Hyperpolarized relaxometry based nuclear T1 noise spectroscopy in hybrid diamond quantum registers
Ashok Ajoy, Ben Safvati, Raffi Nazaryan, J. T. Oon, Ben Han, Priyanka Raghavan, Ruhee Nirodi, Alessandra Aguilar, Kristina Liu, Xiao Cai, Xudong Lv, Emanuel Druga, Chandrasekhar Ramanathan, Jeffrey A. Reimer, Carlos A. Meriles, Dieter Suter, Alexander Pines
HHyperpolarized relaxometry based nuclear T noise spectroscopy in hybrid diamond quantum registers A. Ajoy, ∗ B. Safvati, R. Nazaryan, J. T. Oon, B. Han, P. Raghavan, R. Nirodi, A. Aguilar, K. Liu, X. Cai, X. Lv, E. Druga, C. Ramanathan, J. A. Reimer, C. A. Meriles, D. Suter, and A. Pines Department of Chemistry, and Materials Science Division Lawrence BerkeleyNational Laboratory University of California, Berkeley, California 94720, USA. Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA. Department of Chemical and Biomolecular Engineering, and Materials Science Division LawrenceBerkeley National Laboratory University of California, Berkeley, California 94720, USA. Department of Physics and CUNY-Graduate Center, CUNY-City College of New York, New York, NY 10031, USA. Fakult¨at Physik, Technische Universit¨at Dortmund, D-44221 Dortmund, Germany.
Understanding the origins of spin lifetimes in hybrid quantum systems is a matter of current importance inseveral areas of quantum information and sensing. Methods that spectrally map spin relaxation processes pro-vide insight into their origin and can motivate methods to mitigate them. In this paper, using a combinationof hyperpolarization and precision field cycling over a wide range (1mT-7T), we map frequency dependent re-laxation in a prototypical hybrid system of C nuclear spins in diamond coupled to Nitrogen Vacancy centers.Nuclear hyperpolarization through the optically pumped NV electrons allows signal time savings for the mea-surements exceeding million-fold over conventional methods. We observe that C lifetimes show a dramaticfield dependence, growing rapidly with field up to ∼ C enrichment levels, weidentify the operational relaxation channels for the nuclei in different field regimes. In particular, we demon-strate the dominant role played by the C nuclei coupling to the interacting P1 electronic spin bath. Theseresults pave the way for quantum control techniques for dissipation engineering to boost spin lifetimes in dia-mond, with applications ranging from engineered quantum memories to hyperpolarized C imaging.
Introduction: – The power of quantum technologies, especiallythose for information processing and metrology, relies criticallyon the ability to preserve the fragile quantum states that are har-nessed in these applications [1]. Indeed noise serves as an en-cumbrance to practical implementations, causing both decoher-ence as well as dissipation of the quantum states [2, 3]. Precise spectral characterization of the noise opens the door to strate-gies by which it can be effectively suppressed [4, 5] – case inpoint being the emergence of dynamical decoupling techniquesthat preserve quantum coherence by periodic driving [6]. In thesecases, quantum control sets up a filter that decouples componentsof noise except those resonant with the exact filter period [7], al-lowing spectral decomposition of the dephasing noise afflictingthe system. Experimentally implemented in ion traps [8], super-conducting qubits [9] and solid-state NMR [10], this has spurreddevelopment of Floquet engineering to enhance T decoherencetimes by over an order of magnitude in these physical quantumdevice manifestations [11–13].Methods that analogously spectrally fingerprint T relaxation processes, on the other hand, are more challenging to implementexperimentally. If possible however, they could reveal the originsof relaxation channels, and foster means to suppress them. Appli-cations to real-world quantum platforms are pressing: relaxationin Josephson junctions and ion trap qubits, for instance, occur dueto often incompletely understood interactions with surface para-magnetic spins [14]. Relaxation studies are also important in thecontext of hybrid quantum systems, such as those built out of cou-pled electronic and nuclear spins. In the case of diamond NitrogenVacancy (NV) center electronic qubits coupled to C nuclei [15],for instance, a detailed understanding of nuclear relaxation canhave important implications for quantum sensing [16]: engineeredNV- C clusters form building blocks of quantum networks [17],are the basis for spin gyroscopes [18], and are harnessed as quan-tum memories in high-resolution nano-MRI probes [19]. Nuclear ∗ [email protected] T lifetimes are not dominated by phonon interactions, but insteadare set by couplings with the intrinsic electronic spin baths them-selves – a complex dynamics that is often difficult to probe exper-imentally. Indeed only a small proportion of C spins can be ad-dressed or readout via the NV centers, as also the direct inductivereadout of these spins suffer from extremely weak signals. More-over, as opposed to T noise spectroscopy carried out in the rotat-ing frame [13], probing of T processes have to be performed inthe laboratory frame. This necessitates the ability to probe relax-ation behavior while subjecting samples to widely varying mag-netic field strengths.In this paper, we develop a method of “hyperpolarized relaxom-etry” that overcomes these instrumentational and technical chal-lenges. We measure T relaxation rates of C spins in diamondsamples relevant for quantum sensing with a high density of NVcenters. Our T noise spectroscopy proceeds with high resolutionand over four decades of noise spectral frequency, revealing thephysical origins of the relaxation processes. While experimentsare demonstrated on diamond, it acts here as a prototypical solidstate electron-nuclear hybrid quantum system, and the results areindicative of relaxation processes operational in other systems, in-cluding Si:P [20], wide bandgap materials such as SiC [21, 22],and diamond-based quantum simulator platforms constructed outof 2D materials such as graphene and hBN [23–25]. These re-sults are also pertinent for producing and maintaining polarizationin hyperpolarized solids, for applications employing hyperpolar-ized nanoparticles of Si or diamond as MRI tracers [26, 27], andin the relayed optical DNP of liquids mediated through nanodia-monds [28], since in these applications T relaxation bounds theachievable polarization levels.Key to our technique is the hyperpolarization of C nuclei atroom temperature, allowing the rapid and direct measurement ofnuclear spin populations via bulk NMR [28]. Dynamic nuclearpolarization (DNP) is carried out by optical pumping and polar-izing the NV electrons (close to 100%) and subsequently trans-ferring polarization to C nuclei (Fig. 1A). This routinely leadsto nuclear polarization levels (cid:38) a r X i v : . [ qu a n t - ph ] F e b E nh a n c e m e n t o v e r T ( a u ) DecayBuildup Detection
60 400 0.5 F i e l d Shuttling Loss637msTime (s)
FID E I n c r e a s i n g h e i g h t relax PortableHyperpolarizer D After FlipBeforeFlip
P1 Center C NucleusFlip C Increasing Magnetic Field
LarmorFrequency E n e r g y slope = γ n Spin-flippingNoise B P1 Center H y p e r p o l a r i z a t i o n LaserIrradiationMWIrradiation
VacancyCarbon CNV CenterDiamond Lattice A Figure 1.
Principle. (A)
System consisting of C nuclear spins in dia-mond hyperpolarized via NV centers allowing their direct measurementby bulk NMR. Lattice also contains electronic spin bath of P1 centers.(B) Changing magnetic field allows probing of spin flipping noise that isresonant with the carbon Larmor frequency. (C) Dominant T relaxationmechanism via three-body flip-flops with pairs of P1 center electrons.(D) Experimental platform.
Portable hyperpolarizer is installed in a rapidfield cycling device capable of sweeping between 10mT-7T in the fringefield of a NMR magnet. (E)
Time sequence.
Lower panel shows theschematic steps of laser driven optical C hyperpolarization for ∼ B pol ≈ < B relax , re-laxation and subsequent high field detection at 7T. Upper panel displaystypical data for 200 µ m microdiamond powder, where B relax = CNMR signal amplitude (points) is quantified by its enhancement over the7T Boltzmann signal. Signal growth and decays are fitted to stretchedexponentials (solid lines). factors exceeding ε ∼ -10 , and resulting in highsingle shot detection SNRs. This permits T spectroscopy exper-iments that would have otherwise been intractable. Hyperpolar-ization is equally efficiently generated in single crystals as wellas randomly oriented diamond powders, and both at natural abun-dance as well as enriched C concentrations. The hyperpolarizedsamples are interfaced to a home built field cycler instrument [29](see Fig. 1D and video in [30]) that is capable of rapid and high-precision changes in magnetic field over a wide 1mT-7T range(extendable in principle from 1nT-7T), opening a unique way topeer into the origins of nuclear spin relaxation. C Hyperpolarized relaxometry: – Fig. 1D-E schematicallydescribe the experiment. Hyperpolarization in the C nuclei isaffected by optical pumping at low fields, typically B pol ∼ B relax wherethe spins are allowed to thermalize (see Fig. 1C), and subse-quent bulk inductive measurement at 7T. Experimentally varying B relax allows one to probe field dependent lifetimes T ( B relax ) ,and through them noise sources perpendicular to B relax and res-onant with the nuclear Larmor frequency γ n B relax (Fig. 1B). Here γ n = 10 . MHz/T is the C gyromagnetic ratio. This allows
Field (T) / T ( m H z ) -1 highfieldultra-lowfield Knee FieldB
K(2) B K(1)
Tsallian 1Tsallian 2ConstantSum (Fit) D E nh a n c e m e n t ( a u ) Time (s)
Field (T) T ( s ) / T ( m H z ) Tsallian 1Tsallian 2ConstantSum (Fit) B Hyperpolarized7T Thermal
Single Shot SNR = 400 020406080100 S i g n a l ( a u ) Frequency (kHz)0 7.5 15Enhancement= 372
20 averages at 7TOne average 0510-5-10 0 5 10Frequency (kHz) S i g n a l ( a u ) +Sweep-Sweep A Figure 2.
