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Coherence of nitrogen-vacancy electronic spin ensembles in diamond P. L. Stanwix,1,2 L. M. Pham,3 J. R. Maze,4,5 D. Le Sage,1 T. K. Yeung,3 P. Cappellaro,6 P. R. Hemmer,7 A. Yacoby,4 M. D. Lukin,4 and R. L. Walsworth1,4,∗ 1Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA 2School of Physics, University of Western Australia, Crawley, Western Australia 6009, Australia 3School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA 4Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA 5Facultad de F´ısica, Pontificia Universidad Cato´lica de Chile, Casilla 306 Santiago, Chile 6Department of Nuclear Science and Engineering Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 7Department of Electrical and Computer Engineering, Texas A&M University, College Station, Texas 77843, USA 1 (Dated: January 25, 2011) 1 We present an experimental and theoretical study of electronic spin decoherence in ensembles 0 of nitrogen-vacancy (NV) color centers in bulk high-purity diamond at room temperature. Under 2 appropriateconditions,wefindensembleNVspincoherencetimes(T )comparabletothatofsingle 2 n NVs, with T > 600µs for a sample with natural abundance of 13C and paramagnetic impurity 2 a density∼1015cm−3. Wealsoobserveasharpdecreaseofthecoherencetimewithmisalignmentof J thestaticmagneticfieldrelativetotheNVelectronicspinaxis,consistentwiththeoreticalmodeling 4 of NV coupling to a 13C nuclear spin bath. The long coherence times and increased signal-to-noise 2 providedbyroom-temperatureNVensembleswillaidmanyapplicationsofNVcentersinprecision magnetometry and quantum information. ] l l a The Nitrogen-Vacancy (NV) color center in diamond h 3E - isapromisingsolidstateplatformforstudyingthequan- s tumdynamicsofspinsystems. Experimentsdemonstrat- e m ing spin-state control and manipulation using single NV 13C at. creecnetnertswaonrdk tthoewirarnduscqleuaarnteunmviroinnfmoremnta1t–i5onhaavpepliincsaptiiorends e-V B0 638-800 nm 532 nm m ofNVdiamond,aswellasapplicationstoprecisionmag- N 13C ms = +1 m= -1 neticfieldsensing,6–8 includingthelifesciences.9,10 Com- s 3A - D = 2.87 GHz 2 d mon to all proposals is a desire for long NV electronic ms = 0 n spin coherence times, ideally at room temperature, and (a) (b) o high signal-to-noise of the NV spin-state-dependent op- c tical fluorescence. Polarization Readout [ Optical excitation Electronic spin decoherence of the NV center (S =1) 2 is governed by interactions with the surrounding bath of τ τ Microwave ESR v nuclear and paramagnetic spins (Fig. 1(a)). In high pu- π/2 π π/2 9 rity diamond, decoherence is dominated by 13C nuclear 1 CCD detection 2 spins (I =1/2),11 which are dispersed through the crys- (c) 4 tal with a natural abundance of 1.1%. Electronic spin 6. coherencetimes(T2)longerthan600µsatroomtemper- FIG.1: (a)NVelectronicspinaxisisdefinedbynitrogenand ature have been observed for individual NV centers in 0 vacancy sites, in one of four crystallographic directions. NV 0 this type of sample.7,12 T2 can be increased by isotopi- orientation subsets in an ensemble can be spectrally selected 1 cally enriching the diamond;12 for example, T2 > 1.8ms by applying a static magnetic field, B0. Also shown are 13C : at room temperature has been observed in isotopically nuclearspinsandotherparamagneticimpuritiessuchassub- v enriched ultra-pure diamond with 0.3% 13C.13 stitutional nitrogen. (b) NV center electronic energy level i X structure. Spinpolarizationandreadoutisperformedbyop- In addition to long electronic spin T , enhancing the 2 ticalexcitationandfluorescencedetection. Groundstatespin r signal-to-noiseofNVspin-state-dependentfluorescenceis a manipulation is achieved by resonant microwave excitation. ofsignificantbenefittoprecisionmagnetometry. Tothat The ground state triplet has a zero magnetic field splitting end,ensemblesofNVcentershavebeenproposed,6,14 for D(cid:39)2.87GHz. (c)Hahn-echoexperimentalsequenceusedin which signal-to-noise ideally