Decoherence-protected transitions of nitrogen vacancy centers in 99% 13C-enriched diamond
Anna J. Parker, Hai-Jing Wang, Yiran Li, Alexander Pines, Jonathan P. King
DDecoherence-protected transitions of nitrogen vacancy centers in99% C diamond
Anna J. Parker,
1, 2
Haijing Wang,
1, 2
Yiran Li, Alexander Pines,
1, 2 and Jonathan P. King
1, 2, ∗ Department of Chemistry, University of California, Berkeley, California 94720 Materials Science Division, Lawrence Berkeley National Laboratory, USA (Dated: November 14, 2018)
Abstract
Nitrogen vacancy (NV − ) color centers in diamond are a prime candidate for use in quantuminformation devices, owing to their spin-1 ground state, straightforward optical initialization andreadout, and long intrinsic coherence times in a room-temperature solid. While the C nuclearspin is often a dominant source of magnetic noise, we observe transitions between electron-nuclearhyperfine states of NV − centers in 99% C diamond that are robust to decoherence. At magneticfield strengths ranging from 550 - 900 G, these transitions are observable by optically detectedmagnetic resonance (ODMR), and exhibit linewidths narrowed by factors as high as ∼
130 at roomtemperature over typical electron-type transitions observed from this spin system. We antici-pate the use of these decoherence-protected transitions, in combination with dynamical decouplingmethods, for storage of quantum information. a r X i v : . [ qu a n t - ph ] J un NTRODUCTION
The nitrogen vacancy (NV − ) center is often regarded as a trapped ion in diamond, withlocalized electronic states resulting from its insulating environment and an optical transitiondistinct from the diamond bandgap. Its ground state has a total electronic spin of 1, andthe spin-spin interaction within the defect results in an energy splitting between the m S = 0and m S = ± − centers, the m S = 0 spin state fluoresces more intensely than the m S = ± S = 0 spin stateis referred to as “bright” and the m S = ± − centers. These properties, in combination with the details of theenergy level structure of the NV − center, allow for its use in optical sensing of temperature,pressure, magnetic and electric fields [1–6]; for atomic-scale nuclear magnetic resonance(NMR) and magnetic field imaging, as well as for quantum information processing (QIP)[7–12]. The NV − center is competitive as a quantum bit (qubit) candidate due to the ease ofmanipulating its optical and spin degrees of freedom as well as the favorable properties andversatile fabrication of the diamond. One significant challenge in the implementation of aworking quantum computer common to many implementations is producing a qubit that issufficiently decoupled from its environment but may also be interrogated efficiently [13]. Inthis work we address the issue of spin decoherence of NV − centers.The sources of decoherence and their interactions with the qubit have been studied formany qubit candidates, including photons, phosphorous and silicon nuclear spins in silicon,quantum dots, trapped atoms and ions, superconducting circuits, and the NV − center indiamond [14, 15]. Methods of extending coherence times generally include physically re-moving the sources of decoherence, for example by isotopic enrichment of the host material[16, 17], and dynamical decoupling (DD) pulse sequences inspired by the Hahn echo thatremove interference from static and fluctuating sources of noise [18]. Another approach toachieving useful coherence times is to use transitions that are inherently protected fromsources of decoherence. One form of these transitions is generally termed Zero First-OrderZeeman (ZEFOZ) for satisfying the condition ∂ν∂B →