Hyperpolarized relaxometry applied to a 10% C enrichedsingle crystal. (A)
Signal gains due to hyperpolarization under optimalconditions at B pol ≈ ≈ ≈
372 over 7T, a time saving by ≈ for equal SNR. Inset:
Exemplary signals at B pol ≈ C hyperpolarization. (B)
Relaxation rate R = 1 /T obtained from relaxometry over a wide fieldrange 20mT-1.5T. We observe a rapid growth in relaxation rate below aknee field of 0.5T, and saturation at higher fields. Inset:
Data can be fitto two Tsallian functions, which we ascribe to be originating from inter- C couplings and interactions to the P1 spin bath. (C)
Spin lifetimes as afunction of field, showing significant boost in nuclear T beyond the kneefield, approaching a lifetime ≈ Inset:
Typical relaxation data attwo representative fields showing monoexponential character. (D)
Loga-rithmic scale data visualization, displaying a more equanimous samplingof experimental points, and the knee fields inflection points B (1 , K . Inset:
Decomposition into the constituent Tsallians. Error bars in all panels areobtained from monoexponential fits. the spectral decomposition of noise processes that spawn T re-laxation. For instance pairs of substitutional nitrogen impurities(P1 centers) undergoing flip-flops (Fig. 1C) can apply on the Cnuclei a stochastic spin-flipping field that constitutes a relaxationprocess.Optical excitation for hyperpolarization involves 520nm irra-diation at low power ( ∼ ) applied continuously for ∼ Cspins (see Fig. 2A) [28, 31]. DNP occurs in a manner that iscompletely independent of crystallite orientation. All parts of theunderlying NV ESR spectrum produce hyperpolarization, with in-tensity proportional to the underlying electron density of states.The polarization sign depends solely on the direction of MWsweeps through the NV ESR spectrum (see Fig. 2A inset). Phys-ically, hyperpolarization arises from partly adiabatic traversals ofa pair of Landau-Zener (LZ) crossings in the rotating frame thatare excited by the swept MWs. For a more detailed exposition ofthe DNP mechanism, we point the reader to Ref. [32].Low field hyperpolarization is hence excited independent of thefields B relax under which relaxation dynamics is to be studied.There is significant acceleration in acquisition time since opticalDNP obviates the need to thermalize spins at high fields where T times can be long (for some samples > ≈ ε T ( ) T ( B pol ) , which in our experimental conditionsexceeds five orders of magnitude. In Fig. 2A for instance on a10% enriched single crystal, we obtain large DNP enhancements ε = ≈ ∼ nuclear spins, which we polarize to a bulk value (averagedover all C nuclei) of 0.37% employing just 3000 MW sweeps,indicating a transfer efficiency of ≈ ultrafast relaxometry of thenuclei.Our experiments are also aided by technological attributes ofthe DNP mechanism. DNP is carried out under low fields and laserand MW powers, and allows construction of a compact hyperpo-larizer device that can accessorize a field cycling instrument [35](see [36] for video of hyperpolarizer operation). The wide range(1mT-7T) field cycler is constructed over the 7T detection magnet,and affects rapid magnetic field changes by physically transport-ing the sample in the axial fringe field environment of the mag-net [29]. This is accomplished by a fast (2m/s) conveyor belt actu-ator stage (Parker HMRB08) that shuttles the sample via a carbonfiber rod (see video in Ref. [30]). The entire sample (field) trajec-tory can be programmed, allowing implementation of the polariza-tion, relaxation and detection periods as in Fig. 1C. Transfer timesat the maximum travel range were measured to be 648 ± T n lifetimes we probe. High posi-tional resolution (50 µ m) allows access to field steps at high pre-cision ( [37] shows full field-position map). The field is primarilyin the ˆ z direction (parallel to the detection magnet), since sampletransport occurs centrally, and the diameter of the shuttling rod(8mm) is small in comparison with the magnet bore (54mm). Results: – Fig. 2 shows representative results of T noise spec-troscopy on C nuclei in diamond, considering here a 10% en-riched single crystal. The intriguing data can be visualized inseveral complementary ways. First, considering relaxation rate R = 1 /T (Fig. 2B), the high-resolution data allows us toclearly discern three regimes: a steep narrow R increase at ul-tralow fields ( < full decay curve atevery field value (for example shown in Fig. 2C). Error bars ateach field value are estimated from monoexponential fits of thepolarization decays. The resulting errors are under a few percent.The solid line in Fig. 2B indicates a numerical fit and remarkablyclosely follows the experimental data. Here we employ a sum oftwo Tsallian functions [38, 39] that capture the decay rates at lowand moderate fields, and a constant offset at high field (see Fig. 2B insets).A second viewpoint of the data, presented in Fig. 2C, is of the T relaxation times and highlights its highly nonlinear field depen-dence. There is a step -like behavior in T ( B relax ) , and an inflec-tion point ( knee field) ≈ T ’s saturate.We quantify the knee field value, B (1) K , as the B relax at which therelaxation rate is twice the saturation R that we observe at highfield. This somewhat counterintuitive dependence has significanttechnological implications. (i) Long C lifetimes can be fash-ioned even at relatively modest fields at room temperature. Thisadds value in the context of C hyperpolarized nanodiamondsas potential MRI tracers [40], since it provides enough time forthe circulation and binding of surface functionalized particles toilluminate disease conditions. (ii)
The step-behavior in Fig. 2Calso would prove beneficial for C hyperpolarization storage andtransport. Exceedingly long lifetimes can be obtained by simplytranslating polarized diamond particles to modest ∼ B (2) K at lower magnetic fields below whichthere is a sudden increase in the relaxation rates. The inset in Fig.2D shows the decomposition into constituent Tsallian fits with anarrow and broad widths.Microscopic origins of this relaxation behavior can be under-stood by first considering the diamond lattice to consist of threedisjoint spin reservoirs – electron reservoirs of NV centers, P1centers, and the C nuclear spin reservoir. P1 centers arise pre-dominantly during NV center production on account of finite con-version efficiency in the diamond lattice. Indeed the P1 centersare typically at 10-100 times higher concentration than NV cen-ters; with typical lattice concentrations of NVs, P1s and C nucleirespectively P NV ∼ P e ∼ P C ∼ η ppm,where η is the C lattice enrichment level. At any non-zero fieldof interest, B relax , the electron and nuclear reservoirs are centeredat widely disparate frequencies and do not overlap. We can sepa-rate the relaxation processes in different field regimes to be drivenrespectively by – (i) couplings of C nuclei to pairs (or gener-ally the reservoir) of P1 centers. This leads to the B (1) K feature atmoderate fields in Fig. 2C; (ii) C spins interacting with indi-vidual P1 or NV centers undergoing lattice driven relaxation ( T e processes); (iii) inter-nuclear couplings within the C reservoirthat convert Zeeman order to dipolar order. Both of the latter pro-cesses contribute to the low field B (2) K features in Fig. 2C; andfinally, (iv) a high-field process >
1T that shows a slowly varying(approximately constant) field profile. We ascribe this to arise di-rectly or indirectly (via electrons) from two-phonon Raman pro-cesses. Since these individual mechanisms are independent, theoverall relaxation rate is obtained through a sum, T = (cid:80) ( J ) 1 T ( J )1 (shown in the inset of Fig. 2D). Effect of electronic spin bath: – Let us first experimentallyconsider the relaxation process stemming from C spins couplingto the interacting P1 reservoir. In Fig. 3 we consider single crys-tal samples of natural C abundance grown under similar con-ditions but with different nitrogen concentrations. Their P1 elec-tron concentrations are P e = P e =
100 101 102 103 104101 102 103 104 105Field (mT)Frequency (kHz)10 R e l a x a t i o n R a t e / T ( m H z ) Sample 1 ([P1] = 17 ± 2 ppm)Sample 2 ([P1] = 48 ± 6 ppm)
T1 = .086 sT1 = 1.5 sT1 = 9.5 minT1 = 19.5 min Ultra-Low Field High Field O r d e r s o f M a g n i t u d e A B
Field (T)0 0.4 0.8 1.2 1.6 20246810 / T ( m H z ) -2 R ( m H z ) C
101 102 103Field (mT)00.40.81.21.6 / T ( H z ) P h a s e N o i s e ( d B c / H z ) Field (mT)-500 0 500-40-30-20-100 132.4 mT46.9 mT
336 338 342 344 346340 I n t e g r a t i o n ( a u ) Field (mT)01234
Sample 2ESR Int. E
336 338 342 344 346340 I n t e g r a t i o n ( a u ) Field (mT)0.4.81.21.6
Sample 1ESR Int. D (i)(ii) Figure 3.
