Quantum properties of dichroic silicon vacancies in silicon carbide. Science and Technology, IQST, Pfaffenwaldring 57, D Stuttgart, Germany

Transcription

1 Quantum properties of dichroic silicon vacancies in silicon carbide Roland Nagy 1 *, Matthias Widmann 1 *, Matthias Niethammer 1, Durga B.R. Dasari 1, Ilja Gerhardt 1,2, Ö ney O. Soykal 3, Marina Radulaski 4, Takeshi Ohshima 5, Jelena Vučković 4, Nguyen Tien Son 6, Ivan G. Ivanov 6, Sophia E. Economou 7, Cristian Bonato 8, Sang- Yun Lee 9 & Jörg Wrachtrup 1, Institute of Physics, University of Stuttgart and Center for Integrated Quantum Science and Technology, IQST, Pfaffenwaldring 57, D Stuttgart, Germany 2. Max Planck Institute for Solid State Research, Heisenbergstra\ss e 1, D Stuttgart, Germany 3 Naval Research Laboratory, Washington, DC 20375, USA 4 E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, United States 5 National Institutes for Quantum and Radiological Science and Technology, Takasaki, Gunma , Japan 6 Department of Physics, Chemistry and Biology, Linköping University, SE Linköping, Sweden 7 Department of Physics, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061, USA 8 Institute of Photonics and Quantum Sciences, SUPA, Heriot-Watt University, Edinburgh EH14 4AS, UK 9 Center for Quantum Information, Korea Institute of Science and Technology, Seoul, 02792, Republic of Korea *These authors contribute equally. sangyun.lee@kist.re.kr

2 Abstract: The controlled generation and manipulation of atom-like defects in solids has a wide range of applications in quantum technology. Although various defect centres have displayed promise as either quantum sensors, single photon emitters or light-matter interfaces, the search for an ideal defect with multi-functional ability remains open. In this spirit, we investigate here the optical and spin properties of the V1 defect centre, one of the silicon vacancy defects in the 4H polytype of silicon carbide (SiC). The V1 centre in 4H-SiC features two well-distinguishable sharp optical transitions and a unique S=3/2 electronic spin, which holds promise to implement a robust spin-photon interface. Here, we investigate the V1 defect at low temperatures using optical excitation and magnetic resonance techniques. The measurements, which are performed on ensemble, as well as on single centres, prove that this centre combines coherent optical emission, with up to 40% of the radiation emitted into the zero-phonon line (ZPL), a strong optical spin signal and long spin coherence time. These results single out the V1 defect in SiC as a promising system for spin-based quantum technologies. Introduction: Quantum technologies based on solid-state devices can take advantage of wellestablished fabrication and control methods developed over the past century. Several quantum systems, such as superconducting circuits [1], trapped ions [2], and quantum dots [3] have been extensively used for implementing various quantum information processing tasks. In this context, color centres in diamond [4 6] have gained prominence over the past decade as quantum-enhanced nanoscale sensors [7 9], coherent spin-photon-phonon interfaces [10 12] and quantum registers [13] for quantum networks. Despite their success, the limited emission rate of indistinguishable

3 photons of e.g. the nitrogen vacancy centre and the difficulties of diamond nanofabrication currently inhibit the progress towards efficient and scalable spin-photon interfacing devices [14,15]. Defect spins in silicon carbide (SiC) have been studied as an analogue to diamond colour centres, due to their promising complementary properties and the established technologies in growth, doping and device fabrication [16]. As in diamond, defect spins in SiC exhibit long coherence times [17 19] and optically detectable spin signals at room temperatures [20 22], down to the individual spin level [23,24]. SiC hosts several defects with addressable electronic spins, including silicon vacancies [19,24], divacancies [18,23], and transition metal impurities [25 27]. While most defects used in quantum technology, such as the nitrogen-vacancy (NV) centre in diamond [5], and the divacancy [18,23] and chromium impurities [26,27] in SiC, host an S=1 electronic spin, the silicon vacancy (VSi) in hexagonal SiC features a S=3/2 electronic spin in uniaxial crystal lattices. This brings several advantages for quantum technology. First, it enables the reconstruction of all three magnetic field components [28] with a single defect. First, according to Kramers theorem [29], the degeneracy of a half-integer spin system can only be broken by magnetic fields, making it extremely insensitive against fluctuations in strain, temperature and electric field. Furthermore, the same Landé g-factor of ground and excited states makes the optical transition energy independent of the applied magnetic field [30,31]. These factors have led to a theoretical proposal of a robust interface between spin and photon polarization, which is not perturbed by environmental noise [32,33]. Additionally, while other defects exist in several different orientations in the crystal lattice [34,35], VSi at each inequivalent lattice site in both hexagonal and rhombic SiC polytypes exhibit only one single spin orientation along the c-axis of the crystal [20,22], allowing for a deterministic orientation in a fabricated device, which increases the scalability..

4 As shown by experiments on the NV centre in diamond [10] and the divacancy in SiC [36], the establishment of a coherent spin-photon interface based on defects in solids requires cryogenic operation. From this point of view, the properties of the VSi centre at cryogenic temperatures have not been sufficiently studied and need further exploration. In this work, we address this issue by characterising the low-temperature optical and spin properties of one of the two VSi centres residing at two inequivalent lattice sites of in 4H-SiC, known as the V1 centre [30]. In the first part of the paper, based on ensemble spectroscopic measurements, we first extract basic optical properties of the V1 centre. This defect exhibits two sharp ZPLs, known as V1 and V1, which form the basis of the robust spin-photon interface, which was proposed recently [32]. We then show how resonant optical excitation leads to an efficient spin polarization and readout, resulting in nearly 100% relative increase in optically detected spin signal. Further, we investigate the spin properties under different driving radio-frequency fields and show how dynamical-decoupling improves spin coherence time into the millisecond range. In the second part of the paper, we report, a comprehensive characterization of the optical properties of a single V1 centre at low temperatures. We highlight a Debye- Waller factor (DWF), i.e. the fraction of radiation into the ZPL, of up to 40%. We conclude with some considerations about the prospects to realise a robust spin-photon interface [32,33]. Results Optical Properties: Silicon vacancies in 4H-SiC can occur in two inequivalent lattice sites; the negatively-charged states have been associated with two distinct ZPLs, labeled as V1

