Observation of an anomalous decoherence effect in a quantum bath at room temperature

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1 Reeived 5 My 11 Aepted Nov 11 Pulished 6 De 11DOI: 1.138/nomms1579 Oservtion of n nomlous deoherene effet in quntum th t room temperture Pu Hung1, *, X i Ko n g 1, *, N n Zh o, *, Fzhn Shi 1, Pengfei Wng1, Xi ng Ro ng 1, Ren-Bo Liu & Jingfeng Du1 The deoherene of quntum oets is ritil issue in quntum siene nd tehnology. It is generlly elieved tht stronger noise uses fster deoherene. Strikingly, reent theoretil work suggests tht under ertin onditions, the opposite is true for spins in quntum ths. Here we report n experimentl oservtion of n nomlous deoherene effet for the eletron spin-1 of nitrogen-vny entre in high-purity dimond t room temperture. We demonstrte tht, under dynmil deoupling, the doule-trnsition n hve longer oherene time thn the single-trnsition even though the former ouples to the nuler spin th s twie strongly s the ltter does. The exellent greement etween the experimentl nd theoretil results onfirms the ontrollility of the wekly oupled nuler spins in the th, whih is useful in quntum informtion proessing nd quntum metrology. 1 Hefei Ntionl Lortory for Physil Sienes t the Mirosle nd Deprtment of Modern Physis, University of Siene nd Tehnology of Chin, Hefei, Anhui 36, Chin. Deprtment of Physis nd Centre for Quntum, The Chinese University of Hong Kong, Shtin, New Territories, Hong Kong, Chin. *These uthors ontriute eqully to this work. Correspondene nd requests for mterils should e ddressed to R.B.L. (emil: rliu@phy.uhk.edu.hk ) or to J.D. (emil: df@ust.edu.n ). 1

2 T he oupling etween quntum oet nd its environment uses deoherene, whih is key issue in quntum siene nd tehnology 1 3. Suh oupling is usully understood in terms of lssil noise, suh s in the spetrl diffusion theories tht re widely used in, for exmple, mgneti resonne spetrosopy,5 nd optil spetrosopy 6,7. In modern nnotehnology nd quntum siene, the relevnt environment of quntum oet n e of nnometre or even su-nnometre size 8. Therefore, the environment itself is quntum in nture. In reent yers, quntum theories hve een developed to tret the deoherene prolem in mesosopi quntum th 3 8. These quntum theories hve een suessful in studying deoherene in vrious systems nd predited some surprising quntum effets 7. A numer of experiments hve indited the quntum nture of nuler spin ths 8,1 1,16,17. However, up to now, there hs een no experiment expliitly ddressing the fundmentl differene etween lssil nd quntum ths. A reent theoretil study 9 predited n nomlous deoherene effet (ADE) of quntum th on spin higher thn 1 /. Considering spin-1 under unidiretionl mgneti noise z with Hmiltonin z S z, the spin, initilly in superposition of the S z eigensttes ψ () = , will evolve to ψ (t) = e i φ (t) e i φ (t) + with n umulted rndom phse ( t) = t ( t) dt. The oherene of the single-trnsitions ± is determined y the verge of the rndom phse ftor s L, ± = e ± i φ (t), wheres the doule-trnsition hs oherene L, + = e i φ (t). For Gussin noise, s ommonly enountered, L, ± = e φ (t)φ(t) / nd L, + = e φ (t)φ(t) = L, +. Deoherene of the doule-trnsition ehves essentilly the sme s tht of the single-trnsitions, ut is fster s the doule-trnsition suffers from noise tht is twie s strong s tht suffered y the single-trnsitions. Surprisingly, in mesosopi quntum th mde of nuler spins, where the noise z is the th opertor, the predition is tht the oherene time of the doule-trnsition under dynmil deoupling, where the th evolution is ontrolled through the flips of the entrl spin, n e inresed ove tht of the single spin 9. The nitrogen-vny (NV) entre in high-purity dimond, whih hs n eletron spin-1 oupled to nnometre-sized nuler spin th, is n idel system to study the ADE. The NV entre eletron spin (referred to s the entre spin herefter; Figure 1 displys the struture) hs long oherene time ( ~ ms) even t room temperture 1, nd is promising for pplitions in quntum informtion proessing 17,3 35 nd nnometrology36 1. Besides the pplitions, the NV