Hyperpolarized relaxometry at natural abundance C concentration over four decades of field 1mT-7T (lower axis), probing spin-flippingspectral density from 1kHz-75MHz (upper axis). (A)
Relaxation rate on a logarithmic scale, showing steep field dependence that spans four orders ofmagnitude in T , falling to sub-second lifetimes at ultra-low fields below B (2) K , and saturating to lifetimes greater than 10min. beyond B (1) K . Orangeand green data correspond to CVD samples with different concentration of P1 centers [41] (legend). Solid lines are fits to a combination of two Tsallianfunctions. Shaded regions represent error bounds originating from our accelerated data collection strategy (see Supplemental Information [37]). Insets:
X-band ESR spectra. (B)
High field behavior shows saturating knee field B (1) K occurs at higher field for sample with larger P1 concentration. (C) Low field behavior , where intriguingly sample with more P1 centers has a lower relaxation rate. (D)
Calculated relaxation rate R ( ω L ) arising fromthe coupling of the C spins with the interacting P1 reservoir for the case of 17ppm (green) and 48ppm (orange) electron concentrations, showingqualitative agreement with the experimental data. (E)
Comparing effective phase noise S p ( ω ) for the two samples on a semi-log scale. For clarity, datais mirrored on the X-axis and phase noise normalized against relaxation rates at ω =1mT. Solid lines are fits to Tsallian functions. Dashed vertical linesindicate the theoretical widths obtained from the the respective estimates of (cid:104) d ee (cid:105) , 46.7mT and 131.89mT, matching very closely with experiments. ments are taken by an accelerated strategy (outlined in [37]) overa ultra-wide field range from 1mT-7T, with DNP being excitedat B pol =36mT. For relaxometry at fields below B pol , we employrapid current switching of Helmholtz coils within the hyperpo-larizer device. Both the range of fields, as well as the densityof field-points being probed are significantly higher than previousstudies [33, 43]. This aids in quantitatively unraveling the under-lying physics of the relaxation processes. We note that probingrelaxation behavior below ∼ T ’s being probed.Experimental results in Fig. 3 reveal a remarkably sharp R dependence, best displayed in Fig. 3A on a logarithmic scale,showing variation in relaxation rate over four orders of magni-tude. The data fits two Tsallian functions (solid line), and revealsthe B (1) K inflection point (closely resembling Fig. 2B) beyondwhich the lifetimes saturate. The second knee field B (2) K at ul-tralow fields can also be discerned, although determining its exactposition is difficult without relaxation data approaching truly zero-field. Comparing the two samples (Fig. 3A), we observe a clearcorrelation in the B (1) K knee field values shifting to higher fields athigher electron concentration P e . The high field relaxation rates,highlighted in Fig. 3B, increase with P e . Interestingly at low fields (see Fig. 3C), the sample with lower P e has an enhancedrelaxation rate, yielding an apparent “cross-over” in the relaxationdata between the two samples at ≈ µ m sizes(see Fig. 4). This is because the random orientations of the crys-tallites play no significant role in the P1-driven nuclear relaxationprocess. We do expect, however, that for nanodiamond particles < C nuclei. Consider the Hamiltonian of the sys-tem , assumed for simplicity to be a single C spin, and the environment - the interacting bath of P1 centers surrounding it, H = H S + H E + H SE + H EE where, the first two terms capturethe Zeeman parts, the third term is the coupling between reser-voirs, and the last term captures the inter-electron dipolar cou-plings within the P1 bath. Specifically, H = ω L I z + ω e S z + (cid:88) j A jzx S zj I x + (cid:88) j P1 reservoir thatis widely separated in frequency from C spins. In a simplisticpicture, shown in Fig. 1C, relaxation originates from pairs of P1centers in the same N nuclear manifold (energy-mismatched by δ ) undergoing spin flip-flop processes, and flipping a C nuclearspin (when ω L ≈ δ ) in order to make up the energy difference.In reality, the overall relaxation is constituted out of several suchprocesses over the entire P1 electronic spectrum.Let us now assume the stochastic process s z ( t ) is Gaussian withzero mean and an autocorrelation function g ( τ ) = exp( − τ /τ c ) with correlation time τ c = 1 / (cid:104) d ee (cid:105) . The spectral density func-tion S ( ω ) = √ π (cid:82) ∞∞ g ( τ ) e − iωτ dτ that quantifies the powerof the spin flipping noise components at various frequencies isthen a Lorentzian, S ( ω ) = 2 τ c / (1 + ω τ c ) . Going furthernow into an interaction picture with respect to ω L I z , H ( I ) I = (cid:104) A zx (cid:105) s z ( t ) (cid:16) e − iω L I z t (cid:48) I x e iω L I z t (cid:48) (cid:17) . The survival probability ofthe spin is, p ( t ) = Tr (cid:110) I z e i H ( I ) I t I z e − i H ( I ) I t (cid:111) ∼ e − χ ( t ) where inan average Hamiltonian approximation, retaining effectively time-independent terms, the effective relaxation rate χ ( t ) ≈ R t canbe obtained by sampling of the spectral density resonant with thenuclear Larmor frequency ω L at each field point. This is the ba-sis behind noise spectroscopy of the underlying T process [45].We recover then the familiar Bloembergen-Purcell-Pound (BPP)result [46, 47], where the relaxation rate, T (1)1 = R (1)1 ( ω L ) = (cid:10) A zx (cid:11) S ( ω L ) = (cid:10) A zx (cid:11) (cid:104) d ee (cid:105) ω L + (cid:104) d ee (cid:105) (3) The inter-spin couplings can be estimated from the typical inter-spin distance (cid:104) r e (cid:105) = (3 / π ln 2) / N − / e , where N e = (4 × − P e ) /a [m − ] is the electronic concentration in inverse vol-ume units and a =0.35nm the lattice spacing in diamond [33]. Thecouplings are now related to the second moment of the electronicspectra [44] M e = ( gµ B ) (cid:104) r e (cid:105) , where g ≈ is the electrong-factor, and µ B = 9 . × − erg/G the Bohr magneton in cgsunits. This gives (cid:104) d ee (cid:105) ≈ γ e (cid:113) π √ M e [Hz] ≈ P e [mG], whichscales approximately linearly with electron concentration P e [33].For the two samples with P e =17ppm and 48ppm we obtain spec-tral widths (cid:104) d ee (cid:105) =0.5kHz and 1.42kHz respectively, correspond-ing to field-profile widths of 46.9mT and 132.4mT respectively.These would correspond to inflection points B (1) K = (cid:104) d ee (cid:105) γ n in therelaxometry data at fields 23.5mT and 66.2mT respectively. Thesevalues, represented by the dashed lines in Fig. 3A, are in remark-able quantitative agreement with the experimental data. Moreover,we expect that these turning points (scaling ∝ P e ) are independent of C enrichment η , in agreement with the data in Fig. 2 (see alsoFig. 5).From lattice considerations (see [37]), we can also estimatethe value of the effective hyperfine coupling (cid:104) A zx (cid:105) in Eq. (3),which we expect to grow slowly with P e . We make the assump-tion that there is barrier of r ≈ C spins are “unobservable” because their hyper-fine shifts exceed the measured C linewidth ∆ f det ≈ (cid:104) A zx (cid:105) = (cid:2)(cid:10) A zx (cid:11)(cid:3) / , where the second moment [44], (cid:10) A zx (cid:11) = N (cid:2) µ π γ e γ n (cid:126) (cid:3) (cid:80) j (3 sin ϑ j cos ϑ j ) r j with N being therelative number of C spins per P1 spin, and ϑ j the angle be-tween the P1- C axis and the magnetic field, and index j runsover the region between neighboring P1 spins. This gives, (cid:10) A zx (cid:11) ≈ (cid:16) µ π γ e γ n (cid:126) (cid:17) 65 1 (cid:104) r e (cid:105) (cid:32) r − (cid:104) r e (cid:105) (cid:33) (4)For the two samples, we have (cid:104) r e (cid:105) = C hyperfine interaction (cid:10) A zx (cid:11) ≈ ] and (cid:10) A zx (cid:11) ≈ ] respectively.These values are also consistent with direct numerical estimatesfrom simulated diamond lattices (see [37]). The simple modelstemming from Eq. (2) and Eq. (3) therefore predicts that theeffective hyperfine coupling (cid:104) A zx (cid:105) increases slowly with the elec-tron concentration P e , with the electron spectral density width (cid:104) d ee (cid:105) ∝ P e .Finally, from Eq. (12) we can estimate the zero-field rate stem-ming from this relaxation process, R (0) = (cid:104) A zx (cid:105) (cid:104) d ee (cid:105) ≈ − ]and 317.5[s − ] respectively. Fig. 3D calculates the resulting re-laxation rates from this process R ( ω L ) in a logarithmic plot. Itshows good semi-quantitative agreement with the data in Fig. 3Aand captures the experimental observation that the rates of the twosamples “cross over” at a particular field. It is instructive to rep-resent the data in terms of effective “phase noise” (see Fig. 3E),denoted logarithmically as, S p ( ω L ) = 10 log (cid:16) R ( ω ) R ( ω L ) (cid:17) [dBc/Hz],where ω → represents the relaxation rates approaching zerofield. Fig. 3E shows this for the two samples, employing ω = C nuclei is about 15dB lower in the 17ppmsample. 