5 and V2 [30]. In this work, we focus on the less explored V1 centre. The theoretical energy-level scheme, proposed on the basis of group-theoretical analysis, is sketched in Fig.1a. The ground state of V1 is a spin quartet of symmetry 4 A2 (a 2 1 a 1 1 e 2 ) [30] and total spin S=3/2. The notation in brackets here, a 2 1 a 1 1 e 2, denotes the populations of the one-electron levels deduced using group theory, as described in [30]. The ground and sublevels of V1 are split by a zero-field splitting (ZFS) of 4 MHz [30]. Two excited states ( excited from the ground state ) 4 E (a 1 1 a 1 1 e 3 ) and 4 A2 (a 1 1 a 1 2 e 2 ) can be selectively via resonant laser excitation with ev (858 nm) and ev (861 nm), known as V1 and V1 ZPLs, respectively [30,37]. The ZPLs originating from the two excited states are strongly broadened at elevated temperatures, and are clearly visible only at low temperatures as shown in Fig. 1b. The line V2, associated to a VSi on a different inequivalent lattice site, is located at ev (916 nm) [30,31]. At a temperature of 5.5 K, the emission of the V1 ZPL transition ( 4 A2 to 4 A2 ) dominates. With increasing temperature, the intensity of the V1 ZPL decreases, whereas the V1 ZPL emission ( 4 E to 4 A2 ) becomes more prominent and peaks at around 70 K, as shown in Fig. 1e. The energy difference between the ZPLs of V1 and V1 is about 4.4 mev which corresponds to a thermodynamic equivalent temperature of 51 K. The enhanced emission from the V1 transition at elevated temperatures may be understood as a phonon-assisted process, where the temperature-induced dephasing of the orbital excited states leads to thermally-induced excitation transfer among the excited states between V1 and V1 excited states. The transfer reaches a maximum when temperature-induced mixing (dephasing) becomes comparable to the splitting between the energy levels. The experimental observation, however, cannot be explained by a simple thermally induced excitation transfer but rather requires a more involved mixing of excited states. We performed numerical

6 simulations of the Lindblad master equation for a four-level system with two ground and two excited states as shown in Fig. 1f using the experimental decay rates of the V1 and V1 excited states (to be discussed later). In this model, as discussed in the Supplementary Information (see S2), we introduced, in addition to the normal decay channels from the excited states, a temperature-dependent mixing of the excited states at a rate with a power-law dependence on temperature. We find that the best fit to our data occurs for α~1.57. This is consistent with the inter-level phonon coupling (pseudo Jahn-Teller effect) between the V1 and V1 states mediated by the E- symmetry vibrational normal modes of the lattice [38]. To characterise V1 and V1 ZPLs towards their use for a spin-photon interface as proposed in Refs. [32,33], we investigated the polarisation of the emitted photons. Some polarisation studies have already been reported in the literature [30,31], but only at two orientations, with electric field E parallel (E c) and perpendicular (E c) to the crystal c-axis. Fig. 2 shows PL spectra as a function of the rotation angle of a half-wave plate (HWP) and analysed by a polarising beam-splitter in front of a spectrometer. Since the sample has the c-axis perpendicular to its surface, the analysis of both E c and E c polarisations of the emitted photons requires that the luminescence is excited and detected through the side surface of the sample (see the sketch in Fig. 2a). For completeness, polarisation is also measured through the surface of the sample (Fig. 2b). The polar plots in Fig. 2 represent the integrated intensity of each of the V1 and V1 ZPLs as a function of the rotation angle of the HWP. The measurements presented in Fig. 2 were taken at T=5.5 K, where the V1 line dominates. As seen from Fig. 2a, the dominant polarisations of V1 and V1 ZPLs are almost orthogonal to each other. The full orientation analysis results are in qualitative agreement with previously suggested optical selection rules based on group-theoretical analysis within the single group C3v, representing the symmetry of VSi [30,31]. This analysis predicts E c

7 polarization for the V1 transition and E c polarization for the V1 transition. However, while the photons originating from V1 are quite well linearly polarised E c, the V1 transition is not entirely polarised as E c but contains a component E c. Our findings indicate that the simplified selection rules need revision. Since the negatively-charged VSi contains an odd number of electrons (resulting in half-integer spin), the correct symmetry is the double group C 3v, as previously suggested [32]. The derivation of the selection rules for C 3v leads to a better estimate of the relative contribution of the E c and E c polarisations in the optical emission of the V1 and V1 lines. This is outlined in the Supplementary Information S3. We find that for the V1 transition the distribution among the two polarisations is E c : E c = 3:1, whereas for the V1 transition the proportion is E c : E c = 11:1. These estimates are in excellent agreement with the polar plots in Fig. 2. Fig. 2b shows that, as expected, only photons with E c polarisation are registered in this geometry, and all directions in the plane perpendicular to the c-axis are equivalent, as illustrated by the polar plot (see Supplementary information S4 as well). The orbital and spin properties of this defect are different from the well-studied NV centre in diamond and the divacancy in SiC. While both of the ground states of VSi and NV centres feature A2 orbital symmetry, VSi has entirely different electron and spin configuration (quartet versus triplet) with distinct optical selection rules dictated by the C 3v double group representations. The V1 transition with E c polarisation is reminiscent of the NV centre E± A2 transitions, which are circularly polarised and have been used for spin-photon entanglement [10]. In order to estimate the natural linewidth of the V1 and V1 transitions, the lifetimes of both excited states were measured by applying short laser pulses at 805 nm (see Methods). To distinguish between the optical emission of V1 and V1 lines, we

8 measured the excited state lifetime at different temperatures (5.5 K for V1, and 70 K for V1 ). The observed decay curves are illustrated in Fig. 1c. The excited state lifetimes of V1 and V1 are ns and ns, respectively. The lifetime of V1 is in agreement with the previously reported value [37] which determines the lower bound of the natural linewidth to be around 30 MHz. Optical spin state detection. To observe and control the spin-dependent contrast of the PL, the defect spins need to be polarised. It is known that the ground state spin can be polarized into the sublevels [32] by optical pumping [39] with an off-resonant laser ( =730 nm in this work) in a continuous wave mode. At any constant finite magnetic field (B0), all the spin states are energetically split. By applying resonant radio-frequency (RF) fields (B1) one can induce transitions between spin sublevels, as shown in Fig. 3b, resulting in a change in photoluminescence (optically detected magnetic resonance, ODMR). The relative change in ODMR signal is calculated as [I(f) I off ]/I off, where I(f) is the PL intensity at the RF frequency f, and I off is the PL intensity at the offresonant RF frequency. The spin-sublevels are energetically split using a small field of B 0 = 60 G aligned along the symmetry axis of the defects, which corresponds to the crystallographic c-axis (0001), i.e.. For magnetic field alignment, we use the V2 centres in the ensemble, as described in refs. [24,28,40]. The relative ODMR contrast as a function of the driving RF frequency is shown in Fig. 3a. The negative signal at 170 MHz in Fig. 3a, with contrast 0.05%, is attributed to the V1 ground state spin; a similar signal was also reported for V1 and V3 centres in 6H-SiC [39]. By exciting the optical transition V1 resonantly, with a laser-diode tuned to 861 nm, a positive relative ODMR signal with % is achieved (lower plot in Fig. 3a). In contrast, excitation

