Supplementary Figures

Transcription

1 Supplementary Fgures 3.5K B Permanent Magnet Supplementary Fgure 1: Expermental setup. The hgh-na mcroscope objectve s mounted on a nano-postonng pezo stage. The MW-carryng wre placed near the poston of the SIL. The magnetc feld parallel to the axs of the objectve lens s produced by a rng magnet fxed on the objectve housng. The magntude and the drecton of the feld at the focus of the objectve were characterzed by measurng ODMR of NV centres n a damond sngle crystal. The drecton of the magnetc feld was msalgned by a few degrees. The measured magntude was 490 G. 1

2 Fluorescence Intensty (Arb. Unts) Angle (deg) Supplementary Fgure 2: Dependence of optcal pumpng on the exctaton ellptcty. The ellptcty of the exctaton beam s controlled by λ/4 waveplate placed nto lnearly-polarzed laser beam, therefore, the fluorescence sgnal s gven as a functon of the angle between the fast axs of the waveplate and the ntal laser polarzaton. The curve s not purely snusodal ndcatng sgnfcant devaton of the retardaton phase ntroduced by the waveplate from π/2. Ths fact mght also account for the fdelty of ntalzaton beng lower than theoretcally predcted. 2

3 Coherence π 2π 4π 8π 16π 64π 256π 512π 1024π 2048π 4096π 8192π 16384π 24576π Tme (µs) Supplementary Fgure 3: Smulated CPMG-echo sgnals. Only the contrbutons from octahedral 27 Al nucle are taken nto account. 3

4 Fluorescence Intensty (Arb. Unts) Ce:YSO T 2 =17ns Ce:YAG T 2 =2.2ns Tme (ns) Supplementary Fgure 4: Larmor beats of excted Ce 3+ spn n YSO nanopartcles. The Larmor beats of the cerum spns n the excted 5d(1) state are smlar to the ones reported n Ce 3+ : YAG [1]. Due to the absence of dense 27 Al spn bath, the decoherence tme ncreased from 2.2 ns reported for YAG host to 17 ns at room temperature and s beleved to be lmted by spn-lattce relaxaton n the excted 5d(1) state. 4

5 Fgure 1 Tetrahedral and Octahedral local symmetry of alumnum nuclear spn. The blue dots are alumnum Supplementary atom, and Fgure the red 5: dots Tetrahedral are oxygen and Octahedral atoms. local symmetry of alumnum nuclear spn. The blue dots are alumnum atom, and the red dots are oxygen where 7 6 = atoms. dag=1.87,0.91,2.74b s the effectve g-tensor. We have checked that the effectve g-tensor obtaned from the crystal feld parameters s consstent wth that determned n ensemble ESR experment. The Ce 3+ on s n a spn bath whch conssts of surroundng nuclear spns from the host lattce and, possbly, electron spns of other defects. Here, we only calculate the decoherence contrbuted by the nuclear spns. We wll show that the nuclear spn bath s the domnant decoherence source n our measurement. We consder two speces of atoms whch carry nuclear spns, namely, spn- ' of yttrum and. spn-c of alumnum. The Hamltonan of the nuclear spn. bath reads 5 DEFG = HI JK JK L" JK + I M M L" M N ". PF JK + O * HQRS N * PF. F6FU JK + O * HQRS N * F6FU

6 CCE calculaton CPMG-1024 Coherence Tme ( s) Supplementary Fgure 6: CCE calculaton results of the Ce 3+ on spn coherence under 1024-pulse CPMG DD control. The black curve wth crcle symbols ndcates the frst order expanson result, whle the red dashed lne s the second order (takng nto account the nter-nucle nteracton). The shadow areas (n orange, green and blue) ndcate the coherence decay nduced by the nose wth partcular frequences (shown n the correspondng colors n Supplementary Fgure 8). 6

7 1.00 Echo 0.75 Coherence Tme (ns) Supplementary Fgure 7: Cluster expanson results of Ce 3+ on coherence under CPMG dynamcal decouplng control. The control pulse number s ndcated above the fgure. 7

