Abstract
Long coherence times are key to the performance of quantum bits (qubits). Here, we experimentally and theoretically show that the Hahnecho coherence time of electron spins associated with divacancy defects in 4H–SiC reaches 1.3 ms, one of the longest Hahnecho coherence times of an electron spin in a naturally isotopic crystal. Using a firstprinciples microscopic quantumbath model, we find that two factors determine the unusually robust coherence. First, in the presence of moderate magnetic fields (30 mT and above), the ^{29}Si and ^{13}C paramagnetic nuclear spin baths are decoupled. In addition, because SiC is a binary crystal, homonuclear spin pairs are both diluted and forbidden from forming strongly coupled, nearestneighbour spin pairs. Longer neighbour distances result in fewer nuclear spin flipflops, a less fluctuating intracrystalline magnetic environment, and thus a longer coherence time. Our results point to polyatomic crystals as promising hosts for coherent qubits in the solid state.
Introduction
Impuritybased electron spins in crystals, such as the nitrogen vacancy (NV) centre in diamond^{1,2}, donor spins in silicon^{3}, transitionmetal ions^{4} and rareearth ions^{5} have recently attracted great interest as versatile solidstate quantum bits (qubits). Among the key measures for qubit performance, coherence times characterize the lifetime of a qubit. In quantum computing, long spin coherence times are necessary for executing quantum algorithms with many gates^{6}. Qubits with robust coherence are also ideal systems for developing applications such as collective quantum memories^{7} and nanoscale quantum sensors^{8,9}. Nonetheless, interactions between the spin qubit and the bath of paramagnetic nuclei in the crystal eventually limit the qubit’s coherence^{10,11,12}. One of the standard measures of spin coherence time is the ensemble Hahnecho coherence time (T_{2})^{13}. For NV centers in naturally isotopic diamond and for donor spins in natural silicon, T_{2} times have been measured to be 0.63 ms (ref. 14) and 0.5 to 0.8 ms (refs 15, 16, 17), respectively. These are set by the presence of naturally occurring ^{13}C (1.1%, I_{C}=1/2) isotopes^{11,12,18,19,20,21,22} and ^{29}Si (4.7%, I_{Si}=1/2) isotopes^{10,23,24,25}. For Mn:ZnO, a 0.8ms T_{2} time has been reported^{4}, which is set by the ^{67}Zn (4.1%, I_{Zn}=5/2) isotopic concentration.
Several techniques can be used to extend spin coherence, including isotopic purification^{12,25}, dynamical decoupling^{26,27,28} and the use of particular ‘clock transitions’ that are immune to external magnetic perturbations^{29,30,31}. These techniques cannot be used in all applications, however, and moreover, the extent to which spin coherence can be extended is typically correlated to the original T_{2} time. Therefore, the Hahnecho T_{2} time in a naturally isotopic crystal remains an important metric for qubit performance.
Recently, Christle et al.^{32} reported a T_{2} time of 1.2 ms for divacancies in SiC, which are spin1 defects^{33,34,35,36,37,38,39,40,41,42}. However, the spin dynamics underlying this coherence time were not understood. Naturally isotopic SiC contains both ^{29}Si (4.7%) and ^{13}C (1.1%) isotopes. Nevertheless, in spite of having a higher nuclear spin density than natural diamond, SiC was able to host qubits with a much longer T_{2} time than those of NV centers, implying a suppression of nuclear spin bath fluctuations. Yang et al.^{43} recently published an insightful theoretical paper on the nuclearbath driven decoherence of singlesilicon vacancy (V_{Si}) in SiC, a spin3/2 defect^{44,45,46,47,48,49,50}. Using the clustercorrelation expansion (CCE) theory^{51}, they showed that heterogeneous nuclear spin flipflop processes are suppressed in SiC due to the difference between the gyromagnetic ratios of ^{29}Si and ^{13}C nuclear spins (or heterogeneity). Similar heterogeneity and bath decoupling effects were also discussed for GaAs quantum dots^{52}. Based on the bath decoupling effect, Yang et al.^{43}, suggested that the spin coherence time in naturally isotopic SiC would be longer than that of the NV centre in diamond. However, direct experimental verification in SiC has been challenging using single V_{Si} spins^{48,53}, partly because hyperfine coupling to the S=3/2 state gives rise to irregular coherence patterns^{43}.
Here, we combine experiment and theory to study the decoherence dynamics of the S=1 electronic spin ensemble of the neutral (kk)divacancy in 4H–SiC over a wide range of magnetic fields. We use optically detected magnetic resonance (ODMR)^{36} and a firstprinciples microscopic quantumbath model^{54} combined with the CCE method^{51,52} to demonstrate that the T_{2} time of the divacancy spin in 4H–SiC can reach 1.3 ms, an unusually long T_{2} time. Our theoretical results successfully explain all the important features found in our experiment such as the behaviour of T_{2} as a function of magnetic field and the fine details in the electron spin echo envelop modulations (ESEEM)^{13}. In particular, by studying ensembles of S=1 centers instead of single S=3/2 centers, we provide strong evidence that in SiC, the Si and C nuclear spin baths are decoupled at moderate magnetic field (∼30 mT), confirming the predictions of Yang et al.^{43}. In addition to verifying Yang’s predictions, we show that a key factor underlying the long coherence times in SiC is the fact that homonuclear spin pairs in this binary crystal must be at least two lattice sites away from each other. This separation limits the strength, and therefore the flipflop rate, of the most strongly coupled spin pairs.
