High-fidelity spin and optical control of single silicon-vacancy centres in silicon carbide

Scalable quantum networking requires quantum systems with quantum processing capabilities. Solid state spin systems with reliable spin–optical interfaces are a leading hardware in this regard. However, available systems suffer from large electron–phonon interaction or fast spin dephasing. Here, we demonstrate that the negatively charged silicon-vacancy centre in silicon carbide is immune to both drawbacks. Thanks to its 4A2 symmetry in ground and excited states, optical resonances are stable with near-Fourier-transform-limited linewidths, allowing exploitation of the spin selectivity of the optical transitions. In combination with millisecond-long spin coherence times originating from the high-purity crystal, we demonstrate high-fidelity optical initialization and coherent spin control, which we exploit to show coherent coupling to single nuclear spins with ∼1 kHz resolution. The summary of our findings makes this defect a prime candidate for realising memory-assisted quantum network applications using semiconductor-based spin-to-photon interfaces and coherently coupled nuclear spins.

Using a wavemeter (Coherent WaveMaster) with a resolution of 0.1 GHz, we inferred that the A2 optical transition is the one at higher energy. We now want to infer whether this transition links the spin sublevels | ± We now consider both cases of es (positive and negative): Assuming that es is positive, the A2 optical transition links the S = ± 3 2 sublevels, such that the ground state spin initialization procedure (laser excitation on the A2 transition and continuous MW3 excitation in order to depopulate |+ Assuming on the other hand that es is negative, then A2 links the S = ± 1 2 sublevels, such that the ground state spin initialization procedure would lead to a near deterministic population of |− 3 2 〉 gs . Thereafter, MW2 is applied for Rabi , which does not alter the ground state spin population at all. In addition, the population swappulse at MW3 has also no effect. As a consequence, the eventual population readout of the states | ± 1 2 〉 gs should lead to no observable signal. The experimental results in Figure 3(b) in the main text clearly support therefore that the excited state zero field splitting is positive, i.e. 2 • es = 985 ± 10 MHz.
We note that our ab initio calculations 5 resulted in positive zero-field constant both for the ground and excited states that strongly supports the analysis of experimental results.

Supplementary Note 3. Landé g-factor and optical transition energies of the V1 excited state.
The spin Hamiltonian of ground and excited states with an external axial magnetic field is: gs,es = gs,es ⋅ 2 + gs,es B 0 ⋅ .
Here, the subscripts gs and es denote ground and excited states, respectively.
2 • gs,es denotes the zero field splitting, is the spin projection operator in the zdirection, gs,es is the Landé g-factor, B is the Bohr magneton, and 0 the strength of the axial magnetic field.

sublevels.
Here, Δ gs,es ≈ 1.44 eV is the energy difference between ground and excited states.
At zero magnetic field, one expects two pairwise degenerate spin-conserving optical transitions for the sublevels S = ± 1 2 and S = ± 3 2 . For the latter transition (A2 in our notation), we observe a linewidth (full width at half maximum (FWHM)) of 0 =0 G = 87.6 ± 1.6 MHz. At 0 ≠ 0 G, one would expect to see four optical spin-conserving transitions, provided that there is a sizeable difference in the Landé factors gs and es . As shown in Figure 1(c) in the main text, we do not observe any additional transitions at 0 = 92 G. The linewidth of the A2 transition remains essentially unchanged, i.e. 0 =92 G = 87.7 ± 1.6 MHz. Consequently, we conclude that the change in lin- We assume now that the optical transitions between the sublevels S = ± Using this equation, we infer a displacement of 0 = 0.2 ± 11.6 MHz. By using this result, we now infer the difference in ground and excited state Landé g-factors: Since previous studies 3 have already reported gs = 2.0028, we determine the excited state Landé factor to be es = 2.0033 ± 0.0300.
In addition, as the ground and excited state g-factors have been determined to be nearly identical, if spin-flipping optical transitions (|Δ S | = 1) were allowed, they should appear at ± B 0 ≈ ±258 MHz compared to the spin-conserving transitions.
However, such transitions have not been observed as shown in Figure 1(c).