9 of the V1 optical transition (laser-diode tuned to 858 nm) reveals a negative signal with a relative signal intensity only four times larger than for off-resonant optical excitation (see Supplementary information S5). A similar substantial enhancement of the ODMR signal was reported for the V2 centre in 6H-SiC [39]. Although the underlying mechanism is not yet completely understood, we attribute it to the enhanced spin polarization in the ground state resulting from resonant optical excitation. Resonant excitation of V1 ZPL results in the excitation into the lowest vibrational level of the V1 excited state. This efficiently suppresses the phonon-assisted spin-mixing between the V1 and V1 excited states leading to an improvement in the ODMR signal at sufficiently low temperatures (kbt<4.5 mev). On the other hand, resonant excitation of V1 does not result in such an improvement as it still involves the excitation of V1 (lower in energy) and its vibrational levels. This observation may also indicate that optical polarisation is mainly established by the intersystem-crossing (ISC) between the 4 A2 excited state and the 2 E metastable state while the 4 E state has a less efficient ISC due to competition between the allowed 2 A1 and 2 E metastable states [32] favoring different spin states. The ZFS of the V1 ground state (2DV1=4 MHz) is not resolved due to inhomogeneous broadening of the spin ensemble. A strong spin-signal obviously allows for faster and high-fidelity readout. We expect further enhancement of the ODMR signal when a single V1 centre is isolated owing to the suppressed inhomogeneous broadening. The enhancement signal also has an advantage on the signal-to-noise ratio (SNR) over the quenching signal of, e.g., the NV centre in diamond and divacancy in SiC. If an identical ODMR contrast, namely C, and an identical photoluminescence (PL) intensity without spin resonance are assumed, a positive signal leads to a SNR larger by a factor of (1 C) 0.5 (See Methods). Coherent spin control

10 In order to demonstrate coherent control of the electronic spins, we investigated spin dynamics of the V1 centre ensemble under an applied RF pulse. The pulse sequence is illustrated in Fig. 4b: the spin, polarised by an optical pulse (duration 1 μs), is driven by a radio-frequency (RF) pulse of a fixed duration and frequency frf and subsequently read by a second optical pulse (duration 1 μs). The resulting Rabi oscillations, obtained by repeating this sequence with increasing RF pulse duration, are shown in Fig. 4a. In order to get the distribution of the Rabi oscillations of the four-level system, frf was swept in the range of MHz at B0=60 G. The resulting dynamics is strikingly different from the single-frequency Rabi oscillations typical of a two-level system as shown in Fig. 4a. Further understanding can be obtained by plotting the modulosquare of the Fast-Fourier transform of the Rabi oscillations as a function of frf, for different values of the driving power (Fig. 4c). While for two-level transitions one expects parabolic profiles, corresponding to a Rabi frequency Ω increasing with the detuning Δω as where Ω 0 is the driving frequency determined by the applied B1 field strength, the experimental data reveal richer and more complex dynamics. We explain our observations with the presence of three closely-spaced transitions, corresponding to,, and. While resonantly driving one transition, due to the small ZFS (2DV1=4 MHz), off-resonant excitation of the other two transitions is not negligible. To support this explanation, we developed a theoretical model based on four levels (Sz = ) driven by a single monochromatic radio-frequency field. The system dynamics is investigated assuming initial polarisation into an incoherent mixture of Sz. Further details on the model can be found in the Supplementary Information S6. Our simulations match quite closely the complex structure of the experimental data, supporting our hypothesis on the defect structure as shown in Fig. 4d. When frf is lower than f1 ( the transition f1 is mainly excited, leading to a

11 parabolic profile in Fig. 4c,d. However, off-resonant excitation of the transition f2, coupling to, results in a second weaker Fourier component in the Rabi spectrum with larger Rabi frequency. With increasing RF power (B1 field strength), this component becomes stronger. When frf=f2, one simultaneously drives offresonantly the transitions f1 and f3, resulting in a larger Rabi frequency. This is evident in the plots corresponding to the largest RF power, where the parabolic profile centred around f2 shows a much larger Rabi frequency than the profiles related to f1 and f3. Note that the experimental data can only be explained by assuming the excitation of the Sz Sz transition, which was not reported. Additionally, the assumption for initial polarisation into, which is the case for the V2 centre, does not reproduce the observed signal. The small ZFS poses challenges for high-fidelity coherent spin control, which need to be addressed for the V1 centre to be a serious contender for quantum technology. There are several possibilities to explore: use of (i) a soft (long and weak) pulse for better selectivity [41], (ii) optimal quantum control sequences, (iii) adiabatic passage techniques that restrict the dynamics only to a two-level subspace (e.g. and ), with no leakage to other states of the Hilbert space [42],(iv) pulses designed to avoid a transition by building holes in their frequency spectrum, routinely used by the superconducting qubit community to avoid leakage [43 45], (v) superadiabatic (shortcuts to adiabaticity) control [46], which was recently demonstrated for NV centres in diamond using optical pulses [47]. Alternatively, the V1 in 6H-SiC is known to exhibit a larger ZFS [20,30], which would limit this problem. Decoherence Measurements: It is important to analyse the electron spin decoherence of the V1 ground state since a long coherence time is mandatory to guarantee efficient quantum protocols

12 demanding a long-lived quantum memory such as in spin-photon interfaces [41]. We studied spin coherence at T= 5.5 K with B0= 60 G and 1000 G by Ramsey, Hahn-echo and XY-8 dynamical decoupling pulse sequences. Spin measurements were performed by optical excitation resonant with the V1 ZPL (861 nm). We started with a Ramsey measurement at B0= 60 G to test the electron spin coherent evolution and dephasing time. As discussed above, coherent spin control poses a challenge for a four-level system with a small ZFS (see supplementary information S7). In the following experiments, we take the duration of a -pulse as the minimum time required for transferring the maximal population from ±3/2 to ±1/2 subspace. Experimentally, this is signalled by the maximum contrast in the observed Rabi oscillation signals. The Ramsey pulse sequence can be seen in Fig. 5a. After optical spin polarization, we applied a pulse to create a superposition between and sublevels. Another pulse was applied after a free precession time for a projective readout before the readout laser pulse. We observed an evolution of the coherent superposition, with frequency corresponding to the detuning from the resonant transition, and spin dephasing time T 2 = ns and µs at B0=60 G and 1000 G, respectively, as shown in Fig. 5b. To suppress the inhomogeneous broadening in an ensemble and decouple the spin ensembles from low frequency spin noise sources, we applied a Hahn-Echo sequence. The Hahn-Echo pulse sequence can be seen in Fig. 5a, which adds a pulse between two pulses, to refocus the dephased spin ensemble due to inhomogeneous broadening and slowly fluctuating magnetic noise. Although the applied pulses exhibit limited spin control to a single transition as discussed above, we could see a typical exponential decay with decoherence time T 2 = µs at B 0 = 60 G (Fig. 5c). Since a drastically improved coherence time is expected at a stronger B0 field by suppressing nuclear spin flip-flops