Results
Optically detected spin coherence in SiC
Our experiments use 4H–SiC wafers (purchased from Cree, Inc.) with vacancy complexes intentionally incorporated during crystal growth. The divacancy density is ∼10^{12} cm^{−3} (ref. 37). In this study, we consider the (kk)divacancy^{36,37}, which is schematically shown in Fig. 1. We use a 975 nm laser diode to illuminate the sample, which, through ODMR, polarizes the electronic ground state of the divacancies into their m_{s}=0 state^{36,37}. The divacancies exhibit more intense photoluminescence (PL) in their m_{s}=±1 state^{36,37} than in their m_{s}=0 state, allowing the spin of the defects to be read out via the PL intensity. We use a movable permanent magnet to apply a caxisoriented magnetic field (B)^{36}. To measure the pure spin dephasing rate, we perform standard Hahnecho pulse sequence (π/2 pulse−t_{free}/2−π pulse−t_{free}/2−π/2 pulse)^{13} measurements. The first π/2 pulse creates a superposition of the m_{s}=+1 and m_{s}=0 states, and the following π pulse reverts the spin precession after the t_{free}/2 free evolution. At the end of the Hahnecho sequence, the spin coherence is refocused, removing the effects of static magnetic inhomogeneity. The last π/2 pulse converts the phase difference in the superposition state to a population difference in the m_{s}=+1 and m_{s}=0 states, which we then measure through a change in the PL intensity.
In Fig. 2, we show the measured Hahnecho coherence of the divacancy ensemble at three representative magnetic fields and as a continuous function of magnetic field. At low magnetic fields, for example, 2.5 and 6.5 mT shown in Fig. 2a, the spin coherence rapidly collapses and revives as a function of time. Simultaneously, its envelop decays over time, leading to the loss of coherent phase information within 1 ms. In Fig. 2, we observe that this spin decoherence is largely suppressed and that the coherence is further extended as the static magnetic field is increased. We show the T_{2} as a function of magnetic field in Fig. 3a. We find that T_{2} increases as a function of magnetic field and saturates to 1.3 ms at a magnetic field of roughly 30 mT. There is a dip in T_{2} at a magnetic field of ∼47 mT, which is also visible in Fig. 2c as a coherence drop. This magnetic field converts to 1.31 GHz energy splitting, corresponding to the zerofield splitting of the (kk)divacancy^{37}. The coherence drops at this groundstate level anticrossing as the m_{s}=0 spin state can significantly mixes with m_{s}=−1 spin sublevel.
Quantum bath approach to decoherence
To understand the decoherence dynamics observed in experiment, we use quantumbath theory, which describes the qubit decoherence occurring due to the entanglement between the qubit and the environment^{54}. We apply the same theory to the NV centre and to the (kk)divacancy spin so as to compare results consistently and to understand the underlying physical reasons responsible for their difference. The two defects share many common features^{34,35,36,39}. For example, the caxisoriented (kk) divacancy (Fig. 1a) exhibits the same C_{3v} pointgroup symmetry and ^{3}A_{2} spin triplet ground state as the NV centre in diamond (Fig. 1b). Furthermore, similar to the NV centre, the divacancy ground state is mainly derived from the three carbon sp^{3} orbitals localized around the silicon vacancy site in SiC. The only difference between the divacancyinSiC model and the NVcentreindiamond model is the type of nuclear spin bath along with their lattice structures as shown in Fig. 1a,b, respectively. We note that the dynamics of NVcentre decoherence has been wellunderstood, and that our results are in excellent agreement with those previously reported in the literature^{18,19,22}. In our model, we ignore any possible effects arising from the nuclear and electronic spinlattice relaxation. (See Supplementary Note 1 for further discussions). To solve the central spin model, we use the CCE method^{51,52}, and we systematically approximate the coherence function at different orders. No adjustable parameters are used. Further details on the theoretical methods and the numerical calculations can be found in the methods section and the Supplementary Notes 1–3, together with Supplementary Figs 1–8 and Supplementary Table 1.
In Fig. 2b,d, we show the theoretical Hahnecho coherence functions of the divacancy spin, to be compared with the experimental coherence data shown in Fig. 2a,c, respectively: the agreement between theory and experiment is excellent. In Fig. 3a, we compare the theoretical T_{2} times of the divacancy to the experimentally measured T_{2} times. Both T_{2} curves rapidly increase as a function of the free evolution time (t_{free}) up to a magnetic field of 20 mT. For B>30 mT, they both saturate at a limit of 1.3 ms, although the experimental T_{2} curve appears to saturate more slowly. The dip in T_{2} at a magnetic field around 47 mT is not found in the theory, because in our model, we did not consider spin mixing between m_{s}=0 and m_{s}=−1 near the groundstate level anticrossing. As a verification of our methods, we also compare the computed and measured divacancy T_{2} times with the theoretical T_{2} times of the NV centre in diamond (Fig. 3a). The theoretical limit of the NVcentre T_{2} time is found to be ∼0.86 ms, in agreement with ensembles measurements^{14} and with previous theoretical results obtained by the disjointcluster method^{18} and an analytical method^{22}. Our theoretical results confirm that the divacancy T_{2} time in naturally isotopic 4H–SiC is much longer than that of the NV centre in naturally isotopic diamond.
