PHYSICAL REVIEW B 99, 205423 (2019)

Enhancement of nuclear spin coherence times by driving dynamic nuclearpolarization at defect centers in solids

Girish Sharma,1 Torsten Gaebel,2 Ewa Rej,2 David J. Reilly,2 Sophia E. Economou,1 and Edwin Barnes1

1Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA2ARC Centre of Excellence for Engineered Quantum Systems, School of Physics, University of Sydney,

Sydney, New South Wales 2006, Australia

(Received 23 October 2018; revised manuscript received 3 May 2019; published 17 May 2019)

The hyperpolarization of nuclear spins can enable powerful imaging and sensing techniques provided thehyperpolarization is sufficiently long lived. Recent experiments on nanodiamond 13C nuclear spins demonstratethat relaxation times can be extended by three orders of magnitude by building up dynamic nuclear polarization(DNP) through the driving of electron-nuclear flip-flop processes at defect centers. This finding raises thequestion of whether the nuclear spin coherence times are also impacted by this hyperpolarization process.Here, we theoretically examine the effect of DNP on the nuclear spin coherence times as a function of thehyperpolarization drive time. We do this by developing a microscopic theory of DNP in a nuclear spin ensemblecoupled to microwave-driven defect centers in solids and subject to spin diffusion mediated by internucleardipolar interactions. We find that, similarly to relaxation times, the nuclear spin coherence times can be increasedsubstantially by a few orders of magnitude depending on the driving time. Our theoretical model and results willbe useful for current and future experiments on enhancing nuclear spin coherence times via DNP.

DOI: 10.1103/PhysRevB.99.205423

I. INTRODUCTION

Quantum coherence in many-body systems can give riseto a variety of intriguing phenomena such as interference andentanglement, which do not have any classical analog. Thisproperty lies at the heart of quantum mechanics and is centralto all quantum-based technologies [1–9]. However in reality,a quantum system is never perfectly isolated but is in contactwith its surroundings, to which it loses quantum informationvia a process called quantum decoherence [10–14]. The lossof quantum coherence for nuclear spins remains an importantproblem in diverse fields such as high resolution magneticresonance imaging and quantum computation. While on onehand decoherence can lead to broadening of the NMR peaklinewidth, it also can lead to errors in the processing ofquantum information in systems where nuclear spin statesserve as qubits or quantum memories [15–25]. Decoherencein this case results from the interaction of the nuclear spinwith various environmental degrees of freedom, with the mostprominent channel being the interaction of the nuclear spinwith other neighboring entities such as another nucleus viathe dipole-dipole interaction or electron spin centers via thehyperfine interaction [15,24–26]. Nuclear spin flips resultingfrom these interactions can induce the loss of informationstored in spins and lead to decoherence. Electron-nuclearhyperfine interactions also play an important role in the de-coherence of electron spins [17,27–36]. Therefore it is of fun-damental and technological interest to control and manipulatethe underlying nuclear spin degrees of freedom in order toextend the coherence time of nuclear spins and consequentlyof electron spins.

In addition to being a source of decoherence, the electron-nuclear hyperfine interaction can also be used to manipulate

the state of the nuclear spin ensemble and in particular tocreate dynamic nuclear polarization (DNP) [37]. DNP refersto the generation of a significant nuclear spin polarizationthrough dynamic processes such as driving an electronicsystem with an external field and subsequent spin polariza-tion transfer from electrons to the nuclei. This techniquewas initially suggested by A. Overhauser [38] in 1953. Heproposed that the large Boltzmann polarization of unpairedelectrons could be transferred to neighboring nuclei by satu-rating the corresponding EPR transition, which could result inan enhanced NMR signal by almost two orders of magnitude[38–40]. DNP is currently a powerful method that is appli-cable to a wide variety of physical systems and applications[41–44]. In solids DNP can be achieved by driving the systemwith optical fields or with microwave fields oscillating closeto the electron Larmor frequency, and as a result the largespin polarization of the electrons is transferred to nuclei viaa process known as the solid effect [45–57]. The polarizationcreated in the nuclear spin ensemble via driven DNP processeswill gradually relax and decohere due to internuclear dipolarinteractions and other environmental factors.

Let us consider a simple two level system which consistsof a single spin in the presence of an external magnetic fieldB = (Bx

⊥(t ), By⊥(t ), B‖) in a perfectly isolated environment.

(Bx⊥(t ), By

⊥(t )) = B⊥(cos(νt ), sin(νt )) is a rotating field in thex-y plane with frequency ν, while B‖ is held constant. Thespin Hamiltonian is given by H = −μ · B, where μ = −gμnσ

is the magnetic moment, and σ is the vector of Pauli spinmatrices. Explicitly, the Hamiltonian can be written as

H =(

ω/2 �e−iνt

�e+iνt −ω/2

), (1)

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SHARMA, GAEBEL, REJ, REILLY, ECONOMOU, AND BARNES PHYSICAL REVIEW B 99, 205423 (2019)

where ω = gμnB‖, and � = gμnB⊥/2. The spin dynamicsunder this Hamiltonian is governed by the von Neumannequation for the density matrix ρ̇ = −i[H, ρ]. In the frame co-rotating with the magnetic field, the equations of motion forthe components of the density matrix can be written as iρ̇↑↑ =�(ρ∗

↑↓ − ρ↑↓) and iρ̇↑↓ = δρ↑↓ + �(ρ↓↓ − ρ↑↑) where δ =ω − ν is the detuning frequency of the applied transverserotating field, � is the corresponding Rabi frequency, and thearrow represents the direction of the spin. The time evolutionof this spin system is perfectly unitary since we have assumedthe spin is perfectly isolated from its environment. Conse-quently in this ideal situation, the coherence time is infinite asthe phase information is always maintained. However as wediscussed above, coupling to various environmental degreesof freedom needs to be accounted for in the Hamiltonian,which leads to loss of spin coherence over time. Due tothe complicated and many-particle nature of this problem,which involves several microscopic environmental parame-ters, a simpler phenomenological approach to this problemis to add decay terms in the equations, i.e., iρ̇↑↑ = �(ρ∗

↑↓ −ρ↑↓) − iρ↑↑/T1 and iρ̇↑↓ = δρ↑↓ + �(ρ↓↓ − ρ↑↑) − iρ↑↓/T2.This phenomenological approach successfully describes avariety of experiments [58,59]. Although the Bloch equa-tions do not work well when the environment cannotbe described by classical white noise as in the presentwork, the parameters T1 and T2 still characterize lon-gitudinal and transverse relaxation, respectively, in thetwo level system and can be extracted from experimentalmeasurements.

