Strongly Anisotropic Spin Relaxation in the Neutral Silicon Vacancy Center in Diamond
Brendon C. Rose, Gergo Thiering, Alexei M. Tyryshkin, Andrew M. Edmonds, Matthew L. Markham, Adam Gali, Stephen A. Lyon, Nathalie P. de Leon
aa r X i v : . [ qu a n t - ph ] O c t Strongly Anisotropic Spin Relaxation in the Neutral Silicon Vacancy Center inDiamond
B. C. Rose , G. Thiering , , A. M. Tyryshkin , A. M. Edmonds ,M. L. Markham , A. Gali , , S. A. Lyon , and N. P. de Leon ∗ Dept. of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA Wigner Research Centre for Physics, Hungarian Academy of Sciences, PO Box 49, H-1525, Budapest, Hungary Element Six, Harwell, OX11 0QR, United Kingdom and Department of Atomic Physics, Budapest University of Technology and Economics, Budafoki ´ut 8., H-1111 Budapest, Hungary (Dated: October 10, 2017)Color centers in diamond are a promising platform for quantum technologies, and understandingtheir interactions with the environment is crucial for these applications. We report a study of spin-lattice relaxation ( T ) of the neutral charge state of the silicon vacancy center in diamond. Above 20K, T decreases rapidly with a temperature dependence characteristic of an Orbach process, and isstrongly anisotropic with respect to magnetic field orientation. As the angle of the magnetic field isrotated relative to the symmetry axis of the defect, T is reduced by over three orders of magnitude.The electron spin coherence time ( T ) follows the same temperature dependence but is drasticallyshorter than T . We propose that these observations result from phonon-mediated transitions to alow lying excited state that are spin conserving when the magnetic field is aligned with the defectaxis, and we discuss likely candidates for this excited state. Solid state defects are attractive candidates for quan-tum technologies because they can have long spin co-herence time and can be integrated into nanofabricateddevices. However, interactions with phonons in the solidstate environment can lead to spin decoherence. Colorcenters in diamond with exceptionally long spin coher-ence time have been identified, such as the nitrogen-vacancy center (NV − ), and these are promising candi-dates for a wide range of applications including quantumsensing [1], quantum information processing [2–5], andquantum networks [6–8]. More recently the negatively-charged silicon vacancy center in diamond has beenshown to possess promising optical coherence, but poorspin coherence at 4 K because of a phonon-mediatedorbital relaxation process [9, 10]. We recently demon-strated that the neutral charge state of the silicon va-cancy color center (SiV ) has excellent optical coherence,as well as long spin coherence times at temperatures upto 20 K [11]. These properties make it an ideal candi-date for a single atom quantum memory in a quantumnetwork. However, we also observed that at tempera-tures above 20 K both T and T decrease exponentiallywith temperature. Understanding the origin of this pro-cess is crucial for extending the operation range of SiV to higher temperatures, to enable new applications inquantum information processing and nanoscale sensing.In this letter we investigate the spin-lattice relaxationof SiV in detail. The exponential temperature depen-dence of T above 20 K is consistent with an Orbachprocess [12–14] with an activation energy ( E a ) of 16.8meV, and we observe that the relaxation rate has a sharpdependence on the angle ( θ ) of the magnetic field ( B ) rel-ative to the symmetry axis of the defect (Fig. 1a). Asthe angle of the magnetic field is rotated away from thecrystallographic axis of SiV by just 5 degrees, T de- creases by almost two orders of magnitude. In additionto the unusual orientation dependence of T , T followsthe same temperature dependence as T but at a ratethat is three orders of magnitude faster when the mag-netic field is aligned with the defect symmetry axis.We propose that the strong intrinsic anisotropy inthe spin-lattice relaxation of SiV and the significantlyshorter spin coherence time originate from the presenceof phonon-mediated transitions to an excited state thatare spin-conserving when the magnetic field is alignedwith the quantization axis of the center. This is fun-damentally similar to previous observations in SiV − at4 K, in which a fast orbital relaxation ( T ,orbital = 38ns) is spin-conserving but spin-dephasing, giving rise toa relatively long T ,spin = 2 ms, while T is limited bythe orbital relaxation rate [9, 15]. For SiV − , spin relax-ation arises from differing spin-orbit coupling in the twolow-lying orbital states, and when the external magneticfield is aligned with the SiV − axis, this spin relaxation issuppressed, leading to T ,spin ≫ T ,orbital . However, in alarge off-axis magnetic field, the eigenstates mix, and thespin relaxation rate increases rapidly with angle [15].In SiV , the identity of the low-lying excited state at16.8 meV implied by the Orbach activation energy is un-known. Likely candidates are a low lying singlet state ora triplet vibronic mode [16]. For both candidates, we pro-pose models for the suppression of spin relaxation whenthe external magnetic field is aligned with the quantiza-tion axis, which capture the temperature and orientationdependence of T and the orientation-dependent ratio be-tween T and T . We show that a model incorporatinga singlet excited state closely reproduces our data, andwe outline the physical requirements for a triplet excitedstate that would account for the experimental observa-tions.