In Fig. 3b, we compare the theoretical and experimental coherence functions at two different magnetic fields (12.5 and 17.5 mT). We find that the measured oscillation pattern of the coherence is also well reproduced by the theory, including the relative peak height and width, further verifying our microscopic model comprising ^{29}Si and ^{13}C nuclear spins. In the presence of a static magnetic field, the ^{29}Si and ^{13}C nuclear spins precess at their respective Larmor frequencies and induce ESEEM^{13,55}. In Fig. 3c,d, we compare the Bnormalized fast Fourier transform (FFT) spectra of the full experimental and theoretical coherence functions shown in Fig. 2c,d, respectively. Twopeak structures are clearly seen, centered at the ^{29}Si and ^{13}C nuclear gyromagnetic ratios, which are 8.7 and 10.9 MHz T^{−1} in experiment, and 8.5 and 10.7 MHz T^{−1} in theory, respectively. In addition to the Larmorfrequency peaks, we observe faint, but appreciable hyperbolic features both in experiment and theory as denoted by dotted arrows in Fig. 3c,d, respectively.
Since the ESEEM spectrum is derived from the independent precession of nuclear spins, the generic features of the spectrum may be understood using the analytical solution of an independent nuclear spin model (see Supplementary Fig. 5)^{13,55}:
where i labels individual ^{29}Si and ^{13}C nuclear spins in the nuclear spin bath, k_{i} is a modulation depth parameter, w_{i} is the frequency of the ith nuclear spin and a_{i} is a frequency that depends on the hyperfine coupling parameters and the nuclear frequency (Supplementary Note 3). When the electron spin is in the m_{s}=0 state, the hyperfine field on the nuclear spins is zero, leading to coherence oscillations at the bare nuclear frequencies. For the electron spin in the m_{s}=+1 state, each nuclear spin experiences a different hyperfine field depending on its position relative to the electron spin, giving rise to the hyperfinefrequency term (a_{i}) in equation (1). We note that these a_{i}terms in equation (1) due to weak hyperfine interactions give rise to the hyperbolic features found in the FFT spectra shown in Fig. 3c,d. We find similar hyperbolic features in the computed FFT spectrum of the NV centre in diamond (not shown), although less pronounced compared with that of the SiC divacancy FFT spectrum. The modulation depth parameter, k_{i} in equation (1) is inversely proportional to the magnetic field (Supplementary Note 3), explaining the suppression of the oscillation amplitude at a large magnetic field found both in experiment and theory, as shown in Fig. 2a,b, respectively. The FFT intensities also diminish as B is increased for the same reason as shown in Fig. 3c,d.
Suppressed qubit decoherence in silicon carbide
We now turn our attention to the microscopic origin of the longer T_{2} time of the divacancy (1.3 ms at B=30 mT) compared with that of the NV centre (0.8 ms at B=30 mT), in spite of the much larger number of nuclear spins in the SiC lattice. By comparing calculations performed at different CCE orders (Supplementary Fig. 3), we find that for both NV and the divacancy the computed Hahnecho coherence time is numerically converged at the CCE2 level of theory. This finding indicates that the dominant contribution to decoherence comes from pairwise nuclear transitions induced by nuclear dipole–dipole couplings. The decoherence of the NV centre in diamond is mainly caused by pairwise nuclear spin flipflop transitions (↑↓↔↓↑), which induce magnetic noise at the NV centre through the hyperfine interaction. Other pairwise nuclear spin transitions, such as coflips (↑↓↔↓↑), are suppressed at magnetic fields larger than roughly 10 mT. These results agree well with those previously reported for NV centers in diamond^{18,19,22}.
In 4H–SiC, the nuclear spin interactions can be grouped in two categories: heterogeneous, between ^{13}C and ^{29}Si, and homogeneous interactions between nuclear spins of the same kind. The Hahnecho coherence function of the divacancy can then be written as:
where is a singlecorrelation term from the ith nuclear spin and is an irreducible paircorrelation contribution from the i−j nuclear spin pair. The product over {i,j}_{hetero} include all ^{13}C–^{29}Si nuclear spin interactions, while the product over {i,j}_{homo} include all ^{13}C–^{13}C and ^{29}Si–^{29}Si spin pairs. We define the following heterogeneous and homogeneous coherence functions:
To investigate the effect of the heterogeneity, we vary the gyromagnetic ratio of ^{29}Si (γ_{Si}) as a theoretical parameter while that of ^{13}C (γ_{C}) is fixed at the experimental value. In Fig. 4, is shown at four different γ_{Si} values at a magnetic field of 30 mT. We find that there would be a significant decay of if the ^{29}Si and ^{13}C gyromagnetic ratios were hypothetically the same (Δ_{γ} ≡ γ_{C}−γ_{Si}=0), while small differences in the gyromagnetic ratios (Δ_{γ}=0.03 MHz T^{−1} and 0.16 MHz T^{−1} for the two middle plots in Fig. 4a) are sufficient to significantly suppress the decay. Furthermore, when using the experimental values of γ_{Si} and γ_{C}, does not show any envelop decay, indicating no contribution from pairwise heterogeneous nuclear spin transitions for B>10 mT. Due to the sign difference between the gyromagnetic ratios of ^{29}Si and ^{13}C (γ_{Si}<0, γ_{C}>0), when B>10 mT, the lowestenergy ^{29}Si  ^{13}C pairwise spin transition is the coflip of the nuclear spins (↑↓↔↓↑). In addition to the hyperfine field difference on the order of few kHz, the difference between γ_{Si} and γ_{C} gives an extra Zeeman contribution to the energy gap (∼0.2 MHz at B=10 mT) for the coflips, which is larger than the typical heterogeneous dipole–dipole transition rate (∼kHz) in 4H–SiC.