Supplementary Note 4. Inhomogeneous broadening of optical resonance
In the main text, we report 60 MHz linewidth of single silicon vacancy resonant optical transition. Although this is close to the lifetime-limit, we still observe a small residual inhomogeneous broadening. We attribute the broadening to local electric field fluctuations originating from free carriers injected by ionized defects in proximity. Although the change in dipole moments by optical pumping is minimal, since it still can be cou- charged silicon vacancy centre in 4H-SiC.
In the main text, we report studies on resonant optical excitation spectra that show an outstanding spectral stability in contrast to the nitrogen-vacancy (NV) centre in diamond 6 . We attribute the small inhomogeneous distribution (see Figure 1(d) in the main text) to a low sensitivity of the defects to surrounding electric field fluctuations originating from other defects. Since this may be related to a small dipole moment of the V1 centre in 4H-SiC, we test this hypothesis by performing theoretical calculations as described in the following. We calculate the change in the polarisation for the excitation process between 4 A2 ground and 4 A2 excited states of the negatively charged V1 centre in 4H-SiC (silicon vacancy defect on a hexagonal lattice site). We compare these results with the ones obtained for the nitrogen-vacancy (NV) centre in diamond.

Computational details
We apply density functional theory (DFT) for electronic structure calculation and geometry relaxation using the plane-wave-based Vienna Ab initio Simulation Package (VASP) [7][8][9][10] . The core electrons are treated in the projector augmented-wave formalism 11 . For the 4H-SiC supercell, calculations are performed with 420 eV plane wave cut-off energy and with centred 2 × 2 × 2 k-point mesh to sample the Brillouin zone.
For the diamond supercell, we use 420 eV plane wave cut-off energy and -point to sample the Brillouin zone. We apply Perdew-Burke-Ernzerhof functional in these calculations 12 . The model for the silicon vacancy defect in bulk 4H-SiC is constructed using a 432-atom hexagonal supercell, whereas we use the 512-atom simple cubic supercell to model the NV centre in diamond. The excited state electronic structure and geometry is calculated by constraint occupation of states, or Delta Self-Consistent We calculate the permanent polarisation in ground and excited states, and their difference, in order to infer the coupling to the optical transition. To this end, we use the VASP implementation of both Born effective charge calculation using density functional perturbation theory (DFPT) 14 and the Berry phase theory of polarization [15][16][17] . In a DFT calculation, one can define the change in macroscopic electronic polarisation ( ) as an adiabatic change in the Kohn-Sham potential ( ) where is the occupation number, the elemental charge, e the electron mass, the cell volume, the number of occupied bands, ⃗ the momentum operator, is the adiabatic parameter, is the band energy. The first part of the equation corresponds to the electronic part of the permanent polarisation ( el ), whereas the second part corresponds to the contribution of ions ( ion ) to the permanent polarisation. In a periodic gauge, where the wavefunctions are cell-periodic and periodic in the reciprocal space, the permanent polarisation takes a form similar to the Berry phase expres- Using DFPT, | 〉 can be calculated from the Sternheimer equations with similar self-consistent iterations as in DFT: We determine the radiative transition rate between the ground and excited 4 A2 states by calculating the energy dependent dielectric function r ( ). The spontaneous transition rate is given by the Einstein coefficient where is the refractive index, ℏ is the transition energy, is the optical transition dipole moment, 0 is the vacuum permittivity, and is the speed of light. is proportional to the integrated imaginary dielectric function ( ) of the given transition: where is the volume of the supercell.