8 ra to b j / j Y A l (O ) A l (T ) A l (T ) A l (T ) A l (T ) F lte r fu n c to n t = 2 0 µs F lte r fu n c to n F lte r fu n c to n t = 4 0 µs t = 6 0 µs t = µs F lte r fu n c to n F re q u e n c y ω / (2π) (M H z ) Supplementary Fgure 8: Nose spectrum of the nuclear spn bath and flter functon of CPMG-1024 DD control. 8

9 ~ 1 MHz I z = ± 5 2 ~ 24 MHz ~ 36 MHz ~ 1 MHz I z = ± 3 2 ~ 12 MHz ~ 1 MHz I z = ± 1 2 Supplementary Fgure 9: Energy level structure of alumnum nuclear spn on tetrahedron [Al (T)] stes. The arrows n green, orange, and black ndcates the transton between sublevels wth dfferent I 2 z, whle the purple arrows ndcates the transtons wthn Zeeman doublets. The thckness of the arrows s proportonal to the transton ampltudes. 9

10 F lte r fu n c to n C P M G t = 2 0 µs π = C P M G t = 2 0 µs π = n s F re q u e n c y ω / (2π) (M H z ) F re q u e n c y ω / (2π) (M H z ) Supplementary Fgure 10: Flter functon wth zero and fnte pulse duraton (τ π =20.6 ns). The peak heght s greatly suppressed by the pulse duraton nduced modulaton effect. 10

11 C o h e re n c e C o h e re n c e π = π = n s tm e µs Supplementary Fgure 11: Coherence calculated accordng to Eq. (10) wth the flter functons consderng (bottom panel) / not consderng (top panel) the fnte pulse duraton. The coherence dps are greatly suppressed when takng nto account the 20 ns pulse duratons. 11

12 N o s e s p e c tru m S ( ) µs p e a k = N π / t = π / t n o s e s p e c tru m flte r fu n c to n flte r fu n c to n F re q u e n c y / (2π) (M H z ) Supplementary Fgure 12: Typcal Lorentzan nose spectrum of electron spn bath (blue curve, parameters taken from Ref. [5]), and flter functon of CPMG control (red curve). The nose spectrum decays as ω ( 2) for ωτ C 1, and flter functon has a strong peak located at ω peak = Nπ/t, wth the peak wdth nversely proportonal to t. 12

13 2.0 T 2 tm e (m s ) c a lc u la te d T 2 tm e fttn g T 2 tm e (µs ) P u ls e N u m b e r Supplementary Fgure 13: Lnear dependence of the coherence tme on the CPMG pulse number. The nset shows the coherence tme for small pulse numbers. 13

14 1.0 c a lc u la te d d a ta fttn g 1.0 c a lc u la te d d a ta fttn g C o h e re n c e 0.5 N = N = 2 k = 3.8 k = tm e (µs ) tm e (µs ) 1.0 c a lc u la te d d a ta fttn g 1.0 c a lc u la te d d a ta fttn g C o h e re n c e 0.5 N = k = N = k = tm e (µs ) tm e (µs ) Supplementary Fgure 14: Fttng some of the theoretcal [ calculated decoherence ( ) k ] curves shown n Supplementary Fgure 3 by exp for dfferent pulse numbers. The power ndex ncreases from k 4 for small pulse number to k 8 for large pulse number. t T (N) 2 14