increases as the square root present measurements. of the number of NVs in the ensemble. The use of large spin ensembles with good coherence times is also impor- tantforcollectivequantummemories.15–17Increasingthe density of NV centers, however, is accompanied by an nuclearspinsalone.18–20 Furthermore,thelargevariation increased density of residual nitrogen paramagnetic im- ofcoherencetimesforindividualNVelectronicspins,due purities, which can become the dominant source of NV to the random distribution of 13C near each NV center, spin decoherence, reducing T below the limit set by 13C canalsoaffectensemblecoherence. Therefore, adetailed 2 2 understanding of decoherence mechanisms affecting NV same frequency and thus their modulations of the NV centers – both single NVs and ensembles – accompanied spin coherence do not rephase at the same time in the by the development of control techniques to mitigate de- Hahn-echo sequence, inducing NV spin decoherence of coherence is of great importance for both precision mea- theformexp(cid:2)−α(θ)bτ2(cid:3).11 Hereα(θ)describesthemis- surement and quantum information applications. alignment angle dependence of the imperfect Hahn-echo Only limited studies of electronic spin decoherence in rephasing, and b is proportional to the square of the hy- ensembles of NV centers have been previously reported. perfine interaction between the NV electronic spin and At room temperature, ensemble T times of 58µs were the nearest 13C nuclear spin. For small θ, α(θ)(cid:39)θ2.24 2 measured in CVD diamond;21 whereas at low temper- Notethatthephysicsofthisangle-dependentNVspin ature (<2K) and high magnetic field (>8T), NV en- decoherence is fundamentally different from previously semble T of 250µs was measured in high-temperature observed effects for donor electron spins in semiconduc- 2 high-pressure type 1b diamond, decreasing to T <10µs tors such as Si:P.25,26 For Si:P and similar S = 1/2 sys- 2 for temperatures >20K.22 tems, the electronic spin quantization axis points along the external magnetic field; whereas for NV centers, InthepresentRapidcommunication, wereportanex- the anisotropy of the electronic distribution causes spin perimentalandtheoreticalstudyofthecoherenceproper- quantization along the NV axis for small magnetic fields tiesofNVelectronicspinensemblesinroomtemperature (<500G). In Si:P and similar systems, electron spin diamondsamplesofdifferentparamagneticnitrogen(and decoherence is dominated by fluctuations of a dipolar- consequently NV) concentrations. For a lower nitrogen coupled nuclear spin bath, and is maximized when the densitysample(≈1015cm−3)withnaturalabundanceof magnetic field is aligned along the [111] axis, due to the 13C,wefindNVensembleT inexcessof600µs,compara- 2 enhanced flip-flop rate of the nuclear spins arising from bletothebestresultsforsingleNVcentermeasurements the angular dependence of their dipolar coupling. As in natural isotopic abundance diamond and an order of noted above, NV spin decoherence is minimized when magnitude greater than previous room-temperature en- the magnetic field is aligned with the NV axis. semblemeasurements. Forahighernitrogendensitysam- ple(≈5×1015cm−3)withnaturalabundanceof13C,we In diamond samples where the concentration of NV centers is low enough that they do not interact signifi- findanNVensembleT ≈300µs. Furthermore, forboth 2 cantly with each other and with nitrogen paramagnetic samples we find a sharp decrease in the NV ensemble T 2 impurities, the ensemble Hahn-echo signal can be con- withmisalignmentofthestaticmagneticfieldrelativeto sidered as the average of many independent signals from the NV electronic spin axis being studied, which is con- individual NV centers. In this case, the ensemble Hahn- sistent with our theoretical modeling of an ensemble of NV electronic spins interacting with a 13C spin bath.11 echo signal envelope, E(τ), can be significantly different frommeasurementsofsingleNVcenters,27,28 becausean For high-purity