The absence of heterogeneous nuclear spin transitions amounts to a decoupling of the nuclear spin bath in SiC and therefore the Hahnecho coherence function is given by:
where and are the Hahnecho coherence functions of the divacancy spin coupled to ^{29}Si nuclear spins only and to ^{13}C nuclear spins only, respectively. Since only transitions between homonuclear spins contribute to , the density of nuclear spins contributing to the electron spin decoherence turns out to be similar to that found in diamond^{53}, in spite of the total density of spins being much higher. However, this socalled dilution effect by itself would point to a similar electron spin decoherence rate in SiC and in diamond^{53}, contrary to what is found experimentally (1.3ms and 0.63ms T_{2} time in SiC and diamond, respectively).
To better understand the nature of the nuclear spin baths in SiC, we compare in Fig. 4b the ensembleaveraged numbers of homogeneous nuclear spin pairs that are contributing to the decoherence of the divacancy in 4H–SiC and of the NV centre in diamond. In the former case, the homogeneous ^{29}Si (4.7%) spin pairs are the dominant source of the qubit decoherence, and their number is larger than that of the ^{13}C (1.1%) spin pairs in diamond. However, being further apart, their contribution is weaker than that of the homonuclear spin pairs in diamond. In Fig. 4c the distributions of nuclear spin pairs shown in Fig. 4b, are reported as a function of nuclear–nuclear distance. In the case of the NV centre in diamond, there is a small but significant number of nuclear spin pairs at a distance <3.0 Å, including first, second and third nearest C–C neighbours. These spins exhibit strong secular dipole–dipole transition rates, ranging from 0.24 kHz to 2.06 kHz: while they are minority spin pairs in number, they account for more than 90% of the coherence decay for the NV centre in diamond (Supplementary Fig. 2e). In contrast, in 4H–SiC, the smallest distance between homogeneous spins is 3.1 Å, corresponding to the Si–Si or C–C neighbours in SiC. As a result, the secular dipole–dipole transition rates for all the homogeneous nuclear spin pairs in 4H–SiC turn out to be <0.08 kHz. Our results show that the absence of strongly coupled nuclear spin clusters in SiC plays a key role in explaining the surprisingly long divacancy T_{2} times.
Isotopic purification to lengthen T_{2}
We showed that the coherence time of the divacancy in our naturally isotopic, semiinsulating 4H–SiC is 1.3 ms. In principle, the ^{29}Si or ^{13}C nuclei can be removed by isotopic purification, which is available in SiC (refs 56, 57), and a longer qubit coherence time could be achieved^{12,18,24,58}. In Fig. 5, we report the Hahnecho T_{2} of the divacancy ensemble in 4H–SiC computed as a function of the ^{13}C concentration, while that of ^{29}Si was fixed at given values, and we compare the results with those for the Hahnecho T_{2} of the NV centre in diamond. In the case of the NV centre (Fig. 5f), we find that T_{2} scales as 1/n_{c} , where n_{c} is the concentration of the ^{13}C isotopes, in excellent agreement with previous theoretical^{18} and experimental^{11} findings.
In 4H–SiC, we observe that the divacancy T_{2} time increases as both ^{29}Si and ^{13}C concentrations are reduced. However, this increase does not appear to follow a simple powerlaw scaling behaviour. For example, in Fig. 5a, where the ^{29}Si concentration is fixed at the experimental value of 4.7%, T_{2} is nearly constant as the ^{13}C concentration is lowered below 1.1%. The behaviour of T_{2} is also significantly dependent on the applied magnetic field. We note that even if the ^{13}C concentration is reduced, ^{29}Si nuclear spins are still the majority ones, and thus responsible for limiting the coherence time. As the ^{29}Si concentration is reduced from 4.7 to 0% (Fig. 5a–e, the behaviour of T_{2} as a function of ^{13}C concentration becomes linear, similar to that of the NV centre in diamond. To rationalize the scaling behaviour of the divacancy T_{2}, we compute the dependence of and on the ^{13}C and ^{29}Si concentrations using equation (5), respectively, which we then fit with the compressed exponential decay function, . We find that T_{2} time of and follows a simple scaling law as a function of nuclear spin concentration: and , with a_{Si}=4.27 ms, N_{Si}=−0.74, a_{C}=3.31 ms and N_{C}=−0.86, and the stretching exponent (n) is ∼2.6 for both C and Si when B>30 mT. This exponent is the same as that of the total coherence function, and although in good agreement with experiments (2.3), it is slightly larger. Using equation (5), we thus find that the divacancy T_{2} scales as follows:
Equation (6), plotted as a dashed line in Fig. 5a–f, describes very accurately our full numerical simulation results at magnetic fields >20 mT. As noted above, however, the scaling behaviour significantly changes as the magnetic field is decreased under 20 mT and it cannot be described by equation (6). The inadequacy of equation (6) at low magnetic fields stems from the fact that heterogeneous nuclear spin transitions may occur, further limiting the T_{2} times. Therefore, the decoupling effect leading to equation (5) and thus, the scaling law in equation (6) are invalid at low magnetic fields.