15 Supplementary Methods Here, we nvestgate the spn decoherence of a sngle Ce 3+ on doped n YAG crystal. We perform both cluster-expanson calculatons and nose spectrum analyss. Wth the fnte pulse duraton taken nto account, the theoretcal results are n good agreement to the observed coherence data under CPMG dynamcal decouplng (DD) control. The concluson of the theoretcal analyss s that the observed Ce 3+ on spn decoherence under DD control s domnated by the Al nuclear spn on octahedral stes. System Hamltonan We start from the lattce structure of the YAG crystal. In pure YAG crystal, a cubc unt cell contans 160 atoms, ncludng 96 oxygen atoms (zero-spn), 40 alumnum atoms (spn-5/2), and 24 yttrum atoms (spn-1/2). Notce that the 40 alumnum atoms are dvded nto 2 groups accordng to ther local symmetry, ether tetrahedral (24 atoms) or octahedral (16 atoms) [denoted by (T) or (O) respectvely]. Schematcally, the local surroundngs of both alumnum speces are ndcated n Supplementary Fgure 5. The Ce 3+ on replaces one of the yttrum atom n the lattce, formng a center spn. The level structure of the doped Ce 3+ on s stll atom-lke. The presence of the lattce affects the level structure through the crystal feld. The Hamltonan of the Ce 3+ on s H 4 f on = λl s + A lm O m l + µ B B (l + 2s) = l,m 14 =1 ω ψ ψ (1) where we only consder the ground state 4f manfold, λ s the spn-orbtal couplng strength, O m l s the crystal feld operators represented by sphercal harmoncs, A lm are the strengths of the correspondng symmetry components of the crystal feld, and the thrd term s the Zeeman energy. Here, we have dagonalzed the Hamltonan H 4 f on wth the egen-frequency ω (n ncreasng order, ω > ω j f > j) and the correspondng egen-functon ψ. Now we constran the spn dynamcs n the subspace spanned by the lowest doublet of Hamltonan (1),.e. { ψ 1, ψ 2 }. Usng the Paul matrces σ defned as σ x = ψ 2 ψ 1 + ψ 1 ψ 2, σ y = ( ψ 1 ψ 2 ψ 2 ψ 1 ), and σ z = ψ 2 ψ 2 ψ 1 ψ 1, we project the Hamltonan n Eq. 1 n the two-dmensonal subspace as H eff on = µ BB g eff σ (2) where g eff =dag[1.87,0.91,2.74] s the effectve g-tensor. The Ce 3+ on s n a spn bath whch conssts of surroundng nuclear spns from the host lattce and, possbly, electron spns of other defects. Here, we only calculate the decoherence 15

16 contrbuted by the nuclear spns. We wll show that the nuclear spn bath s the domnant decoherence source n our measurement. We consder two speces of atoms whch carry nuclear spns, namely, spn-1/2 of yttrum and spn-5/2 of alumnum. The Hamltonan of the nuclear spn bath reads as follows: H bath = B + + j oct α,β=al,y, j (g (Al) n µ (Al) n I (Al) Q oct j (I z (Al) j ) 2 + I (α) D αβ j j tetr + g (Y) n µ (Y) n I (Y) ) Q (tetr) j (I z (Al) j ) 2 I (β) j, (3) where the frst lne s the Zeeman energy of alumnum nucle I (Al) and yttrum nucle I (Y). The second and the thrd terms are the quadrupole nteractons of the alumnum nucle wth the crystal feld of octahedral and tetrahedral local symmetry, respectvely. The thrd lne descrbes the nter-nuclear dpolar nteracton. The quadrupole nteracton strengths are obtaned from NMR measurement [2] n YAG crystal,.e. Q (oct) j =0.6 MHz and Q (tetr) j =6.0 MHz. In YAG, local octahedral and tetrahedral envronments have 4 and 3 magnetcally nequvalent prncpal axes, respectvely [3]. Wth the external magnetc feld appled, ths leads to the fact that nuclear spn par flp-flops due to the magnetc dpole nteracton between two alumnum nuclear spns are greatly suppressed by the spn energy msmatch. In ths sense, the nuclear spn bath of YAG crystal can be regarded as non-nteractng (.e., each nuclear spn evolves ndependently). The hyperfne (hf) nteracton between the Ce 3+ on and the nuclear spns, constraned n the lowest doublet subspace, are descrbed by [4] H eff hf = ψ H hf ψ j ψ ψ j σ 2, j=1,2 α=al,y, j A (α) I (α) σ 2 ˆb (4) where we have defned a nose operator ˆb, consstng of the nuclear spn operators. In the nteracton pcture drven by the bath Hamltonan (3), the operator ˆb(t) s tme-dependent. Seen by the central on spn, t s the nose produced by the nuclear spn bath. In Eq. (4), the hyperfne couplng tensor A (α) s defne as A (α) = g eff T (α) T (α) = µ g 0 e µ B g (α) 4π n, µ(α) n, r 3 (3n n 1), (5) 16