102 103 0 0 0 0 Field (mT) 102 103 Field (mT) / T ( H z ) A B / T ( H z ) 50 μm Figure 4. C nuclear relaxation in microdiamond powder . Relax-ation field maps for the randomly oriented natural abundance C micro-diamond powders of size (A) 200 µ m and (B) 5 µ m with accompanyingSEM images (insets). Data is obtained by measuring the full relaxationcurve at every field point, and is quantitatively similar to the single crystalresults in Fig. 3. While Eq. (3) is the dominant relaxation mechanism opera-tional at moderate fields, let us now turn our attention to the thebehavior at ultralow fields in Fig. 3. Eq. (2) provides the frame-work to consider the effect of single P1 and NV electrons to therelaxation of C nuclei. In this case the stochastic process s z ( t ) arises not on account of inter-electron couplings, but due to in-dividual T e processes operational on the electrons, due to forinstance coupling to lattice phonons. The width of the spectraldensity is then given by T e , T (2)1 = R (2)1 ( ω L ) = (cid:10) A zx (cid:11) T e ω L T e (5)While T e is also field-dependent, and dominated by two-phononRaman processes at moderate-to-high field, typical values of T e ∼ ≈ B (2) K ≈ γ n T e = Effect of C enrichment: – To systematically probe this low-field behavior as well as consider the effect of couplings withinthe C reservoir, we consider in Fig. 5 diamond crystals withvarying C enrichment η and approximately identical NV andP1 concentrations. With increasing enrichment, a third relax-ation mechanism becomes operational, wherein at low fields itbecomes possible to dissipate Zeeman energy into the dipolarbath. The field dependence of this process is expected to bemore Gaussian, centered at zero field and have a width ∼ (cid:104) d CC (cid:105) the mean inter-spin dipolar coupling between C nuclei. Wecan estimate (see [37]) these couplings from the second mo-ment, (cid:104) d CC (cid:105) = N (cid:80) j (cid:104)(cid:80) k (cid:0) µ π (cid:126) γ n (3 cos ϑ jk − (cid:1) /r jk (cid:105) / where in a lattice of size (cid:96) , N = N C (cid:96) refers to the number of C spins, and the spin density N C = 0 . η spins/nm . Here ϑ jk = cos − (cid:16) r jk · B relax r jk B relax (cid:17) is the angle between the inter-nuclearvector and the direction of the magnetic field. In the numericalsimulations (outlined in [37]), we evaluate the case consistentwith experiments wherein the single crystal samples placed flat,i.e. with B relax (cid:107) [001] crystal axis. As a result, for C spins onadjacent (nearest-neighbor) lattice sites, ϑ jk = ◦ is the magicangle and d CC jk = 0 .We find (cid:104) d CC (cid:105) ≈ (cid:104) d CC (cid:105) ∝ η / with increasing enrichment. This is in goodagreement with the experimentally determined linewidths (see Field (T) / T ( m H z ) A S e c o n d D e r i v a t i v e ( a u ) Field (T)10 K(1) B K(2) / T ( H z ) Field (T)10 -2 -1 -3 -2 -1 3% Enriched 10% Enriched 100% EnrichedB K(1) B K(2) P o l . B u i l d u p T i m e ( s ) C Enrichment (%) E C CharacteristicBuild Time Monoexp.Biexp. DNP Time (s)0 20 40 60 80 I n c r e a s i n g C C o n c e n t r a t i o n N o r m a l i z e d S i g n a l E nh a n c e m e n t C Enrichment (%) F D i ff u s i o n C o n s t a n t ( n m / s ) D i ff u s i o n l e n g t h ( n m ) Figure 5. Variation with C enrichment . Experiments are performedon single crystal samples placed so that all the NV center orientations areidentical at 54.7 ◦ to B pol =36mT. (A) Relaxation rates on linear and (B)logarithmic field scale, making evident an increase in relaxation rate withincreasing C enrichment at low and high fields. Solid lines are Tsal-lian fit. Error bars are obtained from the relaxation data at various fields.Characteristic knee field B (1) K (dasher vertical line) at moderate fields isindependent of enrichment, evident in the inset. Knee field at ultra-lowfields B (2) K qualitatively is indicated by the dashed line that serves as aguide to the eye. Inset: Second derivative of the fitted lines, showing theknee fields at the zero-crossings. (C) DNP polarization buildup curves also reflect differences in the nuclear spin lifetimes, displaying saturationat much shorter times upon increasing enrichment. DNP in all curves areperformed at 36mT sweeping the entire m s = +1 manifold. (D) Polar-ization buildup times extracted from the data showing that faster nuclearspin relaxation limits the final obtained hyperpolarization enhancementsin highly enriched samples. (F) Spin diffusion constant and diffusionlength for C nuclei numerically estimated from the data as a functionof lattice enrichment. Dashed line indicates the mean inter-electron dis-tance (cid:104) r NV (cid:105) ≈ C enrichment. [37]). We thus expect a turning point at low fields, B (2) K ∼ (cid:104) d CC (cid:105) γ n ,for instance ≈ µ T for natural abundance samples, but scaling to ≈ T s, evidentboth at low (Fig. 5A) and high (Fig. 5B) fields. R rates for thehighly enriched samples (10% and 100%) are obtained by takingthe full relaxation decay curves at every field point, while for thelow enriched sample (3%) enrichment, we use an accelerated datacollection strategy (see [37]) on account of the inherently long T lifetimes. On a logarithmic scale (Fig. 5B), we observe theknee field B (1) K is virtually identical across all the samples, indi-cating it is a feature independent of C enrichment, originatingfrom interactions with the electronic spin bath. This is in goodagreement with the model in Eq. (3). A useful means to evalu-ate the inflection points from the zeros of the second derivative ofthe Tsallian fits, as indicated in the inset of Eq. (3)A. Moreover,the lower inflection field B (2) K scales to higher fields with increas-ing enrichment η , pointing to its origin from internuclear dipolareffects. At the low fields, we also notice that the samples withlower enrichment have higher relaxation rates, and with steeperfield-profile slopes (Fig. 5B). This is once again consistent withthe model that the spectral density height and width being probedscales with (cid:104) d CC (cid:105) .Changes in the nuclear lifetimes are also reflected directly inthe DNP polarization buildup curves, shown in Fig. 5C. We per-form here hyperpolarization of all the samples under the sameconditions, sweeping the entire m s =+1 manifold at B pol =36mT,sweeping over the full NV ESR spectrum. We notice that polar-ization buildup is predominantly mono-exponential (dashed linesin Fig. 5C), except for at natural abundance C, where a biex-ponential growth (solid line) is indicative of nuclear spin diffu-sion. Data demonstrates that highly enriched samples have pro-gressively smaller polarization buildup times (see Fig. 5E) on ac-count of limited nuclear lifetimes at B pol .Moreover, the experimental data allows us to quantify the ”‘ho-mogenization”’ of polarization in the lattice. We assign a spin dif-fusion coefficient D = (cid:104) r n (cid:105) T n (see Fig. 5F) where the T n are eval-uated here by only taking the dipolar contribution to the linewidth, T n ≈ / (cid:104) d CC (cid:105) [49]. Given a total time bounded by T , we cancalculate the rms overall diffusion length [50] as σ = √ DT that is displayed as the blue points in Fig. 5F. Also for reference isplotted the mean NV-NV distance ≈ C re-laxation in enriched samples can have several technological ap-plications. Enrichment provides an immediate means to realizequantum registers and sensing modalities constructed out of hy-brid NV- C spin clusters, and as such ascertaining nuclear relax-ation profiles is of practical importance for such applications. Low η ( ≤ C pairs that can form quan-tum registers [51–53]. The nuclear spin can serve as an ancillaryquantum memory that, when employed in magnetometry applica-tions, can provide significant boosts in sensing resolution [19, 54].With increasing C concentrations η (cid:38) 10% a single NV cen-ter can be coupled to several C nuclei forming natural nodesfor a quantum information processor, and where the nuclear spinscan be actuated directly by hyperfine couplings to the NV elec-tron [55, 56]. Approaching full enrichment levels ( η = -1 R e l a x a t i o n R a t e / T ( H z ) A CD Blue laserNo blue laserIrradiation 20 E nh a n c e m e n t ( a u ) Field = 20 mT / T ( H z ) S a m p l e F i e l d Time M W S w ee p L a s e r B pol relax B DNP RelaxationTime F r e q . 520 nm~ 80 MW2/mm2 465 nm Figure 6. Dynamic optical engineering of electron spin density. Paneldenotes C relaxation rate on a logarithmic scale for 17ppm P1 concen-tration sample employed in Fig. 3 with relaxation dark (red points) andunder low power (80mW/mm ) blue (465nm) irradiation (blue points).Shaded regions represent error bounds (see [37]). (B) Schematic time se-quence of the experiment. (C) Exemplary decay curves obtained at 20mT.