13 due to a large mismatch of nuclear spin Zeeman levels of 29 Si and 13 C [17], we applied a high magnetic field along the c-axis (0001). The spin decoherence time increases at B 0 = 1000 G to T 2 = µs. The observed T2 is, however, shorter than the theoretical expectations [17] and the value measured for a single V2 centre at room temperature [24]. This could be related to the imperfect pulses and to the inhomogeneity of the B0 field for the case B0 = 1000 G (see Supplementary information S7 and S8). These observations, however, do support the findings by Carter et al. [48], related to the fact that the initial state, dephased during the free-precession, cannot be refocused by a pulse due to the oscillating local fields produced by coupled nuclear spins. Thus, the shorter T2 could be related to electron spin echo envelope modulation (ESEEM) [49]. The four sublevels of a S=3/2 electronic spin have four different nonzero values of the hyperfine coupling to nearby nuclear spins and thus result in more complex ESEEM than S=1 systems, whose sublevels have only two different non-zero coupling values [17,24]. Furthermore, as reported by Carter et al., the ensemble inhomogeneous broadening induces beating oscillations among the various modulation frequencies, leading to a shortening of T 2 measured by Hahn-echo [48]. To further suppress decoherence, we applied the XY-8 dynamical decoupling sequence, which acts as a filter for the environmental magnetic noise [19]. This sequence has proven to be effective to extend the coherence time of the S=3/2 spin ensemble associated to the V2 centre from the nuclear spin bath in 4H-SiC [19]. A repeated decoupling pulse scheme leads to a better suppression of noise, increasing the spin decoherence time with N=10 and N=50 repetitions to a value of respectively T 2 = µs and T 2 = ms (Fig. 4c). These results suggest that the heteronuclear spin bath in SiC itself provides a diluted spin bath for not only the V2 centre [24], and divacancy defects [23], but also the V1 centre when the beating due to coupling to nuclear spins at various distances

14 can be efficiently suppressed by dynamical decoupling. Therefore, one can expect a reduced spin decoherence rate from isolated single V1 defects which barely experience B0 field inhomogeneity and have coupling to a minimal number of nuclear spins. Isolated single V1 centres: Most quantum applications require addressing of single defect centres. We isolated single V1 centres in nanopillars fabricated on a 4H-SiC sample [47], as shown in the confocal microscope scan in Fig. 6a. The pillars act as a broadband optical waveguide, which provides enhanced photon collection efficiency [50]. The defect centres in the pillars are randomly distributed between V1 and V2 centres. Addressing of a single centre is proven by the autocorrelation measurement in a Hanbury Brown and Twiss (HBT) configuration as in Fig. 6b, with g (2) (0) < 0.5. The optical saturation curve of the single V1 centre is plotted in Fig. 6c. The saturation power of a single defect inside a pillar is 190 µw and the saturated count rate is 14 kcps. The spectrum at T=4K shows both V1 and V1 ZPLs, a further proof that they correspond to two different excited states of the same defect. To determine the DWF of V1, the contribution of V1 to the PSB has to be minimized. At 4 K, the intensity of the V1 ZPL is weak and we suppose that the contribution to the PSB is also negligible in comparison to V1. Then, the conservatively estimated DWF of the V1 is 40 6 %. To determine the DWF of V1 line, we increased the temperature to 80 K thereby minimize the influence of V1 to the PSB. The largest measured DWF of the V1 is 19 8 % (see Supplementary Information S9). Conclusion and Outlook

15 Featuring S=3/2 Kramers doublets with an identical Landé g-factor in both ground and excited states and two well-distinguishable ZPLs, the V1 centre in 4H-SiC is a promising system to implement a robust spin-photon interface. In this paper, we have experimentally investigated the optical and spin properties of this defect at low temperature. Optical spectroscopy and polarization measurements confirm the symmetry properties of the V1 centre, substantiating the theoretical model leading to the proposed robust spin-photon interface [32]. Further work on a single defect is necessary to identify individual optical transitions and the associated selection rules which are essential for testing the proposed scheme. A spin-photon interface requires narrow optical transitions, small spectral diffusion and slow spin-flip rates by optical pumping cycles. Recent results on the divacancy in SiC shows that the material quality is sufficiently high to satisfy these requirements [36]. Resonant optical excitation of the V1 line leads to a substantial increase in spindependent photoluminescence emission, up to 100% of relative intensity, indicating an efficient spin-dependent transition. There are challenges associated with high-fidelity coherent control of the associated electronic spin due to the small ZFS, and we suggest several routes to resolve these issues. We have shown extension of spin coherence time, through dynamical decoupling sequences, up to 0.6 ms. While the leading contenders for defect-based quantum spintronics, such as the NV centre in diamond and divacancy in 4H-SiC, suffer from low optical emission into coherent zero-phonon lines (with Debye-Waller factors ~3% and 5-7% [36,51], respectively), the V1 centre in 4H-SiC features a significantly higher Debye-Waller factor, on the order of 19-40%. The high ZPL emission could guarantee a high event rate for the proposed generation of spin-photon entanglement. The weak overall photon emission rate of the V1/V1 transition may be circumvented by using photonic

16 structures fabricated on SiC, which recently have shown progress towards high-q cavities and efficient photon collection [37,50,52]. This can further be used to generate strings of entangled photons [12,53]. Absorption-based spin-photon entanglement, which has been recently tested on the NV centre in diamond [11,54], can also be considered. In both cases, spin-photon entanglement would be generated exploiting transitions to the 4 E excited state (V1 line) while the high-contrast ODMR associated with the V1 line could be used for spin intiialziation and read-out. Methods Experimental setup: All measurement were done in a closed-cycle cryogenic system. We used for our measurement a high purity 4H-SiC chip purchased from Cree, Inc. The sample was electron irradiated with a dose of 5x10 17 electrons/cm 2 to create a high density of VSi defect centre. The size of the sample is 2 mm x 1 mm x 0.5 mm. We applied during our measurement a magnetic field from a permanent magnet positioned outside of the chamber with a magnetic field of B = 60 G. In the experiments, we used a resonant and off-resonant 858/861 nm and 730 nm diode lasers with 40 mw and 600 mw maximum output power, respectively. The polarisation of the laser was parallel to the crystal c-axis of the SiC sample. We used for the polarisation measurements an HWP and a PBS to create linear polarised light for excitation. To analyse polarisation of the emitted photons we used an HWP and a PBS in front of the spectrometer. The RF/MW fields were created by a vector signal generator (Rhode & Schwartz SMIQ06B), subsequently amplified by a 30 W amplifier (Mini-Circuits, LZY 22+). The phase for the dynamical decoupling sequence XY-8 was controlled using the vector mode of the SMIQ. The laser was pulsed by an acousto-optical modulator. For both CW and pulsed ODMR experiments using the resonant optical excitation, we