310 370eti Fiel (mT)-1001020 o l a r i z a t i o ( % ) Magnetic Field (mT)010310 340 370 [111] [111] P o l a r i z a t i on ( % ) (a)(c) SimulationExperimentθ = 0 o Dθ = 109.5 o [111][111][111] [111][111][111] F r equen cy ( G H z ) FIG. 1. (a) Ball and stick model of the silicon vacancy centerin diamond. Gray spheres are carbon atoms. The intersti-tial Si atom (blue sphere) and split vacancy (red spheres) arealigned along the h i directions in the diamond lattice, andthe magnetic field ( B ) forms angle θ with the defect axis. (b)Energies of the three ground state spin sublevels for the twoinequivalent orientations of SiV with B k [111]. (c) PulsedESR spectrum of SiV (zero field splitting, D = 0 .
94 GHzor 33.5 mT) measured at X-band frequency (9.7 GHz) withthe magnetic field slightly misaligned from [111] by θ = 2 . ◦ (black), along with the simulated transitions according to Eq.1 (red). The four sets of lines correspond to the two transi-tions m s = 0 ↔ +1 and m s = − ↔ Two high purity { } diamonds grown by chemicalvapor deposition were used in these experiments. Thefirst diamond (D1) was doped during growth with bothboron ( & cm − ) and silicon ( ∼ cm − ) andsubsequently HPHT annealed, resulting in a SiV con-centration of 4 · cm − [17]. The silicon precursorwas isotopically enriched with 90% Si, and all mea-surements in D1 were conducted on a Si hyperfine line.The second diamond (D2) was doped during growth withboron ( ∼ cm − ) and implanted with Si (6 . · cm − ), and was previously described and characterizedin reference [11]. After Si ion implantation and high tem-perature annealing, the resulting SiV concentration was5 . · cm − within the implanted region. Pulsed X-band (9.7 GHz) electron spin resonance (ESR) was per-formed in a standard dielectric volume resonator (BrukerMD5) with a quality factor of Q ≈ [111][111] T T [111] [111] -1 ) T , T ( s ) -5 -3 -1 -4 -2 Temperature (K) FIG. 2. Arrhenius plot of the temperature dependence of T (blue) and T (red) for SiV in diamonds D1 (circles) and D2(squares) for two inequivalent orientations with B k [111].[111]: θ = 2 . ◦ (D1), θ = 0 . ◦ (D2) and [¯11¯1]: θ = 106 . ◦ (D1), θ = 108 . ◦ (D2). The lines correspond to the best fitof Eq. 2. so that the excitation pulse bandwidth was greater thanthe bulk linewidth of spin transitions in all experiments( ∼ T utilized a standard two-pulse Hahn echo sequence with an initial 100 ms pulse of ∼
200 mW of green laser light (532 nm) to optically en-hance the spin polarization. Under these conditions at 5K, we achieve 11 .
5% optical spin polarization into m s = 0(Fig. 1c). T was measured using a three-pulse inversionrecovery sequence [19].The ground state electron spin Hamiltonian of SiV isgiven by [17]: ˆ H = ˆ S † ˜ D g ˆ S + µ B ˆ S † ˜ g B , (1)with electron spin S = 1, zero field splitting tensor ( ˜ D g )with axial part D g = 0 .
94 GHz (at T = 4.8 K), elec-tron g tensor (˜ g ) with parallel and perpendicular com-ponents g k = 2 . g ⊥ = 2 . µ B . The ˜ D and ˜ g tensors are bothaligned along the h i directions. With the field alignedalong [111], there are two inequivalent orientations (Fig.1c): one orientation aligned with the magnetic field sothat θ = 0 ◦ ([111], outer ESR peaks), and three equiva-lent orientations aligned off-axis with θ = 109 . ◦ ([¯1¯11],[1¯1¯1],[¯11¯1], inner ESR peaks).We performed time-resolved measurements for both in-equivalent orientations to study spin relaxation and de-coherence. Below 20 K, T and T are independent oftemperature (Fig. 2). In sample D2, we previously re-ported that SiV exhibits a spin coherence time at lowtemperature that is dominated by spectral diffusion fromthe 1.1% abundance of C nuclei, with T = 0 . ± . in sample D1 is TABLE I. Summary of the rate prefactors, A ( θ ), extractedfrom the curves in Fig. 2 using Eq. 2.Sample A( θ ) (kHz) T , [111] T , [¯11¯1] T , all orientationsD1 2 . ± .
28 378 ±
33 1260 ± . ± .
02 365 ±
53 1180 ± large enough that the spin coherence time is limited byinstantaneous diffusion, with T = 0 . ± .
03 ms [20, 21].At low temperature, T is independent of temperature forboth samples, with T = 46 ± T = 45 ± T at low temperature is similarto previous observations of NV − [22].Above 20 K, both T and T decrease exponentiallywith increasing temperature. In this high temperatureregime the two inequivalent orientations ([111] and [¯11¯1])exhibit similar T but significantly different T . T and T exhibit the same Arrhenius slope for both orienta-tions. The data ( T , [111] , T , [¯11¯1] , T , [111] , T , [¯11¯1] ) werefit according to the equation:1 T , = 1 T sat + A ( θ ) e − E a /kT , (2)where T sat is the saturated decay time at low temper-ature, A ( θ ) is the orientation-dependent rate prefactor, E a is the activation energy, and kT is the thermal en-ergy. The activation energy is the same for all curves, E a = 16 . ± . A ( θ ) varies significantly (Ta-ble I).Unlike T , T exhibits a weak orientation dependenceabove 20 K ( T , [111] ≈ T , [¯11¯1] ). However, since T dis-plays the same activation energy as T , the two decaytimes likely result from the same physical process. Thisis surprising since T is not T -limited; in fact T is 4000times shorter than T when θ = 0 ◦ . We can rule out thatthe decoherence is caused by magnetic noise from nearbycenters with short T because we do not observe a den-sity dependence in T when comparing samples D1 andD2, and we are unable to extend T with further dynami-cal decoupling [21][11]. Moreover, numerical simulationsof ensemble dipolar interactions fail to account for theobserved temperature dependence of T [21].In order to understand the anisotropy in detail we mea-sured the full orientation dependence of T and T at 30 K(Fig. 3), where the Orbach process dominates the spinrelaxation. At this magnetic field ( ∼ . .