17 where T (α) s the dpolar couplng tensor, r s the dsplacement of the -th nuclear spn to the Ce 3+ on, n s the correspondng drecton vector, g e (g (α) n, ) and µ e(µ (α) n, ) are the electron (nuclear) spn g-factor and electron (nuclear) Bohr magneton, respectvely. The hyperfne couplng n Eqs. (4) and (5) s equvalent to a descrpton that the on s a magnetc moment µ = µ B g eff σ/2 nteractng wth the nuclear spns. Notce that we have assumed that the Ce 3+ on s a pont spn. Ths s a good approxmaton for the nuclear spns wth dstance much larger than the on radus, whle for the a few nuclear spns close to the on (wth dstance Å), ths approxmaton only gves a rough estmaton of the couplng strength. Snce the decoherence we concern s manly caused by a large number of nuclear spns, the naccuracy of the hyperfne couplng of the adjacent nuclear spns does not strongly affect the overall decoherence behavor. Ce 3+ Ion spn decoherence The system s subject to a magnetc feld of 490 Gauss along the [110] drecton. For convenence, ths drecton s chosen to be z-axs unlke n the man text. In such a magnetc feld, the electron spn Zeeman splttng ( GHz) s several orders of magntude larger than the nuclear spn splttng ( MHz). The hyperfne nteracton (also on the order of MHz, at the most) between the on electron and the nuclear spns are not strong enough to compensate the energy msmatch and to flp the on electron states. In ths case, we neglect the flppng operators σ x and σ y of the on electron n the effectve hyperfne Hamltonan (4). Wth ths approxmaton, the dynamcs of bath spns s condtoned on the on electron states. For on electron n the ψ 2 (or ψ 1 ) state, the bath spns wll evolve accordng to the condtonal Hamltonan H + (or H ) H ± = H bath ± ê z A (α),α I (α) = H bath ± ˆb z (6) where ê z s the unt vector of z axs, and only the z component of the nose b s retaned. If the on electron s prepared n a superposton state ( ψ 1 + ψ 2 )/ 2 by a π/2-pulse, t wll get entangled wth the condtonal bath spn states drven by H ±. Assumng the bath spns are ntally n a thermal equlbrum state ρ bath, under N-pulse dynamcal decouplng, the on center spn coherence L(t) s expressed as [5] L(t) = Tr[...e H +t 3 e H t 2 e H +t 1 ρ bath e H t 1 e H +t 2 e H t 3...]. (7) 17

18 Correlated-cluster-expanson (CCE) calculatons In prncple, n order to calculate the center spn coherence accordng to Eq. (7), one must solve the many-body dynamcs nvolvng, e.g., hundreds of nuclear spns. However, as shown n Ref. [5], the center spn coherence L(t) can be factorzed nto the contrbutons L C (t) of dfferent correlated nuclear spn clusters C,.e., L(t) = L C (t), (8) C where L C (t) = 1 f the cluster C can be dvded nto two non-nteractng subclusters C 1 and C 2. Partcularly, n our case, the nter-nucle dpolar couplng s neglgble small comparng to ther energy dfference. As we dscussed above, the nuclear spns wll evolve ndependently. Thus, the coherence L C (t) 1 for any clusters C wth more than 1 nuclear spn. The coherence s the product of the contrbutons form sngle nuclear spns, as shown n the numercal calculaton results n Supplementary Fgure 6. The calculated coherence by cluster-expanson method shown n Supplementary Fgure 7 has a smlar overall decay tme than the observed data. Ths confrms that, n our samples, the nuclear spn bath s the domnatng decoherence source to the Ce 3+ on. However, the cluster-expanson results show strong modulatons n addtonal to the overall coherence decay (see Supplementary Fgure 6). These coherence modulatons are not observed n the measurement. In the followng, we perform an analyss of the nose spectrum produced by the nuclear bath. The nose spectrum analyss wll reveal the underlyng physcs of the CCE calculaton results. Also, t explans the reason for the absence of strong coherence modulaton n our experment. Nose spectrum analyss The on spn coherence defned n Eq. (7) can be approxmately expressed n terms of the nose spectrum of the nuclear spn bath and the flter functon due to the DD control as L(t) = e 1 2 χ(t), (9) where the functon χ(t) s determned by the overlap between the nose spectrum S (ω) and the flter functon F(ωt) χ(t) = o dω [ S (ω) ] π ω 2 F(ωt). (10) 18