0, in other words when the first-ordermagnetic field ( B ) dependence of a transition with frequency ν goes to zero. These are2ypically found in systems that possess internal interactions (i.e. hyperfine interactions orzero field splittings) that do not commute with the Zeeman interaction. For spin qubits withsuch internal structure, magnetic field conditions exist where the Zeeman energy cancels thehyperfine or zero-field energy, resulting in an avoided crossing or level anti-crossing (LAC).Approaching the regime of the LAC, the magnetic field dependence of spin states becomesnonlinear, resulting in ZEFOZ conditions where the transition frequency is desensitized toperturbations in the magnetic field. Such phenomena may also be understood by realizingthat the first-order Zeeman shift of a transition frequency is its effective gyromagnetic ratio, γ eff . Spin transitions with weak γ eff may be characterized by long coherence times becausethey couple weakly to local magnetic field fluctuations that result in dephasing. This is thesame reasoning used to explain why nuclear spin coherence times generally exceed electronspin coherence times by orders of magnitude: their gyromagnetic ratios are weaker thanthat of the electron by a factor of 10 -10 .The use of ZEFOZ transitions to extend coherence was originally introduced by Longdelland coworkers with rare-earth metal ion doped materials [19]. Extended coherence timeswere demonstrated for Pr :Y SiO and Pr :La (WO ) for enhancements of ∼
630 usingthe ZEFOZ method alone [19, 20]. Building upon these gains has led to some of the longestquantum memory times currently known [21, 22]. Furthermore, the ZEFOZ method isbroadly applicable. In trapped Be + ions, the ZEFOZ transition between hyperfine levels | F = 2 , m F = 0 (cid:105) ↔| F = 1 , m F = 1 (cid:105) has a coherence time of ∼
10 s, which is an improvementof 5 orders of magnitude over transitions in the same system whose magnetic field dependenceis linear [23]. Use of a ZEFOZ transition in bismuth-doped natural silicon results in overtwo orders of magnitude increase in coherence time to ∼
90 ms [24]. Such transitions havealready been applied to extend coherence of NV − centers in 1.1% C diamond: Lesik andcoworkers reported an 8-fold increase in coherence time for bulk NV − centers by tuningto a ZEFOZ point at zero magnetic field induced by the hyperfine interaction with N ofthe defect [25]. This gain in coherence was shown to diminish to a factor of 2 for shallowimplanted NV − centers or those in nanodiamonds where the defect is subject to strongerelectric field fluctuations. Lastly, Xu et al. employed continuous-wave DD of dressed NV − center spin states, which essentially utilized ZEFOZ transitions in the rotating frame, in1.1% C diamond to achieve enhancements of ∼
20 in spin coherence time [26].In the current study we apply the principles of the ZEFOZ technique to the mixed states3f NV − centers in C-enriched diamond and observe a maximum reduction of linewidth by afactor of 133.7 from pure electron-type spin transitions typical of the system. We are able toexplain much of the spectral behavior of this spin system observed by continuous-wave (CW)-ODMR using a simple model of four spins: the NV − and its three nearest-neighbor C nuclei.Due to our method of detection, we cannot observe transitions directly at ZEFOZ pointsfound within this spin system, meaning that while coherence is enhanced significantly, thetransitions are protected from decoherence (“deocherence-protected transitions” or DPTs)rather than true ZEFOZ transitions.
DECOHERENCE OF NV − CENTERS IN A NUCLEAR SPIN BATH
The spin dynamics of NV − centers are highly sample dependent, determined largely bythe spin content of the diamond host as well as by the morphology and treatment of thematerial. Tuning these conditions may result in coherence times as short as hundreds ofnanoseconds and as long as milliseconds [16, 27–31]. Consequently, extensive effort has beeninvested in the development of diamond samples ideal for applications that require long spincoherence time.In bulk single crystalline diamond, the main sources of decoherence for NV − centers maybe C nuclear spins, P1 centers (a spin- / electronic defect arising from a substitutionalnitrogen atom in the diamond lattice), or neutral NV centers [31]. Because our sample haslow nitrogen content and high C content, we consider only the effect of the