94 GHz). The relative orientationof the magnetic field was varied by rotating the crystalabout a h i axis from θ = 0 ◦ ( B k D ) to θ = 90 ◦ ( B ⊥ D ). The ESR spectrum (Fig. 1c) was measured Time (ms)0 500 I n t eg r a t ed E c ho ( a . u . ) -2 o o (b) 0 2500 5000 7500 100000 589.0 o Time (ms) I n t eg r a t ed E c ho ( a . u . ) -2 -1 T , T ( m s ) (a)
30 75 -1 -1 -1 FIG. 3. (a) Orientation dependence of T (blue dots) and T (red dots) in sample D1 at T = 30 K, measured on them s = 0 ↔ +1 transition. Lines show the theoretical orien-tation dependence of the two characteristic spin relaxationtimes T ,a (solid line) and T ,b (dashed line), as well as T (long dashed line) predicted for an Orbach process with anexcited singlet state. (b) Selected decay curves with theircorresponding fits (red) showing the biexponential behaviorof the spin relaxation at particular magnetic field orientations( θ ), indicated by the vertical dashed lines in panel (a). to determine the crystal orientation to within 1 ◦ . Therelaxation time exhibits dramatic anisotropy, and as thecrystal is rotated away from θ = 0 ◦ , the spin relaxationbecomes clearly biexponential (Fig. 3). Near θ = 0 ◦ , T drops rapidly, and rotating by just 5 ◦ increases therelaxation rate by almost two orders of magnitude. Near θ = 55 ◦ , the decay is a single exponential with a shorttimescale that is insensitive to small rotations. Beyond55 ◦ the two timescales diverge and differ by over 3 ordersof magnitude at θ = 90 ◦ (Fig. 3b).We propose a model that captures the four salient fea-tures of the data: (1) the strong anisotropy of T , (2) thebiexponential nature of T , (3) the temperature depen-dence of T , and (4) the large ratio between T and T .Generically, an Orbach process is a two-phonon relax-ation process [14] that connects the ground state spinsublevels m s = − , , +1 through a low-lying excitedstate ( | Ψ i ) with amplitudes t − , t , and t +1 , respec-tively (Fig. 4a). The amplitudes ( t − , t , and t +1 ) areoverlap parameters between the ground triplet states andthe excited state, t m s = h m s | Ψ i [21]. This gives rise tothree possible relaxation rates between distinct pairs ofthe ground state triplet spin sublevels m s ↔ m s ′ = − ↔ , ↔ +1 , − ↔ +1:1 T ,m s ↔ m s ′ = C (cid:12)(cid:12) t m s t m s ′ (cid:12)(cid:12) e − E a /kT , (3)where C is a constant.If the excited state Ψ is a singlet state ( S = 0), it isinvariant under magnetic field orientation, so the behav-ior of T can be captured by considering the mixing ofthe ground state [21]. The mixing of the spin sublevelsin the presence of a large off-axis magnetic field leads to: | t − | | t | | t +1 | = cos θ sin θ sin θ sin θ cos θ sin θ sin θ sin θ cos θ (cid:12)(cid:12) t − (cid:12)(cid:12) (cid:12)(cid:12) t (cid:12)(cid:12) (cid:12)(cid:12) t (cid:12)(cid:12) ,(4)where (cid:12)(cid:12) t m s (cid:12)(cid:12) are the overlap parameters at zero magneticfield. Substituting Eq. 4 in Eq. 3 and solving the 3x3relaxation rate matrix equation for the ground state spin(S=1) provides the T relaxation times [21]. If (cid:12)(cid:12) t (cid:12)(cid:12) = (cid:12)(cid:12) t − (cid:12)(cid:12) = (cid:12)(cid:12) t (cid:12)(cid:12) , then Eqs. 3 and 4 predict that the spinrelaxation is isotropic. However, if (cid:12)(cid:12) t (cid:12)(cid:12) ≫ (cid:12)(cid:12) t − (cid:12)(cid:12) , (cid:12)(cid:12) t (cid:12)(cid:12) ,the spin relaxation is strongly anisotropic with two char-acteristic times approximated as:1 T ,a = 38 C (cid:12)(cid:12) t (cid:12)(cid:12) sin (2 θ ) e − E a /kT T ,b = 12 C (cid:12)(cid:12) t (cid:12)(cid:12) sin ( θ ) e − E a /kT . (5)In this limit the model captures the observed angular de-pendence of the two timescales in T as shown in Fig. 3a.By comparing numerical calculations of the orientationdependence of T for different ratios of (cid:12)(cid:12) t /t ± (cid:12)(cid:12) (Fig. 4b),we can place a lower bound on the imbalance betweenthese rates, (cid:12)(cid:12) t /t ± (cid:12)(cid:12) >
100 [21].We can also predict the effect of this Orbach processon T . Customarily, the Orbach process is viewed as aspin relaxation process [14]. However, transitions to the (a)(b)m s = -1 m s = 0 m s = +1 m s = -1 m s = 0 m s = +1t /t ±1 -1 t +12 t +1 t t -12 t + t t +12 +t -1 t T ( m s ) FIG. 4. (a) Level diagram of the Orbach process with a sin-glet excited state. Spin relaxation (left) occurs in two stepsthrough the excited state and depends on the product of theoverlap parameters, while decoherence (right) can arise froma single step, and depends on the sum. (b) Plot of T ,a (solidcurves) and T ,b (dashed curves) for selected values of (cid:12)(cid:12) t /t ± (cid:12)(cid:12) using the rate coefficient C extracted from experiment. excited state via absorption and emission of phonons canalso lead to decoherence even when the spin projectionis preserved, similar to what has been observed for or-bital relaxation in SiV − [15]. While the spin relaxationrate relies on a spin flip and therefore the product