19 The flter functon F(ωt) s the Fourer transform of a sgn functon f (t), whch toggles between +1 and -1 whenever a π pulse s appled. For the N-pulse CPMG DD control, the flter functons F(ωt) reads (see Supplementary Fgure 8) [ ( ωt ) ] 2 ωt F(ωt) = 4 sec 1 cos, for odd N, 2N 2 [ ( ωt ) ] 2 ωt F(ωt) = 4 sec 1 sn, for even N. (11) 2N 2 The nose spectrum S (ω) s the Fourer transform of the nose correlaton functon C(t 2 t 1 ), whch s defned as C(t 2 t 1 ) = Tr bath [ρ bath ˆb z (t 1 )ˆb z (t 2 )] (12) The nose spectrum S (ω) can be expressed n the egen-representaton of the bath Hamltonan H bath as S (ω) = P b j 2 δ(ω ω j ), (13) > j where ω j = ω ω j s the transton frequency between the egen-states φ and φ j of H bath, b j = φ ˆb z φ j s the scatterng matrx element of the nose operator ˆb z, and P s the populaton on the egen-states φ. Performng the ntegral n Eq. (10), we obtan P b j 2 χ(t) = F(ω j t) (14) > j Equaton (14) shows that each quantum transton between two egen-states φ and φ j for j contrbutes a nose component n the frequency ω j wth the spectrum weght W j b j 2 /ω 2 j. Supplementary Fgure 8 shows the nuclear spn bath nose spectrum weght as functons of the frequency. The nose that arses from the quantum transton of dfferent nuclear spn speces [yttrum, alumnum on octahedron stes (O) and alumnum on tetrahedron stes (T)] of the bath are ndcated by dfferent symbols (damond, crlcle, and star symbols, n turn). For yttrum (Y) nuclear spn-1/2 s (red damond symbol), they emt nose wth frequency correspondng to ther Zeeman frequency ( 100 khz n 490 Gauss magnetc feld). For alumnum nuclear spn-5/2 s on octahedral stes [Al (O)], the transton frequences are determned by the quadrupole nteracton 0.6 MHz, and the Zeeman frequency 500 khz. Wth dfferent orentatons of the prncple symmetry axes, the nose from Al (O) spns spreads n the range below 5 MHz. ω 2 j 19

20 For alumnum nuclear spn-5/2 s on tetrahedral stes [Al (T)], due to the large quadrupole nteracton 6.0 MHz, the spectrum can be separated nto four groups (denoted by dfferent colors) around 1 MHz, 12 MHz, 24 MHz, and 36 MHz. Ths can be understood by consderng the energy levels of an Al (T) n an external magnetc fled (see Supplementary Fgure 9). As shown n Supplementary Fgure 8, the flter functon F(ωt) of the N-pulse CPMG control has sharp peaks whose heght s proportonal to N 2. As ncreasng the evoluton tme t, the flter functon peaks sweep from the hgh frequency sde. Whenever the peak gets across the non-zero nose spectrum components (.e. the symbols n Supplementary Fgure 8), the coherence drops accordng to Eq. (14). For example, through Supplementary Fgure 8, we fnd that the nose components around 36 MHz s too weak to contrbute sgnfcant decoherence. Furthermore, at t 20µs, the flter functon peak overlaps wth the nose components around 24 MHz produced by the Al (T) nuclear spns, and ths gves rse to the frst coherence dp n Supplementary Fgure 6 (.e., the dp hghlghted by the n the frst shadow n orange). When t 40µs, the flter functon peak gets across the 12 MHz nose components, and generates the dps n the green shadow n Supplementary Fgure 6. Smlarly, at t 60µs, the second flter functon peak s responsble for the coherence dp n the second orange shadow n Supplementary Fgure 6. Fnally, when t 100µs, the frst peak of the flter functon overlaps wth the strong nose produced by the Al (O) nuclear spns. Ths nose s so strong and s extended to the lower frequency, formng a quas-contnuous band, that causes the coherence decay wthout recovery. The low frequency nose produced by the yttrum nuclear spn and the 1 MHz nose component by Al (T) nuclear spn, whch are bured n the band, does not take effect before the total coherence decays out. Wth above nose spectrum analyss, we gve a physcal understandng of the coherence dynamcs shown n Supplementary Fgure 6. Moreover, the nose spectrum method also provdes a convenent way for analyzng the mperfect pulse effect (such as the fnte duraton of the π pulses n the CPMG control), whch explans the absence of the coherence dps produced by Al (T) nuclear spns n experments. Pulse duraton effect In the cluster-expanson calculaton, the mcrowave pulses are assumed to be deal,.e., δ-pulse wth zero duraton. Here, we demonstrate that, the fnte π-pulse duraton ( 20 ns n our measurement) could cause the absence of the strong coherence dps (see n Supplementary Fgure 6) n experments. The net effect of a π pulse s changng the sgn of the nose (from +1 before 20