diamond, decoherence of single NV ensemble contains a broad distribution of spin coherence electronic spins is dominated by hyperfine interactions lifetimes due to variations in the location of 13C nuclei with nearby 13C nuclear spins. The contact term of this proximal to individual NVs in the ensemble: interaction decreases exponentially with separation; after a few lattice sites it is not larger than a few MHz.23 Meanwhile, the dipolar part of this interaction decreases E(τ)=(cid:90) dbexp(cid:0)−α(θ)bτ2(cid:1)f(b). (1) as r−3 and is responsible for the collapses and revivals observed in Hahn-echo measurements of single NV centers.1Whentheexternallyappliedstaticmagneticfieldis Here, f(b) is the probability distribution for the mag- aligned with the NV axis, the dipolar field contributions nitude of the NV–proximal 13C hyperfine-interaction- of all 13C nuclei cancel after each 2π Larmor precession squared in an ensemble. Depending on details of this periodofthe13Cnuclearspins,ascanbemeasuredbythe distribution,theensembleHahn-echosignalenvelopecan Hahn-echosequence.1–3 Overlongertimescales(>600µs have either a single exponential or Gaussian decay at for natural abundance 13C), weak dipole-dipole interac- large times τ.27,28 tions between 13C nuclei and between 13C and paramag- Inourexperimentsweusedacustom-built, wide-field- netic impurities induce fluctuations (“flip-flops”) in the of-view fluorescence microscope to measure the coher- NV–13C hyperfine interaction, leading to NV spin de- ence properties of large ensembles of NV centers. The coherence. The predicted form of this decoherence for electronic spin state of negatively charged NV centers the Hahn-echo signal of an individual NV is exp[−βτn], canbepolarized,manipulatedandread-outusingoptical where n is between 3 and 4 and the constant β depends and microwave excitation (see Figure 1(b)). Spin-state- on details of the relative location of nearby 13C nuclei.11 dependent radiative-relaxation enables NV polarization However, when the magnetic field makes a small angle θ and readout: NVs initially in m =0 optically cycle be- s withtheNVaxis, theLarmorprecessionfrequencyΩ of tween the ground and an electronically excited “bright” i individual13Cnucleiismodifiedbythehyperfineinterac- state, while NVs initially in |m | = 1 have a significant s tion with the NV center to become dependent on θ and branching to a long lived metastable “dark” state that the relative NV–13C position r : Ω = Ω +δΩ(θ,r ). non-radiatively relaxes to m = 0. Optical excitation is i i 0 i s As a consequence, 13C nuclei do not all precess with the providedbyaswitched532nmlasersource,focussedonto 3 1 (cid:101) = 180 ± 5° 1 (cid:101) = 175.4 ± 0.4° T2= 631 ± 9 µs T2= 282 ± 7 µs n = 1.20 ± 0.09 n = 1.08 ± 0.04 0.5 0.5 p) 1 (cid:101) = 171.3 ± 0.1° p) 1 (cid:101) = 22.7 ± 0.4° bility ( Tn2== 311.835 ± ± 5 0 µ.0s8 bility ( Tn2== 102.936 ± ± 1 00. 0µ9s a a b b o o Pr0.5 Pr0.5 1 (cid:101) = 159.6 ± 0.1° 1 (cid:101) = 64.2 ± 0.1° T2= 184 ± 8 µs T2= 23 ± 4 µs n = 1.04 ± 0.06 n = 0.42 ± 0.04 0.5 0.5 0 200 400 600 800 0 200 400 600 800 Spin Echo Time 2(cid:111) (µs) Spin Echo Time 2(cid:111) (µs) (a)Sample A (b)Sample B FIG. 2: Example Hahn-echo measurements of ensemble NV electronic spin decoherence and dependence on static magnetic field orientation: (a) sample A with NV density ≈ 3×1013cm−3 and N density ≈ 1015cm−3; and (b) sample B with NV density ≈ 1×1014cm−3 and N density ≈ 5×1015cm−3. Clearly seen in this data is a sharp decrease of the ensemble NV coherence lifetime T as the B field is misaligned from the NV electronic spin axis, and as the paramagnetic impurity level 2 0 increases. For each data plot, the left vertical axis indicates the measured probability for NVs in the ensemble to be in the m =0 state at the end of a Hahn-echo experiment, as a function of the echo time τ; the black line is a fit of exp[−(τ/ )n] s T2 to the overall Hahn-echo signal envelope, where the exponent n is a fit parameter