Discussion
We used a combined experimental and theoretical study to investigate the decoherence dynamics of divacancy spin qubits in 4H–SiC. We showed that, for B>30mT at T=20K, the T_{2} time of the divacancy reaches 1.3 ms almost two times longer than that of the NV centre. Using a combined microscopic quantumbath model and a CCE computational technique, we found that 1.3 ms corresponds to the theoretical limit imposed by the presence of nuclear spins from naturally occurring ^{29}Si and ^{13}C isotopes. This limit is much longer than the corresponding one for the NV centre, which is ∼0.86 ms. The long spin coherence in SiC stems from the combination of two effects: the decoupling of the ^{13}C and ^{29}Si spin baths at a finite magnetic field, and the presence of active spins much further apart than those in diamond (for example, the closest ones belong to second neighbours in SiC and to first neighbours in diamond). We showed that, while the coherence of the NV centre is mainly limited by a few strongly interacting nuclear spin pairs belonging to nuclei within ∼3.0 Å of each other, in SiC, the homonuclear spin pair interactions are much weaker as they belong to second or further neighbours (see Fig. 1a). We note that the absence of strongly interacting nuclear spins in SiC is not a simple dilution effect. For example, the nuclear spin density in natural diamond is very low (1.1%), that is, it can be considered a diluted bath. Nevertheless, the distance between nuclei is such that strong nuclear spin interactions may arise, contributing to the decoherence of the NV centre in diamond. In SiC, Si and C spins have a much larger minimal distance from each other.
All experiments were performed at a low temperature (T=20 K) to exclude thermal effects and to focus on the pure dephasing of the divacancy spin (see Supplementary Note 1 for further discussions). Upon an increase of temperature, however, the divacancy T_{2} time would decrease significantly, as demonstrated in previous work^{37}. In ref. 37, at low field, the T_{2} time of the divacancy spin was observed to decrease from 360 μs at 20 K to 50 μs at room temperature. In contrast, the NVcentre coherence has been known to be relatively insensitive to a temperature change, thus a long coherence time can be measured even at room temperature^{14}. The insensitivity of the NVcentre coherence to temperature has been mainly attributed to the high Debye temperature and small spin–orbit coupling in diamond. However, the origin of the temperature dependence of the divacancy coherence in SiC is yet unknown.
Although overall, our theoretical and experimental results are in excellent agreement, we did find a few minor discrepancies. First, the ESEEM frequencies in experiment are blueshifted by ∼0.2 MHz T^{−1} from the free ^{13}C and ^{29}Si frequencies. The blueshift effect becomes prominent in the appearance of the coherence oscillation at a low magnetic field such as B=2.5 mT in Fig. 2a. When compared with the corresponding theoretical plot in Fig. 2b, the ESEEM peaks appear slightly faster in the experiment. Two possible reasons for the blueshift of the ESEEM frequencies could be the presence of a stray transverse magnetic field^{18} and the presence of nonsecular Zeeman and hyperfine interactions^{21}, which our theory does not consider (see Supplementary Note 1 for further details). Second, we found that the stretching exponent, determined from fits of the coherence decay is 2.3 in experiment, and 2.6 in theory. For the NV centre, our model yields 1.9, which is in a good agreement with previous analytical calculations^{22}. Experimentally, in diamond, decay exponent ranging from 1.2 to 2.7 were reported^{14}, depending on the sample and the Bfield misalignment. Finally, the theoretical divacancy T_{2} times also saturate at a smaller B field than the experimental T_{2} times, for reasons we do not understand.
In this study, we considered the coherence of divacancy spin ensembles. However, the divacancy decoherence dynamics at the singlespin level is also of interest. In Supplementary Fig. 4, we show the variation of the divacancy singlespin T_{2} time in random nuclear spin environments compared with that of the NV centre in diamond. We find that the divacancy singlespin T_{2} ranges from 0.6 to 1.7 ms at a magnetic field of 11.5 mT, while it ranges from 0.4 to 1.4 ms at B=11.5 mT for the NV centre in diamond. Similar to the NV centre in diamond, the divacancy singlespin coherence dynamics could show a rich complex dynamics depending on individual local nuclear spin environments. Other important factors for the singlespin coherence in SiC may include the effects of strain, thermal, magnetic and electric inhomogeneities.
Our combined experimental and theoretical work lays a solid foundation to understand the robust divacancy spin coherence. The essential physics should apply to other potential spin qubits in SiC as well, thus providing a benchmark for future implementation of other spin qubits in this material^{59,60,61}. Moreover, our model has implications beyond the crystal studied in this effort. The dynamics responsible for the coherence found in SiC, a binary crystal, may allow qubits in ternary and quaternary crystals to have even longer spin coherence times. For example, our results suggest that alloying the SiC lattice with larger elements such as Ge may further extend the coherence time of the divacancy spins. Since substitutional Ge would replace some ^{29}Si atoms, it could serve as an alternative path to isotopic purification, especially for applications that require a large number of coherent spins. In addition, interesting host crystals with useful functionalities are normally found in binary or ternary crystals such as carbides, nitrides and oxides^{59,62}. The piezoelectricity in AlN is one example. Complex oxides can exhibit exotic collective behaviours such as ferroelectricity, ferromagnetism and superconducting behaviour. Combining these collective degrees of freedom with coherent spin control in complex materials would be a promising route to hybrid quantum systems.
Methods
Experimental methods
As described in the main text, the 4H–SiC samples are highpurity semiinsulating wafers purchased from Cree, Inc (part number: W4TRD0R0200). Since they contain ‘offtheshelf’ neutral divacancies, we dice them into chips and measure them without any further sample preparation. The SiC samples are 3–4 mm chips attached to coplanar microwave striplines with rubber cement. In turn, the microwave stripline is soldered to a copper cold finger, which is cooled by a Janis flow cryostat.