nuclear spinbath on NV − linewidth (∆ ν ). We use this linewidth as an indicator of coherence time ( T )by the relation T ∗ ∼ (∆ ν ) − , where T ∗ , the inhomogeneous dephasing time, is less than T .Mizuochi et al. reported NV − resonances as narrow as 18 kHz by isotopic purification to99.9997% C diamond. In comparison, a dense nuclear spin bath such as that in 99% Cdiamond can lead to linewidths ranging from 50-70 MHz (shown in blue, Figure 1B).Thisdifference is the consequence of broadening by a range of hyperfine interactions from Cnuclei occupying all sites surrounding the NV − defect in the diamond lattice.We explain some of the behavior exhibited by an ensemble of NV − centers in C-enricheddiamond (Figure 1A) in this paper. The ensemble is approximated as a four-spin “molecule”composed of the NV − center and the three nearest-neighbor nuclei, called “first-shell nuclei”.The operators S z , S x , and S y describe the electron spin angular momentum, individual4uclear spin angular momenta are described by the operators I nx , I ny , and I nz , and thetotal nuclear spin operators are given by K x , K y , and K z . With the applied field alongthe NV − axis, the ground state spin Hamiltonian ( H gs ) includes the electron spin Zeemanterm ( γ NV B z S z ), the ground state zero-field splitting interaction ( D gs = 2.87 GHz), andthe ground state hyperfine Hamiltonian ( H HF,gs ) of the NV − center interacting with threefirst-shell spin- nuclei, H gs = γ NV B z S z + D gs (cid:18) S z − S (cid:19) + H HF,gs where H HF,gs = ( S · A · I ) + R ( iφ ) z ( S · A · I ) R ( − iφ ) z + R ( i φ ) z ( S · A · I ) R ( − i φ ) z . The gyromagnetic ratio of the NV − ( γ NV ) is ∼ − spin state along that axis. With the NV − - C hyperfine tensor ( A ) projected onto the NV − axis, taking the nucleus C to lie in the xz plane, the hyperfine term for one nuclear spinin H HF,gs becomes( S · A · I ) = A xx S x I x + A yy S y I y + A zz S z I z + A xz ( S z I x + S x I z )with hyperfine tensor elements [32]: A xx = 166.9 MHz, A yy = 122.9 MHz, A zz = 90 MHz,and A xz = -90 MHz. We obtain the hyperfine interactions with the remaining two nuclei byincremental rotations of φ = π about the NV − axis ( R ( iφ ) z ).The magnetic field dependence of the eigenvalues of the four-spin model is shown inFigure 2. This system has a total of 24 eigenstates where each electron spin state is splitinto eight electron-nuclear spin states. We will refer to the sets of eigenstates where the zprojection of the NV − spin angular momentum is approximately 0, -1, or 1 as the m S =0, -1, or +1 “manifolds”. In the limit of B = 0, the three hyperfine interactions from thefirst-shell spin- nuclei split the m S = ± , there is only one combination that gives a total nuclear spinof ± , but three possible combinations each that give a total nuclear spin of ± . This givesrise to the 1:3:3:1 intensity ratio of resonances observed in the zero field spectrum, shownin Figure 1B. The transitions leading to this quartet structure (given in blue in the energy-level diagram in Figure 1C) are inhomogeneously broadened by the various configurationsof weak hyperfine interactions of nuclei occupying sites at a distance of 3-8 angstroms533, 34] from the defect. In contrast, the zero field spectrum of an ensemble of NV − centersin 1% C diamond (shown in black, Figure 1B) has one primary resonance, the zero fieldresonance (ZFR), split by crystal strain of approximately 3 MHz. The primary resonance isaccompanied by two broadened lines of weakened intensity at D gs − . D gs + 70 . − center coupled to a first-shell C nucleus [35]. It should be noted that the linewidth of this particular ensemble of NV − centers in 1% C diamond is large for NV − centers in diamond of this composition at ∼ S = 0 manifold are also split into four two-fold degenerate levels(see Figure 2B) in the limit of B = 0 due to the transverse terms of the hyperfine tensor.As a result the structure of the m S = 0 manifold cannot be described by the z-projection oftotal nuclear spin angular momentum. Even at magnetic fields far from any LAC regime,these eigenstates exhibit nonlinear magnetic field dependence and sufficient mixing for theobservation of narrowed DPTs by CW-ODMR spectroscopy. RESULTS AND DISCUSSION