entre eletron spin system is lso good model system for fundmentl reserh on deoherene 8,9 nd dynmil deoupling ontrol. The deoherene of NV entre eletron spins in type-ii smples is used y oupling to the quntum spin th formed from the 13 C nuler spins tht lie within severl nnometres from the entre 8,9. Here we report the experimentl oservtion of the ADE using n NV entre system t room temperture. The oserved entre spin deoherene is in exellent greement with the mirosopi theory. The omined experimentl nd theoretil results demonstrte the pility of mnipulting the evolution of the surrounding 13 C nuler spins y ontrolling the entre spin. This mnipultion pves the wy for exploiting spin ths for quntum informtion proessing 5 nd nnometrology 39,1. Results System nd model. We demonstrte the ADE using prmgneti resonne mesurements nd mirosopi lultions. The experiments re sed on optilly deteted mgneti resonne 6 of single NV entres in type-ii dimond t room temperture. After eing polrized into the stte y 53-nm lser pulse, the entre spin stte is mnipulted y the resonnt mirowve pulses nd the spin oherene is red out through the mesurement of fluoresene I > L +, (t) I+> L,+ (t) I> NATURE COMMUNICATIONS DOI: 1.138/nomms1579 N V B I > 1N I> 1N I+> 1N Initiliztion 53 nm Lser Mirowve Trigger for reding intensity ( Fig. 1 ). The lultion is sed on quntum mnyody theory 6, whih tkes into ount the hyperfine oupling etween the entre spin nd the 13 C th spins nd the dipolr intertion etween 13 C th spins (see Methods). Under zero field, the entre spin hs three eigensttes quntized long the z diretion (the NV xis, [111] diretion), nmely, ± nd. In the experiment, wek mgneti field ( < guss) is pplied long the NV xis to lift the degenery etween + nd. Eh energy level of the spin sttes ± is split into three su-levels owing to the hyperfine oupling with the 1 N nuler spin in the NV entre ( Fig. 1 ). The oherene L, + (t) of the single-trnsition + nd the oherene L +, (t) of the doule-trnsition + ( Fig. 1 ) re mesured for single NV entre. The system hs Hmiltonin H = H NV + H B + H hf. The entre spin Hmiltonin is H NV = Δ S z γ e BS z, where Δ denotes the zerofield splitting nd γ e = s 1T 1 11 is the eletron gyromgneti rtio. The th Hmiltonin H i Ii z B = w + i, Ii D I i ontins the nuler spin Zeemn splitting ( ω ) nd the dipolr intertion etween nuler spins (with oeffiients D i ). The entre spin ouples to the nuler spins through Hhf = Sz A I Szˆ z, where A is the hyperfine oupling to the th nuler spin I. Here the trnsverse omponents of the hyperfine oupling hve een dropped euse they re too wek to use the entre spin flip (with A Δ ). Owing to its dipolr form, the hyperfine oupling strength depends inverse-uilly on the distne of the nuler spin from the entre. The relevnt th spins re those loted within few nnometres from the entre ( Fig. 1d ). Outside this rnge, the nuler spins hve too wek hyperfine oupling to ontriute to the entre spin deoherene 9. Thus, within the deoherene timesle ( < ms), the entre spin together with ~ 1 th spins form reltively losed quntum system ( Supplementry Fig. S1 ). d Mnipultion Red out Figure 1 System nd methods for mesuring NV entre spin deoherene in 13 C nuler spin th in dimond. ( ) Atomi struture of nitrogen (N) - vny (V) entre in dimond nd the mgneti field B. ( ) Pulse sequenes for the optilly deteted mgneti resonne mesurement. The entre spin is initilized y the 53-nm lser pulse, mnipulted y mirowve pulses, nd red out through the fluoresene. ( ) Energy levels of n NV entre spin. The eletron spin nd 1 N-nuler spin sttes re denoted y m s nd m I 1 N for m s,m I =, ±, respetively. The level splitting is used y the hyperfine oupling to the 1 N nuler spin. The single-trnsition oherene L, + nd the doule-trnsition oherene L +, re mesured in the experiments. The solid (dshed) vertil rrows denote the nonseletive (seletive) mirowve exittions of the entre spin. ( d ) Shemti of 13 C-nuler spin th (enlosed y the irle), whih together with the NV entre spin form reltively losed quntum system. NV 13 C