For ODMR measurements, we use a 300 mW, 1.27 eV (975 nm) diode laser, purchased from Thorlabs, Inc. 60 mW reaches the sample. We focus the laser excitation onto the sample using a 14 mm lens and collect the PL using that same lens. We then focus the collected PL onto an InGaAs photoreceiver, which was purchased from FEMTO, a German electronics manufacturer. Although we did ensemble measurement, it may be worth commenting on the count rates achieved in asreceived samples. When single defects were considered in our previous study^{32}, we observed count rates of 3–5 kcts. However, because we were using a lower efficiency measurement apparatus than the avalanche photodiodes used for diamonds, this should not be directly compared with the 20–30 kcts of a typical NV centre. To gate the laser during the Hahnecho measurements, we use an acoustooptical modulator.
The radio frequency (RF) signals in this paper were generated by an Agilent E8257C source, whose output was gated using an RF switch (MiniCircuits ZASWA250DR+). These signals were then combined, amplified to peak powers as high as 25 W (Amplifier Research 25S1G4A), and then sent to wiring in the cryostat. The RF and optical pulses were gated with pulse patterns generated by a digital delay generator (Stanford Research Systems DG645) and an arbitrary waveform generator (Tektronix AWG520). The phase of the Rohde & Schwartz signal was also controlled by the AWG520 through IQ modulation.
We used lockin techniques to take all of the Hahnecho data in this paper. Specifically, we alternated the phase of the final π/2 microwave pulse of the Hahnecho sequence between +π/2 and −π/2. This alternation causes the spin coherence, at the end of the Hahnecho sequence, to be projected alternatively to opposite poles of the m_{s}=+1/m_{s}=+0 Bloch sphere. Because the (kk)divacancy’s PL from the m_{s}=+1 pole of the Bloch sphere is stronger than that from the m_{s}=+0 pole, this alternation induces a change in PL (ΔPL) between the two pulse sequences. Without spectrally filtering the PL, the ODMR contrast (ΔPL/PL) is roughly 0.5%. When spectrally filtering the PL (which we did not do in this work), the ODMR contrast is 20% for the (kk)divacancy. To transform the ΔPL signals to a spin coherence measurement, we simply normalized the ΔPL−t_{free} traces, by dividing them by the maximum of the ΔPL trace.
Theoretical methods
To calculate the Hahnecho coherence of the (kk)divacancy in 4H–SiC and the NV centre in diamond, we considered a central spin model in which an electron spin with total spin 1 is coupled to an interacting nuclear spin bath through the secular electronnuclear hyperfine interaction. Given the dilute nature of the nuclear spin density both in 4H–SiC (4.7% of ^{29}Si and 1.1% of ^{13}C) and diamond (1.1% of ^{13}C), we only considered the direct dipole–dipole interaction for the nuclear–nuclear spin coupling. We calculated the full timeevolution of the combined qubit and nuclearbath system, and computed the offdiagonal elements of the reduced qubit density matrix by tracing out the bath degrees of freedom at the end of the Hahnecho sequence (π/2 pulse−t_{free}/2−π pulse−t_{free}/2−echo). We considered randomly generated nuclear spin bath ensembles. A heterogeneous nuclear spin bath in 4H–SiC has ∼1,500 nuclear spins within 5 nm from the divacancy site, while the nuclear spin bath of diamond has ∼1,000 nuclear spins within 5 nm form the NV centre. We used the clustercorrelation expansion theory to systematically approximate the coherence function. Further details are found in Supplementary Notes 1–3.
Code availability
The codes that were used in this study are available upon request to the corresponding author.
Data availability
The data that support the findings of this study are available upon request to the corresponding author.
Additional information
How to cite this article: Seo, H. et al. Quantum decoherence dynamics of divacancy spins in silicon carbide. Nat. Commun. 7, 12935 doi: 10.1038/ncomms12935 (2016).
References
 1
Gruber, A. et al. Scanning confocal optical microscopy and magnetic resonance on single defect centers. Science 276, 2012–2014 (1997).
 2
Jelezko, F. et al. Observation of coherent oscillation of a single nuclear spin and realization of a twoqubit conditional quantum gate. Phys. Rev. Lett. 93, 130501 (2004).
 3
Zwanenburg, F. A. et al. Silicon quantum electronics. Rev. Mod. Phys. 85, 961–1019 (2013).
 4
George, R. E., Edwards, J. P. & Ardavan, A. Coherent spin control by electrical manipulation of the magnetic anisotropy. Phys. Rev. Lett. 110, 027601 (2013).
 5
Xia, K. et al. Alloptical preparation of coherent dark states of a single rare earth ion spin in a crystal. Phys. Rev. Lett. 115, 093602 (2015).
 6
DiVincenzo, D. P. The Physical implementation of quantum computation. Fortschr. Phys. 48, 771–783 (2000).
 7
Schuster, D. I. et al. Highcooperativity coupling of electronspin ensembles to superconducting cavities. Phys. Rev. Lett. 105, 140501 (2010).
 8
Hong, S. et al. Nanoscale magnetometry with NV centers in diamond. MRS Bull. 38, 155–161 (2013).
 9
Rondin, L. et al. Magnetometry with nitrogenvacancy defects in diamond. Rep. Prog. Phys. 77, 056503 (2014).