In CW-ODMR of NV − centers, a transition in NV − spin state is saturated and observedas a decrease in fluorescence intensity. The contrast, or normalized difference in fluorescenceintensity, of these transitions may be explained as a depopulation of the m S = 0 spinstate [36]. Since mixed eigenstates are not characterized by the pure Zeeman basis, butlinear combinations of these states, the notion of “bright” and “dark” NV − spin statesmust therefore be redefined as the spin states with the most or least m S = 0 spin statecharacter. Due to small mixing of the electronic spin state in the hyperfine states of the m S = 0 manifold, transitions that are nominally forbidden become weakly allowed and havesufficient contrast to be detected by ODMR.At magnetic field strengths where the influence of electron-nuclear hyperfine interactionsand the zero-field splitting remains significant, all transitions in the four-spin system arepossible. In order to gain insight into the nature of the decoherence protected transitionsobserved, we reduce the number of possible transitions to consider by characterizing them6ith an intensity factor κ , which evaluates the probability of their observation by CW-ODMR. κ is determined by the product of the transition matrix element (TME), a termdescribing the difference in population of final and initial states (∆ (cid:104) ρ (cid:105) ), and a term describingthe optical contrast (∆ (cid:104) S z (cid:105) ) of the transition: κ = ( T M E )(∆ (cid:104) ρ (cid:105) )(∆ (cid:104) S z (cid:105) )Where T M E = (cid:104) ϕ f | ( γ NV ( S x + S y + S z ) + γ I ( K x + K y + K z )) | ϕ i (cid:105) ∆ (cid:104) ρ (cid:105) = (cid:104) ϕ f | ρ | ϕ f (cid:105) − (cid:104) ϕ i | ρ | ϕ i (cid:105) ∆ (cid:104) S z (cid:105) = (cid:104) ϕ f | S z | ϕ f (cid:105) − (cid:104) ϕ i | S z | ϕ i (cid:105) Here, because optical contrast in ODMR is determined by a change in electron spin state,we define the optical contrast term as the difference in expectation value of S z in the final( ϕ f ) and initial ( ϕ f ) eigenstates. The transition matrix element is calculated as the innerproduct of the final and initial eigenstates of the spin angular momentum operators with theelectron ( S x , S y , S z ) and total nuclear ( K x , K y , K z ) spins, weighted by their gyromagneticratios. We consider all projections of angular momentum because the orientation of ourmicrowave excitation is not exactly known. In this way we do not unintentionally excludetransitions that could explain the behavior in the ODMR spectrum. Finally, ∆ (cid:104) ρ (cid:105) is thedifference in expectation value of the density matrix ( ρ ) of the eigenstates involved in thetransition. To describe the optical pumping, we construct a density matrix where the eight m S = 0 sublevels are equally populated and the nuclear spin states are thus unpolarized( ρ = ( E − S z ) NV ⊗ E C ⊗ E C ⊗ E C ).In Figure 3, the transitions observed by CW-ODMR spectroscopy are summarized withtransitions we predict using κ greater than 10 − , which was determined empirically to bestfit the data. A more detailed view of the DPTs is given in the inset to the right. Roughly twotypes of transitions may be distinguished: high intensity m S = 0 to m S = ± − spin states due to weakened effective gyromagnetic ratios of the eigenstatesnear LACs, indicating longer dephasing (T *) and potentially coherence (T ) times.Two main sets of LACs where DPTs may be found are observed upon inspection ofthe eigenvalues of the NV − -(3) C system in Figure 2. Set 1 of LACs (Figure 2C) results7rom the Zeeman interaction canceling the hyperfine interactions with the first-shell Cnuclei, and occurs at magnetic field strengths 0- ∼