3 NATURE COMMUNICATIONS DOI: 1.138/nomms1579 ARTICLE Free-indution dey of the entre spin oherene. Vie we d f rom the entre spin, the hyperfine oupling provides quntum noise field ˆ z. Sine ˆz in generl does not ommute with the totl Hmiltonin H, ertin noise-field eigenstte will evolve to superposition of different eigensttes of ˆ z, whih leds to quntum flututions of the entre spin splitting. The Hmiltonin n lso e expressed s ( ) ( ) H= w + H, =, ± (1) where ω α = Δ α γ e B is the eigenenergy of α, nd H ( ) = H + ˆ B z governs the th dynmis onditioned on the entre spin stte. Viewed from the th, the hyperfine oupling is k tion, onditioned on the entre spin stte. Thus, the entre spin deoherene is used y onditionl th evolution, whih reords the whihwy informtion of the entre spin 5,9. Besides the quntum flututions, there re lso lssil therml flututions due to the rndom orienttions of nuler spins t room temperture. Indeed, the therml flututions (lso lled inhomogeneous rodening) re muh stronger thn the quntum flututions nd use the free-indution dey of entre spin oherene within severl miroseonds. However, the inhomogeneous rodening effet n e removed y spin eho 7. The oexistene of lssil nd quntum flututions nd their different effets under spin eho ontrol enle the in-situ omprison etween the lssil nd quntum noises. Figure shows the free-indution dey of the entre spin oherene. Both single- nd doule-trnsition deoherene hve Gussin dey envelopes exp( t /T* ), with the dephsing time of the doule-trnsition ( T * = ±.11 μ s) out hlf tht of the singletrnsition ( T * =.33 ±.39 μs). This verifies the sling reltion L+, ( t)= L, + ( t), () s predited in ref. 9 for therml flututions. The experimentl dt re in good greement with the numeril results otined y onsidering the inhomogeneous rodening of 13 C nuler spin th. Hhn eho of the entre spin oherene. The quntum flututions eome relevnt when the inhomogeneous effet is removed y spin eho 7. Figure 3 shows the Hhn eho signls under n externl mgneti field B = 13.5 guss. The single-trnsition oherene presents periodi revivls. In ontrst, the doule-trnsition oherene deys to zero within severl miroseonds nd does not reover. Suh qulittive differenes result diretly from mnipultion of the quntum th on the entre spin flip Figure Free-indution dey of the NV entre spin oherene. (, ) Mesured (olour lines with symols) nd lulted (envelopes in lk lines) free-indution dey of the single- ( ) nd doule-trnsition ( ) oherene, respetively. The osilltions nd the symmetri envelopes re due to oupling to the 1 N nuler spin. A mgneti field B = 13.5 guss is pplied long the NV xis. Error rs re s.d Time (μs) Time (μs) Figure 3 Hhn eho of the NV entre spin oherene. (, ) Mesured (olour lines with symols) nd lulted (lk lines) Hhn-eho signls of the single- ( ) nd the doule-trnsition ( ), respetively. ( ) Close up of the initil dey in (, ) for single- (red symols for experiments nd line for theory) nd doule-trnsition (lue symols for experiments nd line for theory). A mgneti field B = 13.5 guss is pplied long the NV xis. Error rs re s.d. Under wek mgneti field, the entre spin deoherene is minly indued y the single 13 C nuler spin dynmis 8,39,8. The dipolr intertion etween nuler spins n e negleted for the moment. Thus, the th Hmiltonin H B only ontins the nuler 7 Zeemn energy (with γ C B / π ~1. khz, γ C = s 1 1 T eing the gyromgneti rtio of 13 C nulei). The hyperfine field α A (with A / π ~5 khz for nuler spin I loted 1.5 nm from the entre) is omprle to the Zeemn frequeny. Eh nuler spin preesses out different lol fields = γ C B + α A, onditioned on the entre spin stte α. The entre spin deoherene is expressed s 8,39,8. ( ) ( ) L, () t I () t I () t, where I (t) is the preession of the th nuler spin out the lol field strting from rndomly set initil stte I. The onditionl evolution of th spins reords the whih-wy informtion of the entre spin nd uses the deoherene. Under flip opertion of the entre spin α α, the nuler spin preession is mnipulted s ) I () t =exp[ ih I( t t) ]exp[ ih I t] I, tht ( ) ( ) ( is, the nuler spin hnges its preession