It is the transverse dephasing time T2 which characterizeshow long a quantum state remains coherent. In the fields ofmagnetic resonance imaging and quantum computation, pulsesequence techniques have been developed as a method toreduce spin dephasing and thereby increase the T2 coherencetime [60–67]. Some well known examples of these includeHahn’s spin echo (SE) (single π pulse), the Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence (multiple π pulses),and periodic dynamical decoupling (PDD). These controltechniques extend coherence times by effectively decouplingthe system from its environment [60–67].

In semiconductors, it has been shown that electron spindephasing times (T ∗

2e) can be extended by programming thenuclear spin ensemble as this effectively reduces the num-ber of degrees of freedom and, consequently, the variancein effective fields sampled by an ensemble of experiments[43,68,69]. It has also been shown that polarizing electronspins in ensembles can improve electron spin coherence times(T2e) [70]. DNP is also expected to enhance electron spincoherence times (T2e) [71], although this has yet to be demon-strated in experiment. Recent experiments have demonstratedan increase in the relaxation time of nanodiamond 13C spinsby dynamically polarizing the nuclear bath via microwaveassisted DNP [56]. The nuclear spin relaxation time has beenobserved to increase with the hyperpolarization time by threeorders of magnitude. This raises the question of whether thenuclear spin coherence time is similarly impacted by thishyperpolarization process. Preliminary experimental resultson nanodiamond 13C spins suggest that the hyperpolarizationprocess can in fact enhance the nuclear spin coherence time[72,73].

In this work we theoretically examine the effect of DNP onnuclear spin coherence and calculate T2 as a function of thedriving time. We start with a central spin model of an electronlocated at a paramagnetic site interacting with the surroundingnuclear bath via the hyperfine interaction. Driving this systemat a microwave frequency close to the ESR frequency then in-duces a large DNP in the surrounding nuclear spin bath, whichtypically spreads out to a few nanometers around the electronsite. Using Liouville’s equation, we then calculate the spatialdistribution of the nuclear polarization around the electron siteas a function of the driving time. The DNP induced effectivemagnetic field produced by polarization of the nuclear spinsin the crystal however also undergoes its own dynamics due tothe internuclear dipole-dipole coupling [74,75]. These dynam-ics lead to local fluctuations in the nuclear spin field. Sincethe exact quantum mechanical treatment of this mechanism(which is a many-particle dipolar interaction) is prohibitivelycomplicated, in this work we treat the dynamical evolution ofnuclear spins caused by such an interaction using a stochasticdiffusion model for an effective nuclear spin field [76,77],which remains a valid approximation for the time scalesconsidered in this work. The use of stochastic models todescribe diffusion in dipolar-coupled spin lattice systems hasa long history dating back to the seminal works of Ander-son and Klauder [78,79]. Here, we extend these approachesto include the diffusion of inhomogeneous DNP originatingfrom a driven electronic spin. Using an effective Gaussianmodel for the DNP solution, we then analytically calculatethe nuclear spin-spin correlation function, which is crucialfor the evaluation of the T2 time. Finally, we calculate theT2 coherence time and study its dependence under dynamicaldecoupling pulse sequences (like SE, CMPG) as a functionof the driving time. We find that the nuclear T2 coherencetime substantially increases by a few orders of magnitudedepending on the DNP drive time. We also apply our results tocalculate the T2 time of 13C nuclear spins for experimentallyrelevant parameters. Our theoretical model and results will beuseful for current and upcoming experiments on enhancingthe coherence time of nuclear spins via DNP.

This paper is organized as follows: In Sec. II we discussdynamic nuclear polarization in solids and obtain an exactnumerical solution for the spatial distribution of nuclear po-larization for a microwave driven DNP process enabled byelectron-nuclear hyperfine interaction. In Sec. III we discussthe problem of diffusion of the spin polarization into the bulkof the sample within the framework of a stochastic model andalso calculate the two point correlation function. In Sec. IVwe evaluate the nuclear spin coherence time as a function ofthe polarization time. We conclude in Sec. V.

II. DYNAMIC NUCLEAR POLARIZATION

Let us begin by first considering the case of a singleunpaired electron (at a paramagnetic site) in a solid locatedat the origin which is coupled to the surrounding nuclear bathin the presence of a magnetic field. The total Hamiltonian forthis system is given by

H = He + Hn + Hen + Hnn, (2)

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FIG. 1. (a) A schematic representation of polarization transfervia DNP. The electron center constitutes an ESR line at a frequencyω0s. The nuclear spins in a radius of r < a around the electroncenter are strongly polarized while the nuclei within a < r < bparticipate in the DNP process. The spin polarization is transferredfurther away to b < r < c via nuclear spin diffusion mediated bydipole-dipole interactions between the nuclei. In red is shown apossible nuclear process where there is a spin flip-flop between pairsof nuclei, leading to local fluctuations in the nuclear field. (b) Thecorresponding energy level diagram. The ESR line is at frequencyω0s and the NMR line is at frequency ω0I . Driving the system at amicrowave frequency ωm = ω0s − ω0I results in a spin polarizationtransfer between electron and nuclei (DNP).

where He = ω0sSz and Hn = −∑i ω0I I i

z are the Zeeman en-ergies of the electron and the nuclei, respectively, in the pres-ence of an external magnetic field. The summation i is overall the nuclei, which are assumed to be of the same species forsimplicity. Hen is the hyperfine interaction between electronand nuclei, and Hnn is the dipole-dipole interaction betweenthe nuclei. The hyperfine coupling between the electron andnuclei comprises an isotropic contact hyperfine interaction∼aisoS · I (which requires a nonzero overlap between the elec-tron and nuclear wave functions and is therefore localized nearthe electron site), and the anisotropic interaction given by thedipolar coupling between the electron and nuclear magneticmoments. For our purposes we will only consider the dipolarcoupling, retaining parts which are significant under the highmagnetic field approximation [58]:

Hen =∑

i

Ai1SzI

iz + Ai

2SzIix, (3)

where Ai1 and Ai

2 are the secular and nonsecular parts of theinteraction, respectively [58],

Ai1 = T (3 cos2 θi − 1), (4)