Theoretical results
The results of the Berry phase evaluation for macroscopic dipole moment calculation are shown in Supplementary Tables 1 and 2 for the V1 centre in 4H-SiC and the NV centre in diamond, respectively. The change in the total dipole moment is about 20 times larger for NV centre in diamond with respect to that for V1 centre in 4H-SiC. This means that the V1 centre has intrinsically low coupling strength between optical transition and stray electric fields. Supplementary

Experimental results and discussion
Preliminary studies have been performed to constrain tot via Stark shift control of optical transition frequencies.
To this end, we spanned two parallel copper wires over the sample in order to apply an electric field. The wires were separated by approximately 100 µm and voltages up to ±200 V were applied. We note that by applying higher electric fields led to electrical breakdown in the cryostats low-vacuum atmosphere. Considering the relative permittivity of 4H-SiC ( r ≈ 10), this results in an estimated in-crystal field of about ±200 kV m . Two experiments were performed, one in which the electric field was applied along the crystal's c-axis, and a second one in which the field was orthogonal to the c-axis. We performed resonant excitation studies at zero magnetic field ( 0 = 0 G) as a function of the electric field strength, in analogy to the studies shown in Figure 1 This is approximately two orders of magnitude smaller than reported for NV centres in diamond (≈ 6.3 GHz

MV m
) 18 , such that we estimate tot < 0.009. However, theory implies that the coupling coefficient is about an order magnitude smaller for V1 centre in SiC than that for NV centre in diamond. We show below that a compensating field can be de- The small spectral diffusion may be understood by assuming that another donor defect lies near the V1 centre with about the same distance but another location where the illumination will activate that donor, and the resulting electron, free carrier, will be captured by the previously positively charged donor defect. According to a previous study 20 , low energy Si atoms will produce vacancy defects in about 10 nm region; thus Si atoms that are created by 2 MeV electron irradiation have much higher kinetic energy and should produce vacancies, antisites and interstitials at larger distances, around 40 nm and larger distances. By applying an external electric field that is parallel to the symmetry axis of the V1 centre, illumination will again ionize a donor defect but the electric field will drag the electron in the opposite direction of the electric field. One of the carbon vacancies around V1 centre will capture this electron, and the positive donor and negative carbon vacancy will form an electric field that mostly screens the external electric field. We find that if these defects are both 40 nm apart from the V1 centre along the symmetry axis then they shield the external electric field to about 10% of its magnitude. As a consequence, the resulting Stark-shift agrees with the experimental data ( tot < 0.09). Although, this estimation is crude as the statistics about the point defects around V1 centre is not known but our scenario still explains all the experimental findings. We think that both the small coupling constant of the V1 centre and the shielding effects created by the donor and acceptor point defects around the V1 centre are responsible for the spectral stability of the V1 centre.