diretion nd frequeny. Thus, the nuler spin th is mnipulted through ontrol of the entre spin. The oherene t the eho time is lulted s 8. ( ) ( ) h t h t L, ( t )= 1 sin sin. (3) () When the entre spin is in the stte, ll the nuler spins preess out the sme lol field B. This ft leds to the singletrnsition oherene reovery when the eho time is suh tht the nuler spins omplete full yles of preession in period of free evolution under the mgneti field (tht is, γ C B = n π for integer n ). This effet is shown in Figure 3, whih is onsistent with previous oservtions 17. The height of the reovery peks deys owing to the nuler nuler spin intertion in the th 8,39,8 (Supplementry Fig. S ). For the doule-trnsition oherene, however, the hyperfine ouplings re non-zero nd, therefore, the nuler spins hve different lol fields for oth of the entre spin sttes ±. 6 3

4 NATURE COMMUNICATIONS DOI: 1.138/nomms1579 d PDD PDD PDD-3 PDD- PDD Figure Deoherene of the NV entre spin under PDD ontrol. ( ) Mesured single- (lk line with squre symols) nd doule-trnsition oherene (red line with irle symols), under the one- to five-pulse PDD ontrol (PDD-1 to PDD-5, from top to ottom). The sled single-trnsition oherene L, + (lue line with tringle symols) is plotted for omprison. ( ) The lulted dt, plotted in the sme formt s in ( ). () Comprison etween the mesured (symols) nd the lulted (solid lines) single-trnsition oherene. ( d ) The sme s, ut for the doule-trnsition. A mgneti field B = 5 guss is pplied long the NV xis. Error rs re s.d. Consequently, the doule-trnsition oherene hs no full reovery under the eho ontrol in the wek field. Figure 3, show exellent greement etween the theory nd the experimentl oservtion. A lose up of the initil oherene ollpse (for γ C B < π ) shows tht the single-trnsition oherene still deys slower thn the doule-trnsition oherene ( Fig. 3 ). Atully, in the initil spineho dey under wek mgneti field, the short time ondition 1 is stisfied for most nuler spins oupled to the entre spin. The short time expnsion of eqution () gives L,+ ( t) ( 1 g CB A t 8) nd L+ (, t) ( 1 g CB A t 8), whih stisfy the sling reltion for lssil Gussin noise in eqution () ( Supplementry Fig. S3 ). Thus, in the reltively short time rnge, the ADE is not yet fully developed. Dynmil deoupling ontrol of the entre spin oherene. To further explore the quntum nture of the nuler spin th, we employ the multi-pulse dynmil deoupling ontrol, to elongte the entre spin oherene time nd to mke the ontrol effets on the quntum th more pronouned. Figure ompres the single- nd doule-trnsition oherene under the periodi dynmil deoupling (PDD) ontrol y eqully sped sequenes of up to five pulses (pplied t, 3, 5, lled Crr Purell sequenes for two or more pulses). To fous on the initil-stge deoherene, we use wek field ( B = 5 guss) so tht the susequent revivl of the entre spin oherene is suppressed sine the revivl period (out.37 ms in Hhn eho nd 1.85 ms in PDD-5) is long s ompred with the overll deoherene time 8. In the Hhn eho (PDD-1), where the short time ondition pproximtely holds, the singletrnsition oherene nd the doule-trnsition oherene pproximtely stisfy the sling reltion in eqution (). With inresing the numer of ontrol pulses, the doule-trnsition oherene time inreses more thn tht of the single-trnsition. Surprisingly, under the five-pulse ontrol, the doule-trnsition hs signifintly longer oherene time thn the single-trnsition. Suh ounter-intuitive phenomen unmiguously demonstrte the quntum nture of the nuler spin th. The different dependene on dynmil deoupling of the singlend doule-trnsition deoherene, though ounter-intuitive, n e understood with geometril piture of nuler spin preession onditioned on the entre spin stte ( Fig. 5 ). By repeted flip ontrol α α of the entre spin, nuler spin I preesses lterntively out the lol fields nd ( α ). The deoherene is used minly y the 13 C spins tht re loted lose to the entre spin, whih hve hyperfine fields muh stronger thn the wek externl field ( A γ C B ). The lol fields ( ± ) = γ C B ± A, orresponding to the entre spin sttes ±, re pproximtely nti-prllel nd the ifurted nuler spin preession pthwys hve smll distne δ +, t eho time ( Fig. 5 displys shemti of the evolution pths on the Bloh sphere, nd Supplementry Movie 1 displys the nimtion). Thus, under dynmil deoupling ontrol of the douletrnsition, the entre spin deoherene due to the losely loted nuler spins is lrgely suppressed. Figure 5 shows the ontriutions to the doule-trnsition deoherene of three strongly oupled 13 C spins in the rndomly generted spin th (the sme th s used in lultion for Figures ), whih re lrgely suppressed y the dynmil deoupling ontrol. On the ontrry, this deoherene suppression mehnism does not exist in the single-trnsition se

5 NATURE COMMUNICATIONS DOI: 1.138/nomms1579 δ,+ δ +, (+) t = 3 (+) () t = 3 t = ( ) t = (Fig. 5 displys shemti digrm of the evolution pths on the Bloh sphere nd Supplementry Movie displys the nimtion). Figure 5d shows the ontriutions to the single-trnsition deoherene of the sme three strongly oupled 13 C spins s in Figure 5, whih re muh greter thn in the doule-trnsition se. Thus, the overll oherene of the single-trnsition deys fster thn tht of the doule-trnsition. This explins the ADE. Figure shows exellent greement etween the mesured entre spin deoherene under PDD nd the lultion. Some fetures of slow osilltions or shoulders in the lulted deoherene do not perfetly mth those in the mesured dt, espeilly for the higher order dynmil deoupling. This differene in the detil is understndle s suh fetures re sensitive to the speifi rndom positions of few losely loted 13 C spins, whih re not determined. Nevertheless, the experiments unmiguously onfirm the predition tht the doule-trnsition oherene time grows to e longer thn tht of the single-trnsitions. Disussion Our oservtion of the ADE using NV entre oherene estlishes the quntum nture of nuler spin ths t room temperture. Previous studies of oherene ontrol of NV entre spins in eletron-spin ths 3 3, show tht the deoherene is well desried y lssil noise theory. The fundmentl differene etween nuler d Figure 5 Physil piture for understnding the ADE. ( ) Bifurted nuler spin preession out the lol fields (lue rrow) nd (red rrow) under the -pulse PDD ontrol of the doule-trnsition. The lue (red) pth shows the nuler spin preessing out ( ) from time to for the entre spin stte + ( ), then preessing out ( ) from time to 3 fter the entre spin is flipped to ( + ) t time, nd then preessing out ( ) from time 3 to fter the entre spin is flipped k to + ( ) t time 3. ( ) The lulted deoherene of doule-trnsition under PDD-5 ontrol used y three losely loted 13 C spins in the rndomly generted th used in lultion for Figures. The oordintes (in units of lttie onstnt.357 nm) re ( 3, 3, 5) /, (, 3 /, 1 /), nd (1, 1, ) for the lk, red, nd lue urves in turn. ( ) The sme s ( ) ut for the single-trnsition. δ +, nd δ, + denote the distnes etween the ifurted pths t the eho time ( ) for the doule- nd single-trnsitions, respetively. As the two lol fields ( ± ) re lmost nti-prllel, δ +, is muh smller thn δ, +. (d ) The deoherene of the single-trnsition due to the sme three spins in ( ). Notie the different sles of the vertil xes in (,d ). The ontriutions to the doule-trnsition of these spins re muh smller thn those to the single-trnsition. ARTICLE spin ths nd eletron spin ths lies in the intr-th intertion strength reltive to the th-entre spin oupling 9. For nuler spin ths, the dipolr intertion etween nuler spins t verge distne ( ~ 1 Hz) is muh weker thn the typil hyperfine oupling (>khz) 8,9. With suh wek intr-th intertion, the diffusion of oherene mong nuler spins is muh slower thn the deoherene (of timesle ~ ms). Thus, the entre spin nd the th n e regrded s reltively losed quntum system. For eletron spin ths, however, the oupling etween th spins t verge distne is muh stronger thn the typil th-entre oupling. As result, the oherene will rpidly diffuse from losely loted th spins to those t distne during the entre spin deoherene. Therefore, n eletron spin th ehves like mrosopi open system nd the lssil noise theory is