Ai2 = 3T sin θi cos θi, (5)

where T = (μ0/4πr3)γeγN h̄2 is the bare hyperfine strength,r is the distance between the electron center and nuclearspin, θi is the polar angle, and γe and γN are the electronand nuclear gyromagnetic ratios. As depicted in Fig. 1(a),the nuclear spins surrounding the electronic center can bedivided into three regions according to their role in the DNPprocess. The nuclei very close to the electronic spin (i.e., thosewithin radius r < a, where a is typically 0.1 nm to 0.3 nm[80]) undergo rapid electron-nuclear spin flip-flops [second

term in Eq. (3)] and become polarized. However, they alsoexperience a strong Knight shift [first term in Eq. (3)] thatchanges their effective Zeeman energies. This prevents thesenuclei from participating in nuclear spin diffusion because thediffusion process is driven by energy-conserving nuclear spindipolar flip-flop processes originating from the internucleardipole-dipole interaction Hnn. This is known as the frozen-core or diffusion-barrier effect [81–83]. The nuclear dipolarflip-flops are suppressed by the energy mismatch betweenthe nuclei close to the defect center and those in the bulk.Thus, it is the nuclei which are further away from the center(a < r < b) but still hyperfine coupled to the electronic spinthat are responsible for the diffusion process. The distance b,which depends on the transition probabilities of the hyper-polarization mechanisms and relaxation, can typically extendup to a few nanometers [82,84]. The nuclei in this range arestill close enough to the center to undergo electron-nuclearhyperfine flip-flops and become polarized, but they are farenough away that their Knight shifts are small and do notprevent dipolar flip-flops with nuclei in the bulk. In Fig. 1,we represent the bulk nuclei as those lying in the radial spanb < r < c.

In this section, we focus on quantitatively understandinghow the nuclei in the intermediate region a < r < b becomepolarized. We are particularly interested in the case where thisDNP process is accelerated by external microwave driving.Note that the rate and amount of DNP will vary across theintermediate region since the hyperfine interaction Hen scaleslike r−3 and has an angular dependence. Here, we determinethe distribution of nuclear spin polarization across this regionas a function of the driving time. In the next section, we thenstudy how this nuclear polarization gets transferred to the bulkvia spin diffusion.

We first consider the case of an electronic center coupledto a single nuclear spin. Diagonalizing the Hamiltonian H0 =He + Hn + Hen

H̃0 = UH0U† = ω0sSz − ω̃0I Iz + A′SzIz, (6)

where

ω̃0I = ω0I

2(cos η− + cos η+) − A1

4(cos η− − cos η+)

− A2

4(sin η− − sin η+), (7)

A′ = −ω0I (cos η− − cos η+) + A1

2(cos η− + cos η+)

+ A2

2(sin η− + sin η+), (8)

tan η∓ = A2

A1 ∓ 2ω0I, (9)

and U = exp[i(η−(I/2 + Sz )Iy + η+(I/2 − Sz )Iy)] is the uni-tary transformation. To study DNP in this system, we nowintroduce the microwave field HM = BeSx cos(ωmt ), whichcan drive the nuclear spin flip transitions for particular valuesof the drive frequency ωm. Here, we neglect the direct effect ofthis oscillating field on the nuclear spins because their Zeeman

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splitting is on the order of MHz and thus far detuned from thedrive. However, the microwave drive can still indirectly causenuclear spin flips due to hybridization of the electronic andnuclear spin states under the hyperfine flip-flop interaction.These hybridized states are the eigenstates of H0. Althoughthese states are hybridized, we can still label them in terms ofseparate spin quantum numbers for the electron and nucleussince the hybridization is weak owing to the smallness ofthe hyperfine coupling constant. Thus we can denote themas |sesn〉, where se and sn each assume the values ↑, ↓corresponding to the spin. We can see how these states areaffected by the drive by transforming HM into the eigenstatebasis [85]:

H̃M = UHMU †

= Be cos(ωmt )(Sx cos δ − 1

2 (S+I− + S−I+) sin δ

+ 12 (S+I+ + S−I−) sin δ

), (10)

where 2δ = η− − η+. The S±I± terms in the above Hamilto-nian are the spin-flip terms that couple different eigenstates ofH0 and enable the generation of DNP. In particular, they drivetransitions between the states |↑↑〉 and |↓↓〉 and between |↑↓〉and |↓↑〉. When A2 = 0, these spin flip terms in H̃M vanish,indicating the absence of any microwave driven nuclear po-larization. Note that from Eq. (5), A2 is generically nonzeroexcept when θ = nπ/2, where n = {0, 1, 2, 3}. When A2 �= 0,either the S+I− + S−I+ or the S+I+ + S−I− terms facilitatethe transfer of nuclear polarization. In the interaction frameof H̃0, one can derive an effective Hamiltonian Heff when themicrowave frequency is tuned to drive either of the nuclearspin flip transitions. In the remainder of the paper, we focuson the latter choice, and the corresponding energy levels anddrive frequency are depicted in Fig. 2(b). Specifically, whenωm = ω0s − ω0I the double spin-flip term S+I+ + S−I− in H̃M

is selected and Heff becomes

Heff = Be

4sin δ(S+I+ + S−I−). (11)

In the interaction frame of reference, microwave driven nu-clear dynamics can then be described by the Liouville-vonNeumann equation ρ̇ = −i[Heff, ρ] + L[ρ], where we cansolve for the evolution of the density matrix ρ correspond-ing to the four-component electron-nuclear spin system. TheLindblad operator L[ρ] accounts for relaxation processes (likedecay of the electron spin to the ground state), which gives anonunitary evolution of the quantum system and consequentlya steady state solution for ρ(t ) when t → ∞. SpecificallyL[ρ], which is the relaxation superoperator term, can beexpressed as

L[ρ] =∑

k

(LkρL†

k − 12 [L†

k Lkρ + ρL†k Lk]

), (12)

where Lk are the Lindblad operators. The index k runs from 1to 4, with the nontrivial elements of the Lindblad operators be-ing 〈↑↑|L1|↓↑〉 = √

γ2, 〈↑↓|L2|↓↓〉 = √γ2, 〈↑↑|L3|↓↓〉 =√

γ1, 〈↑↓|L4|↓↑〉 = √γ1, where the first (second) arrow in-

dicates the direction of the electron (nuclear) spin. L1 and L2

describe processes conserving nuclear spin, while L3 and L4

describe (much slower) processes involving flipping of the nu-clear spin as well. The Liouville-von Neumann equation can

FIG. 2. Spatial profile 〈Pz(r, θ, t )〉 of the microwave driven nu-clear polarization around the electron center at the origin as ob-tained from the solution of the Liouville-von Neumann equation atvarious times. (a) At time t = 0.1 s. (b) At time t = 1 s. We usethe parameters ω0s = 10 μeV, Be = 0.1 μeV, ω0I = 0.0004ω0s. Therelaxation parameters for the Lindblad operator used were γ1 = 1 μsand γ2 = 1 s. Note that this so far does not account for nucleardiffusion (which is the subject of Sec. III) but rather provides aninitial condition to that problem.

be solved analytically as ρ̃(t ) = S (t )ρ̃(0), where ρ̃(t ) is thedensity matrix ρ(t ) written as a single column vector, and thematrix S (t ) is S (t ) = e(H+G )t , where H = i(Heff ⊗ I − I ⊗Heff ), G = ∑

m [Lm ⊗ Lm − 12 I ⊗ L†

mLm − 12 L†

mLm ⊗ I]. Thusfor an arbitrary initial condition ρ(0), the density matrix ρ(t )at a later time can be evaluated exactly.