of theoverlap parameters T ∝ | t m s | (cid:12)(cid:12) t m s ′ (cid:12)(cid:12) (Fig. 4a, left), thedecoherence rate depends on the sum of overlap param-eters, T ∝ | t m s | + (cid:12)(cid:12) t m s ′ (cid:12)(cid:12) (Fig 4a, right), if we assumethe spin coherence is completely lost within a single cy-cle. The observed ratio of T to T will therefore dependon both the angle of the magnetic field and the ratio ofoverlap parameters. More accurately, the model predicts(Fig. S4): T ,a T , ↔± = (cid:16) | t ± | + | t | (cid:17) (cid:16)(cid:12)(cid:12) t (cid:12)(cid:12) + 2 (cid:12)(cid:12) t ± (cid:12)(cid:12) (cid:17) | t ± t | . (6)The orientation dependence of T predicted from thismodel is plotted in Fig. 3a, where we also includedthe effect of instantaneous diffusion in sample D1, andis plotted in detail in Fig. S3. The anisotropy in T ismostly canceled in the sum | t m s | + (cid:12)(cid:12) t m s ′ (cid:12)(cid:12) . The modelprovides the best fit for both the T and T data when (cid:12)(cid:12) t /t ± (cid:12)(cid:12) ≈
125 (Fig. 3a).If the excited state Ψ is instead a triplet state (S=1),then the overlap parameters cannot be written in a com-pact form, but we analyze this case in detail in the sup-plementary information [21]. Briefly, phonon-mediatedorbital relaxation to a vibronic excited state is gener-ally spin conserving, but differences in the ground andexcited state spin Hamiltonians can lead to mixing dur-ing the time spent in the excited state. Specifically, forSiV the ground and excited states can have differentzero field splitting tensors ( D e and D g ). Since the Zee-man splitting in these measurements is 9.7 GHz, the zerofield splittings must differ by a comparable scale in or-der to reproduce the observed ratio of T to T , and wefind that the data can be qualitatively reproduced when D e ∼ − D g = 0 .
94 GHz. It is unlikelythat the zero field splittings differ by such a large mag-nitude. Alternatively, the small ratio of T to T couldalso arise from incomplete spin dephasing. If the excitedstate lifetime is short compared to the spin precessiontime ( τ < π ~ /E Zeeman ∼
50 ps), then the spin coher-ence is partially preserved in an excitation cycle [23]. Amodel involving a triplet excited state would thereforerequire either that D e ≫ D g or that the excited statelifetime is short enough to partially preserve coherence.In summary, we have shown that spin relaxation inSiV at high temperature is dominated by an Orbachprocess that is strongly dependent on the magnetic fieldorientation, and T exhibits the same temperature de-pendence as T , but at a significantly faster rate. Theseobservations can be explained by a model for the Orbachprocess where the overlap parameters from the m s = 0and m s = ± m s = 0 (Fig. 1c) [11]. Alternatively, these observationscan be qualitatively reproduced by a model with a tripletexcited state that either exhibits a much larger zero fieldsplitting than the ground state or a very short excitedstate lifetime. Although our present results cannot defini-tively identify the excited state, detailed spectroscopycan help distinguish between these two cases. For exam-ple, absorption spectroscopy of different isotopes couldelucidate the vibronic structure [24, 25], and the nature ofthe singlet state can be explored using time-resolved pho-ton correlation measurements, as well as temperature-dependent intersystem crossing rates [26, 27]. Further-more, at temperatures well above the activation energy,there should be enough population in the excited stateto observe spin resonance transitions associated with aspin-triplet state with different zero field splitting. Wehave not observed the existence of additional transitionswith large zero field splitting, but on-going work includesincreasing our measurement sensitivity at higher temper-atures to search thoroughly for such states.The strong intrinsic anisotropy in the spin-lattice re-laxation of SiV stands in contrast to prior studies ofNV − , in which spin relaxation is mostly insensitive to the magnetic field orientation (except in cases where thedefect density is high enough that the relaxation is dom-inated by dipolar interactions [28]). To the best of ourknowledge, there has not been a detailed study of the ori-entation and temperature dependence of spin relaxationin NV − at high magnetic fields, and it would be interest-ing to perform such measurements in light of our work.Similarly, a more detailed orientation and temperaturedependence of T ,spin in SiV − would further elucidateanalogous spin and orbital relaxation processes, and re-cent measurements at dilution refrigerator temperatureshave started to explore the mechanisms for spin relax-ation and decoherence [29, 30].Additionally, these observations point to a promisingavenue of exploration for high temperature operation ifthe excited state involved in the Orbach process is a spinsinglet state. The imbalance of the overlap parameters tothe excited state implies that superpositions of m s = ± T . Future experiments includedouble quantum spin resonance measurements to inter-rogate the coherence of such superposition states. ACKNOWLEDGEMENTS