Supplementary Note 6. Electronic fine structure and spin polarization of the V1 defect.
A simplified electronic fine structure model of the V1 defect is shown in Supplementary back to gs1 and from ds2 back to gs2 are labeled as 3 and 4 , respectively, and they are responsible for the shelving lifetime of this defect.
We obtain the relationships between 1 and 2 , as well as 3 and 4 , by evaluating the corresponding DSO matrix elements, = 2 ℏ |〈 gs |Σ . | ds 〉| 2 , in the symmetry adapted ds/gs wave-functions basis of = 5 active electrons 27 . We find that 2 metastable doublet states (with symmetries A1 and 2E) have non-zero DSO matrix elements that can participate in the ISC. The E-symmetry ds (e 3 ) can couple to both es and gs by the orthogonal component of the DSO ⊥ . ⊥ (with respect to the c-axis).
The remaining A1 ds (ve 2 ) is strongly hybridized with the E ds (e 3 ) by the ∥ . ∥ component of the DSO. As a result, the overall ISC can be simplified into two doubly degenerate metastable states, labeled as ds1 and ds2, which can only couple to the spin ± 1 2 ⁄ or the ± 3 2 ⁄ states of the es and gs, respectively.
Resonant optical excitation along the A2 transition (gs 2 → es 2 ) will lead to an optical pumping into the gs1. Therefore, to observe a measurable signal during PLE, the ground state spin must be able to relax. Alternatively, one can use MW pulses on the gs spin states for coherent control as well as to overcome this optical pumping. During a continuous broadband MW pulse where all three gs spin transitions shown in Figure   2(a) are allowed to relax equally, the PLE amplitude mismatch between A1 and A2 is directly determined by the ISC rates of 1 and 2 . We find that the faster 1 rate causes more population to be removed non-radiatively from |± The es ZFS sign, which is determined as in the above section (S3), can also be confirmed by comparing this model to a series of MW schemes in the presence of a magnetic field 0 = 96 G (i.e. B 0 z ≫ 2 gs ). In these schemes, a continuous broadband MW is scanned from 252MHz to 272MHz and at each frequency the MW centered the A2/A1 peak ratio is determined from the corresponding PLE spectra. At 258 MHz, all three spin transitions of the ground state (see Figure 2(a)) are allowed to depopulate and the A2/A1 ratio is solely determined by the 1  In the main text, we report a spin decoherence time of 2 = 0.85 ± 0.12 ms, measured by Hahn echo. Although this result is better than the previously reported values 4,21-23 , one may anticipate reduced decoherence rates owing to the use of an isotopically purified nuclear spin free 4H-SiC sample, in analogy to previous experiments with isotopically purified diamond and silicon 24,25 .
As we used a rather low dose of electron beam irradiation to create defect centres, the concentration of paramagnetic defects is small (∼ 10 13 cm −3 ) as explained in Methods section, which cannot explain the observed 2 times. Shallow nitrogen donors are also discarded as a major decoherence source as their concentration is also too low to be significant (∼ 3.5 ⋅ 10 13 cm −3 ) since the equivalent spin dipole-dipole interaction in electronic spin bath requires higher total impurity concentration ( ∼ 6 • 10 14 cm −3 ) 21 .
We therefore attribute the main decoherence source to be undesired defects near surface paramagnetic defects created by cutting with a typical concentration in the low 10 13 cm −2 range in the region about ~1 µm from the surface 26 . Before irradiation, samples were annealed to 1130 °C in N2 gas flow to reduce the concentration of paramagnetic surface defects to below detection of Electron Paramagnetic Resonance (EPR) (below 10 12 cm −2 ). Since the N2 annealing is known to reduce only 10% of the surface defects and the EPR experimental conditions are not optimized for the detection of the surface defects, it is safe to assume that the surface defect concentration within ~1 m range from the surface has an upper limit of 10 16 cm -3 . As the optical transition dipole of the investigated defect centres is parallel to the crystal's c-axis, we had to flip the sample by 90°. This comes with the trade-off that all observed defects are located close to the cutting surface. The cutting surfaces are also expected to have structural defects induced by micro cracks caused by cutting. In the future, this issue can be addressed by improved sample processing 26 or by growing SiC layers on a-plane substrate so that solid immersion lens can be fabricated on as-grown surfaces, which contain no such defects.

Supplementary Note 8. Additional data for orientation and polarisation of the optical transitions.
Previous studies have already shown that the silicon vacancy centre at a hexagonal lattice site (V1 centre) is most effectively excited using a linearly polarised off-resonant laser whose polarisation is parallel to the crystal's c-axis 3,4 .
Here, we investigate the behaviour of the individual optical transitions A1 and A2 under resonant excitation. As we have shown in the main text, when applying a magnetic field, no additional optical transitions are observed, supporting the absence of circularly polarised optical transitions.
In the following, we show that both transition dipoles are indeed linearly polarised and parallel to each other. To this end, we applied broadband microwaves in order to continuously mix the ground state populations, and performed resonant optical excitation along either A1 or A2 at an intensity of about 1 W cm −2 . The polarisation of the excita-