7 2 = 14 state basis set (the six sublevels in

18 the NV GS and ES plus the metastable level) (N down and up) and corresponding coupled rate equations governing the density operator ρ. This satisfies b} ρ = i H \]\,^_, ρ + γ D σ ρ, where γ describes the NV decay, excitation rates and Γ G. The Lindblad superoperator is D σ ρ = σρσ À ' 5 (σà σρ + ρσ À σ), and population decays and excitations are the only source of dephasing. In Supplementary Figure 5 we show the steady state numerical solution of p G (blue full circles) and p GH (orange full circles) as a function of Ω for Γ G = 10 khz. We can see that p GH is not strongly perturbed even for large Ω. The grey lines in Supplementary Figure 5 are the constant p GH extracted from Eq. S3 and p G from Eq. S2. These follow the numerical result closely indicating the approximations we make are valid. Time dependent NV-N master equation: So far we have dealt with the NV-N system in the steady-state. We extend the results from the previous section to the time-dependent case in order to model the experiments we perform involving pulsed laser excitation. We use Torrey s method which provides a general form for the time dependent solution of a master equation [8]: p O t = Ae &üý + Be &þý cos(bt) + C b e&þý sin(bt) + D. The terms a, b and c are dependent on the rates of the master equation and are independent of initial conditions, whereas coefficients A, B, C and D depend only on initial parameters. In the limit of γ GH,G Ω, Γ G we have:

19 a = γ GH,G + Γ G, 2 b = i γ GH,G Γ G 2 f!ω 5 γ GH,G and c = γ GH. Exponents including c decay quickly and hence can be ignored and the oscillatory terms are heavily damped due to large γ GH resulting in the following time dependent expression for our NV-N system under optical excitation: p G t ) + t = F + p G t ) F e & # $À# Ô ý where F = % ÔÕ# $ # $ À# Ô, Γ < = & 'Ò n ( ÔÕ and p G t ) is the initial N spin polarization. With the laser turned off (leading to Ω = 0) the equation is simply: p G t ) + t = p G t ) e &# Ôý. During our IDSE sequence we repeatedly apply the optical sequence shown in Supplementary Figure 6. t read t int t IDSE t wait IDSE measurement

20 Supplementary Figure 6: the optical pulse sequence used during the IDSE measurement. We find the polarization p G at the time of the IDSE measurement by nesting the time-dependent equations above to give p G t ;d\,, t â_^, t b[:, t )[;\ = ('&ô * +,ìå (-. /- 0) Àô (* +,ìå /* 1ìŸí )-. /* +,ìå -0 &ô * 1ìŸí -. /(* Ÿ í /* +,ìå )(-. /- 0) )%ÔÕ # 0 ('&ô (* 23ÜÛ /* Ÿ í /* 1ìŸí )-. /(* Ÿ í /* +,ìå )-0 )(#. À# 0 ). (S4) 1 Polarisation of N spin Initialisation time (μs) Supplementary Figure 7: Degrees of polarization for each N spin N 1, N 2 and N 3 in black, blue and red, respectively, extracted from fits to data in Supplementary Figure 3 shown as circles with corresponding error bars. Solid lines are fits to the rateequation-based polarization model. We fit the polarization buildup data shown in Supplementary Figure 7 using the result of Equation S4 to extract Ω ; couplings. The optical excitation time for readout is given by t b[: = 0.8 μs initialisation time, t ;d\ is 2, 6 and 20 μs, the time taken for the IDSE sequence is t â_^ =

21 4 μs and the waiting time, t )[;\, is adjusted to keep the total cycle length at 21.4 μs. Under the assumption of Gaussian noise on p ; values, we extract a distribution of Ω ; couplings and rates Γ G from the fits. This distribution is represented in spatial coordinates in Fig. 3 of the main text (where the dependence of Ω on r and θ is given in section 2 of the Supplementary).

22 5. N spin lock about y axis We perform a control measurement to test that the difference of D + and D - seen in the inset of Fig. 4b in the main text indeed arises from the optical polarisation step and not from rotation errors during the sequence. Here we perform a measurement identical to the one shown in the inset of Fig. 4b but this time we lock the N spins about the y axis instead of the y axis. The data and fits to a Cos 2πντ e &ý ý a C e &(ý ý 8579) + C are shown in Supplementary Figure 8. By changing the locking direction, we exchange the two traces D + and D - and the measured traces invert. 1 Contrast D + D - A τ(μs) Supplementary Figure 8: Hartmann-Hahn locking with N spins locked about -y axis

23 6. IDSE with optical and Hartmann-Hahn polarization The concatenation of optical and Hartmann-Hahn polarization allows for targeted polarization transfer to individual cluster spins. We apply an optical initialization pulse for 2 μs, followed directly by a spin lock of 280 ns = 1/(2Δ ' ), which leads to a spin exchange between the NV and spin N1. The data is shown in Suppl. Fig. 9 in green circles (lock up HH, as defined in the main text), red circles (lock down HH ) and black circles (no spin lock, only optical polarization). The NV phase evolution due to polarized spin N1 produces a clear oscillation on the overall shape, and this oscillation is reversed when the lock is performed in the opposite direction. The black data is similar to Fig. 2d of the main text, with the shape dominated by the slower phase evolution of spins N2 and N3. The parameters extracted from the fits (see section 3 of the Supplemental Material) are shown in Suppl. Table 2. The polarization of N1 is raised from 5% (purely optical polarization) to 17% (optical and Hartmann-Hahn polarization), an almost three-fold increase, while the polarisations of N2 and N3 remain little affected by the Hartmann-Hahn scheme (8% to 10% for N2 and 51% to 53% for N3). This result demonstrates

24 that choosing the lock time to match the coupling 1/(2Δ) of a particular spin enables the targeting of enhanced polarization transfer to individual spins. 50 Phase difference(degrees) τ (ns) Supplementary Figure 9: IDSE measurement of cluster state after optical and Hartmann- Hahn polarization Fit values for N=3 Standard error p ':: (%) 17 1 p ':: (%) p ']<\ (%) 5 1 p 5:: (%) 10 1 p 5:: (%) 5 1 p 5]<\ (%) 8 1 p B:: (%) 53 1 p B:: (%) 42 1 p B]<\ (%) 51 1 Supplementary Table 2: Best fit parameters of 3-spin model with standard errors.

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