21 the pulse to -1 after the pulse, or vce versa). Whle durng the π pulse, we assume that the nose can be effectvely taken to be zero. Thus, wth fnte pulse duraton τ π, the flter functon n Eq. (11) should be modfed to [6] [ ( ωt ) ( ωτπ ) ] 2 ωt F(ωt) = 4 sec cos 1 cos, for odd N, 2N 2 2 [ ( ωt ) ( ωτπ ) ] 2 ωt F(ωt) = 4 sec cos 1 sn, for even N. (15) 2N 2 2 Notce that a cos ωτ π 2 factor appear n the slow varyng envelope functon n the bracket. In other words, the fnte pulse duraton τ π provdes an addtonal cosne functon, whch modulates the envelope functon. For a gven pulse duraton τ π, the peaks of the flter functon wll be greatly suppressed f they are close to the zero ponts of cos ωτ π 2,.e., ω π τ π. Supplementary Fgure 10 shows the flter functon wth zero and fnte pulse duraton. The peak heght s reduced by about 3 orders of magntude. Wth the modfed flter functon, the coherence dps become much narrower and shallower, as shown n Supplementary Fgure 11, and are hardly observed n experments due to the low tme resoluton. Here, we pont out that we have used a smplfed model to descrbe the fnte pulse duraton effect, and to demonstrate the suppresson of the coherence dps. More sophstcated model could be adapted (for example, ncludng the pulse shape, and randomness of the pulse tmngs) to descrbe the non-deal pulse effect. These factors would also reduce the flter functon peaks, and cause the absence of the coherence dps n the real measurement. The detaled nvestgaton of these effects goes beyond the scope of ths paper. Decoherence nduced by Al (O) nuclear spns Wth all the above dscussons, we conclude that the observed Ce 3+ on decoherence s domnated by the nose produced by the Al (O) nuclear spns. We sngle out the effect of Al (O) nuclear spns, and calculate the decoherence accordng to Eq. (10) wth only the nose components from Al (O) nuclear spns ncluded (.e., the blue dots n Supplementyra Fgure 8). Also, n the calculaton we dropped the nose components wth the spectrum weght b j 2 /ω 2 j > 0.2. For strong nose, the spectrum approxmaton Eqs. (9)-(14) s not vald. Fortunately, as shown n the rgorous CCE calculatons n Supplementary Fgure 7 and our prevous study [7], these strong nose components do not sgnfcantly contrbute to the overall decoherence behavor.the calculated coherence agrees the observed coherence qute well (see Fgure 3). The decay tme s slghtly dfferent wth expermental observed data, whch s possbly due to the naccuracy of the hyperfne couplng strength and other mperfectons n the real measurement. Wth ths, we conclude that the nose 21