(D) Relaxation rates on linear field scale. We observe that the blue radi-ation leads to a decrease in C nuclear lifetimes, which we hypothesizearises from fluctuations introduced in the electronic spin bath upon re-capture after P1 center ionization. This illustrates that the electronic spinspectral density can be optically manipulated, and potentially ultimatelyalso narrowed under sufficiently high-power ionization irradiation. of C sensor spins ( ∼ /cm ), as much as > times thenumber of NV centers, can be harnessed to increase sensitivity. Discussion: – Experimental results in Fig. 3 and Fig. 5 sub-stantiate the C relaxation pathways operational at different fieldregimes, and potentially highlight the particularly important roleplayed by the electronic reservoir towards setting the spin life-times. Our work therefore opens the door to a number of in-triguing future directions. First, it suggests the prospect of in-creasing nuclear lifetimes by raising the NV center conversionefficiency [59]. More generally, it points to the efficacy of ma-terials science approaches towards reducing paramagnetic impu-rities in the lattice. Finally, it opens the possibility of employingcoherent quantum control for dissipation engineering, to manip-ulate the spectral density profile seen by the nuclei and conse-quently lengthen their T . Applying a “ pulse sequence to increase T ” has been a longstanding goal in magnetic resonance [60, 61],but is typically intractable because of inability to coherently con-trol broad-spectrum phonon interactions. Instead here since thenuclear T stems from electronic T e processes, these can be“echoed out”; In particular, the application of electron decoupling(such as WAHUHA [62] or Lee-Goldburg [63] decoupling) on theP1 spin bath would suppress the inter-electron flip-flops, narrowthe noise spectral density, and consequently shift the knee field B (1) K to lower fields. Such T gains just by spin driving at roomtemperature and without the need for cryogenic cooling, and con-sequent boosts in the hyperpolarization enhancements – scalingby the decoupling factor – will have far-reaching implications forthe optical DNP of liquids under ambient conditions.Given the multi-frequency microwave control driving each ofthe N manifolds would entail [64], an attractive alternate all-optical means is via the optical ionization of P1 centers, for in-stance by irradiation at blue ( (cid:46) T . Fig. 6 shows prelim-inary experiments in this direction, where we study the changein the relaxation rate under 465nm blue irradiation. Due to tech-nical limitations (sample heating) we limit ourselves to the lowpower ∼ regime. We observe a comparative decrease in nuclear T with respect to decay in the dark. Note that, in con-trast, we do not observe significant change in the lifetimes under520nm excitation. Under weak blue excitation the P1 centers arenot ionized fast enough, and we hypothesize that upon electronrecapture, the P1 centers can affect the C nuclei over a longerdistance in a lattice. The blue irradiation thus causes a “ stirring ”of the electronic spin bath and an increase in the nuclear relaxationrate. While the experiments in Fig. 6 unambiguously affirm thatinteractions with the electronic bath set the low field nuclear T ,the exact interplay between optical ionization and recapture ratesrequired for T suppression is a subject we will consider in futurework. Conclusions: – Employing hyperpolarized relaxometry, wehave mapped the C nuclear spin lifetimes in a prototypical di-amond quantum system over a wide field range, in natural abun-dance and enriched C samples, and for both single crystals aswell as powders. We observe a dramatic and intriguing field de-pendence, where spin lifetimes fall rapidly below a knee fieldof ∼ C hyperpolarizationefficiency in diamond, with applications to hyperpolarized imag-ing of surface functionalized nanodiamonds and for the DNP ofliquids brought in contact with high surface area diamond parti-cles. Acknowledgments: – We gratefully acknowledge discussionswith A. Redfield, D. Sakellariou and J.P. King, and techni-cal contributions from M. Gierth, T. McNelly, and T. Virtanen.C.A.M. acknowledges support from NSF through NSF-1401632and NSF-1619896, from Research Corporation for Science Ad-vancement through a FRED award, and research infrastructurefrom NSF Centers of Research Excellence in Science and Tech-nology Center for Interface Design and Engineered Assembly ofLow-Dimensional Systems (NSF-HRD-1547830). 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Department ofChemical and Biomolecular Engineering, and Materials Science Division Lawrence Berkeley National Laboratory University of California, Berkeley,California 94720, USA. Department of Physics, and CUNY-Graduate Center, CUNY-City College of New York, New York, NY 10031, USA. Fakultat Physik, Technische Universitat Dortmund, D-44221 Dortmund, Germany. CONTENTS References 8I. EPR Measurements in Fig. 3 10II. Field Cycling 10III. Data Processing 10A. Fit models 10B. Accelerated data collection strategy 12C. Error estimates 12IV. Model For Hyperpolarized Relaxometry 13A. Lattice estimates for electron reservoir 13B. Lattice estimates for C reservoir 14C. Lattice estimates for hyperfine couplings to NV and P1reservoirs 15 I. EPR MEASUREMENTS IN FIG. 3 EPR spectra of the two samples in Fig. 3 were examined witha microwave power 6mW, averaging over 50 sweeps, with mod-ulation amplitudes of 0.1mT and 0.01mT and at sweep fields of3350G - 3500G and 3300G - 3600G for the two samples respec-tively. Concentrations of P1 centers were estimated by using aCuSO reference outlined in Ref. [41].In order to determine the linewidths of the EPR spectra, a scriptwas written to determine the data range at which Tsallis fits shouldbe applied by first finding the indices where the spectral maximaand minima occured. Midpoints were then determined betweenthe maximum and minimum indices and the first derivative of theTsallis function was fit to the ranges between the calculated mid-points. Because the baseline was not perfectly zeroed, jumps inthe fit values occurred between each range. Applying fits to eachindividual peak rather than applying one Tsallis function to mul-tiple peaks produced a better baseline correction since the offsetsdiffered between ranges. Each peak was corrected by subtract-ing the median y-value over the fit range and then making manualcorrections if necessary. Once the corrections were completed,the first integrals over each individual range were obtained usingtrapezoidal integration. The resulting integral arrays were thenconcatenated and a second integral was obtained. The resultingfirst integral allowed us to find the line widths of each P1 peak(FWHMs), and the second integral resembled a step function fromwhich the relative step heights of each P1 peak could be found. Toaccount for the hyperfine splittings of the P1 spectra an averageover all peaks linewidths was taken and weighted by the height ofeach peak. The ratio of the averaged linewidths between the twosamples in Fig. 3 was found to be 2.97, consistent with the ratio of the P1 concentration of the two samples up to the accuracy ofthe concentration estimates. II. FIELD CYCLING T noise spectroscopy relies on our ability to rapidly vary themagnetic field experienced by a test sample using a homemadeshuttling system built over a 7T superconducting magnet [29].Samples are held in an NMR tube (Wilman 8mm OD, 1mm thick-ness) (seeFig. S1D) and pressure-fastened from below the magnetonto a lightweight, carbon fiber shuttling rod (Rock West com-posites). Using a high precision (50 µ m) conveyor actuator stage(Parker HMRB08) (see Fig. S1B), we are able to repeatedly andconsistently shuttle from low fields ( ∼ 30 mT) below the magnetfor polarization to high fields (7T) within the magnet for NMRdetection at sub-second speeds ( < T lifetimes (see Fig. S2) – particularly at fields above 100mT –our resulting NMR signals are recorded with minimal loss in en-hancement. High precision shuttling allowed for the measurementof a full z-direction field map (see Fig. S3) ), where the field wasmeasured as a function of position using an axial Hall probe forfields less than 3.5T. To accommodate the fast shuttling technique,the conventional NMR probe was modified to be hollow, allowingfor shuttling through the probe to low magnetic fields below themagnet. Custom made “printed” coils (see [67]) are employed fordirect inductive detection of the NMR signals [29]. III. DATA PROCESSINGA. Fit models Nuclear T at a given magnetic field is determined by measur-ing the decay of NMR signal ε ( t ) with respect to time t spentdecaying at that field. By measuring the change in signal overvarious times, relaxation decay curves are determined, and T ( B ) estimated. We find that all the data can be fit to a stretched expo-nential of the form (see Fig. S4A), ε ( t ) = ε e − ( tT B ) ) p , (6)where p ∈ (0 , is a stretch factor [48], and (cid:15) represents the baresignal enhancement obtained from DNP and assuming no loss dur-ing shuttling. For certain samples, such as the 10% C sample inFig. 