vlid. For NV entre spin deoherene in eletron spin ths, we expet the ADE e sent nd the sling reltion in eqution () e oserved insted. The quntum nture of nuler spin ths n lso e understood y the k-tion of the entre spin to the th. For the trnsition α α, the Hmiltonin in eqution (1) n z e expressed in pseudo-spin form s Hps = 1 s z + H, where σ z = α α α α is the pseudo-spin opertor, α α = H H ( α ) is the effetive noise field to the pseudo-spin nd H = 1 ( ( ) ( ) H + H ) is the effetive th Hmiltonin. For the single-trnsitions ±, the effetive noise field is ˆ ˆ ± = z, nd the doule-trnsition + hs twie stronger noise s + = z. For the doule-trnsition, the effetive th Hmiltonin is H + = H B, ut for the single-trnsition the effetive th Hmiltonin is H H 1 ± = ± ˆ B z with the extr term owing to the hyperfine oupling, whih is the k-tion of the entre spin to the th. For the nuler spin th, the hyperfine oupling is typilly stronger thn the intr-th intertions nd the k-tion strongly modifies the effetive th Hmiltonin. In prtiulr, for this work, the hyperfine oupling provides muh stronger lol field thn the pplied mgneti field for nuler spins lose to the entre spin. In the single-trnsition se, euse of the k-tion, the nuler spins hve enhned preession frequenies in omprison to the doule-trnsition se. Thus, viewed from the entre spin, the effetive th for the doule-trnsition produes noise with lower frequenies thn in the single-trnsition se nd, therefore, the entre spin oherene is etter proteted y the dynmil deoupling ontrol. This explins the ADE oserved in nuler spin ths. In ontrst, for eletron spin ths, the oupling strength within the th Hmiltonin H B is muh lrger thn the k-tion term 1 ˆ z (refs 3 3,, 9 ). For different entre spin trnsitions the modifition of the th dynmis due to the k-tion is negligile. In this sense, the eletron spin th ehves s lssil th nd the ADE should not our. Finlly, we point out tht, in this work, the ADE is demonstrted in the wek mgneti field regime ( < guss) in whih the quntum flututions is used minly y single nuler spin dynmis. The ADE ws predited in ref. 9 in the strong field regime, where the flututions re used minly y nuler spin pir dynmis. These works indite tht the ADE is roust ginst the detils of the deoherene mehnisms ut is universl phenomen due to the quntum nture of mesosopi ths. Methods Experimentl setup. All the experiments re rried out t room temperture. The type-ii dimond single rystl smple hs nitrogen density less thn 5 pp nd the nturl undne of the 13 C isotope. Individul NV entres re optilly ddressed y onfol mirosope mounted on piezoeletri snner, nd re identified y the mesurement of the nti-unhing effet through the seond-order orreltion funtion ( Supplementry Fig. S ). To void the influene of the surfe, n NV entre loted 1 μ m elow the surfe is used in the experiments. The wek mgneti field is generted y three pirs of Helmholtz oils with n ury of 5

6 d I PL (.u.).96 3π/ 3π/ π/ π π π/ π/ / π 3π/ 3π π/ π π π π 1 degree for the diretion nd 1 guss for the mgnitude. All the pulse signls used in the experiments re synhronized y pulse genertor with time resolution of out ns. Centre spin initiliztion nd redout. The entre spin stte is initilized nd red out y 53 nm ontinuous-wve lser, whih is gted using n ousto-optil modultor. A 1 μ s optil pulse with n extr 5 μ s witing time pumps the system into the stte. To red out the spin stte, ns ounting pulse is pplied with 6 ns witing time fter turning on the lser. Centre spin mnipultion. The entre spin is mnipulted y resonnt mirowve pulses. A liner mplifier oosts mirowve pulse power to the desired mplitude nd μ m dimeter opper wire ouples the mirowve field into the dimond. The doule-trnsition oherene is generted nd ontrolled y omposite pulses ( Fig. 6 ). The π / -rottion of + is relized y π /-rottion of + followed y π -rottion of, nd the π -rottion of + is relized 3π/ π/ π π π π n π π/ / /,88,885,89 Freq. (MHz) e Contrst C (%) Contrst C (%) Contrst C (%) / 15 π π/ π/ Pulse durtion (µs) Figure 6 Mirowve mnipultion of the entre spin. ( ) Mirowve