and the long term baseline is taken to be 0.5 (the mixed spin state); and the errors in θ, T and n are given by one-standard-deviation confidence intervals of the fit. 2 Theredlineisafittotheobservedecho-signalmodulations,whicharecollapsesandrevivalsofNVspincoherenceinducedby precession of the natural abundance 13C nuclear spin bath in applied, hyperfine and dipolar magnetic fields. thesamplewitha20xobjective. Fluorescenceemanating pure diamond substrate (3×3×0.2mm) with <1ppm from the sample is collected back through the same ob- of ambient nitrogen impurity and <1012cm−3 NV den- jective, then filtered and imaged onto a CCD array. The sity, which was overgrown with a high NV content layer useofaCCDarrayallowsthesampleresponsetobespa- ≈2µmthick. ThedensityofNVcentersinthethinlayer tially resolved with resolution < 1µm and field-of-view regionofsampleAandBisapproximately3×1013cm−3 > 100µm. The ensemble of NVs is coherently manipu- and1×1014cm−3 respectively,asdeterminedbyfluores- lated on a ground-state electronic spin transition with a cence measurements normalized to single NV measure- resonantmicrowavemagneticfieldgeneratedfromaloop ments made on each sample using a confocal microscope antenna placed close to the diamond sample, which pro- optimized for NV measurements, similar to that used in duces a homogenous microwave B field over the region Ref. 7. Thedensityofnitrogenimpuritiesinthesamples 1 of interest. The degeneracy of the m = 0 to m = ±1 is approximately 50 times larger, as determined by the s s ground-statespintransitionsisliftedbyauniformstatic CVDgrowthprocedure. AHahn-echopulsesequence(il- B magnetic field, applied using a permanent magnet lustratedinFig.1(c))wasusedtomeasuretheelectronic 0 trimmedbya3-axissetofHelmholtzcoils. Precisealign- spincoherencetimeoftheNVensembleineachofthetwo ment of B allows a subset of NVs in the ensemble, cor- samples. NV centers were first prepared in the m = 0 0 s responding to one of four crystallographic orientations, groundstatebyapplyinga532nmlaserexcitationpulse. to be spectrally distinguished by their spin transitions. TheHahn-echosequence(π/ −τ−π−τ−π/ )wasthen 2 2 Theorientationθ andmagnitudeB oftheappliedmag- repeated for a range of free spin evolution periods τ (the 0 netic field relative to the axis of one class of NV centers finalπ/ pulseinthissequenceprojectsacoherentsuper- 2 is determined from the Zeeman shifts of the NV ground- position of NV spin states into a population difference, state electronic spin resonances (ESR) together with the which is detected via spin-state-dependent fluorescence). measured Hahn-echo revival frequency, B =ω /γ .29 The data presented here is the integrated response of re- 0 L 13C gionswitharea≈30µm2 and2µmdepthineachsample, We employed two CVD diamond samples in our mea- averagedover∼105identicalexperiments. Thus,NVen- surements of NV ensembles. Each sample consists of a 4 semblesconsistedof∼103 NVspinsofasinglecrystallo- function of θ. In all these measurements, the Hahn-echo graphic class in the detection region. Over macroscopic signal envelope was fit to exp[−(τ/ )n] to determine T2 distances (∼ 1mm), each diamond sample was found to thecoherence lifetime T (Fig.3(a)) anddecay exponent 2 have uniform spin coherence properties. n (Fig. 3(b)). Good agreement is found between our experimentally-determined values for T as a function of 2 θandtheoreticalpredictions(seesolidcurveinFig.3(a)), 700 which include the decoherence contributions from proxi- s) 600 SSaammppllee AB mal 13C nuclear spins via the form exp(cid:2)−α(θ)bτ2(cid:3).11,30 µ e ( 500 Decoherence due to 13C “flip-flops” was included phe- m Ti nomenologically by multiplying the simulated NV echo ence 340000 signal by exp(cid:104)−(cid:0)τ/T2(θ=0)(cid:1)4(cid:105). The low values for the her decay exponent n in ensemble experiments is explained Co 200 by the large dispersion of the dominant NV–13C dipole- T 2100 dipole interaction (see Eq. 1). The distribution of this