80 G. Set 2 (Figure 2D) results fromthe Zeeman interaction canceling the energy of the spin-spin coupling that leads to theZFS. While a third set of LACs is known to occur from the spin states of the electronicexcited state near 500 G, the hyperfine tensors of the NV − center and neighboring nuclearspins in the electronic excited state are largely unknown. In this study, we develop a simplebasis for understanding the DPTs that become observable by CW-ODMR at magnetic fieldsapproaching but not directly coinciding with the conditions of Set 2 of LACs ( ∼
565 - 950G), due to the complexity of the behavior of eigenstates in that magnetic field regime (seethe inset of Figure 3). The majority of these transitions have energies corresponding totransitions between eigenstates in the m S = 0 manifolds.It is important to distinguish the transitions we are able to observe in this system us-ing CW-ODMR from ZEFOZ transitions described thus far in the literature. In previousstudies, ZEFOZ transitions are observed in spherically symmetric systems only perturbedby isotropic hyperfine interactions. For these systems, there exists a “ZEFOZ point” atthe LAC where the magnetic field dependence of at least one eigenstate involved in thetransition approaches zero [21]. In contrast, our system is perturbed both by hyperfineinteractions and the spin-spin coupling of the NV − . The transitions we study, which wedistinguish by terming them decoherence-protected, occur between hyperfine sublevels ofthe m S = 0 manifold, whose magnetic field dependence originates only from mixing withm S = -1 eigenstates due to the transverse terms of the hyperfine interaction. This meansthat although a ZEFOZ point exists for these transitions at zero magnetic field where thecoherence properties are optimal, there is a broad range of magnetic fields where transitionswith enhanced coherence properties may be observed. Furthermore, our detection method(CW-ODMR), though highly advantageous for its simplicity, is limited in the sense that itrequires a transition to have a nonzero change in the z projection of electron spin angularmomentum (∆ m S =) for observation. Transitions with the best coherence properties, where∆ m S is closest to 0, are thus not directly observable via optical contrast.Nevertheless, we find DPTs between m S = 0 sublevels with sufficient optical contrastfor CW-ODMR at magnetic fields as far from Set 2 of LACs as ∼
550 G, where ∆ m S is on the order of 10 − . The enhancement of coherence is gauged by the linewidths ofthese transitions, which are significantly narrowed in comparison to typical electron-type8ransitions of this spin system. The reduction in linewidth becomes less dramatic as themagnetic field approaches Set 2 of LACs, where the m S = 0 eigenstates become increasinglymixed with those in the m S = -1 manifold and transitions involve increasing ∆ m S . Thistrend is reflected in the nonlinearity of transitions near Set 2 of LACs illustrated in the insetof Figure 3, as well as in Figure 4, which shows spectra of DPTs selected over a range ofmagnetic fields (608-871G). It is clear that transitions with increasing optical contrast arealso more sensitive to magnetic noise.The observed transitions are assigned according to energy and the intensity factor κ of the predicted transition (see Figure 5). Transitions occurring between the same twoeigenstates were grouped according to these assignments, and their first- (cid:0) γ eff = ∂ν∂B (cid:1) andsecond-order (cid:16) curvature, C = ∂ ν∂ B (cid:17) magnetic field dependence were determined empiricallyusing quadratic fits. These parameters for the transitions shown in Figure 4 are givenin Table I, and may be used to gauge the transition’s sensitivity to axial magnetic fieldfluctuations. In this class of transitions, linewidths as narrow as 527 kHz as well as effectivegyromagnetic ratios and curvatures as low as 13.62 kHz/G and 0.51 kHz/G are estimated,respectively. Comparing with the maximum linewidth of 70.46 MHz for an electron-typetransition we observe between the m S = 0 and m S = +1 eigenstates away from any setof LACs, these transitions are narrowed up to a factor of 133.7. The dephasing