This work was also supported by the Princeton Centerfor Complex Materials, a NSF Materials Research Sci-ence and Engineering Center (grant No. DMR-1420541)and the NSF EFRI ACQUIRE program (grant No.1640959). G. Thiering and A. Gali were supported bythe NKP-17-3-III New National Excellence Program ofthe Ministry of Human Capacities and the EU Commis-sion (DIADEMS Project Contract No. 611143). Theauthors gratefully acknowledge Jeff Thompson, ShimonKolkowitz, and Ashok Ajoy for helpful discussions. Theauthors would also like to thank Sorawis Sangtawesin andZihuai Zhang for help in proofreading the manuscript. ∗ [email protected][1] G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hem-mer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Brats-chitsch, F. Jelezko, and J. Wrachtrup, “Nanoscale imag-ing magnetometry with diamond spins under ambientconditions,” Nature , 648–651 (2008).[2] T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. 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D.Lukin, “State-selective intersystem crossing in nitrogen-vacancy centers,” Phys. Rev. B , 165201 (2015).[41] G. Thiering and A. Gali, “Ab initio calculation of spin-orbit coupling for an NV center in diamond exhibitingdynamic Jahn-Teller effect,” Phys. Rev. B , 081115(2017). STRONGLY ANISOTROPIC SPIN RELAXATIONIN THE NEUTRAL SILICON VACANCYCENTER IN DIAMOND: SUPPLEMENTARYMATERIALInstantaneous diffusion in sample D1
At low temperatures, the coherence time in sample D1is limited by instantaneous diffusion, which arises whena microwave pulse induces spin flips on a dense bath ofparamagnetic centers. If we consider a central spin sur-rounded by its neighbors, then the pulse will induce rota-tions of the neighbors as well as the central spin. Phaseresulting from the pulse-induced change in the dipolarmagnetic field is not refocused during a Hahn echo se-quence, limiting the coherence time to T ID ) .The effect of instantaneous diffusion can be mitigatedby using a smaller rotation angle ( θ ) for the second mi-crowave pulse of a Hahn echo sequence, since the changein the net dipolar magnetic field scales as sin ( θ / T ∝ T − ID ) sin ( θ ). Using a smaller rotating angle will en-hance T , but it will also decrease the bulk echo signalby the same factor. In sample D1, the apparent deco-herence rate increases linearly with sin ( θ /
2) (Fig. S1).The data were fit according to the following:1 T = 1 T SD ) + 1 T ID ) sin ( θ / T SD ) is the spectral diffusion decay time. Thefit results in T SD ) = 0 . ± .
22 ms, most likely arisingfrom the 1 .
1% of C nuclei [11] and T ID ) = 0 . ± .
056 ms.We note that the Hahn echo spin coherence times re-ported for sample D1 in Figs. S1, S5c, and S6 ( T = 0 . T = 0 .
48 ms). This arises fromthe nonuniform population distribution of SiV centersin this sample over the four inequivalent crystal orienta-tions ([111],[1¯1¯1], [¯11¯1], [¯1¯11]), which has been reportedpreviously as sample C in reference [31]. The data inFigs. 3, S3, S5, S6, and S1 is taken using the [¯11¯1] ori-entation (smaller SiV concentration), while the data inFig. 2 is taken using the [111] orientation (larger SiV concentration). Decoherence arising from T -induced spin flips offast relaxing neighbors An alternative hypothesis for the observed tempera-ture dependence of T (Fig. 2 in the main text) and itsrelative magnitude with respect to T is that rapid de-phasing arises from dipolar interactions with other SiV / T ( k H z ) ( θ / 2)01234 FIG. S1. Decoherence rates for SiV centers in sample D1measured at 5 K as a function of the rotation angle ( θ ) of thesecond pulse in a Hahn echo sequence. The linear dependenceconfirms that T is limited by instantaneous diffusion. Theblack curve is a fit according to Eq. S7. spins in the bath, such as those misaligned with the exter-nal magnetic field. We can immediately rule out spectraldiffusion from SiV spin flip-flops and instantaneous dif-fusion mechanisms arising from dipolar interactions be-tween SiV centers since these mechanisms would be in-dependent of temperature. Instead, we consider the con-tribution of spectral diffusion arising from the fast T relaxation of nearby SiV centers [37]. This decoherencemechanism is strongest when T of the spin bath is com-parable to the T of the central spin under consideration.In our samples, T ∼ . T ∼ orientations misaligned with the magnetic field( θ ≈ ◦ ) at temperatures above 20 K. We numer-ically model the contribution to the Hahn echo decayfrom these three equivalent off-axis sites for the range ofdensities in samples D1 and D2 [33]. The electron spinHamiltonian describing a pair of SiV spins, S and S ,is given by:ˆ H = ~ ω ˆ S z + ~ ω ˆ S z + ~ A ( r ) ˆ S z ˆ S z + ˆ V ( t ), (S8)with dipolar interaction between the SiV spins A ( r ) = g z g z µ B ~ − (cid:0) − ( θ ) (cid:1) r − , (S9)where ω and ω are the transition frequencies of thespins, r is the distance between the spins, θ is theangle between r and B , g z and g z are the longitudinalcomponents of the g tensors. For our model we considerthat S is a slow relaxing spin ( θ = 0 ◦ ) whose coherencetime is being measured, and S is a fast relaxing spin ( θ =109 ◦ ) whose spontaneous T flips induce decoherence of S . The term ˆ V ( t ) accounts for the fast Orbach spinrelaxation rate of S spins by inducing random spin flipsat a rate W . The contribution to the echo signal decayfor S is [33]: V (2 τ ) = "(cid:18) cosh( Rτ ) + WR sinh( Rτ ) (cid:19) + A ( r )4 R sinh( Rτ ) (cid:21) exp ( − W τ ), (S10)where τ is the inter-pulse delay in a Hahn echo sequence, W = 1 /T , h ¯11¯1 i (fast relaxing sites, Fig. S2 blue line), R = W − A ( r ) /
4, and r = n − / is the aver-age inter-spin distance. This expression is averaged overall angles θ and added to the Hahn echo decay thatarises from C spectral diffusion alone ( T SD ) = 0 . densities in and abovethe range of the two samples studied here, which haveSiV concentrations of less than 5 · cm − . The den-sity required to account for the data would need to be 100times higher. Furthermore, at high temperatures, mo-tional narrowing should lead to an increase in T , whichdoes not qualitatively agree with the observed tempera-ture dependence (Figs. 2 and S2). Dynamical Decoupling using CPMG
We previously reported dynamical decoupling mea-surements using the Carr-Purcell-Meiboom-Gill sequenceon sample D2 [11, 34]. The Hahn echo T displays aplateau below 20 K corresponding to C spectral dif-fusion, but is limited by an Orbach process above 20 K.We observed that T ,CP MG is unchanged above 20 K andfollows the temperature dependence of T . However be-low 20 K T ,CP MG becomes substantially longer than T and follows the extrapolated temperature dependence ofthe Orbach process. We hypothesize that the CPMGexperiment refocuses slow spectral diffusion that arisesfrom the C nuclei, but it does not refocus fast effectsfrom the Orbach process, as expected. All of the pointsin the CPMG measurement lie along the same curve T − ,CP MG = A exp ( − E a /kT ), where A = 1180 ±
210 kHzand E a = 16 . ± . Orientation dependence of T and T measured on m s = − ↔ In the main text we presented the orientation depen-dence of the T and T times for SiV for measurementson the m s = 0 ↔ +1 transition. We also repeated thesame measurements on the m s = − ↔ -1 )10 −5 −4 −3 −2 −1 T , T ( s ) T T FitSimulated5x10 FIG. S2. Arrenhius plot of simulations of T (red lines) re-sulting from spectral diffusion arising from fast relaxing SiV centers. The blue dashed line is a fit of Eq. 2 to the temper-ature dependence of T (blue squares) of fast relaxing SiV sites. This fit is incorporated in Eq. S10 to simulate T for arange of defect densities (labels in units of cm − ). The sim-ulations indicate that for the range of densities studied, thedecoherence arising from spin flips of nearby SiV centers isnot significant and is inconsistent with the observed temper-ature dependence of T (red dots). find that it gives a nearly identical orientation depen-dence (Fig. S3). Since T , − ↔ ≈ T , ↔ , we can con-clude that (cid:12)(cid:12) t (cid:12)(cid:12) ≈ (cid:12)(cid:12) t − (cid:12)(cid:12) . Additionally, because of the 1GHz zero field splitting of SiV , the measurements on the m s = − ↔ ∼
300 G larger (when aligned with the [111] direction)compared to the measurements on the m s = 0 ↔ +1transition in Fig. 3. This implies that the Orbach processhas a weak dependence on the magnetic field strength. Ratio of T to T The singlet model predicts that the observed ratio of T to T in Figs. 2 and 3 is strongly dependent on theratio of the overlap parameters at zero field. The ana-lytical form of this dependence is shown in Eq. 6 whichis plotted in Fig. S4. This figure shows that this ratiois strongly dependent on the orientation of the magneticfield, indicating that the best way to extract the ratioof the zero field overlap parameters is by performing aglobal fit across all orientations (Fig. 3).0 θ (degrees)0 22.5 45 67.5 90 T , T ( m s ) -1 FIG. S3. Orientation dependence of T (blue) and T (red)measured on the m s = − ↔ (cid:12)(cid:12) t /t ± (cid:12)(cid:12) = 125: (solid line) T ,a , (dashed line) T ,b , and (long dash) T . | / |t ±1 |10 T / T o o o o o o FIG. S4. Plot of T ,a /T , ↔± vs. | t | / | t ± | from Eq. 6for several values of θ . Model for spin relaxation: Orbach process with asinglet excited state