17 sent in another laser (730 nm), off-resonant to the ZPLs, which increased the total PL intensity in the phonon side band slightly by up to a factor of two. However, the ODMR contrast was nearly independent of the use of the additional off-resonant laser. The full experimental method description can be found in the Supplementary Information. Signal-to-Noise ratio of enhancement and quenching ODMR signal: The contrast of ODMR signals is defined by, C p = (I p I 0 ) I p or C m = (I 0 I m ) I 0 for the signal which is enhanced and quenched by spin resonance, respectively. Here, I 0, I p, and I m are the PL intensity without applying magnetic resonance, the total PL intensity which is enhanced and quenched by magnetic resonance, respectively. Note that 0 < C < 1, and C is different from the relative ODMR intensity used in Fig.3. If we assume identical contrast (C p = C m = C), and shotnoise limit, the relative ratio of the signal-to-noise ratio (SNR) of the enhancement signal with respect to that of the quenching signal is, by using uncertainty propagation, Cp C C p C C m I I I m 0 3 I0 II I 2 m 2 m p 0 p 0 Cm 3 I p 1 1 C 1

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21 Acknowledgement We acknowledge funding by the ERA.Net RUS Plus Program (DIABASE), the DFG via priority programme 1601, the EU via ERC Grant SMel and Diadems, the Max Planck Society, the Carl Zeiss Stiftung, the Swedish Research Council (VR ), the Carl-Trygger Stiftelse för Vetenskaplig Forskning (CTS 15:339), the Knut and Alice Wallenberg Foundation (KAW ), the JSPS KAKENHI (A) 17H01056, the KIST Open Research Programs (2E27231), the National Science Foundation DMR Grant Number , the U.S. Office of Secretary of Defense Quantum Science and Engineering Program, the COST Action MP1403 Nanoscale Quantum Optics, funded by COST (European Cooperation in Science and Technology), and by EPSRC (grant EP/P019803/1), and the Army Research Office under contract W911NF We thank Roman Kolesov, Rainer Stöhr, and Torsten Rendler for fruitful discussions and experimental aid. We also acknowledge motivating discussions with Michel Bockstedte, Adam Gali, Thomas L. Reinecke, and Jingyuan Linda Zhang. Author Contributions R.N., M.W., D.D., I.G., C.B., S.L., and J.W. conceived and designed the experiment; R.N., M.W., M.N., C.B., and S.L. performed the experiment; R.N., M.W., D.D., I.G., Ö.S., I.I., S.E.E., C.B., and S.L. analyzed the data; T.O., N.S., and I.I., contributed materials and electron irradiation; M.R., and J.V. contributed to fabrication of SiC nanopillars; D.D., Ö.S., I.I., and S.E.E. provided theoretical support; R.N., M.W., D.D., I.G., Ö.S., N.S., I.I., S.E.E., C.B., S.L. & J.W. wrote the paper. All authors commented on the manuscript. Competing financial interests The authors declare no competing financial interests.

22 Figure 1: Optical transition of the V1 centre in 4H-SiC. (a) The energy level scheme of the V1 centre exhibits a spin 3/2 ground state and a ZFS of 2D=4 MHz. There exist two optical transitions V1 and V1 at respectively 861 and 858 nm. They have different optical selection rules and only the V1 excited state experience fast ISC to the metastable state which also decays to the ground state. (b) PL spectra of a VSi ensemble at 5.5 and 70 K. V1 and V1 ZPLs are at 858 nm and 861 nm, respectively, while V2 ZPL is at 916 nm. (c) The optically excited state lifetime is measured with an 805 nm picosecond laser. (d) Suggested scheme for the spin-photon interface. Simultaneous absorption of two V1 transitions after formation of a superposition 3/2 and + 3/2 leads to spin-photon entanglement. The cycling transition of V1 can be used for high-fidelity readout. (e) The temperature dependence of both V1 and V1 ZPLs from 5 K to 130 K. (f) A model to describe the temperature dependence in (e). g 1 and g 2 represent the ground state and V 1 and V 1 the excited states. λ is representing the energy splitting of 4.4 mev. Increasing the temperature leads to thermal induced mixing between the two excited states. If the thermally induced mixing rate is comparable to λ, a population transfer from V1 excited state to V1 excited state occurs. The normalised intensity defined as the population difference (ρ V1 + ρ g2 ρ V1 ρ g1 ) in the asymptotic limit is plotted as a function of temperature, where the temperature-dependent mixing rate of the excited states,, has a powerlaw dependence ( ~1.57) on the temperature T. The observed maximum is at =. See text for details.

23 Figure 2: The optical polarisation of the V1/V1 transitions. (a, b) The polarisation of the V1/V1 ZPLs analysed with an HWP and PBS at two different sample orientations. Left: the polar plot of the normalised V1 and V1 intensities. Middle: the density plots showing the absolute intensities of the V1/V1 ZPLs. Right: Schematic diagrams depicting the sample orientation with respect to the laser incident orientation. In (a), 0, equivalently 180, corresponds to the c-axis orientation.

24 Figure 3: Optical detection of the spin state. (a) ODMR with a 730 nm laser at 60 G. The peaks at 110 and 250 MHz are spin resonant transitions from the V2 centre ensemble in SiC (ZFS=70 MHz). The central dip at 180 MHz comes from the V1 centre ensemble. In the lower panel is the measurement with a resonant laser (resonant to V1, 861 nm) showing the relative ODMR intensity close to 100 %. (b) Zeeman effect of the spin 3 ground state of the V1 2 centre. Note that D is negative, and ZFS=2D=4 MHz. f1, f2, and f3 represent possible resonant transitions. The inset show the Zeeman splitting at B0<4G.

25 Figure 4: Coherent control of the V1 spin ensemble. (a) Rabi measurement with detuned RF driving frequencies. (b) Pulse scheme for a Rabi measurement. The first laser pulse (Init.) is polarising the spin state. The RF pulse is manipulating the spin state followed by the last laser pulse (Read) for the spin state readout. (c) Fast Fourier transformed Rabi oscillations at different RF powers. (d) Simulated Rabi oscillations. The dotted lines indicate three resonant RF frequencies shown in Fig. 3b. The strong zero frequency intensities in both (c) and (d) are removed for better distinguishability of the Rabi frequencies. See text for details.

26 Figure 5: Spin decoherence and dephasing measurements. (a) Pulse sequences for Ramsey, Hahn-Echo and XY- 8 sequence. (b) Ramsey measurement at two different magnetic fields B 0 = 60 and 1000 G. (c) The spin decoherence measured at two magnetic fields B 0 = 60 and 1000 G. The measured spin decoherence time T 2 is 4.4 µs and 83 µs at 60 and 100 G, respectively. The spin decoherence times by the XY-8 decoupling sequence are µs and 0.6 ms with N=10 and 50, respectively.