22 from alumnum nuclear spns on the octahedral stes are the domnate decoherence source for the observed data. Electron spn bath vs. nuclear spn bath In ths secton, we compare the dfferent decoherence behavor of the center spn n an electron spn bath and a nuclear spn bath, and show that the lnear dependence of the decoherence tme on the CPMG control pulse number s a sgnature of the nuclear spn bath. For the electron spn bath, the ntra-bath nteracton (between bath electron spns) are typcally much stronger than the couplng between the center spn and the bath spns. Meanwhle, each bath electron spn s coupled to the envronment of tself (e.g., phonon bath). In ths sense, the electron spn bath behaves more lke a classcal macroscopc envronment, whose nose spectrum s descrbed by a contnuous functon. As demonstrated n Ref. [8], for the electron spn bath of NV center n damond, the nose from the electron spn bath s well modelled by a spectrum of Lorentzan shape S (ω) = b 2 /τ C ω 2 + ( 1 τ C ) 2 (16) Supplementary Fgure 12 shows an electron spn bath nose spectrum usng the typcal parameters n Ref. [5], (.e., the nose ampltude b 2µs 1 and the correlaton tme τ C 25 µs). One fnds that the nose spectrum of electron spn bath decays as ω 2 wth a soft tal extendng to the hgh frequency regon. Ths s n sharp contrast to the case of nuclear spn bath, as shown n Supplementary Fgure 8, where the nose spectrum s lmted by an upper-bound frequency, and exhbts a hard cutoff at about 5MHz (gnorng the nose components produced by Al (T) nuclear spns, whch s not resolved due to the fnte pulse duraton). In the electron spn bath case, accordng to Eq. (10), the decoherence functon s estmated as χ(t) = 0 = b2 τ c π [ dω S (ω) π ω 2 ( t Nπ ] F(ωt) ω 2 peak S (ω peak)n 2 ω ) 4 N 2 π t = b2 t 3 N 2 π 4 τ C ( t T (N) 2 ) 3 (17) Notce that we have used the fact that the ntegral s domnated by the flter functon peak wdth heght N 2 and wdth ω = π/t. Equaton (17) shows that 22

23 [ ( the center spn decoherence n an electron bath follows exp t T (N) 2 ) 3 ] wth the decoherence tme under CPMG-N control T (N) 2 N 2 3. The decoherence behavor s qute dfferent for the center spn n a nuclear spn bath. Snce we have shown that the nuclear spn bath nose spectrum has a hard cutoff ω cutoff (n our case, ω cutoff 2 π 5 MHz), above whch the nose spectrum drops to zero quckly (nstantaneously, n the deal case) and do not contrbute to the decoherence. Consequently, decoherence only occurs when the flter functon peak approaches the cutoff frequency,.e., ω cutoff = ω peak = Nπ/t, whch mples the decoherence tme s lnear dependent on the pulse number T (N) 2 = Nπ/ω cutoff (see Supplementary Fgure 8). Ths explans the lnear dependence of decoherence tme on the pulse number n nuclear spn bath (see Supplementary Fgure 13). Based on ths pcture, the tme-dependence of the exponental functon χ(t) of the center spn coherence s much more complcated than a smple unversal powerlaw χ(t) t 3. Indeed, the tme dependence of χ(t) s gven n Eq. (14), whch s a weghted sum of the flter functons F(ω j t) wth dfferent exctaton frequences ω j. If we ft the theoretcal results by functon χ(t) t k wth the power k beng the free fttng parameter, we fnd that the power ncreases from k 4 for small pulse number (e.g. for Hahn echo N = 1) to k 8 for large pulse number (see Supplementary Fgure 14), much larger than k = 3 n electron spn bath. The N-dependent decay power s another evdence for the nuclear spn bath nduced decoherence. Supplementary References [1] R. Kolesov, K. Xa, R. Reuter, M. Jamal, R. Stöhr, T. Inal, P. Syushev, and J. Wrachtrup, Phys. Rev. Lett. 111, (2013) [2] K. C. Brog, W. H. Jones, and C. M. Verber, Phys. Lett. 20, 258 (1966) [3] C. Alff and G.K. Werthem, Phys. Rev. 122, 1414 (1961) [4] A. Abragam and B. Bleaney, Electron Paramagnetc Resonance of Transton Ions (Dover Publcatons, Inc., 1970) [5] W. Yang and R.-B. Lu, Phys. Rev. B 78, (2008) [6] M. J. Bercuk, H. Uys, A. P. VanDevender, N. Shga, W. M. Itano, and J. J. Bollnger, Nature 458, 996 (2009) [7] N. Zhao, Z.-Y. Wang, and R.-B. Lu. Phys. Rev. Lett. 106, (2011). 23

24 [8] G. de Lange, Z. H. Wang, D. Rste, V. V Dobrovtsk, and R. Hanson, Scence 330, 60 (2010) 24