2C, we observe that p ≈ , while for most samples with low C enrichment (including at natural abundance), p ∈ (0 . , .We ascribe this stretch factor to be arising from spin diffusion of1 Figure S1. Field cycling device interfaced with portable diamond hyper-polarizer. (A) Mechanical shuttler is connected to a 7T magnet and in-terfaced to a portable hyperpolarizer. Sliding rails attached to the bottomof the device allow for adjustment of hyperpolarizer box and centering ofsample above coil. (B) The carbon-fiber shuttling rod is moved along aconveyor belt through use of a twin-carriage actuator. (C) The 8mm shut-tling rod is centered in the 38mm magnet bore, with a Teflon guide forself-alignment. (D) Diamond sample is held within an 8mm wide NMRtube, and fitted with a plunger and mirror to prevent excess movement ofsample and bolster efficacy of optical pumping. the inhomogeneous polarization in the lattice that is driven by theDNP process.By measuring the relaxation rate R ( B ) = 1 /T ( B ) over arange of magnetic fields allowed by the field cycler, a relaxationfield map R ( B ) can be obtained, as shown in Fig. 2B. Theserelaxation profiles are then fit to a sum of two Tsallis distributions[36], a generalization of Gaussian and Lorentzian functions thatallows greater flexibility in representing the relaxation rate as afunction of field. Additionally our model assumes a constant off-set to account for the saturation of the relaxation rate at high field,with functional form of a single Tsallian with respect to field B , R ( B ) = C (cid:34) q − − (cid:18) BC (cid:19) (cid:35) − q − + C (7)where fitting parameters C , C , C describe the amplitude, widthand vertical offset of the function respectively, and q regulates the 642 646 650 654 6580100200 Shuttle Time (ms) N u m b e r o f S h u tt l e s ±2.612 ms Figure S2. Sample shuttling repeatability. Shuttler operation (1400runs) between polarization ( ∼ ± Field (T) P o s i t i o n ( mm ) Outside (below)7T Magnet Magnetometer saturation ~3.5T FIDDNP A B 928 mm Figure S3. Field map (A) Measurement of longitudinal (z axis) magneticfield over full field cycler range using a sensitive magnetometer. Datapoints were attained by shuttling magnetic field probe through center ofthe magnetic bore while held within the hollow carbon fiber shuttlingrod, limiting accuracy to the 50 µ m precision of the actuator. Positionof magnet entrance is shown to demonstrate fringe field profile. Due tomagnetometer constraints, high field measurements saturate at 3.5T. (B)Polarization is generated ∼ effective contribution of the function’s tail to the overall area un-der the function, with pointwise limits q = 1 and q = 2 denot-ing Gaussian and Lorentzian functions respectively. Originallythe fitting models were limited to either Lorentzian/Gaussian line-shapes, and the model was susceptible to deviate from the experi-mental relaxation estimates at high field. By allowing variation ofthe parameter q , qualitatively better fits to the relaxation profilescan be found and analyzed in relation to one another.2 100 101 102 103 104 Magnetic Field (mT) / T ( H z ) 102 103 104 Field (mT) S i gna l ( au ) Time (s) S i gna l ( au ) DataFitted Curve A BC DataFitted CurveError RangeError Bounds Figure S4. Data processing. (A) Spin polarization decay curves are ac-quired by repeated hyperpolarization of the diamond sample followed bytime-dependent relaxation at a given field. By varying wait time and mea-suring the resulting NMR signal, relaxation parameters at this field can beestimated by fitting the data to a stretched exponential function. Becausethe relaxation rate equation incorporates a phenomenological stretch fac-tor to account for T1 heterogeneity at different fields, decay experimentsare done at varied fields and the fitted parameters are used for differentfield regimes. (B) enhancement data is also taken at varying fields withwait time kept constant, providing a 1D slice of the relaxation dynam-ics at wide field ranges. To maximize signal contrast the wait times aredynamically adjusted to account for different T1 regimes. (C) Using thetwo previous experiments, a relaxation field map is constructed using theestimated rate equation parameters and the 1D enhancement data. Er-rors result from the quality of the decay curve fits and inaccuracies in themeasured magnetic field. B. Accelerated data collection strategy Due to long relaxation times at high field, occasionally ap-proaching ∼ 20 minutes, production of enhancement decay dataat an array of magnetic fields is time-intensive. In order to hastenmeasurement times, and to obtain a denser map of nuclear T es-timates at a large number ( ∼ ε ( t w ) after somefixed wait time t w (typically 30s) at a certain field is measured (seeFig. S4B). Because the sample decays for the same time at eachfield, this set of enhancement values provides a hint as to the re-laxation mechanisms throughout the full field range. To estimate T from this data, however, requires knowledge of the enhance-ment generated before relaxation begins. To estimate this quan-tity, hereafter referred to as ε , decay curves are experimentallyacquired at certain fields using several averages per experiment,ensuring low error when fitting this curve to a stretched exponen-tial model. Using the fit parameters T and p , ε can then beestimated as ε = ε ( t w ) e (cid:16) twT (cid:17) p (8)This estimate allows us to reconstruct the relaxation rate at eachfield for which enhancement measurements were acquired. Byreordering the relaxation equation, the estimate of R at field B becomes R ( B ) = ln (cid:104) ε ε ( t w ) (cid:105) p t w (9)The quality of this reconstruction is improved by doing multipledecay curve experiments at varying fields so that the appropriatestretch factor p can be determined for different field regimes. Forthe two natural abundance C samples in Fig. 3 we used decaycurve data at fields of 20mT, 35mT, 150mT, and 7T for the relax-ation field reconstructions, with stretch factors p ≈ p ≈ t w for all points, thesensitivity of the decayed enhancement readings is maximized byusing dynamically varied wait times t w at different fields; the lossin enhancement then becomes approximately 50% of the initialpolarization value. This process mitigates errors in the measuredenhancement values by creating sufficient contrast between theinitial and decayed enhancement values, without excessively di-minishing the signal relative to the noise.Let us now quantify the time savings resulting from this datacollection strategy. By removing the need to explicitly plot thesignal decay over time at every magnetic field point, the effec-tive dimensionality of our T ( B ) measurement process is reduced,which allows determination of T at a large number of field pointsrapidly. To develop an intuition for the accelerated in the aver-aging time gained as a result, we assume an even sampling ofthe signal decay, in time increments ∆ t across n steps. To ob-tain estimates of T at N field values, this would require at thevery least a total time t D = N ∆ t (cid:80) ni i = N ∆ t n ( n +1)2 . Whileemploying the accelerated 1D measurement strategy in contrast,signal enhancement is measured after a fixed wait time t w at eachfield. These measurements are obtained at all N field points, af-ter sampling with high accuracy the signal decay curves at N d overlapping fields to construct estimates of the initial enhance-ment and stretch factor at varied fields. The experiment wouldtherefore expend a minimum time of t = N t w + N d ∆ tn ( n +1) / . This measurement strategy incurs a theoretical time gainof t t = N ∆ tn ( n +1)2 Nt w + N d ∆ tn ( n +1) , with the simplifying assumption thatzero time is spent moving between fields as well as during signaldetection. To demonstrate the possible time gains of this method,assume signal decay measurements at ∆ t = 10 s increments fora total of n = 40 points in time, across N = 100 field points.This may then be compared to the accelerated 1D measurementstrategy, with signal enhancement measurements after a fixed hy-perpolarization time of t w = 30s at each field. If N d = 4 decaycurves are used to estimate the relevant relaxation properties atfour separate fields, the time gain of the 1D strategy is t t ≈ . C. Error estimates Let us now outline the error estimation in the T ( B ) data. Theprimary sources of error come from the tightness of the decaycurve fits to estimate ε and p at different fields, the shot-to-shoterror in the measured enhancement, and the error in the wait timespent relaxing at a given field. Because of the high averaging doneto generate relaxation decay curves, the error in ε and p , taken3from the fitting function confidence intervals, is very small ≈ + z direction. This allows us to probefields lower than what is covered by the field cycler. At the polar-ization location and with no current driven through the coil, the7T magnet produces a field of 20.8mT, but fields as low as 1mTand even further can be attained with use of the coil. To accountfor the build-up of magnetic field due to the coil, we attribute anerror of 2s to all points found by this process. In combining bothshuttled and coil-generated field points there was a constant off-set of 15mT added to all shuttled field points to make the curvesconsistent with the low field relaxation rate points. IV. MODEL FOR HYPERPOLARIZED RELAXOMETRY We now provide more details of the model employed to capturethe relaxation mechanisms probed by our experiments. We hadidentified from the experiments three relaxation channels that areoperational