27 Figure 6: Single V1 centres. (a) Confocal fluorescence raster scan showing single silicon vacancy V1 and V2 centres in SiC nanopillars at 5.5K. (b) The g (2) autocorrelation measurement indicates clearly a single photon emission character, g (2) (0) < 0.5. (c) The saturation curve of a single V1 centre PL emission. (d) A single V1 defect PL spectrum with the V1 /V1 ZPLs at 858 and 861 nm, respectively. The Debye-Waller factor (DWF) calculated by V1 V1+ PSB is 40 %.

28 Supplementary information for Quantum properties of dichroic silicon vacancies in silicon carbide S1. Experimental Method: All measurements were performed on a high purity 4H-SiC substrate purchased from Cree, Inc (2 mm x 1 mm x 0.5 mm). The sample was electron irradiated (2 MeV) with a dose of 5x10 17 electrons/cm 2 to create a high density of VSi defect centres. In order to remove contaminations on the surface acetone in an ultrasonic bath was used. All the spin measurements at low magnetic field (B0 = 60 G) were done with the sample flipped on the side, with the c-axis perpendicular to the optical excitation/detection axes. Figure S7: Experimental setup. The measurements were done with a home-built confocal setup. A closed-cycle cryostat from Montana Instruments was used to perform the lowtemperature experiments. See text for details. The sample was placed in a closed-cycle cryostat from Montana instruments, at a temperature around 5K. For measurements at low magnetic field, a static magnetic field B 0 = 60 G was applied parallel to the c-axis by a permanent magnet outside the cryostat chamber, mounted on a x-y-z-linear translation stage. The magnetic field splits the energy levels of the spin states so that we could address them individually with radio-frequency (RF) electromagnetic wave. For a high magnetic field (B 0 = 1000 G),

29 a smaller permanent magnet was placed between the sample and the heat sink in the cryostat. Optical excitation was performed either resonantly, by a 858/861 nm laser diode using a Littrow external cavity (900 µw on the sample, 300 GHz linewidth) or offresonantly by a 730nm laser diode (usually 500 µw on the sample, or being varied from 0 to 500 µw for the optical saturation as in Fig.6c of the main text). The light was focused on the sample by a high NA (0.9) air objective (Zeiss). Photoluminescence spectra were recorded by a spectrometer (Princeton Instruments, Acton SP2300, grating: 300 g mm 1 ). For the determination of the Debye-Waller factors (Figure 6 of the main text and the section S7), the changes in the responsivity of the CCD camera and the efficiency of the grating are considered. For both CW and pulsed ODMR experiments using the resonant optical excitation, we sent another re-pumping laser (660 nm), off-resonant to the ZPLs simultaneously into the sample. Re-pumping increased the total PL intensity in the phonon side band slightly by up to a factor of two. However, the ODMR contrast was independent of the use of the additional offresonant laser. For polarisation and magnetic resonance measurements, the photoluminescence was detected by near infrared enhanced APDs from Perkin Elmer (SPCM-AQRH-15). Photon polarisation was analysed by a half-wave plate (HWP) and a polarising beam splitter (PBS) in front of the spectrometer. The HWP was rotated by a stepper-motor (M101, LK-instruments) with 0.3 resolution. A 887 nm tunable long pass filter (VersaChrome TLP x36) was placed before the detectors. The RF/MW fields were created by a vector signal generator (Rhode & Schwartz SMIQ 06B), amplified by a 30 W amplifier (Mini-Circuits, LZY 22+). Radio-frequency (RF) fields were delivered by a copper wire with a diameter of 20 µm, which was spanned on the sample surface. For the creation of RF pulses, a RF switch (ZASWA- 2-50DR, Mini-Circuits), controlled by a home-built FPGA-based pulse generator was used. The laser was pulsed by an acousto-optical modulator (EQ Photonic ) whose driving RF wave was pulsed by an RF switch, also controlled by the home-built pulse generator. The phase for the dynamical decoupling sequence XY-8 was controlled using the vector mode of the SMIQ 06B. The input of the I/Q modulator of the SMIQ 06B was controlled by RF switches from Mini-Circuits and the home-built electronic device for the voltage generation. See Fig. S1 for schematic description.

30 S2. Temperature dependence model In this section, we discuss a model for the observed temperature dependence of the photoluminescence, shown in Fig. 1b,e,f in the main text. Starting from the molecular orbits of the defect centre, one can form irreducible representation for the symmetry operator of the defect in which the ground and excited states are constructed. By incorporating the spin-orbit and spin-spin interactions, one can find the splitting among various states resulting in the formation of energy subspaces 4 A2 ground and excited states separated by 1.44 ev, and the 4 E excited state split from the 4 A2 excited state by 4.4 mev. At any finite temperature T, these energy states are broadened; thermal energy acts as a dephasing operation that introduces incoherent mixing among the energy states. As the 4 A2 ground and excited states are split by 1.44 ev, the energy gap exceeds way above the thermal broadening for the working temperatures and is not relevant to the present discussion. On the other hand, the excited states 4 A2 and 4 E which are split by only 4.4 mev corresponding to 51K should mix in a fashion that changes the optical emission properties. We see similar effects in the experiment as shown in Fig. 1 of the main text and below we give a simple quantum mechanical model that explains this temperature behaviour. Instead of considering all the 32 states of the excited subspace and ground state subspace, we consider a simpler case of a four-level system that explains the observed temperature behaviour of the V1 excitation. The four-level system is formed by two excited states and two ground states. The Hamiltonian governing the dynamics in the dressed basis given by H = E + g 1 e + + E g 2 e + λ( e + e + e e ) + h. c., where the optical fields with an intensity E ±, couple the ground and excited states and λ, is the spin-orbit splitting of the excited states. The dressed excited states are given by e ± = 1 [ e 2 1 ± e 2 ]. Additionally, the excited states decay to the respective ground states and also suffer from a temperature dependent dephasing that causes incoherent mixing within the dressed basis. Including these non-unitary terms the total dynamics is then by a master equation ρ (t) = i[h, ρ(t)] + γ k k {L k ρl k 1 2 (ρl kl k + L k L k ρ)}