 10
de Sousa, R. & Das Sarma, S. Theory of nuclearinduced spectral diffusion: spin decoherence of phosphorus donors in Si and GaAs quantum dots. Phys. Rev. B 68, 115322 (2003).
 11
Mizuochi, N. et al. Coherence of single spins coupled to a nuclear spin bath of varying density. Phys. Rev. B 80, 041201 (2009).
 12
Balasubramanian, G. et al. Ultralong spin coherence time in isotopically engineered diamond. Nat. Mater. 8, 383–387 (2009).
 13
Schweiger, A. & Jeschke, G. Principles of Pulse Electron Paramagnetic Resonance Oxford University Press (2001).
 14
Stanwix, P. L. et al. Coherence of nitrogenvacancy electronic spin ensembles in diamond. Phys. Rev. B 82, 201201 (R) (2010).
 15
Tyryshkin, A. M. et al. Coherence of spin qubits in silicon. J. Phys. Condens. Matter 18, S783–S794 (2006).
 16
George, R. E. et al. Electron spin coherence and electron nuclear double resonance of Bi donors in natural Si. Phys. Rev. Lett. 105, 067601 (2010).
 17
Morley, G. W. et al. The initialization and manipulation of quantum information stored in silicon by bismuth dopants. Nat. Mater. 9, 725–729 (2010).
 18
Maze, J. R., Taylor, J. M. & Lukin, M. D. Electron spin decoherence of single nitrogenvacancy defects in diamond. Phys. Rev. B 78, 094303 (2008).
 19
Zhao, N., Ho, S.W. & Liu, R.B. Decoherence and dynamical decoupling control of nitrogen vacancy center electron spins in nuclear spin baths. Phys. Rev. B 85, 115303 (2012).
 20
Doherty, M. W. et al. Theory of the groundstate spin of the NV^{−} center in diamond. Phys. Rev. B 85, 205203 (2012).
 21
Childress, L. et al. Coherent dynamics of coupled electron and nuclear spin qubits in diamond. Science 314, 281–285 (2006).
 22
Hall, L. T., Cole, J. H. & Hollenberg, L. C. L. Analytic solutions to the centralspin problem for nitrogenvacancy centers in diamond. Phys. Rev. B 90, 075201 (2014).
 23
Witzel, W. M., de Sousa, R. & Das Sarma, S. Quantum theory of spectraldiffusioninduced electron spin decoherence. Phys. Rev. B 72, 161306 (R) (2005).
 24
Abe, E. et al. Electron spin coherence of phosphorus donors in silicon: Effect of environmental nuclei. Phys. Rev. B 82, 121201 (2010).
 25
Tyryshkin, A. M. et al. Electron spin coherence exceeding seconds in highpurity silicon. Nat. Mater. 11, 143–147 (2011).
 26
Ryan, C. A., Hodges, J. S. & Cory, D. G. Robust decoupling techniques to extend quantum coherence in diamond. Phys. Rev. Lett. 105, 200402 (2010).
 27
De Lange, G., Wang, Z. H., Riste, D., Dobrovitski, V. V. & Hanson, R. Universal dynamical decoupling of a single solidstate spin from a spin bath. Science 330, 60–63 (2010).
 28
Ma, W.L. et al. Uncovering manybody correlations in nanoscale nuclear spin baths by central spin decoherence. Nat. Commun. 5, 4822 (2014).
 29
Wolfowicz, G. et al. Atomic clock transitions in siliconbased spin qubits. Nat. Nanotechnol. 8, 561–564 (2013).
 30
Mohammady, M. H., Morley, G. W., Nazir, A. & Monteiro, T. S. Analysis of quantum coherence in bismuthdoped silicon: a system of strongly coupled spin qubits. Phys. Rev. B 85, 094404 (2012).
 31
Balian, S. J., Wolfowicz, G., Morton, J. J. L. & Monteiro, T. S. Quantumbathdriven decoherence of mixed spin systems. Phys. Rev. B 89, 045403 (2014).
 32
Christle, D. J. et al. Isolated electron spins in silicon carbide with millisecond coherence times. Nat. Mater. 14, 160–163 (2014).
 33
Baranov, P. G. et al. EPR Identification of the triplet ground state and photoinduced population inversion for a Si–C divacancy in silicon carbide. JETP Lett. 82, 441–443 (2005).
 34
Son, N. T. et al. Divacancy in 4HSiC. Phys. Rev. Lett. 96, 055501 (2006).
 35
Gali, A. Time‐dependent density functional study on the excitation spectrum of point defects in semiconductors. Phys. Status Solidi (B) 248, 1337–1346 (2011).
 36
Koehl, W. F., Buckley, B. B., Heremans, F. J., Calusine, G. & Awschalom, D. D. Room temperature coherent control of defect spin qubits in silicon carbide. Nature 479, 84–87 (2011).
 37
Falk, A. L. et al. Polytype control of spin qubits in silicon carbide. Nat. Commun. 4, 1819 (2013).
 38
Klimov, P. V., Falk, A. L., Buckley, B. B. & Awschalom, D. D. Electrically driven spin resonance in silicon carbide color centers. Phys. Rev. Lett. 112, 087601 (2014).
 39
Falk, A. L. et al. Electrically and mechanically tunable electron spins in silicon carbide color centers. Phys. Rev. Lett. 112, 187601 (2014).
 40
Falk, A. L. et al. Optical polarization of nuclear spins in silicon carbide. Phys. Rev. Lett. 114, 247603 (2015).