The spatial profile (which arises from the spatial depen-dence of the hyperfine constants A1 and A2) of the inducednuclear polarization around the electron center can be stud-ied as a function of the microwave driving time by evalu-ating the mean value of nuclear polarization 〈Pz(t, r, θ )〉 =Tr(Pzρ(t, r, θ )) where the operator Pz is the nuclear spinoperator Iz written in the interaction representation. Explicitly,Pz = eiH̃0tU IzU †e−iH̃0t . In Fig. 2 we plot the spatial profile ofthe microwave driven nuclear polarization around the elec-tron center at the origin as obtained from the solution ofthe Liouville-von Neumann equation at two different times.Clearly the spread of DNP with increasing drive times isevident. It is evident from the figure that the buildup of DNPis not isotropic. The origin of this pattern can be tracedback to the anisotropy of the nonsecular part (A2) of thehyperfine interaction [see Eq. (5)]. We noted earlier that A2

is responsible for DNP, and its θ dependence precisely gives

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0 100 200 300 400 500t (ms)

0

0.2

0.4

0.6

0.8

1 P

z(t,

,r)

d = 6 nm, etc.

d = 10 nm

d = 8 nm

FIG. 3. Time evolution of DNP around the electron center at theorigin as obtained from the solution of the Liouville-von Neumannequation at various distances from the origin. We use the parame-ters ω0s = 10 μeV, Be = 0.1 μeV, ω0I = 0.0004ω0s, θ = π/4. Therelaxation parameters for the Lindblad operator used were γ2 = 1 μsand γ1 = 1 s. Also note that this solution so far does not account fornuclear spectral diffusion (which is the subject of Sec. III) but ratherserves as its initial condition.

rise to the nonuniform pattern in Fig. 2. Even though thespatial profile around the electron center is not isotropic, theschematic representation presented in Fig. 1 provides us witha good approximation to the problem. This DNP solution willbe used as an initial condition for the problem of nuclear spindiffusion (which is the subject of Sec. III). Figure 3 shows thetime evolution of DNP at various distances from the origin asobtained from the solution of the Liouville equation.

III. NUCLEAR SPIN DIFFUSION

We have so far focused on the problem of DNP in solids,which produces a large spin polarization of the nuclei aroundthe paramagnetic electron center. In this section we willdiscuss the problem of diffusion of the spin polarization intothe bulk of the sample. The large nuclear Overhauser fieldproduced via DNP in the vicinity of this electron defect sitedynamically evolves through mutual dipole-dipole couplingof the nuclei. This also causes temporal fluctuations of thenuclear spin field due to processes such as a pair of spinflips. Since the exact quantum mechanical treatment of thismany-particle dipolar interaction is quite complicated, we willdescribe the dynamical evolution of the nuclear spin field withthe following stochastic diffusion model [76]

∂I (x, t )

∂t= D

∂2

∂x2I (x, t ) + ζ (x, t ). (13)

In the above equation I (x, t ) is the nuclear field, which ona coarse-grained scale encompasses several atomic sites, Dis the isotropic diffusion constant, and ζ (x, t ) is an effectivestochastic field which models the randomness associated withthe nuclear spin flips. The validity and the derivation ofthe above model has been a subject of many earlier works[74,77,86], but for the sake of completeness we will brieflysketch the derivation of the above diffusion model.

The Hamiltonian for the mutual dipole-dipole couplingbetween the nuclei can be written as a sum of the Ising and

the flip-flop terms:

Hnn = −∑i �= j

(Bi jI

i+I j

− − 2Bi jIizI j

z

), (14)

where Bi j is the coupling between two nuclear spins I i andI j , and the summation is over all pairs of nuclear spins. Weare interested in the evolution of the expectation value for thez component of the nuclear spin at site k, which will evolveaccording to [74]

∂

∂t

⟨Ikz

⟩ = − 1

ih̄Tr

{ρ(t )

[H ′

nn, Ikz

]}, (15)

where H ′nn = −∑

i �= j Bi j I i+I j

− is the nuclear flip-flop interac-tion, and ρ(t ) is the nuclear spin density matrix. This can berewritten as [74,77,86]

∂

∂t

⟨Ikz

⟩ =∑i �=k

Wik(⟨

I iz (t )

⟩ − ⟨Ikz (t )

⟩), (16)

where

Wik =∫ t

0Tr

{[H ′

nn(t ), Ikz

][H ′

nn(t − t ′), I iz

]}dt ′ (17)

has a physical meaning as the flip-flop rate between nuclearspins at sites i and k. The parameter Wik can be approximatelyanalytically calculated when the upper integration limit istaken to be large [86]. We can then Taylor expand 〈Iz(t )〉 forsite i around site k:

⟨I iz (t )

⟩ ≈ ⟨Ikz (t )

⟩ + ∂⟨Ikz (t )

⟩∂xα

(xα

k − xαi

)+ 1

2

∂2⟨Ikz (t )

⟩∂xα∂xβ

(xα

k − xαi

)(xβ

k − xβi

) + · · · , (18)

where Einstein’s summation convention is implied for thespatial indices α and β. Substituting the above equation inEq. (16), and identifying D = Dαα = ∑

i Wik (xαk − xα

i )2/2,one can obtain the first term on the right-hand side of Eq. (13).Note that the linear in x term vanishes. The connection be-tween the diffusion constant D and the dipole-dipole interac-tion is now transparent. Note that in Eq. (13), Iz(t ) is simplywritten as I (t ), for ease of notation.