times T ∗ estimated from these linewidths may be extended by use of dynamical decoupling methods[18].Upon inspection of Figures 3 and 5, it is clear that the system is too complex to un-ambiguously assign each transition. Furthermore, direct comparisons between the observedlinewidths and the effective gyromagnetic ratios characterizing a transition are not alwaysconsistent. This may indicate homogeneous broadening, which is not taken into account inthe four-spin model. Inherent in this simplification of the spin system under considerationis that all sources of noise causing spectral broadening are axial magnetic fields that arestatic on the time scale of the NV − -(3) C spin dynamics. In reality, the NV − experiencescoupling to nuclei occupying all surrounding lattice sites in a volume with a radius as greatas 5 angstroms from the defect. Experimentally, 15 nuclear sites with interaction strengthsranging from 400 kHz to 14 MHz have been measured, whereas 9 nuclear sites with a totalof 39 symmetrically equivalent positions have been calculated to have interaction strengthsranging from 1.5-19.4 MHz [33, 34]. Such a range of coupling provides various relaxation9athways induced by coherent interactions, i.e. nuclear spin flips, or energy conservingflip-flops, the understanding of which is beyond the scope of this manuscript.Here the general behavior of a class of transitions in a novel spin system has been pre-dicted. Whereas the amplitude of the transition observed by CW-ODMR becomes strongernear Set 2 of LACs, the coherence properties of DPTs approach an optimum far from Set2 of LACs, where first- and second-order magnetic field dependence approaches a mini-mum. These trends are only true, however, for the DPTs we observe using CW-ODMRspectroscopy due to Set 2 of LACs. As is evident from Figures 2 and 3, this spin system issufficiently complex to yield DPTs at a variety of magnetic field strengths and orientations.Narrow features at low magnetic fields (0.5-600 G, some examples found in the Earth’s fieldCW-ODMR spectrum of NV − centers in 99% C diamond in Figure 1) are discussed furtherin a study by Jarmola et. al. [37]. Given the nature of DPTs, where eigenstates are leastsensitive to inhomogeneous broadening in the limit of (cid:104) S z (cid:105) →
0, alternate detection schemessuch as raman heterodyne spectroscopy [38, 39] may be required to achieve the maximumpossible extension of coherence time in this spin system.
CONCLUSION
We have shown that the complex behavior of nitrogen vacancy centers in C-enricheddiamond may be approximated by considering the simple spin molecule of the NV − centercoupled to the three nearest-neighbor C nuclei. The internal interactions of this spinsystem, i.e. the spin-spin coupling of the NV − center and hyperfine interactions with threeneighboring nuclei result in two regimes of magnetic field where spin states undergo avoidedcrossings and DPTs may be found. Such transitions involve electron-nuclear mixed spinstates where sufficient projection of NV − spin onto the mixed state as well as optical pumpingto the m S = 0 manifold allow for their observation by CW-ODMR spectroscopy. In this studywe find a subset of DPTs whose linewidths are narrowed as a consequence of transitions inspin state being desensitized to the fluctuating magnetic field environment of C-enricheddiamond. Of these transitions, we estimate effective gyromagnetic ratios as low as 13.62kHz/G, with linewidths narrowed by factors as high as 133.7. These results demonstrate amethod by which NV − coherence times may be significantly extended, which is a necessarystep in the development of NV − centers for use in quantum information processing.10 XPERIMENTAL