Here we present a detailed analytical derivation of thespin relaxation of SiV for an Orbach process mediatedby a spin singlet excited state. The neutral silicon va-cancy center has D d symmetry with a ground spin-triplet state ( A g ), and the first excited singlet state isexpected to be E g . The splitting between these statesis unknown. At zero magnetic field the triplet and sin-glet states can mix through spin-orbit coupling assistedby phonons [38–41]: | ¯ A m s =02 g i = | A m s =02 g i + t | E g i| ¯ A m s =+12 g i = | A m s =+12 g i + t | E g i| ¯ A m s = − g i = | A m s = − g i + t − | E g i| ¯ E g i = | E g i + X m s t m s | A m s g i , (S11)where t m s are state mixing coefficients. In the main textwe refer to them as overlap parameters that connect thesinglet and triplet subspaces since t m s = h ¯ A m s g | ¯ E g i .The t m s coefficients arise from spin-orbit coupling andthus depend only on the orbital symmetry of the involvedzero-field states, which is independent of the applied mag-netic field.The triplet eigenstates in the presence of a magneticfield can be found using a Wigner rotation to transformthe eigenstates of the zero field splitting term from themolecular frame to the laboratory frame (the frame inwhich the Zeeman interaction is diagonal). This modelassumes that in a magnetic field the eigenstates of thespin Hamiltonian have mostly Zeeman character and thezero field splitting term can be neglected ( gµ B B/h ≫ D ).A general rotation, R , can be expressed in terms of Eulerangles: R ( α, β, γ ) = R ˆ z ( γ ) R ˆ n ( β ) R ˆ z ( α ), (S12)where ~ Ω = ( α, β, γ ) is the set of Euler angles following the“passive” convention. Under this rotation the irreducibletensors in the spin Hamiltonian T J,m transform to ρ J,m as: ρ J,m = R ( α, β, γ ) T J,m R − ( α, β, γ ) = X m ′ D Jm ′ ,m ( α, β, γ ) T J,m ′ , (S13)where D Jm ′ ,m (Ω) is the Wigner matrix of rank J . Theelements of this matrix are: D Jm ′ ,m ( α, β, γ ) = exp( − im ′ α ) d Jm ′ ,m ( β ) exp( − imγ ),(S14)with d Jm ′ ,m ( β ) = Z θ,φ d Ω Y ∗ Jm ′ ( θ, φ ) e − i ~ βJ ˆ n Y Jm ( θ, φ ) ,(S15)where Y Jm ′ ( θ, φ ) are the standard spherical harmonicfunctions and J ˆ n is the component of the total angularmomentum along ˆ n k h i . Then for J = S = 1:1 D S =1 m ′ ,m ( α, β, γ ) = β )2 e − i ( α + γ ) − √ sin( β ) e − iα − cos( β )2 e − i ( α − γ )1 √ sin( β ) e − iγ cos( β ) − √ sin( β ) e iγ − cos( β )2 e i ( α − γ ) 1 √ sin( β ) e iα β )2 e i ( α + γ ) .(S16)If we specifically define R as the rotation away from h i about the h i axis, so that α = ϕ , β = θ , and γ = 0 ◦ define the orientation of the magnetic field, the mixingof the transition amplitudes is given by: t m ′ = X m D S =1 m ′ ,m ( ϕ, θ, t m , (S17)From this we obtain the transition rates ( | t m | ) by in-voking the random phase approximation to neglect thecross terms (averaging over ϕ ). The physical origin of therandom phase approximation can arise from taking anensemble average over a bath of phonons that randomlyinduce transitions to the excited state through spin-orbitcoupling. The result is: | t m ′ | = X m h (cid:12)(cid:12) D S =1 m ′ ,m ( ϕ, θ, (cid:12)(cid:12) i ϕ (cid:12)(cid:12) t m (cid:12)(cid:12) , (S18)where h (cid:12)(cid:12) D S =1 m ′ ,m ( ϕ, θ, (cid:12)(cid:12) i ϕ = cos ( θ/ sin ( θ ) sin ( θ/ sin ( θ ) cos ( θ ) sin ( θ )sin ( θ/ sin ( θ ) cos ( θ/ . (S19)In the main text, Eqns. 3 for the overlap coefficients inthe presence of an off-axis magnetic field are obtained bysubstituting Eq. S19 into Eq. S18.Next, the transition rate matrix (Eq. S19) can be usedto model the spin relaxation processes for S = 1 wherethe populations P = ( P − , P , P +1 ) evolve according to: d P ( t ) dt = ˜ R P ( t ), (S20)where the rate matrix ˜ R is given by:˜ R m,m ′ = C (1 − δ m,m ′ ) µ m − m ′ | t m | | t m ′ | − Cδ m,m ′ X m ′′ = m | t m | | t m ′′ | µ m ′′ − m , (S21)where δ m,m ′ is the Kronecker delta function and µ =exp( hf /kT ) is the Boltzmann factor at T = 30 K and f = 9 . (cid:12)(cid:12) t (cid:12)(cid:12) = (cid:12)(cid:12) t − (cid:12)(cid:12) and µ = 1,this results in two distinct rate eigenvalues λ , λ corre-sponding to T ,a = λ − e − E a /kT and T ,b = λ − e − E a /kT : T ,a = 2 e − E a /kT C | t | (cid:16) | t +1 | + | t − | (cid:17) T ,b = e − E a /kT C | t − | (cid:16) | t +1 | + | t | (cid:17) . (S22)Eqns. S22 were used to simulate the angular dependenceof T as a function of | t | / | t ± | in Figs. 3a and 4b. Ifwe assume that (cid:12)(cid:12) t (cid:12)(cid:12) ≫ (cid:12)(cid:12) t ± (cid:12)(cid:12) then Eqns. S22 reduce toEqns. 5. Model for spin relaxation: Orbach process with atriplet excited state
The excited state can also be a spin triplet state, suchas a quasilocalized vibronic mode or a low lying electronicstate. For this model we define two S=1 spin Hamiltoni-ans for the ground state ( ˆ H g ) and excited state ( ˆ H e ) thatdiffer only in their zero field splitting tensors ( ˜ D g = ˜ D e ):ˆ H g = ˆ S † ˜ D g ˆ S + µ B ˆ S † ˜ g B ˆ H e = ˆ S † ˜ D e ˆ S + µ B ˆ S † ˜ g B , (S23)with eigenstates | m s i g and | n s i e , respectively. The ratematrix describing the spin relaxation is given by: R m.m ′ = C (1 − δ m,m ′ ) X n | g h m | n i e e h n | m ′ i g | µ m − m ′ − Cδ m,m ′ X m ′′ = m µ m ′′ − m X n | g h m ′′ | n i e e h n | m i g | .