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Yuki Nakamura, Shunsuke Nishimura, [Takuya Iwasaki](https://orcid.org/0000-0002-1103-2433), [Shu Nakaharai](https://orcid.org/0000-0002-6329-3942), Shinichi Ogawa, Yukinori Morita, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Kento Sasaki, Kensuke Kobayashi

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[Systematic investigation of dynamic nuclear polarization with boron vacancy in hexagonal boron nitride](https://mdr.nims.go.jp/datasets/b758984c-5455-49fd-a9fc-97fb15ec2ba8)

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Systematic investigation of dynamic nuclear polarization with boron vacancy inhexagonal boron nitrideYuki Nakamura,1 Shunsuke Nishimura,1 Takuya Iwasaki,2 Shu Nakaharai,3 Shinichi Ogawa,4 YukinoriMorita,4 Kenji Watanabe,5 Takashi Taniguchi,2 Kento Sasaki,1 and Kensuke Kobayashi1, 6, 71Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan2Research Center for Materials Nanoarchitectonics,National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan3Department of Electric and Electronic Engineering, Tokyo University of Technology,1404-1 Katakuramachi, Hachiohji, Tokyo 192-0982, Japan4National Institute of Advanced Industrial Science and Technology,1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan5Research Center for Electronic and Optical Materials,National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan6Institute for Physics of Intelligence, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan7Trans-scale Quantum Science Institute, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan(Dated: January 29, 2025)Dynamic nuclear polarization (DNP) using the boron vacancy (V−B ) in hexagonal boron nitride(hBN) has gained increasing attention. Understanding this DNP requires systematically investi-gating the optically detected magnetic resonance (ODMR) spectra and developing a model thatquantitatively describes its behavior. Here, we measure the ODMR spectra of V−B in h10B15N overa wide magnetic field range, including the ground state level anti-crossing (GSLAC), and com-pare them with the results of the Lindblad-based simulation that considers a single electron spinand three neighboring 15N nuclear spins. Our simulation successfully reproduces the experimen-tal spectra, including the vicinity of GSLAC. It can explain the overall behavior of the magneticfield dependence of the nuclear spin polarization estimated using the Lorentzian fitting of the spec-tra. Despite such qualitative agreement, we also demonstrate that the fitting methods cannot giveaccurate polarizations. Finally, we discuss that symmetry-induced mechanisms of V−B limit themaximum polarization. Our study is an essential step toward a quantitative understanding of DNPusing defects in hBN and its quantum applications.I. INTRODUCTIONSpin defects in wide-bandgap semiconductors have at-tracted attention as a platform for developing quantumtechnologies [1–4]. For example, spin defects in dia-mond [5–8] and SiC [9] are extensively studied for ap-plications in quantum sensing, quantum simulation, andquantum communication due to their optical addressabil-ity and long coherence times from cryogenic to room tem-peratures. Recently, the boron vacancy (V−B ) in hexag-onal boron nitride (hBN) was discovered to have elec-tron spins that can be manipulated at room tempera-ture [10, 11]. Since hBN is a van der Waals (vdW)crystal, V−B has the following advantages in its quan-tum sensor application to materials: (i) V−B can bebrought within a few atomic layers of a measurementtarget through stamping methods [12]. (ii) V−B can becreated at arbitrary two-dimensional positions by irradi-ating hBN with a focused ion beam [13, 14]. (iii) hBN isthin, small, easy to process, and readily integrated intovarious devices. Currently, V−B is beginning to be usedas a quantum sensor to explore the physical properties ofmaterials [12, 15–17].Both nitrogen and boron atoms have nuclear spins,which couple to the V−B electron spin via hyperfine in-teractions. In general, nuclear spins in solids have longercoherence times than electron spins. In diamonds, forinstance, they are frequently utilized as quantum memo-ries [18–22]. In such cases, the nuclear spins are manip-ulated and read out via electron spins. Realizing suchnuclear spin control and