All spectra were acquired using simple continuous-wave application of laser and microwaveexcitation to detect changes in steady-state populations using a homebuilt confocal mi-croscopy system. Magnetic fields as high as 1175 G are applied in these measurements bymounting a neodymium permanent magnet in a goniometer for control of field orientation.The relevant microwave frequencies are applied using waveform generators. The waveformsare chopped using a microwave switch and amplified before being sent to a broadband mi-crowave loop resting directly beneath the diamond sample. Optical excitation is performedwith a 200mW pumped solid state 532 nm laser. The beam is switched by driving anacousto-optic modulator and focused with a 0.7 NA microscope objective to an approximatewaist of 1.4 µ m (confocal length ∼ µ m, excitation volume ∼ µ m ). After necessaryfiltering and loss due to optics, the power applied to this volume is 4 kW/cm . The fluores-cence that results is separated from the excitation beam using a dichroic mirror, long-passfiltered, and fiber-coupled to a single photon counting module for detection. To describethe acquisition of a CW-ODMR spectrum, we refer you to the inset from Figure 1B. Thetime period for microwave excitation and fluorescence acquisition is 1 ms. The fluorescenceacquired during microwave excitation is normalized by fluorescence acquired after the NV − ensemble is reinitialized by optical pumping. The sequence for each microwave frequency inthe spectrum is averaged 2000 times.The diamond sample used in these measurements was an electronic grade wafer of di-mensions 1 x 2 x 0.5 mm grown by Element Six. The sample was grown by chemical vapordeposition to a thickness of 500 µ m, where 30 µ m of the thickness on one face was grownfrom 99% C-enriched methane. The estimated concentration of nitrogen impurity in theenriched layer is < − centers distributed unevenly acrossthe sample. These naturally-formed NV − centers in the C-enriched layer were studied inthe present experiments. For determination of magnetic field alignment along the NV − axis,an ensemble of nitrogen vacancy centers was created in the opposite face of the diamondwhere the abundance of C nuclei is 1.1%. The creation of this ensemble of NV − centerswas accomplished by an implant by Innovion Corporation of 35 keV and 60 keV N ions toa fluence of 10 ions/cm , followed by an anneal under nitrogen atmosphere for 2 hours at800 ◦ C. The sample is situated over the broadband microwave loop and the focal point of11aser beam leaving the microscope objective. A stepper motor with 1 µ m stepping resolutionin X, Y, and Z dimensions is used to position the implanted ensemble to align the field atthe focal point of the beam, and then stepped to position the C-enriched layer at the focalpoint for studying DPTs. The quality of field alignment was confirmed by a difference indisplacement of the m S = 0 to -1 and +1 transitions from the ground state zero field splittingof no greater than 5 MHz, allowing us to estimate an error of ± ∼
10 MHz. In addition,we estimate the error associated with using this implanted ensemble of NV − centers as amagnetometer by the following equation [40]: δB (cid:39) γ NV (cid:112) N S t m T ∗ Where γ NV is the NV − gyromagnetic ratio, N S is the number of NV − spins, t m is themeasurement time, and T ∗ is the dephasing time. For the concentration and volume ofexcitation given above, N S (cid:39) . . When approximating t m (cid:39) T ∗ (cid:39) (∆ ν ) − , where(∆ ν ) − is the linewidth of the ensemble given above, we obtain a δB of 0.05G, which isnegligible in comparison to the field alignment error. ACKNOWLEDGEMENTS
This work was supported by the U.S. Department of Energy, Office of Science, BasicEnergy Sciences under Contract No. DE-AC02-05CH11231. We would like to acknowledgeDaniel Twitchen and Matthew Markham at Element 6 as well as Dmitry Budker for con-tribution of the sample used in this work. Thermal annealing of the diamond sample wascarried out at the Molecular Foundry at Lawrence Berkeley National Laboratory, which issupported by Office of Science, Office of Basic Energy Sciences, of the U.S. Department ofEnergy under Contract No. DE-AC02-05CH11231. We are grateful to Tevye Kuykendallfor help in performing this task. ∗ [email protected][1] G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D.Lukin, Nature , 54 (2013).