Our main focus is on natural diamond, which has a 1.1%concentration of nuclear spins, although our analysis can beapplied to the case of enriched diamond as well. A theo-retical estimate for the diffusion constant was provided inRefs. [84,87,88]. We summarize the argument here. We startby assuming that the 13C spins are distributed in a randomway in the lattice. A typical distance between two spinsmay be obtained using the Poisson distribution by puttingthe probability, exp(−(4πnCr3/3)), of finding no other spinswithin a distance r of a spin placed at the origin, equal to1/2 [84]. Here nC is the concentration of 13C. This givesr = 0.55n−1/3

C , which for natural diamond yields r = 4.42 Å.The 13C spins form a dilute magnetic system for which itis possible to calculate the second moment of the frequencydistribution using the Van Vleck expression [84,87,88]

M2 = 3

4γ 4

N h̄2I (I + 1) f∑

k

(1 − 3 cos2 θ jk )2

r6jk

, (19)

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where γN is the 13C gyromagnetic ratio, I = 1/2, and fis the fraction of lattice sites occupied by magnetic spins(which is 0.011 for natural diamond), and θ jk is the angle theinternuclear vector r jk makes with the magnetic field. For thediamond lattice, this yields M2 ∼ 1.43 × 106 s−2 (performingthe summation for a single crystal over all neighbors andwith magnetic field parallel to the [014] direction [84]). Thediffusion constant is given by D = 1

30

√M2r2, giving D ≈

7.8 nm2/s as a theoretical estimate. The diffusion constant canbe obtained experimentally by substituting M2 = π/(2T 2

2 ) inthe above formula for the diffusion constant to obtain D =1

30

√π2 r2/T2, where r is again the internuclear separation, and

T2 is the measured coherence time (see, e.g., Ref. [74]). Usingthe above estimate for r and taking the coherence time tobe ≈ 1 ms [72,73] yields D ≈ 8.1 nm2/s. The experimentaland theoretical estimates agree reasonably well. We employthese estimates to evaluate the coherence time in the nextsection.

The scale on which I (x, t ) varies is on the order of thediffusion length, which can be estimated from the formulaL = √

2DT1. Using our estimate of the diffusion constant,D ≈ 8.1 nm2/s, and our previously measured result for T1

(1 hour—see Ref. [56]), we find L ≈ 240 nm. The fact thatthis scale is orders of magnitude larger than the internuclearseparation (r ≈ 5 Å) justifies the use of a coarse-grainedapproximation for I (x, t ).

Following Ref. [76], we treat the field ζ (�x, t ) as behavinglike white noise, which has the following correlation functions

〈ζ (x, t )〉 = 0, (20)

〈ζ (x, t )ζ (y, s)〉 = �0δ(x − y)δ(t − s). (21)

White noise is a reasonable assumption since the dipolarflip-flops conserve energy, and the occurrence of one flip-flopevent should not alter the probability of a later event. In theabove equations, averaging is done over all possible noiserealizations. The average noise is zero, and choosing �0 =−ηD∂2/∂x2 leads to the conservation of the order parameter[89] (which is the total nuclear spin polarization in the presentcase), in any possible noise realization. The parameter η

determines the noise strength. It is convenient to switch toFourier space, where Eq. (13) becomes

∂

∂tI (q, t ) = −Dq2I (q, t ) + ζ (q, t ). (22)

The above equation has the following general solution

I (q, t ) = I (q, 0)e−Dq2t +∫ t

0dse−Dq2(t−s)ζ (q, s). (23)

Fourier transforming Eq. (21) we find that the noise correla-tions in Fourier space take the following form

〈ζ (q, t )ζ (k, s)〉 = (2π )3ηDq2δ(q + k)δ(t − s). (24)

Note that the above correlation function vanishes for thezeroth (q = 0) Fourier mode, implying a strict conservationof the total nuclear spin polarization. The spin-spin correlation

function then becomes

〈I (q, t )I (k, s)〉= e−Dq2(t+s)〈I (q, 0)I (k, 0)〉− η

2(2π )3δ(q + k)(e−Dq2(t+s) − e−Dq2|t−s|),

(25)

which can be evaluated once the initial correlation function〈I (q, 0)I (k, 0)〉 at time t = 0 is known. In the absence ofany DNP or any other dynamic processes, this function canbe drawn from a stationary equilibrium distribution, howeverin the present case this initial condition is determined bythe solution of DNP (which has been treated in detail inSec. II). The exact functional form I (x, 0) of the spatialdistribution of DNP (see Fig. 2) is not a simple mathematicalfunction with a precise closed form. Hence to make our modelanalytically tractable for subsequent analysis, we will assumethe following simplified form for I (x, 0) centered around thedefect site (at the origin):

I (x, 0) = I0 exp(−α2(x2 + y2 + z2)). (26)

At this point, a few comments are in order. First, note thatwe are neglecting the role of spin diffusion during the DNPprocess for the sake of tractability. Including this effect shouldlead to an enhancement of DNP in the bulk, suggesting thatthe results we obtain below for the buildup of DNP and itsinfluence on T2 times of bulk nuclear spins may be conserva-tive. However, preliminary experimental results for T2 versusdriving time T suggest that this is not a big effect [72,73]. Wealso note that we are neglecting nuclear spin relaxation. Thisis reasonable since our focus is on diffusion and decoherence,both of which occur on timescales much shorter (on the orderof seconds) than the relaxation time, which is on the order ofminutes or hours [56].

In Eq. (26), the parameter α has a crucial dependence onthe DNP driving time. Specifically, for longer driving times,the spread of the nuclear polarization around the origin shouldincrease, and therefore α should decrease. For a fixed I0 thetotal nuclear spin polarization (z component) in the sampleis I0π

3/2α−3, which again highlights the fact that the nuclearpolarization increases with driving time. Since we have nowapproximated the DNP distribution to be of Gaussian forminvolving the parameter α, we must numerically determineα as a function of driving time T using our exact DNPsolution from Sec. II. We do this by calculating the fullwidth at half maximum (FWHM) along a chosen directionfrom our actual DNP solution (see Fig. 2) and relating thecalculated FWHM to the standard deviation of our Gaussianapproximation, i.e., FWHM = 2

√ln2/α. Figure 4 shows the

plot of the numerically evaluated α as a function of drivingtime (T ) using the Gaussian approximation. Since the plot oflog(α) vs log(T ) has a linear fit, one can easily obtain α forarbitrary driving times.

Before we proceed with the diffusion calculation, it isworth commenting on the consequences of choosing differentinitial values of α. In Fig. 5 we plot the numerically evaluatedα as a function of θ for different driving times T . We find thatthe average value 〈α〉θ /αθ=π/4 ≈ 1.24 for all T ’s, where 〈α〉θis obtained by averaging over all θ values for a particular T .Further, we also note that the value of α between θ = π/4 and

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ENHANCEMENT OF NUCLEAR SPIN COHERENCE TIMES … PHYSICAL REVIEW B 99, 205423 (2019)

FIG. 4. Logarithm (natural) of the numerically evaluated α (inthe units m−1) as a function of the logarithm (natural) of the drivingtime T (in the units s) using the Gaussian approximation along aparticular direction (θ = π/4). The plot shows the data (in bluedots) for log(α) as a function of log(T ). The data shows a linearbehavior shown in the red line (at least for the timescales we areconcerned with) and thus can be extrapolated to obtain α for arbitrarydriving times. The parameters chosen are the same as in Fig. 2.We find that for those parameters, log(α(T )) = a1 log T + a2, witha1 = −0.16632, and a2 = 18.8096.