(S24)In the triplet model spin flips can occur through any ofthe three spin sublevels of the excited state (Fig. S5b),increasing the complexity of the rate matrix. Spin relax-ation arises from the overlap between the eigenstates ofthe two triplet states, and slight variations in the charac-ter of the states become important. Thus the zero fieldsplitting terms for both the ground state and excitedstate cannot be neglected when calculating the tripletstate overlap coefficients and the rate matrix ( R m,m ′ ).For these reasons the analytical solution for the Orbachmodel with a triplet excited state is not compact, andinstead we numerically simulate the spin relaxation bydiagonalizing the rate matrix Eq. S24.In general ˜ D e can differ from ˜ D g in either its quantiza-tion axis, magnitude of the axial component, or magni-tude of the rhombicity parameter. In the case where the2 E g3 A m s = -1 m s = 0 m s = +1S=1S=0 m s = -1 m s = 0 m s = +1S=1S=1(a)(b) n s = -1 n s = 0 n s = +1 −1 T , T ( s ) (c) FIG. S5. (a) Model for the Orbach process of SiV with asinglet excited state. The transition rates ( | t m s | ) are de-termined by spin orbit coupling and depend on the overlapbetween the electronic wavefunctions of the ground tripletstate and the excited singlet state. (b) Model for the Orbachprocess of SiV with a triplet excited state. The transitionrates between the ground state spin sublevels depend on theoverlap between the spin eigenstates of the ground state andexcited state, and must be summed over all spin sublevelsin the excited state. (c) T and T orientation dependencefrom Fig. 3a, plotted against calculated fits from the singlet(black) and triplet (red) Orbach models: (solid line) T ,a ,(dashed line) T ,b , and (long dash) T . The singlet modelfit assumes (cid:12)(cid:12) t /t ± (cid:12)(cid:12) = 125. The triplet model fit assumes acoaxial excited state ZFS tensor with D e = 5 GHz. quantization axis of the excited state is not aligned withthe quantization axis of the ground state (e.g. due toan E type quasilocalized vibronic mode that breaks D d symmetry) the resulting orientation dependence qualita-tively disagrees with the T data. The same disagreementwas found to be true for the case where rhombicity wasintroduced into the excited state spin Hamiltonian. How-ever, the orientation dependence of T can be partiallyreproduced by assuming that ˜ D e is axial (no rhombicity)and also coaxial with the ground state ˜ D g , thus preserv-ing D d symmetry. Focusing on just fitting the T orien-tation dependence (ignoring T ), the closest fit to the T data was found with D e = 1 GHz. However, this excitedstate zero field splitting tensor predicts that T ∼ T data, D e needs to becomparable to the Zeeman energy, and the T data isreproduced best by the triplet model when D e ≈ − T for this excited state zero field splitting is quali-tatively similar to the data, but lies outside of the errorbars for both time constants (Fig. S5c). Furthermore,such a large difference in zero field splitting between theground and excited states is unlikely. Orientation dependence of T Our model for the Orbach process predicts a weak ori-entation dependence of T . The orientation dependencefits shown in Fig. 3a utilize the same overlap amplitudes( t , t ± ) to explain both T and T . The actual expres-sion used in fitting the T dependence in Fig. 3a is givenby: 1 T , ↔± = 13 C (cid:16)(cid:12)(cid:12) t (cid:12)(cid:12) + 2 (cid:12)(cid:12) t ± (cid:12)(cid:12) (cid:17) (cid:16) | t | + | t ± | (cid:17) +1 T ID ) + 1 T SD ) , (S25)which in addition to the Orbach process also includesinstantaneous diffusion and C spectral diffusion mech-anisms. We used T ID ) = 0 .
319 ms and T SD ) = 0 . T is shown in Fig. S6with the simulated fits according to the singlet (Fig. S6a)and triplet (Fig. S6b) models, using the t m s and C pa-rameters determined from the T data. The singlet modelhas no other free fitting parameters, and we plot Eq. S25for the singlet model assuming that (cid:12)(cid:12) t /t ± (cid:12)(cid:12) = 125 asdetermined from the fit of the T orientation dependence(Fig. 3a). The singlet model predicts the magnitude of T with reasonable accuracy.3 T ( m s ) θ (degrees)0.100.150.200.25 T ( m s )
30 75600 15 45 90θ (degrees)30 7560 D e (a) FIG. S6. Orientation dependence of the SiV electron spincoherence time, T (replotted from Fig. 3a). (a) Predictedorientation dependence using an Orbach model with a singletexcited state (black curve, Eq. S25) with no free parameters.Measurements were made on both the m s = 0 ↔ +1 tran-sition (red points) and the m s = − ↔ D e (labeledin units of GHz). The triplet model has four free parameters, two an-gles that set the quantization axis of the excited state,the axial part of the zero field splitting tensor, and therhombic part of the zero field splitting tensor. We onlyconsider the case where the zero field splitting tensor ofthe excited state is axial and aligned with the symmetryaxis of the defect since this is the case that best producesthe measured T orientation dependence (Fig. S5c). Thedependence for several values of D e is shown and the bestfit occurs with D e ≈ − T2