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Parameters of assigned decoherence protected transitions at 608 G, 739 G, and 871G. Center frequencies ( ν ) and linewidths (∆ ν ) are given in MHz. γ eff (cid:0) ∂ν∂B (cid:1) and C (cid:16) ∂ ν∂ B (cid:17) aregiven in kHz/G and kHz/G , respectively. For comparison, γ NV ∼ (cid:15) ) are calculated using the average ∆ ν of electron-type transitionsobserved at magnetic fields away from any set of LACs (65.19 MHz). B ν ∆ ν γ eff C κ (cid:15)
608 22.30 1.24 13.62 0.57 6.04 x 10 − − − − − − − − − − − − − V - NV - -(3) C m S = 0m S = ±1 m K = + 3/2m K = + 1/2m K = - 1/2m K = - 3/2~120 MHz ... MWSignalLaser
Abs Ref C C C A)B)C) N N ............ Frequency (MHz) A m p li t ud e FIG. 1:
The NV-(3) C electron-nuclear spin system. (A) Schematic of the NV − center in 1.1%and 99% C diamond. N, C, C, and the lattice vacancy are depicted as blue, gray, darkgray, and transparent cyan spheres, respectively. In fully-enriched C diamond, all lattice sitesaround the NV − center are occupied by C nuclei. (B) Earths field ( 0.5G) CW-ODMR spectraof ensembles of NV − centers in 1.1% (black) and 99% (blue) C diamond. The inset describesthe experiment used to acquire all spectra presented in this study (See Experimental). (C)Energy level diagram of the NV − center in the absence of C nuclei in the first shell (black) andwith three C nuclei in the first shell (blue). The spin quantum number of the NV − is denotedby m S , whereas the spin quantum number of the total nuclear spin of the three first-shell Cnuclei is denoted by m K .
400 800 E n e r g y A) C)
550 MHz
800 900 1000 1100 120050 100 1500
15 MHz9 MHz 2 MHz
Magnetic Field (G) D) A zz ~ 130 MHz D = B) m S = +1, m K m S = -1, m K m S = 0, m K FIG. 2: (A) Calculated eigenvalues of the NV − -(3) C spin system as a function of magneticfield strength along the NV − axis. (B) A more detailed view of the m S = 0 manifold is shown.The transverse terms of the hyperfine tensor result in four doubly-degenerate sublevels at zeromagnetic field. This degeneracy is broken upon application of a magnetic field, and the eightsublevels exhibit a very weak, nonlinear magnetic field dependence from approximately 0-800 G.Two sets of LACs exist for the electronic ground state spin Hamiltonian of the NV-(3) Csystem. (C) Set 1 (0.5 - 80 G) occurs between the m S = -1 and m S = +1 manifolds and (D) Set2 (800 - 1200G) occurs between the m S = -1 and m S = 0 manifolds. r e qu e n cy ( M H z ) Magnetic Field (G)
600 800 1000 120002000400060008000
Weakly-AllowedElectron-typeObserved
600 800 1000 1200050100150
FIG. 3:
Predicted and observed transitions of the NV − -(3) C spin system. Predictedtransitions have intensity factors ( κ ) greater than 10 − . This is 67.5 % of all possible transitions.The frequency range of the DPTs is magnified at the right for a more detailed view. Manyweakly-allowed transitions with first-order magnetic field dependence terms (i.e. ∂ν∂B , or γ eff orders of magnitude lower than γ NV are predicted in this magnetic field regime. .0000.9921.0000.9921.0000.992 A m p li t ud e Frequency (MHz)
B = 871 GB = 739 GB = 608 G Frequency (MHz) Strong Amplitude Long Coherence W W W W W W W W W W W W W FIG. 4:
CW-ODMR spectra of DPTs for three different strengths of magnetic field are given todemonstrate how the nature of the transitions changes as the magnetic field is increased towardsthe regime of Set 2 of LACs. Gaussian fits of transitions occurring between the same sets ofeigenstates (W i ) are color-coded. Unassigned transitions are not fit. Though the amplitude oftransitions detectable by CW-ODMR far from Set 2 of LACs is low, the best coherence propertiesare exhibited in this magnetic field regime. agnetic Field (G)
600 800 1000 F r e qu e n cy ( M H z ) W W W W W W W W FIG. 5:
Assigned DPTs (W i ) of the NV − -(3) C spin system compared to all predictedtransitions at magnetic field strengths near Set 2 of LACs. The observed transitionscorresponding to the same sets of eigenstates, as well as quadratic fits of their magnetic fielddependence, are color-coded consistently with those in Figure 4. Quadratic fits are used forempirical estimation of first- and second-order magnetic field dependence. The predictedtransitions corresponding to κ > − are underlaid in grayscale.are underlaid in grayscale.