θ ≈ 0 differs by just a factor of two. Since the entire curve inFig. 5 shifts vertically as T increases, it follows that changingthe initial value of α is tantamount to shifting T . This in turnwould lead to a horizontal shift of the T2 versus T curvesthat we obtain in the next section, but it will not affect theshape of these curves. In what follows, we average α over allvalues of θ to obtain quantitatively accurate results that canbe compared with experimental observations. However, firstwe will examine qualitative trends in T2 versus T , and for thispurpose it is sufficient to fix the value of α (here we chooseα = αθ=π/4) to reduce the computational cost.

0.1 0.2 0.3 0.4/

0.1

0.15

0.2

0.25

0.3

T = 1s

T = 10s

FIG. 5. Numerically evaluated α (in the units nm−1) as a functionof θ for ten different driving times T . The black diamond on eachcurve indicates the average value 〈α〉θ (obtained by averaging overall θ values for a particular T ). The dotted line indicates that theaverage value for each driving time T collapses on to the value at aparticular θ . We find that 〈α〉θ /αθ=π/4 ≈ 1.24. We also note that thevalue of α between θ = π/4 and θ ≈ 0 differs by just a factor of two.

Fourier transforming Eq. (26), we have the initial condition

I (q, 0) = π3/2 I0

α3exp

(−(qx

2 + q2y + q2

z

)/4α2). (27)

We can then calculate the correlation function at finite timesusing Eq. (25):

〈I (q, t )I (k, s)〉 = π3 I20

α6e−(q2+k2 )/4α2−Dq2(t+s)

− η

2(2π )3δ(q + k)(e−Dq2(t+s) − e−Dq2|t−s|).

(28)

The effect of the net nuclear spin magnetization on a singlenuclear spin can be captured using an effective magnetic fielddescription. Consider a nuclear spin at position xk interactingwith all the other nuclear spins via the dipolar interaction,which has the form shown in Eq. (14). The terms that involvethe spin operator for the kth spin, Ik , are

Hknn = −

∑i �=k

Bik (I i+Ik

− + I i−Ik

+) + 2Ikz

∑i �=k

BikI iz . (29)

Multiplying by the reduced density matrix for all the nuclearspins other than the kth one and performing a partial trace, wearrive at an effective Hamiltonian for the kth spin:

Hkeff =−

∑i �=k

Bik (〈I i+(t )〉Ik

− + 〈I i−(t )〉Ik

+) + 2Ikz

∑i �=k

Bik〈I iz (t )〉.

(30)

We now assume that only the z component of the nuclear spinbath is nonzero on average so that we retain only the last termabove, and we replace the expectation value by the continuousfield I (x, t ):

Hkeff = 2Ik

z

∑i �=k

BikI (xi, t ). (31)

This is essentially the same as performing a semiclassicaltreatment in which bath correlators are replaced by classicalfields (see, e.g., Ref. [90]). The coefficients Bik decay likea power law in the distance between the ith and kth nuclei.When we go to the continuum limit, we are assuming thatI (x, t ) is approximately constant over distances correspondingto the average internuclear separation, which is on the orderof a few Angstroms as estimated above. Since Bik is close tozero for distances much larger than this, it is effectively like adelta function on the scale on which I (x, t ) varies appreciably.Thus, in the continuum limit, we may write

Hkeff = γN Ik

z

∫nk (x)I (x, t ), (32)

where n(q) is the Fourier transform of n(x). This result can beinterpreted in terms of an effective magnetic field B(t ) seen bythe kth nuclear spin:

B(t ) = γN

∫d3xn(x)I (x, t )

= γN

∫d3q

(2π )3n(q)I (−q, t ), (33)

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SHARMA, GAEBEL, REJ, REILLY, ECONOMOU, AND BARNES PHYSICAL REVIEW B 99, 205423 (2019)

where n(x) is the spatial nuclear density profile of the nucleus,and n(q) is its Fourier transform. For a given single nucleuslocated at (x0, 0, 0), we assume a Gaussian profile:

n(x) =(

β2

π

)3/2

exp(−((x − x0)2 + y2 + z2)β2), (34)

n(q) = exp(−(

q2x + q2

y + q2z

)/4β2

)exp(−iqxx0). (35)

The parameter β related to the spatial spread of the nucleuscan be adjusted to represent a scenario close to the actualphysical case such that n(x) is close to a Dirac-delta function.To obtain physically relevant results, an ensemble averagingover many nuclear sites can be performed by varying theposition of the center of the nucleus. For a particular angle θ

the value of the parameter α can be calculated as describedpreviously, and for that chosen angle the radius x0 can bevaried within a range. From the definition of I (k, t ) the expec-tation value of the effective magnetic field can be calculated:

〈B(t )〉 = I0

(α)3

√1

A3e−x2

0/A, (36)

where we have defined A = α−2 + β−2 + 4Dt . We are in-terested in the correlation function for the magnetic field〈B(t + s)B(s)〉 which is calculated to be

〈B(t + s)B(s)〉 = I20 e−x2

0/F1 e−x20/F2

(α)6√

F 31 F 3

2

− η

2π3/2

(1

F 3/23

− 1

F 3/24

),

(37)

where we have defined the following constants

F1 = α−2 + β−2 + 4D(t + 2s), (38)

F2 = α−2 + β−2, (39)

F3 = 2β−2 + 4D(t + 2s), (40)

F4 = 2β−2 + 4Dt . (41)

Note that the additional time dependence due to ‘s’ in theabove equations arises because the nuclear diffusive dynamicsstart from a nontrivial source term (which comes from theDNP solution) at time ‘s.’ This specifically means that we arefirst driving the system from time 0 to time s, resulting in aninitial DNP distribution as discussed earlier. The parameterα is therefore a function of driving time [α = α(s)], as alsodemonstrated in Fig. 4. We now define the two point correla-tion function

C(t ) = 〈B(s + t )B(s)〉 − 〈B(s + t )〉〈B(s)〉, (42)

where the effect of the overall increase of the effective mag-netic field 〈B〉 with the driving time has been removed bysubtracting the mean value of the effective magnetic field.We point out that the Gaussian approximation allows us toanalytically calculate the two point correlation function C(t )in Eq. (42), which otherwise would not be feasible for a morecomplex form of I (x, 0). The first spectral density is given by

FIG. 6. Numerically obtained normalized C(ω) for a particulardrive time of s = 15 s. The inset shows the analytically evaluatedcorrelation function C(t ).

the Fourier transform of C(t )

C(ω) =∫ +∞

−∞dte−iωtC(t ). (43)

Since the functional form of C(t ) given by Eqs. (42), (37),and (36) does not allow an analytical expression for C(ω), wewill resort to a numerical evaluation of the Fourier transformC(ω) in our calculations. In Fig. 6 we plot the numerically ob-tained C(ω) for a particular drive time s. The inset shows theanalytically evaluated correlation function C(t ). Qualitativelywe expect the correlation function C(t ) to decay as t → ∞.Numerically we find that C(ω) is a monotonically decreasingfunction of ω. Also note that it is the nontrivial variation ofC(ω) with respect to s which is important for determining howχ (t ) and in turn T2 behave, as evaluated in the next section.