interactionsquared,f(b),wasdeterminedbycalculating 0 0 45 90 135 180 the effect of 13C nuclei placed randomly in the diamond Misalignment Angle (cid:101) (degrees) lattice at natural isotopic abundance. For the dominant (a) NV–13C dipole-dipole interaction, we find that this distribution scales as f(b) ∼ b−3/2 for 13C nuclei within 3 about 10nm of an NV center. Thus, f(b) more closely Sample A resembles a Lorentzian than a Gaussian distribution, re- 2.5 Sample B sulting in Hahn-echo signal envelopes that decay more n nt 2 like a single exponential than a Gaussian at large times ne (seeFig.2). NotethatforsampleB,theexperimentally- o xp1.5 determined decay exponent is n (cid:39) 1 even for θ (cid:39) 0 E y (Fig.3(b)),whichcouldresultfrominteractionsbetween a ec 1 NV spins and the higher density of paramagnetic impu- D rities (nitrogen and NVs) in this sample. 0.5 In summary, we demonstrated experimentally that 0 0 45 90 135 180 large ensembles of NV centers in high-purity diamond Misalignment Angle (cid:101) (degrees) with natural abundance of 13C can have electronic spin (b) coherence lifetimes at room temperature that are com- parable to the best measured for single NV centers FIG. 3: Summary of Hahn-echo measurements as a function (T > 600µs). We also found a sharp decrease in NV ofmisalignmentangleθbetweenthestaticmagneticfield,B , 2 0 T as the applied magnetic field is misaligned from the andtheNVelectronicaxis:(a)coherencelifetimeT and(b) 2 2 NV axis, consistent with the predictions of our model decay exponent n, determined from fits of the signal envelope to exp[−(τ/ )n]. Sample A (lower NV and nitrogen for an ensemble of NV electronic spins (S =1) coupled concentration) waTs2measured for θ ranging from 90 to 180 via position-dependent hyperfine interactions to a 13C degrees; sample B (higher NV and nitrogen concentration) nuclear spin bath, leading to imperfect 13C spin-echo re- was measured for θ from 0 to 180 degrees. Solid curve in (a) vivals and hence NV decoherence. Our results demon- is the prediction of the NV ensemble decoherence model, as strate the potential of NV ensembles for applications in described in the main text. precisionmagnetometryinboththephysicalandlifesci- ences, combining long electronic spin coherence times at As shown in Figs. 2 and 3, we measured ensemble NV room temperature with the enhanced signal-to-noise ra- coherencetimescomparabletothatforsingleNVsineach tio provided by many NVs in the detection volume.6 In sample: T2 > 600µs for sample A and T2 ≈ 300µs for addition, the demonstrated techniques could be used to sample B, with the static magnetic field B0 well aligned increasethecoherencetimeofsolid-statequantummem- withtheNVelectronicspinaxis. Ourmeasurementsalso ories, such as an NV electronic spin ensemble coupled to confirm the predicted11 sharp decrease of NV T2 with a superconducting resonator15,16 or flux qubit.17 small misalignment of B from the NV axis. Example 0 Hahn-echo measurements for three representative mis- This work was supported by NIST, NSF and DARPA. alignment angles θ are shown in Fig. 2 for samples A Wegratefullyacknowledgetheprovisionofdiamondsam- and B; whereas Fig. 3 summarizes the results of a larger ples by Apollo Diamond and technical discussions with set of systematic measurements of ensemble NV T as a Patrick Doering and David Glenn. 2 5 ∗ Electronic address: [email protected] 22 S. Takahashi et al., Phys. Rev. Lett. 101, 047601 (2008). 1 L. Childress et al., Science 314, 281 (2006). 23 A. Gali et al., Phys. Rev. B 77, 155206 (2008). 2 T. Gaebel et al., Nat. Phys. 2, 408 (2006). 24 This form of decoherence can be found by expanding Ω(0) n 3 M. V. G. Dutt et al., Science 316, 1312 (2007). in Eq. (7) of Reference11 for small angle θ. 4 P. Neumann et al., Science 320, 1326 (2008). 25 A. Tyryshkin et al., J. Phys.: Condens. Matter 18, S783 5 L. Jiang et al., Science 326, 267 (2009). (2006). 6 J. Taylor et al., Nat. 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