31 where the Lindblad operators L 1 = g 1 e +, L 2 = g 2 e, L 3 = ( e 1 e 1 e 2 e 2 ), with respective decay rates γ 1, γ 2, γ d. While the optical decay is described by the operators L 1, L 2 the dephasing is described by L 3. We assume a power-law temperature dependence for the dephasing rate, i.e., γ d (T α ). By taking γ 2 > γ 1, and solving the master equation we find the temperature dependence (see Fig. 1 of the main text) of the relative intensity of the two emission lines by evaluating the steady state populations in the ground states that are initially equally populated. At very high temperaturesi. e., γ d λ, the dressed basis is completely dephased resulting in no population exchange among the two excited states. Similarly, in the other extreme limit γ d λ, there is no population exchange between e ± states as the detuning is much larger than the thermal dephasing. Only in the intermediate regime, there is population exchange between e ±, allowing for maximal population transfer to g 2, i.e., enhanced emission in the second decay channel, V1. Moreover, the physical origin of the mixing between the 4 A2 and 4 E states in our model can be determined by examining the molecular vibrations of the Si monovacancy centre. These Raman and infrared active vibrational normal modes in the vicinity of the defect belong to the 2 A1+ 2 E irreducible representations of the C3v symmetry group. Therefore, the first (V1) and second (V1 ) excited states with 4 A2 and 4 E symmetry, respectively, are only allowed to couple each other by the E symmetry vibrational normal modes. Such inter-level coupling between nearly degenerate electronic states that are linear in lattice displacements is also known as the pseudo- Jahn-Teller effect. Using an electron-lattice coupling Hamiltonian of 0 θ η H e p = G [ θ 0 0] + Δ η [ 0 1 0] K(θ2 + η 2 ) given in the electronic basis states { A ; E x, E y } of 4 A2 and 4 E states, we obtain a minimum Jahn-Teller energy configuration of E JT = Δ 6 ε 0 ( Δ 2 ) 4ε 0 at normal mode displacement Q = θ 2 + η 2 = [Q 2 0 (Δ 2G) 2 ] 1/2. The ε 0 = G 2 2K and Q 0 = G K correspond to the Jahn-Teller energy and normal mode displacement,

32 respectively, when the energy splitting between the excited states is zero (Δ = 0). They are given in terms of the electron-phonon coupling G and lattice K force constants. From Q > 0, the condition for the Jahn-Teller instability to occur reduces to the wellknown order-of-magnitude rule Δ (4ε 0 ) < 1 [1]. It is important to note that due to small energy splitting (4.4 mev) between the excited states of this defect, this condition gets satisfied and the wave function at A2 excited state minimum becomes a linear combination of the electronic A2 and E excited states. To illustrate the temperature dependence of this Jahn-Teller mixing, we define the coupling strength of the E symmetry normal mode vibrations to the defect as g(ω) = ρ D (ω)f(q(ω))ξ(ω) in terms Debye density of states ρ D = 3ω 2 2υ 3 n π 2 and phonon coupling coefficient ξ(ω) = (ħω n 2 2Mυ n ) 1/2 for the n th mode with velocity υ n. This leads to a T 3/2 (ω 3/2 ) temperature dependence consistent with our findings above. The defect wave functions extended over many lattice sites are reflected in a cut-off function f(q) = (1 + r B 2 q 2 /4) 2 using an effective mass approximation within an effective Bohr radius of r B. We now have a quick look at one of the E symmetry transverse phonon modes around the defect propagating along [100] and polarized along [001] with sound velocity of υ = m/s in 4H-SiC. We use an effective Bohr radius of 2.7 [2] nm to include most of the charge density around the V1 defect, comparable to some deep centre acceptor states and 3C-SiC. Resulting electronphonon coupling strength square with respect to temperature is shown in Fig.S2 and it is closely related to what is observed in the experiment (Fig. 1e of the main text) since the contrast between V1 and V1 states is proportional to g 2. Although more sophisticated cut-off functions can be used for this deep centre, they are beyond the scope of the current work.

33 Figure. S2: Normalised electron-phonon coupling square g 2 between the E symmetry transverse vibrational mode (propagating along [001]) and V Si temperature. defect in 4H-SiC as a function of S3. Group theory analysis of the V1 and V1 ZPL emission In this section, we provide a detailed analysis of the observed polarisation dependence of the V1 and V1 ZPLs shown in Fig. 2 of the main text. The individual symmetries of each state belonging to the V1 ground and first excited 4 states (both labelled as A 2 1 as { E 3/2 2 in the manuscript due to their orbital symmetry) are given, E 3/2, E + 1/2, E 1/2 } in the { 3/2 + i 3/2, 3/2 i 3/2, 1/2, 1/2 } spin basis at zero magnetic field. Similarly, the symmetries of each state belonging to the second excited state (labeled as 4 E) are given as {E + 1/2, E 1 1/2, E 3/ E 3/2, E 3/2 2 E 3/2, E + 1/2, E 1/2, E + 1/2, E 1/2 } in the { 3/2, 3/2, 1/2, 1/2, 1/2, 1/2, 3/ 2, 3/2 } spin basis. Note that the previously predicted small spin mixing between some of the 4 E states (due to higher order spin-spin interactions) [3] are omitted here. The optical selection rules between these various symmetries are given in Table S1 in terms of the vector components {(x, y ); z } {E; A 1 } of the electric field within the C 3υ double group. The z axis is parallel to the defect s c-axis.

34 C 3υ E 1 1/2 E 3/2 2 E 3/2 E 1/2 A 1, E E E 1 E 3/2 E - A 1 2 E 3/2 E A 1 - Table S1. Optical selection rules for the symmetries of C 3υ double group. Therefore, by considering all four possible transitions after the spin selection rules are applied, we find that the V1 transition has a mixed polarisation of (E : E )=(3:1) involving the electric field components parallel and orthogonal to the c-axis. For the more complicated V1 transition consisting of 12 possible transitions, we found the polarization ratio to be roughly (1:11). Thus, the V1 ( 4 A2 to 4 A2) transition contains polarizations both parallel and orthogonal to the c-axis whereas the V1 ( 4 A2 to 4 E) is mostly polarized along the basal plane of the defect. These expectations are in very good agreement with our experimental results shown in Fig. 2 of the main text. S4. Additional polarisation dependence data The polarisation dependence data in Fig.2b of the main text is almost independent of the rotation angle of the HWP. It indicates either an unpolarised light or a circularly polarised light. To identify the polarisation status of the V1/V1 ZPL measured when the optical axis (laser incident orientation) was parallel to the c-axis, the additional analysis was done replacing the half-wave plate with a quarter-wave plate (QWP) before the PBS. As can be seen in Fig. S3, we observe the unpolarised light emission from both V1 and V1 ZPL.

35 Figure S3: V1 and V1 ZPL polarisation analysed by a QWP and a PBS. The laser incident orientation is parallel to the c-axis. S5. Optical spin detection by V1 ZPL excitation In this section, we present additional data about optically detected magnetic resonance (ODMR) experiments performed by resonant optical excitation of the V1 ZPL. To resonantly excite V1 we used a home-built tuneable external-cavity laser (λ = 858 nm) in continuous wave mode. Luminescence was detected with the 887 nm tuneable nm long pass (LP) filter. The resultant relative ODMR intensity, calculated according to the explanation in the main text, is only 0.2 % and the sign is negative (Fig. S4). The change in the ODMR intensity by the resonant optical excitation of the V1 ZPL is significantly smaller than what we observed for the resonantly excited V1 line and comparable to the case of the off-resonant optical excitation (Fig. 3a in the main text). These observations could suggest that optical spin polarisation is mainly established by the intersystem crossing (ISC) between the 4 A2 excited state, related to the optical line V1, and the 2 E metastable state. The ISC from the 4 E excited state, related to V1, does not seem to induce efficient optical polarisation.