pulse sequenes for the free-indution dey experiments of singletrnsition (the upper sequene) nd doule-trnsition (the lower sequene) under 13.5 guss mgneti field. The squres in lue (ornge) represent the pulses tht exite the + ( ) trnsition. The pulse res re denoted ove the squres. All the pulses re nonseletive. ( ) The sme s ( ), ut for the Hhn eho mesurements. ( ) Mirowve pulse sequenes used in the PDD- n experiments for the single-trnsition oherene mesurement (the upper sequene) nd those for the douletrnsition oherene mesurement (the lower sequene) under 5 guss mgneti field. The pulses re expressed in the sme mnner s in (, ). Seletive pulses re used to generte, ontrol nd red out the spin stte. ( d ) The optilly deteted eletron spin resonne spetrum of the trnsition of the entre spin mesured under 5 guss mgneti field. The splitting due to the hyperfine oupling to the 1 N nuler spin is resolved. The red, lue nd green rs indite, in turn, the three Ri frequenies used in the experiments,.5 MHz (for seletive exittion of single-trnsition in ), 1.3 MHz (for seletive exittion of doule-trnsition in ), nd 1 MHz (for nonseletive exittions in, ). (e ) From top to ottom, the Ri osilltion with Ri frequenies.5 MHz (seletive exittion), 1.3 MHz (seletive exittion), nd 1 MHz (nonseletive exittion), orresponding to the rs in ( d ). NATURE COMMUNICATIONS DOI: 1.138/nomms1579 y two π -rottion of sndwihed y π -rottion of +. For fir omprison, the durtion of ontrol in the single-trnsition oherene mesurement is kept pproximtely the sme s the totl durtion of the orresponding omposite ontrol in the doule-trnsition oherene mesurement. This is relized either y repling π / nd π rottions with 3 π / nd 3π rottions (mplitude kept the sme ut durtion vried, Fig. 6, ), s in the experiments in Figures nd 3 where B = 13.5 guss, or y using pulses of smller mplitudes in the single-trnsition oherene mesurement ( Fig. 6 ), s in the experiments in Figure where B = 5 guss. With different pulse mplitudes under different mgneti field, the mirowve pulse errors used y oupling to the 1 N nuler spin re gretly suppressed. Beuse of the hyperfine oupling to the 1 N nuler spin, the mirowve ontrol is desried y the Hmiltonin in the rotting-frme referene s H = A S z I Z + B 1 S x, where A =.18 MHz, is the hyperfine oupling strength nd B 1 is the Ri frequeny of the driven field. Errors would result if the π pulses do not fully flip ll the three hyperfine-split lines. Under mgneti field B = 13.5 guss, typil π pulse hs durtion of 5 ns (orresponding to the Ri frequeny of out 1 MHz). Thus, the two entre spin single-trnsitions ± re well resolved wheres the three trnsition lines for different sttes of the 1 N nuler spin-1 re ll spetrlly overed ( Fig. 6d ). Under mgneti field B = 5 guss it is diffiult to spetrlly resolve the two single-trnsitions without seletion of different 1 N sttes. Therefore we use soft pulses (1.1 μs nd.39 μ s for the π -rottion in the single- nd doule-trnsition oherene mesurements, respetively, orresponding to the Ri frequenies of.5 MHz nd 1.3 MHz) to seletively exite the single-trnsitions orresponding to only one of the 1 N sttes (nmely the 1 N stte, Figure 6d ) nd to mximlly suppress the trnsitions when the 1 N nuler spin is in the other two sttes ( ± 1 1 N ). The seletive nd nonseletive exittions re onfirmed y the mesurements of Ri osilltions ( Fig. 6e ). The ontrst mesured y the seletive pulse is one third of tht mesured y the nonseletive pulse. Dt proessing nd error nlysis. The mesured ontrst is given y the reltive hnge in fluoresene intensity s C = ( I M I ) / I, where I M is the signl, I is the fluoresene where the spin oherene hs totlly deyed nd I is the fluoresene when the entre spin is in the stte, ll of whih re mesured independently in every experimentl run. The ontrst is normlized to the verged vlue t the initil deoherene stge (for smll time, where the oherene presents the plteu feture). The results, however, do not depend on the normliztion ( Supplementry Fig. S5 ). The sttistil errors ome minly from the photon-ounting shot