application with quantum de-fects in hBN [23–25] requires understanding the hyperfineinteractions with two-dimensionally distributed nuclearspins and the phenomena arising from them.Dynamic nuclear polarization (DNP) is a phenomenondriven by hyperfine interactions and is used for initial-izing and reading out nuclear spins. Recently, severalpapers [23–27] reported that the high polarization of V−Belectron spins induced by optical pumping is transferredto the three adjacent nitrogen nuclear spins. Gao etal. [23] demonstrated DNP and coherent control of ni-trogen nuclear spins in hBN with natural isotope compo-sition ratios (hnatBnatN). Ru et al. [25] reported DNP inhnatBnatN in the magnetic field of 25–100 mT and 140–200 mT, excluding the ground state level anti-crossing(GSLAC) of V−B . They proposed a model that consid-ers a single electron spin, three nitrogen nuclear spins,and optical transitions and found that the calculated re-sults of the Lindblad equation are qualitatively consis-tent with experimentally estimated polarization. Someof the present authors [26], Clua-Provost et al. [27], andGong et al. [24] reported DNP in isotopically engineeredhBN (hnatB15N or h10B15N). The optically detectedmagnetic resonance (ODMR) spectra of V−B in such iso-arXiv:2501.16715v1  [cond-mat.mes-hall]  28 Jan 20252topically engineered hBN are more straightforward thanthose of hnatBnatN. Based on the simplified ODMR spec-tra, Clua-Provost et al. [27] estimated the polarization ofnitrogen nuclear spins in a wide range of magnetic fields(10–150 mT), including GSLAC, and compared the re-sults with a simulation similar to that of Ru et al. [25].This paper first presents high-resolution and accurateODMR spectra V−B in h10B15N obtained between 10 and150 mT. Second, we show that the estimated nuclearpolarizations using the Lorentzian fitting of the ODMRspectra are qualitatively consistent with the true (gen-uine) polarization in the Lindblad-based simulation of amodel that considers a single electron spin, three adja-cent nitrogen nuclear spins, and optical transitions; thatis, the simulation can explain the overall behavior of thepolarization depending on the magnetic fields. Third,we prove that the simulation considering microwave andinhomogeneous broadening reproduces the experimentalspectra quantitatively over a wide range of magneticfields, including GSLAC. Nevertheless, we also demon-strate that the conventional Lorentzian fitting methodis not sufficiently accurate to estimate the true polariza-tion. Finally, based on the simulation, we discuss thatdynamics related to defect symmetry such as coexistenceof flip-flop and flip-flip interactions and nuclear spin en-tanglement limit the maximum polarization.This paper is organized as follows. Section II intro-duces the fundamental properties of V−B in h10B15N andthe hyperfine interactions with 15N. Then, we explainthe mechanism of DNP in h10B15N. Section III describesour simulation based on the Lindblad equation. SectionIV outlines our experimental setup and sample specifica-tions. Section V discusses the experimental and simula-tion results and their implications. Section VI concludesthe study.II. DNP IN HEXAGONAL BORON NITRIDEhBN is a vdWmaterial composed of a hexagonal latticearrangement of boron and nitrogen atoms, as shown inFig. 1(a). It is a wide-gap semiconductor with a bandgapof ∼ 6.0 eV [28]. The negatively charged vacancy at theboron site is referred to as V−B . The electrons localizedin this defect form multiple levels within the band gap,such as the orbital ground state (GS) and excited state(ES), which are spin triplets, and the metastable sin-glet state (MS) [Fig. 1(b)]. When V−B is in the GS (ES)state, the electron spin levels are split by Dgs = 3.47 GHz(Des = 2.09 GHz) in a zero magnetic field. The electronspin has a quantized axis along the z-direction (the c-axisof the hBN crystal) [Fig. 1(a)], and the magnetic quan-tum number mS is defined as the projection of the spinangular momentum onto the z-axis. We can excite thetransition from GS to ES in a spin-conserving manner(∆mS = 0) by irradiating with a green laser (transitionrate: ΓL), as shown in Fig. 1(b). The relaxation from theES to GS includes spin-conserving radiative relaxation,VB-EnergyBz (mT)Dgs Des 75 124ESLACGSLAC(b)(a)(c)ESGSflip-flop flip-flipISCexcitation& radiative