36 Figure S4: ODMR under the resonant optical excitation of the V1 ZPL. S6. Spin Rabi oscillations Here we discuss a theoretical model for the Rabi oscillations described in Fig. 4 of the main text. System Hamiltonian. We consider a spin S = 3/2 system, driven by a radiofrequency field with amplitude Ω and frequency ω under a static magnetic field B. The system Hamiltonian is H = H 0 + Ω [cos(ωt) S x + sin(ωt) S y], where H 0 is the diagonal Hamiltonian with energy levels ε ±3/2 = ± ( 3 2 ) γb and ε ±1/2 = (D ± γb), γ is the gyromagnetic ratio (γ = 28 MHz/mT) and D the zero-field splitting (2D = 8 MHz for the V1 centre). We set ourselves in a frame rotating at angular velocity ω around the z-axis: e 3iωt/ U = e iωt S z = [ 0 e iωt/ e iωt/2 ] e 3iωt/2 The Hamiltonian in this rotating frame can be calculated as H rot = U HU iu U. Neglecting fast components at 2ω (rotating-wave approximation), we get:

37 H rot = [ ε 3/2 + (3/2)ω ( 3/4)Ω ( 3/4)Ω ε 1/2 + (1/2)ω 0 Ω / Ω/2 0 ε +1/2 (1/2)ω ( 3/4)Ω ( 3/4)Ω ε +3/2 (3/2)ω] System evolution. Let u i be the eigenvectors of H 0, i.e. the eigenvectors corresponding to S z = 3/2, 1/2, +1/2, +3/2. The time-independent Schrödinger equation for H is H v i = α i v i. Let V the matrix of the eigenvectors of : v i = k V ik u k. Let us assume that the initial state is one of the eigenvectors of H 0 : ψ 0 = u k, which can be expressed as a linear combination of eigenvectors of H as u k = V kl 1 l v l. The temporal evolution is described by Schrödinger equation as: ψ(t) = e iht/ħ ψ 0 = e iα lt/ħ V 1 kl v l = e iα lt/ħ V 1 kl V lm u m After a time t, the probability associated to the state u m is: l l,m p mk (t) = e iα lt/ħ V 1 kl V lm l,m Assuming that each spin level u m is associated to a photoluminescence intensity I m, the total photo-luminescence I k (t), assuming initialization into ψ 0 = u k, evolves over time as: I k (t) = I m p mk (t) m For the silicon vacancy, optical excitation results in polarisation into an incoherent mixture of S z = 3/2 and S z = +3/2. This case would require the solution of the Liouville equation for the density matrix. Here, we take a more empirical approach, considering incoherent superposition of pure states, each evolving according to Schrödinger equation, as discussed above. Assuming that the probability of initial spin polarization into the eigenstate u k is described by P k, the temporal evolution of photoluminescence intensity is calculated as: 2

38 I(t) = I m P k e iα lt/ħ V 1 kl V lm m,k l,m 2 Figure S5. Rabi model: Probability of occupation for the four spin levels as a function of time under radiofrequency driving at frequency f and amplitude Ω. Numerical simulations. We investigate the electron spin dynamics assuming perfect initial polarisation in an incoherent mixture of S z = 3/2 and S z = +3/2, corresponding to P +3/2 = P 3/2 = 0.5, P +1/2 = P 1/2 = 0. Results are shown in Fig. S5, for three different values of Ω (Ω/2π=2MHz, 7MHz, and 15MHz). In each case, we plot the dynamics for a driving frequency corresponding to each of the three transitions: f = f1 ( 3/2 1/2 ), f = f2 ( 1/2 + 1/2 ), f = f3 ( + 1/2 + 3/2 ). We

39 include decoherence in the results by an exponential decay with time constant T 2 = 200 ns. The simulation results, presented in the main text (Fig. 4), show good agreement with the experimental data. Simulation accuracy could be further improved by including inhomogeneous broadening and finite bandwidth of the applied RF field. Numerical simulations were also performed solving the Liouville-von Neumann equation for the density matrix: no significant difference was observed with respect to the simplified model illustrated above. S7. Pulsed spin control: Due to the incoherent mixture of various spin states during the Rabi driving, we find that at various driving strengths, there are no perfect revivals of the population in any subspace of the four-level system. This makes the definition of π, and π/2- pulses more complex when compared to the well-studied two-level system. For both experimental and theoretical analysis, we define a π pulse as the minimum time required for transferring the maximal population from ± 3/2 ± 1/2 subspace. Experimentally, this is signalled by the maximum contrast in the observed fluorescence. But to really know how the state of the four-level system looks during these pulses, we consider the following limits (i) Ω D, i.e., when the applied B1 field strength is much stronger than the zerofield splitting. In this limit, the π pulse will result in and similarly the π/2 pulse, ± 3/2 π 1 ([ ± 3/2 ± i 1/2 ]) 2 ± 3/2 π/2 1 ( 2 ± 3/2 i ± 1/2 ) 3 (ii) Ω D, i.e., when the applied B1 field strength is of the same order as that of the zero-field splitting. ± 3/2 π 1 ([ 2 ± 3/2 3/2 ] + 3[ ± 1/2 ± i 1/2 ]) 3 (iii) Ω D, i.e., when the applied microwave power is much smaller than that of the zero-field splitting. ± 3/2 π i ± 1/2

40 The final states after the applied pulses clearly depict the complexity involved incoherently driving and controlling a four-level system. S8. Spin Rabi oscillations at high magnetic field The Rabi measurements were done at B 0 = 60 and 1000 G. The data which were collected at B 0 = 60 G is shown in Fig. 4 of the main text. The magnetic field was applied at B 0 = 60 G with a large cylindrical permanent magnet (30 mm diameter and 50 mm thick) from outside the cryostat chamber and the magnetic field was aligned to the c-axis. Figure S6: Rabi oscillations at B 0 = 1000 G. The Rabi measurements at B 0 = 1000 G reveal seriously broaden Rabi frequency distribution over a large RF frequency range. The density plots: The observed Rabi oscillations at RF power of 23 dbm as a function of the RF frequency (a) and the FFT of them (b). The 2D plots: The Rabi oscillations at the RF frequency of 2820 MHz measured at three different RF powers (a) and their FFT (b). For B 0 =1000 G a small samarium cobalt magnet (1 mm thick and 10 mm diameter) was placed beneath our 4H-SiC sample. The measured Rabi oscillations in time domain and their FFT can be seen in Fig. S6. Seriously broaden Rabi frequency distributions near-independent on the detuning of the RF field frequency over the tested RF frequency range can be seen. We attribute this broadening to the serious inhomogeneous broadening originated from the small magnet placed right below the sample. Note that similar broadening can happen in the disordered spin distribution under a homogeneous magnetic field [4,5]. The large magnet used for 60 G experiment may not produce such serious inhomogeneity since it was placed outside