 41
Ivády, V. et al. Theoretical model of dynamic spin polarization of nuclei coupled to paramagnetic point defects in diamond and silicon carbide. Phys. Rev. B 92, 115206 (2015).
 42
Klimov, P. V., Falk, A. L., Christle, D. J., Dobrovitski, V. V. & Awschalom, D. D. Quantum entanglement at ambient conditions in a macroscopic solidstate spin ensemble. Sci. Adv. 1, e1501015 (2015).
 43
Yang, L.P. et al. Electron spin decoherence in silicon carbide nuclear spin bath. Phys. Rev. B 90, 241203 (2014).
 44
Soltamov, V. A., Soltamova, A. A., Baranov, P. G. & Proskuryakov, I. I. Room temperature coherent spin alignment of silicon vacancies in 4H and 6HSiC. Phys. Rev. Lett. 108, 226402 (2012).
 45
Baranov, P. G. et al. Silicon vacancy in SiC as a promising quantum system for singledefect and singlephoton spectroscopy. Phys. Rev. B 83, 125203 (2011).
 46
Riedel, D. et al. Resonant addressing and manipulation of silicon vacancy qubits in silicon carbide. Phys. Rev. Lett. 109, 226402 (2012).
 47
Kraus, H. et al. Roomtemperature quantum microwave emitters based on spin defects in silicon carbide. Nat. Phys. 10, 157–162 (2013).
 48
Carter, S. G., Soykal, Ö. O., Dev, P., Economou, S. E. & Glaser, E. R. Spin coherence and echo modulation of the silicon vacancy in 4HSiC at room temperature. Phys. Rev. B 92, 161202 (2015).
 49
Soykal, Ö. O., Dev, P. & Economou, S. E. Silicon vacancy center in 4HSiC: electronic structure and spinphoton interfaces. Phys. Rev. B 93, 081207 (2016).
 50
Kraus, H. et al. Magnetic field and temperature sensing with atomicscale spin defects in silicon carbide. Sci. Rep. 4, 5303 (2014).
 51
Yang, W. & Liu, R.B. Quantum manybody theory of qubit decoherence in a finitesize spin bath. Phys. Rev. B 78, 085315 (2008).
 52
Witzel, W. M. & Das Sarma, S. Quantum theory for electron spin decoherence induced by nuclear spin dynamics in semiconductor quantum computer architectures: spectral diffusion of localized electron spins in the nuclear solidstate environment. Phys. Rev. B 74, 035322 (2006).
 53
Widmann, M. et al. Coherent control of single spins in silicon carbide at room temperature. Nat. Mater. 14, 164–168 (2015).
 54
Breuer, H. P. & Petruccione, F. The Theory of Open Quantum Systems OUP Oxford (2007).
 55
Van Oort, E. & Glasbeek, M. Optically detected low field electron spin echo envelope modulations of fluorescent NV centers in diamond. Chem. Phys. 143, 131–140 (1990).
 56
Ivanov, I. G. et al. Highresolution raman and luminescence spectroscopy of isotopepure ^{28}Si^{12}C, natural and ^{13}C  enriched 4HSIC. Mater. Sci. Forum 778–780, 471–474 (2014).
 57
Simin, D. et al. Alloptical dc nanotesla magnetometry using silicon vacancy fine structure in isotopically purified silicon carbide. Phys. Rev. X 6, 031014 (2016).
 58
Witzel, W. M., Carroll, M. S., Morello, A., Cywiński, Ł. & Das Sarma, S. Electron spin decoherence in isotopeenriched silicon. Phys. Rev. Lett. 105, 187602 (2010).
 59
Weber, J. R. et al. Quantum computing with defects. PNAS 107, 8513–8518 (2010).
 60
Koehl, W. F., Seo, H., Galli, G. & Awschalom, D. D. Designing defect spins for waferscale quantum technologies. MRS Bull. 40, 1146–1153 (2015).
 61
Szász, K. et al. Spin and photophysics of carbonantisite vacancy defect in 4H silicon carbide: a potential quantum bit. Phys. Rev. B 91, 121201 (2015).
 62
Seo, H., Govoni, M. & Galli, G. Design of defect spins in piezoelectric aluminum nitride for solidstate hybrid quantum technologies. Sci. Rep. 6, 20803 (2016).
Acknowledgements
H.S. thank Nan Zhao and Setrak Balian for helpful discussions. H.S. is primarily supported by the National Science Foundation (NSF) through the University of Chicago MRSEC under award number DMR1420709. G.G. is supported by DOE grant No. DEFG0206ER46262. D.D.A. was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division. We acknowledge the University of Chicago Research Computing Center for support of this work. This work was supported by Air Force Office of Scientific Research (AFOSR), AFOSRMURI, Army Research Office (ARO), NSF and NSFMRSEC.
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H.S. developed the numerical simulations and performed the theoretical calculations. A.L.F., P.V.K. and K.C.M. performed the optical experiments. D.D.A. and G.G. supervised the project. All authors contributed to the data analysis and production of the manuscript.
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Correspondence to David D. Awschalom.
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Supplementary Figures 18, Supplementary Table 1, Supplementary Note 13 and Supplementary References (PDF 2467 kb)
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Seo, H., Falk, A., Klimov, P. et al. Quantum decoherence dynamics of divacancy spins in silicon carbide. Nat Commun 7, 12935 (2016). https://doi.org/10.1038/ncomms12935
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