noises nd lser-fousing spot drifting. To suppress errors from the lser-fous drifting, the lser is re-foused every min during the mesurement. To redue the effet of the shot noises, eh mesurement hs een repeted out 1 6 times. The kground photon ounting I is mesured y more thn 1 7 times so tht its sttistil flutution n e negleted. Besides the sttistil flututions, there re two mor uses of systemti errors: first, the MW pulse frequenies re determined with n error out ± khz, less thn % of the Ri frequenies; nd seond, the durtions of the pulses re determined up to n error of ± ns, whih indues n error in the Ri frequenies of out % in the se of short ontrol pulse used in the experiments in Figures nd 3 (where the π pulses hve durtions of out 5 ns), ut hs negligile effet in the se of soft ontrol pulse used in the experiments in Figure. The ove ftors ontriute to the error rs estimted nd shown in Figures. Theoretil model nd numeril simultion methods. I n t h e n u m e r i l lultion, the nuler spin th is generted y rndomly pling 13 C toms on the dimond lttie round the NV entre with nturl undne 1.1 %. Inlusion of out 1 13 C nuler spins within nm from the NV entre is suffiient for onverged result of the entre spin deoherene in the timesle onsidered in this pper. The inhomogeneous rodening nd deoherene times depend on the rndom positions of the 13 C toms in the lttie 8,8. In the simultion, the onfigurtion of 13 C tom positions is rndomly hosen to produe single-trnsition dephsing time T * nd single-trnsition deoherene time in Hhn eho ( T ) lose to the experimentl results (under 13.5 guss field). Otherwise, there is no fitting prmeter. Different rndom onfigurtions of the 13 C tom positions do not ffet the essentil results ut result in differenes in the detiled fetures ( Supplementry Fig. S6 ). The generted th does not ontin 13 C in the first few oordinte shells of the NV entre (whih hs hyperfine oupling >1 MHz), to e onsistent with the NV entre under the experimentl oservtion. The hyperfine intertion is ssumed to hve dipolr form with the eletron spin loted t the vny site. The spin oherene is lulted y pplying the luster orreltion expnsion method 6, whih tkes into ount, order y order, the mny-ody orreltions indued y the dipolr intertions etween nuler spins, nd n identify the ontriution of eh nuler spin luster to the totl deoherene. The onverged results re otined y inluding lusters ontining up to 3 nuler spins. The mirowve pulses re modelled y instntneous pulses, whih is good pproximtion onsidering the ft tht the pulse durtions re ll muh shorter t h n t h e d e o h e r e n e t i m e s i n t h e e x p e r i m e n t s. 6

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Eletron spin deoherene in isotope-enrihed silion. Phys. Rev. Lett. 15, 1876 (1 ). Aknowledgements This work ws supported y Ntionl Nturl Siene Foundtion of Chin under Grnt No nd 1835; the Chinese Ademy of Sienes nd the Ntionl Fundmentl Reserh Progrm of Chin under Grnt No. 7CB95; Hong Kong Reserh Grnt Counil / Generl Reserh Fund CUHK1; The Chinese University of Hong Kong Foused Investments Sheme; nd Hong Kong Reserh Grnt Counil HKU1 /CRF /8. Author ontriutions J.D. designed nd supervised the experiments. X.K., F.S., P.W., X.R. nd J.D. prepred the experimentl set-up. P.H. nd X.K. performed the experiments. R.B.L. oneived the effet. R.B.L. nd N.Z. formulted the theory. N.Z. rried out the lultion. R.B.L., N.Z. nd P.H. wrote the pper. All uthors nlysed the dt, disussed the results nd ommented on the mnusript. Additionl informtion Supplementry Informtion ompnies this pper t ntureommunitions Competing finnil interests: The uthors delre no ompeting finnil interests. Reprints nd permission informtion is ville online t reprintsndpermissions/ How to ite this rtile: Hung, P. et l. Oservtion of n nomlous deoherene effet in quntum th t room temperture. Nt. Commun. :57 doi: / nomms1579 (11). Liense: This work is liensed under Cretive Commons Attriution-NonCommeril- Shre Alike 3. Unported Liense. To view opy of this liense, visit retiveommons.org/lienses/y-n-s/3./ 7