IV. ENHANCEMENT OF NUCLEAR COHERENCE TIME

In this section we will use the two point correlation func-tion C(ω) to evaluate the coherence time T2. Let us considerthe quantum state of a single nucleus prepared in an initialstate |ψ〉 = c↑|↑〉 + c↓|↓〉 evolving under the stochastic fieldB(t ). The mean field 〈B(t )〉 and the correlation function ofthis field were obtained in the previous section [Eqs. (36)and (37)]. The corresponding Hamiltonian is given by H =γN B(t )σz/2, where σz is the Pauli matrix. The state at time twill be given by

|ψ (t )〉 = e− i2

∫ t0 γN B(s)dsc↑|↑〉 + e+ i

2

∫ t0 γN B(s)dsc↓|↓〉. (44)

The off-diagonal element of the density matrix characterizesthe nuclear coherence and can be quantified by the functionW (t ) as:

W (t ) = |〈ρ↑↓(t )〉||〈ρ↑↓(0)〉| =

∣∣∣∣⟨exp

(−i

∫ t

0γN B(s)ds

)⟩∣∣∣∣. (45)

The function W (t ) in the above expression can describedecoherence effects corresponding to a free induction decay,i.e., the nucleus is prepared in a quantum state and allowed toevolve freely under the given Hamiltonian. For more complexpulse sequences (like spin echo, CPMG, PDD, etc.), a corre-sponding function f (t, s) can be introduced in the integrand

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ENHANCEMENT OF NUCLEAR SPIN COHERENCE TIMES … PHYSICAL REVIEW B 99, 205423 (2019)

0 0.2 0.4 0.6 0.8t(ms)

0

0.5

1W

(t)

CPMG (n=4)CPMG (n=2)SEFID

FIG. 7. Decoherence under various dynamical decoupling pulsesequences (free induction decay, spin echo, and CPMG) for adriving time of T ∼ 0.01 s. The other parameters used are a1 =−0.16632, a2 = 18.8096, x0 = 10 nm, D = 25 nm2/s, β−1 = 1 Å,η ∼ 10−20 (eVs)2/m3, I0 = √

π h̄/(0.2 μm3). The initial DNP dis-tribution is taken to have α = αθ=π/4.

of the above equation to account for multiple pulses [91]. Thecharacteristic time of decay of W (t ) is denoted as T2, definedby log(W (T2)) = −1. Therefore one can typically write therelation W (t ) ≡ e−χ (t ), where the quantity χ (t ) can be writtenas [91]

χ (t ) =∫ ∞

0

dω

2πC(ω)

F (ωt )

ω2, (46)

where C(ω) is the first spectral density defined in Sec. IIIEq. (43). The function F (z) is the filter function, which de-pends on the type of pulse sequence employed. It encapsulatesthe effect of the pulse sequence on decoherence. For example,the filter functions for spin echo (SE) and CPMG pulses are[91]

F (z) = 8 sin4(z/4); for SE (47)

F (z) = 8 sin4(z/4n) sin2(z/2)/ cos2(z/2n);

for CMPG even n. (48)

In the results we present below (Figs. 7–10), we will focuson the regime where the nucleus is located at a distance x0 =10 nm from the electron center. However, for an ensemblemeasurement, we must average over all possible locations.This is done in Fig. 11 where we calculate the T2 time forexperimentally relevant parameters in nanodiamond. Figure 7shows decoherence under free induction decay and variousdynamical decoupling pulse sequences (Hahn’s spin echoand CPMG pulses) for a chosen driving time of T ∼ 0.01 s.As one would expect, the dephasing slows down as moredynamical decoupling pulses are applied.

In Fig. 8 we show the logarithm of the nuclear T2 as afunction of the logarithm of the driving time for various valuesof the stochastic noise parameter η. For short drive times, weobtain coherence times on the same order of magnitude asprior experimental results for enriched diamond [92,93]. Wesee a clear enhancement of the nuclear T2 time with drivingtime for −3 < log10(T ) < 0. The T2 time is suppressed withincreasing noise parameter η, however we point out that it is

-3 -2 -1 0 1 2log

10(T(s))

-8

-7

-6

-5

-4

-3

log

10(T

2(s))

= 0

= 20

= 40

= 60

= 8 0

= 100

FIG. 8. Logarithm of the nuclear T2 time as a function ofthe logarithm of the driving time for various different values ofthe noise strength η. A clear enhancement (up to ∼3 orders ofmagnitude) of the nuclear T2 time with driving time is seen for−3 < log10(T ) < 0. We also note the suppression of T2 time withincreasing noise strength. Further the T2 time saturates after the DNPdrive time is increased beyond T ∼ 1 s. We use a1 = −0.16632,a2 = 18.8096, x0 = 10 nm, D = 25 nm2/s, η0 ∼ 10−20 (eVs)2/m3,β−1 = 1 Å, and use the n = 4 CPMG pulse sequence. The initialDNP distribution is taken to have α = αθ=π/4.

essential that η �= 0 in order to obtain physically acceptableresults for T2. Further we also note that the T2 time saturatesafter the driving time is increased beyond a certain time Tsat.The exact value of the obtained T2 time and the saturationtime depends on our choice of parameters. Figure 9 shows thelogarithm of the nuclear T2 as a function of the logarithm ofthe driving time for various values of the diffusion constant D.We note that for a higher diffusion constant, the T2 coherencetime saturates more quickly as a function of driving time. Thisbehavior is expected because a higher diffusion rate should in-crease the temporal spread of the driving induced polarization[see Eq. (13)], however η sets the scale for an upper limit on

-3 -2 -1 0 1 2log

10(T(s))

-6

-5

-4

-3

log

10(T

2(s))

D=D0

D=2D0

D=4D0

D=6D0

D=8D0

D=10D0

FIG. 9. Logarithm of the nuclear T2 time as a function ofthe logarithm of the driving time for various different values ofthe diffusion constant D. For larger D values the coherence timesaturates more quickly as a function of driving time. We usea1 = −0.16632, a2 = 18.8096, x0 = 10 nm, η ∼ 10−20 (eVs)2/m3,β−1 = 1 Å, D0 = 0.5 nm2 s−1, and use the n = 4 CPMG pulse se-quence. The initial DNP distribution is taken to have α = αθ=π/4.