at different field regimes, driven respectively by (i) couplings of the C nuclei to pairs (or generally the reservoir)of P1 center, (ii) individual P1 or NV centers, and (iii) due tospin-diffusion effects within the C reservoir. In this section, wedetail lattice calculations that allow the estimation of the spectraldensities in each of these cases.Consider again the three disjoint spin reservoirs in the dia-mond lattice, the electron spin reservoir of NV centers, electronreservoir of substitutional-nitrogen (P1 centers), and the C nu-clear spin reservoir. They are centered respectively at frequencies ω NV ≈ [(∆ ± γ e B relax cos ϑ NV ) + ( γ e B relax sin ϑ NV ) ] / , ω e ≈ [( γ e B relax + m I A P1 (cid:107) cos ϑ P1 ) + ( m I A P1 ⊥ sin ϑ P1 ) ] / and the nu-clear Larmor frequency ω L = γ n B relax ; where ϑ NV , ϑ P1 are anglesof the NV(P1) axes to the field, A P1 (cid:107) ≈ A P1 ⊥ ≈ N nuclear spin, m I = {− , , } is the N manifold, ∆ =2.87GHz is the NV cen-ter zero field splitting, and γ e = 28 MHz/G and γ n = 1 . kHz/Gare the electronic and nuclear gyromagnetic ratios. A. Lattice estimates for electron reservoir In order to determine the relaxation in behavior Eq. (3) quanti-tatively, let us determine typical inter-spin couplings and distancesfor the electron reservoir from lattice concentrations. First, forthe electronic spins, given the relatively low concentrations, andthe fact that the lattice is populated independently and randomly,we make a Poisson approximation following Ref. [33]. An es-timate for the typical inter-spin distance (cid:104) r e (cid:105) is obtained by de-termining the distance at which the probability of finding zeroparticles is . Given the lattice spacing in diamond a =0.35nm,and the fact that there are four atoms per unit cell, we can es-timate the electronic concentration in inverse volume units as, N e = (4 × − P e ) /a [m − ]. Then from the Poisson approx-imation (cid:104) r e (cid:105) = (3 / π ln 2) / N − / e we obtain, for instance, (cid:104) r NV (cid:105) = (cid:104) r P1 (cid:105) = M e = 920 ( gµ B ) (cid:104) r e (cid:105) , (10)where g ≈ is the electron g-factor, and µ B = 9 . × − erg/Gthe Bohr magneton in cgs units. Substituting this leads to, M e = 43 . P e [mG ], and allows us to estimate the electronicline width, ∆ f e = (cid:104) d ee (cid:105) ≈ γ e (cid:113) π √ M e [Hz] ≈ P e [mG],that scales approximately linearly with electron concentration P e .Here we have assumed a Lorentzian lineshape and quantifiedthe linewidth from the first derivative [33]. Typical values are ∆ f NV =29.52kHz and ∆ f P1 =2.95MHz at 1ppm and 100ppm con-centrations respectively.Let us now estimate the effective hyperfine interaction fromthe P1 centers to the C reservoir. Our estimate can be accom-plished by sitting on a P1 spin, and evaluating the mean perpendic-ular hyperfine coupling that contributes to the spin flipping noise, (cid:104) A zx (cid:105) = (cid:2)(cid:10) A zx (cid:11)(cid:3) / , where we setup the second moment sum, (cid:10) A zx (cid:11) = 1 N (cid:104) µ π γ e γ n (cid:126) (cid:105) (cid:88) j (3 sin ϑ j cos ϑ j ) r j (11)where N is the total number of C spins for every P1 center and ϑ j is the angle between the P1- C axis and the magnetic field.Numerically the factor µ π γ e γ n (cid:126) = ]. For sim-plicity, we can approximate the sum by an integral, and includingthe density of C spins N C = 0 . η spins/nm (see Fig. S6B),where η is the C enrichment level, (cid:10) A zx (cid:11) = (cid:16) µ π γ e γ n (cid:126) (cid:17) N C (2 π ) N C V (cid:90) (cid:104) r e (cid:105) r (cid:90) π/ (9 sin ϑ cos ϑ ) r r drdϑ where V = π (cid:104) r e (cid:105) corresponds to the volume of spins consid-ered. We have assumed that the “sphere of influence” of a particu-lar P1 spin notionally extends to the mean distance between neigh-boring P1 centers, for instance (cid:104) r e (cid:105) = P e =10ppm.The integral lower limit is set by the requirement that the hyper-fine shift of the C nuclei is within the detected NMR linewidth ∆ f det ≈ r = [19 . / (∆ f det )] / ≈ r goes to quantify a “barrier” around around each P1 cen-ter, wherein the hyperfine interactions prevent the C nuclei frombeing directly observable in our relaxometry experiments. The an-gle part of the integral evaluates to / , and effectively therefore, (cid:10) A zx (cid:11) = (cid:16) µ π γ e γ n (cid:126) (cid:17) 65 1 (cid:104) r e (cid:105) (cid:32) r − (cid:104) r e (cid:105) (cid:33) (12)For instance, for the two natural abundance single crystal sam-ples that we considered in the Fig. 3 of the main paper with P1concentration 17ppm and 48ppm, we have (cid:104) r e (cid:105) = C hyper-fine interaction (cid:10) A zx (cid:11) ≈ ] and (cid:10) A zx (cid:11) ≈ ]respectively. The simple model predicts that the effective hy-perfine coupling increases slowly with the electron concentration P e , that the electron spectral density width (cid:104) d ee (cid:105) ∝ P e . It also4 Frequency (kHz) N o r m a l i z e d S i g n a l ( a u ) DNPThermal 1% 3% 10% 25% 100%0.510.510 20 Increasing Enrichment 876 Hz 951 Hz 1.5 kHz 2.14 kHz 3.94 kHz Figure S5. Comparison of DNP and thermal C lineshapes. Panels indicate lineshapes under (A) hyperpolarization carried out at low field (1-30mT)and (B) 7T thermal polarization. DNP is excited from the optically polarized NV centers which are ≈ shows that the electron spectral density is independent of C en-richment η to first order. The zero-field relaxation rates stem-ming from this coupled-electron mechanism can now be calcu-lated as R (0) = (cid:10) A zx (cid:11) / (cid:104) d ee (cid:105) ≈ − ] and 317.5[s − ]. Thismatches our expectation for the order of magnitude of the zerofield rate since we expect that the C relaxation time T n matchesthat of the electron T e ≈ (cid:10) A zx (cid:11) = (cid:104) N (cid:80) j ∈ ∆ f det (cid:10) A zx,j (cid:11)(cid:105) within the detection barrier directly fromthe diamond lattice (see Fig. S6F and Sec. IV C). We ob-tain (cid:10) A zx (cid:11) = ] and 2.26[(kHz) ] for the P e =17ppm and48ppm samples respectively, in close and quantitative agreementwith the values predicted from Eq. (12) (considering the approx-imations made in the analysis). Numerics also confirm that thehyperfine values (cid:10) A zx (cid:11) are independent of enrichment η (see Fig.S6F) in agreement with the experimental data. B. Lattice estimates for C reservoir In contrast, since the C reservoir has a much larger spin den-sity, especially at high enrichment levels, we will estimate the in-terspin distances (cid:104) r n (cid:105) and couplings ∆ f n numerically. The exper-imentally obtained C lineshapes and resulting linewidths for allthe samples considered are shown in Fig. S5. We begin by firstsetting up a diamond lattice numerically and populating the Cspins with enrichment level set by η . The numerical calculation istractable since only small lattice sizes typically under (cid:96) =10nm aresufficient to ensure convergence of the various dipolar parameters(see Fig. S6A). To a good approximation, we determine the spindensity of the C nuclei to be N C = 0 . η spins/nm (see Fig.S6B). Next, in order to determine the nuclear dipolar linewidths,we consider the secular dipolar interaction between two nuclearspins j and k in lattice, d CC jk = µ π (cid:126) γ n (3 cos ϑ jk − 1) 1 r jk (13) where ϑ jk = cos − (cid:16) r jk · B pol r jk B pol (cid:17) is the angle between the inter-nuclear vector and the direction of the magnetic field. In the nu-merical simulations we will consider, we evaluate the case of sin-gle crystal samples placed flat, i.e. with B pol (cid:107) [001] crystal axis.As a result, for C spins on adjacent lattice sites, ϑ jk = ◦ isthe magic angle and d CC jk = 0 . We note that Eq. (13) is a good ap-proximation even during the hyperpolarization process. Indeed,although hyperpolarization is performed in the regime where thenuclear Larmor frequency ω L is smaller than the hyperfine inter-action A to the NV center, the hyperfine field is only transientlyon during the microwave sweep. Given the fact that the NV centeris a spin-1 electron, there is no hyperfine field applied to the nucleiwhen the NV is optically pumped to the m s = 0 spin state. Indeedthis constitutes the majority of time period of the DNP process.We now evaluate the effective mean dipolar coupling (cid:104) d CC (cid:105) be-tween the nuclei from the second moment, (cid:104) d CC (cid:105) = 1 N (cid:88) j (cid:34)(cid:88) k (cid:16) µ π (cid:126) γ n (3 cos ϑ jk − (cid:17) r jk (cid:35) / , (14)where N = N C (cid:96) refers to the number of C spins in the lat-tice, and for the convergence, we assign for simplicity, /r jj =0.This simply allows us to sum over all the spins j in the lat-tice. In practice, we evaluate the parameter (cid:104) d CC (cid:105) in Eq. (14)over several ( ≈ 20) realizations of the lattice and take an en-semble average (see Fig. S6C). We report an effective errorbar from the standard deviation of this distribution. The fidelityof the obtained results is evaluated by testing the convergence (cid:15) ( (cid:96) ) = (cid:107) (cid:104) d CC (cid:105) (cid:96) +1 − (cid:104) d CC (cid:105) (cid:96) (cid:107) , where the ( (cid:96) + 1) superscript in-dicates a lattice expanded by 1nm. As is evident in the represen-tative example for η = (cid:15) → ) for (cid:96) ≈ C nuclei.It is instructive to now compare the estimated values with theexperimentally determined nuclear linewidths ∆ f n ( η ) measuredat 7T (see Fig. S5 and blue points in Fig. S6C). The scaling (solidline in Fig. S6C) of the experimental data ∼ η / matches closelywith the estimated result through Eq. (14) (see red line in Fig.5 Lattice range [nm] M ean D i po l a r c oup li ng [ H z ] E rr . D i po l a r c oup li ng [ H z ] Enrichment [%] S p i n den s i t y [/ n m ] Enrichment [%] M ean R M S d i p . c oup l . [ k H z ] Enrichment [%] M ean NN d i s t an c e [ n m ] E ff e c t i v e NN d i s t an c e [ n m ] Enrichment [%] D i ff u s i on c on s t. [ n m / s ] D i ff u s i on l eng t h [ n m ] Enrichment [%] M ean R M S h y pe r f i ne [ k H z ] Enrichment [%] M ean R M S H y pe r f i ne [ k H z ] Enrichment [%] N o . o f s p i n s w i t h h y p > k H z A B C DE F G H pp m e - c on c . pp m e - c on c . Evaluated Figure S6. Calculated interspin parameters pertaining to C and NV reservoirs as a function of lattice enrichment η . (A) Convergence ofnumerical estimates is representatively illustrated by plotting the mean C dipolar coupling (cid:104) d CC (cid:105) (cid:96) and the residual (cid:15) ( (cid:96) ) as a function of consideredlattice size (cid:96) . We evaluated here the case of a 1% enriched diamond single crystal. We observe good convergence beyond a lattice size of about 10nm. (B) Spin density of C nuclei shows, as expected, very close to linear dependence with η . Solid line is a linear fit, whose slope returns thelattice spin density ≈ . η spins/nm . (C) Effective inter-nuclear dipolar coupling (cid:104) d CC (cid:105) evaluated from second moment (red line). Blue points showthe experimentally obtained linewidths. Green line indicates ζ (cid:104) d CC (cid:105) with broadening factor ζ = 2 . , and shows a good numerical agreement withexperimental data. (D) Mean inter-spin distance (cid:104) r n (cid:105) between lattice C nuclei in evaluated from the RMS dipolar coupling (red points) and fromeffective nearest-neighbor lattice distances (blue points). The two estimates show a good match, with the inter-spin distance falling approximately as η / . (E) Diffusion constant and diffusion length numerically estimated with lattice enrichment. Here we employed experimentally obtained valuesof C T1. Dashed line indicates the mean inter-electron distance between NV centers at 1ppm concentration, indicating that spin diffusion canhomogeneously spread polarization in the lattice almost independent of C enrichment. (F) Effective hyperfine coupling (cid:10) A obs zx (cid:11) to P1 centers in caseof single crystal samples with 17ppm (red points) and 48ppm (blue points) electron concentration. Results indicate that (cid:10) A obs zx (cid:11) is independent of Cenrichment η . (G) Estimates of mean RMS NV- C hyperfine interaction (cid:104) A NV (cid:105) with lattice enrichment. (H) Estimation of directly participating Cnuclei in the DNP process, defined as those nuclei for which the hyperfine coupling to the closest NV center is greater than 200 kHz. We obtainan approximately linear increase with enrichment. Error bars in all panels are numerically estimated from standard deviation of lattice parameterdistributions over several realizations of the lattice configuration. S6C). However we find that the numerical value overestimates thelinewidth by an additional broadening factor ζ ≈ . . The greenpoints show a close match between experimental values and nu-merically evaluated ζ (cid:104) d CC (cid:105) .This effective coupling now allows us to estimate the meaninter-spin distance (cid:104) r n (cid:105) as a function of C enrichment (see Fig.S6D), (cid:104) r n (cid:105) = (cid:20) (cid:104) d CC (cid:105) µ π γ n (cid:126) (cid:21) − / (15)We find a scaling ∼ η − / (red line in Fig. S6D). It is also in-teresting to compare these values to those alternatively evaluateddirectly from the lattice (blue points in Fig. S6C). For this, werely on the fact that the (cid:104) r n (cid:105) distances largely reflect the nearest-neighbor (NN) spin distances. We define the NN spin (say k ) tothe spin j as the one which has the dipolar coupling d jk is maxi-mal. Now for every spin j in the lattice, we determine the nearestneighbor inter-spin distance R j = (cid:12)(cid:12)(cid:12) r NN jk (cid:12)(cid:12)(cid:12) , and construct a row ma-trix, R = { R j } , with j th element R j . Finally, repeating and con-tacentating this row matrix for several realizations of the lattice,we finally estimate (cid:104) r n (cid:105) = (cid:104) R (cid:105) for the i th realization of the lat-tice. The comparison between these two metrics is demonstratedin Fig. S6D), and show reasonably good agreement. These inter-spin distances and the coupling values allow us toestimate the spin diffusion coefficient D ( η ) as a function of latticeenrichment (see Fig. S6E). This quantifies the spread of polariza-tion away from directly polarized C nuclei, and also serves as ameans to quantify the homogenization of polarization in the lat-tice. Following Ref. [49], we heuristically assign a spin diffusioncoefficient D = (cid:104) r n (cid:105) T n where the T n are evaluated here by onlytaking the dipolar contribution to the linewidth, T n ≈ / (cid:104) d CC (cid:105) .Given a total time bounded by T , we can calculate the rms overalldiffusion length [50] as σ = √ DT that is displayed as the bluepoints in Fig. S6D. Also for reference is plotted the mean NV-NVdistance ≈ C. Lattice estimates for hyperfine couplings to NV and P1reservoirs Let us finally evaluate, through similar numerical means, detailsof the hyperfine interaction between C reservoir and the electronreservoirs of the P1 centers and NV centers. We draw a distinc-tion between the NV and P1 centers in the fact that the former arespin-1, with a nonmagnetic m s = 0 state (with no hyperfine cou-6pling to first order), while the latter are spin / . When hyperfineshifts exceed the observed 7T NMR linewidth ∆ f det ∼ / P1 centers.In order to perform the estimation, in the generated lattice ofsize (cid:96) = (cid:104) r e (cid:105) , we populate C spins with enrichment η , and in-clude an electron at the lattice origin. The mean perpendicularhyperfine interaction between P1- C spins is calculated from thesecond moment, from the individual hyperfine couplings A zx,j that are smaller than the detection barrier ∆ f det (cid:10) A obs zx (cid:11) = (cid:88) j ∈ obs (cid:10) A zx,j (cid:11) / = N obs (cid:88) j ∈ obs (cid:16) µ π γ e γ n (cid:126) (cid:17) (3 sin ϑ j cos ϑ j ) r j / where N obs refers to the number of spins amongst the total N = N C (cid:96) spins for which (cid:10) A zx,j (cid:11) < (∆ f det ) . Here r j is the dis-tance of the j th C nucleus, and ϑ j the angle of P1- C axis tothe magnetic field, and we have ignored the effect of N hyper-fine interactions intrinsic to the P1 center. This effective hyper-fine field, scaling with lattice enrichment η , is then indicated bythe red (blue) points in Fig. S6F for electron concentrations of17ppm (48ppm) respectively. The error bars indicating the stan-dard deviation of the obtained distributions upon several hundredrealizations of the lattice. We observe that the effective hyperfineinteraction (cid:10) A obs zx (cid:11) is almost independent of η , and is higher forlattices with higher P e electron concentration. This is consistentwith the results obtained through Eq. (12) and matches our experi-mental observations in Fig. 5 of the main paper. For natural abun-dance samples we numerically obtain (cid:10) A obs zx (cid:11) = full hyper-fine interaction to C spins of varying enrichment, consideringno operational detection barrier. (cid:104) A NV (cid:105) = (cid:88) j (cid:10) A j, NV (cid:11) / = N (cid:88) j (cid:16) µ π γ e γ n (cid:126) (cid:17) (cid:2) (3 r jz − + (3 r jx r jz ) + (3 r jy r jz ) (cid:3) r j / where we employed a lattice size (cid:96) = (cid:104) r NV (cid:105) =12nm, and N = N C (cid:96) refers to the number of C spins in the lattice with index j running over all them. Here the angle part of the hyperfine inter-action is evaluated by assigning the direction cosines, for instanceas, r jz = ( (cid:126)r j · ˆ z NV ) /r j , where ˆ z NV is the unit vector aligned alongthe N-V axis, collinear with the direction of the strong zero fieldsplitting that forms the dominant part of the Hamiltonian at lowfields. This effective hyperfine field, scaling with lattice enrich-ment η , is then indicated by the blue points in Fig. S6G. Our DNPmechanism is a low-field one and is primarily effective when thefull hyperfine coupling (cid:104) A j, NV (cid:105) is of the order of greater than thenuclear Larmor frequency ω L = γ n B pol , where B pol is the polar-izing field. We can heuristically measure the number of directlypolarized spins surrounding an NV center as being those for which (cid:104) A j, NV (cid:105) > Cenrichment, with a constant ratio ≈ . ηη