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FIG. 10. Density plot for the nuclear T2 time (in seconds) as afunction of the noise parameter and diffusion constant for a constantdriving time of T = 10 s. The parameters used were D0 = 25 nm2/s,η0 ∼ 10−23 (eVs)2/m3, a1 = −0.1662, a2 = 18.4374, x0 = 10 nm,β−1 = 1 Å, and the n = 4 CPMG pulse sequence was used. Theinitial DNP distribution is taken to have α = αθ=π/4. An increasein noise strength leads to a higher suppression of T2 compared tolowering the diffusion constant D. The different order of the T2

obtained here is due to our different choice of parameters comparedto Figs. 8 and 9.

T2. Figure 10 shows the density plot for the nuclear T2 timeas a function of the noise strength and diffusion constant for aconstant driving time T . An increase in noise strength leads toa higher suppression of T2 compared to lowering the diffusionconstant D. The different order of the T2 obtained in Fig. 10 isdue to our different choice of parameters compared to Figs. 8and 9.

FIG. 11. Plot of the logarithm of the nuclear T2 time as a func-tion of logarithm of the DNP polarization time for experimentallyrelevant parameters for nanodiamond [56], i.e., D = 5 nm2/s, ω0s ∼300 μeV. Note that for this plot we perform the ensemble averagingto obtain the T2 time over a wide range of radii from the center0.5 nm < x0 < 10 nm and also over the angle θ . For the DNP Gaus-sian model for each θ we have calculated α separately from the exactDNP solution. The top inset shows the saturation of the T2 time forarbitrarily higher DNP driving times and the bottom inset shows thedecay of the signal with time for increasing DNP driving times. Theother parameters we chose were β−1 = 1 Å, η ∼ 10−23 (eVs)2/m3

[76], and the n = 4 CPMG pulse sequence.

The two processes that play a central role in our analysisare DNP induced by driving electron-nuclear flip-flops anddiffusion caused by nuclear dipole-dipole interactions. Forfinite magnetic fields the dipolar interaction between twonuclei can be effectively written as a sum of Overhauserand flip-flop terms Hdip ≈ t ′(I+iI− j + I+ j I−i − 2Iz j Izi ), wheret ′ is the energy scale of the interaction, and i, j representnuclei indices. When the nuclear bath is completely unpo-larized (m = 0), the distribution of the nuclear spins in theconfiguration space has the maximum entropy, while for afully polarized nuclear bath (m = 1) the configurational en-tropy is zero as there is only one way to arrange the spins.Therefore when we consider processes in which the totalspin is conserved, such as a pair of spin flips which causesfluctuations in the magnetic field (noise), an unpolarized bathis expected to result in maximal noise and therefore the lowestcoherence time T2, while a fully polarized bath should resultin minimal noise and a maximal (ideally infinite if otherdecoherence processes are ignored) T2 time. For intermediatebath polarizations, the phase space for flip flops is reducedand is sharply peaked around m = 0. Therefore physically wealso expect that longer driving times, which causes higheraverage bath polarization, should result in an enhanced T2

coherence time. Further we note that all the curves in Fig. 9saturate to the same value of T2 indicating the fact that thenuclear polarization itself saturates to the same distributionregardless of the diffusion constant. The magnitude of thediffusion constant only affects how quickly the DNP reachesits saturation value. The T2 time for a nucleus at x = 10 nmonly improves once the DNP has propagated until x0 fromthe electron defect center, after which the T2 time remains aconstant as the DNP wave continues to propagate outwards,and the DNP becomes uniform over the sample. The fact thatall the curves in Fig. 8 also saturate near the same value of T isagain a consequence of the fact that the DNP saturation pointonly depends on T and not on η. However, the different curvesin Fig. 8 do not all saturate to the same value of T2 because thedephasing caused by η diminishes the gains in T2 afforded bythe DNP.

In Figs. 7–10 we demonstrated the generic behavior ofthe T2 time as a function of the driving time as obtainedwithin our theoretical model. In actual hyperpolarization ex-periments, the driving time can be increased up to a fewminutes or even hours. In recent experiments [56] performedon nanodiamond, an increase of the relaxation time of 13Cspins up to three orders of magnitude has been observed bydynamically hyperpolarizing the nuclear bath via microwavedriving. In Fig. 11 we plot the T2 coherence time usingexperimentally relevant parameters for nanodiamond and notethat the nuclear coherence time increases up to three ordersof magnitude (∼1 ms–1 s) as the driving time is increased.Note that for this plot we perform the ensemble averaging toobtain the T2 time over a wide range of radii from the center0.5 nm < x0 < 10 nm for a particular θ and also averageover the angle θ . Since the DNP solution is anisotropic andhas a C4 symmetry, for the Gaussian model for each θ wehave calculated α separately from the exact DNP solution.Our theoretical results suggest that nuclear hyperpolarizationvia microwave driving not only enhances relaxation timesbut also nuclear spin coherence times by several orders of

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ENHANCEMENT OF NUCLEAR SPIN COHERENCE TIMES … PHYSICAL REVIEW B 99, 205423 (2019)

magnitude, as also suggested by preliminary experimentalresults on nanodiamond 13C spins [72,73].

V. CONCLUSION

In this work, we calculated the nuclear spin coherencetime for an ensemble of dipolar-coupled nuclear spins in thevicinity of a driven defect center in a solid. We showed thatwhen electron-nuclear spin-flip transitions are driven withmicrowave fields, nonsecular terms in the electron-nuclearhyperfine interaction can generate a large dynamic nuclearpolarization. We then analytically calculated the spatial distri-bution of this nuclear polarization as a function of the drivingtime and used this as a starting point to study the subsequentdiffusion and fluctuations of the polarization. To study theevolution nuclear T2 time in an analytically tractable manner,we approximate the spatial distribution of the nuclear polar-ization around the defect center by a Gaussian distribution.Using these results, we then obtained the coherence times

of nuclear spins far from the defect center as a function ofthe driving time, fluctuation strength, and speed of diffusion.We found that the coherence generically increases by severalorders of magnitude as the driving time is increased up untila saturation point that depends on the strength of the dipolarinteraction. In the case of 13C nuclear spins, this translates toa nearly three orders of magnitude coherence time increase,a result that parallels a similar enhancement in relaxationtimes seen in recent experiments [56]. Our theoretical modeland results will be therefore useful for current and upcom-ing experiments on enhancing the coherence time via DNP.Preliminary experimental results on nanodiamond 13C spinsalso suggest that the nuclear spin coherence time is in factenhanced by the hyperpolarization [72,73].

ACKNOWLEDGMENT

The work of E.B. and S.E.E. was supported by NSF (GrantNo. 1741656).

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