decay(d)ES Des Dgs GSMSVB-FIG. 1. (a) Schematic of V−B in h10B15N. The x-axisis defined along the a-axis and the z-axis along the c-axisof the h10B15N crystal. (b) Energy level structure and op-tical transitions of V−B . The relevant transition rates ΓL,Γ0, γe0, γe1, γg0 , and γg1 are depicted. (c) Magnetic field de-pendence of V−B energy levels. In the present experiment,the zero-field splitting is estimated as Dgs = 3.47 GHzfor the GS and Des ∼ 2.09 GHz for the ES. Level anti-crossings (LAC) between mS = 0 and mS = −1 occur atBz ∼ 124 mT (GS) and Bz ∼ 75 mT (ES). (d) DNP mech-anism. Green arrows indicate optical excitation from GS toES and radiative transitions from ES to GS. Red arrows rep-resent ISC from ES mS = ±1 to GS mS = 0. Light bluearrows correspond to flip-flop transitions |mS = 0,mI⟩ ↔|mS = −1,mI + 1⟩. Gray dashed arrows indicate flip-fliptransitions |mS = 0,mI + 1⟩ ↔ |mS = −1,mI⟩.which involves a direct transition accompanied by thephotoluminescence (PL) with a transition rate of Γ0, andspin-nonconserving non-radiative relaxation via intersys-tem crossing (ISC) through MS. The transition rates inthis non-radiative paths are γe0, γe1, γg0 , and γg1 , as definedin Fig. 1(b). Since γe0 < γe1 [29–31] and γg0 > γg1 [30, 31],the non-radiative relaxation process allows us to polarize,or initialize, the electron spin state into mS = 0 throughcontinuous optical excitation. Furthermore, due to theserelaxation processes, the PL intensity for the mS = ±1states is weaker than that of themS = 0 state. Therefore,we can read out the electron spin state by referencing thedifference in the PL intensity.The electron spin of V−B in h10B15N is coupled to thethree adjacent 15N nuclear spins via hyperfine interac-tions. In this paper, the three adjacent 15N atomic nucleiare indexed as i = 1, 2, and 3, as shown in Fig. 1(a). TheHamiltonian considering the electron spin and the three3nuclear spins is given by:Ĥv = DvŜz2+ γeBzŜz −3∑i=1γnBz Î(i)z +3∑i=1ŜĀ(i)v Î(i),3∑i=1ŜĀ(i)v Î(i) =3∑i=1∑j=x,y,z∑k=x,y,zA(i)v,jkŜj Î(i)k , (1)where v = gs or es denotes the orbital levels, Bz is themagnetic field along the z-axis, Ŝ and Ŝz are the electronspin operators of V−B with S = 1, and Î(i) and Î(i)z are thenuclear spin operators of the nearest-neighbor 15N nucleiwith I(i) = 1/2. γe = 28 MHz/mT is the gyromagneticratio of the electron spin, and γn = −4.3 kHz/mT is thegyromagnetic ratio of the 15N nuclear spin. The term∑3i=1 ŜĀ(i)v Î(i) represents the hyperfine interactions be-tween V−B and the nearest-neighbor 15N nuclei, whereA(i)v,jk (j = x, y, and z, k = x, y, and z) are the compo-nents of the hyperfine interaction tensor Ā(i)v .In most of the magnetic fields used in this study, thecondition |Dv − γeBz| ≫ |A(i)v,jk| ∼ 100 MHz is satisfied,allowing the secular approximation:∑3i=1 ŜĀ(i)v Î(i) ≈∑3i=1 A(i)v,zzŜz Î(i)z = Av,zzŜz(∑3i=1 Î(i)z ). Thus, the states∣∣∣mS ,mI =∑3i=1 m(i)I〉form a good basis. As a re-sult, the energy levels of |mS = ±1⟩ are split by |Av,zz|for each of the four total nuclear spin states mI ={+3/2,+1/2,−1/2,−3/2}.On the other hand, when the condition |Dv − γeBz| ∼|A(i)v,jk| is satisfied, the off-diagonal components of theHamiltonian in Eq. (1) cannot be neglected. This condi-tion is called the level anti-crossing (LAC) [red circles inFig. 1(c)]. The hyperfine interaction term in Eq. (1) canbe rewritten as [24]:3∑i=1ŜĀ(i)v Î(i) =3∑i=1{Av,zzŜz Î(i)z + (A(i)v,+Ŝ+Î(i)− + h.c.)+ (A(i)v,−Ŝ+Î(i)+ + h.c.)}, (2)where Ŝ± ≡ Ŝx ± iŜy, Î(i)± ≡ Î(i)x ± iÎ(i)y , andA(i)v,+ =A(i)v,xx +A(i)v,yy4,A(i)v,− =A(i)v,xx −A(i)v,yy4+A(i)v,xy2i. (3)In the derivation of Eqs. (2) and (3), the mirror symme-try of V−B concerning the x-y plane is considered, lead-ing to A(i)v,xz = A(i)v,yz = A(i)v,zx = A(i)v,zy = 0 [24, 26].The second term of the right-hand side of Eq. (2) repre-sents flip-flop interactions and causes transitions between|mS = 0,mI⟩ ↔ |mS = −1,mI + 1⟩ [light blue arrows inFig. 1(d)]. The third term represents flip-flip interac-tions, causing transitions between |mS = 0,mI + 1⟩ ↔|mS = −1,mI⟩ [gray dashed arrows in Fig. 1(d)].Considering the D3h symmetry of the hyperfine inter-actions in GS [23], the following simple expression can beobtained (see Appendix A for details):A(i)gs,+ =Ags,xx +Ags,yy4,A(i)gs,− =Ags,xx −Ags,yy4e2iϕ(i),ϕ(1) = 0, ϕ(2) = −2π/3, ϕ(3) = −ϕ(2), (4)where Ags,xx ≡ A(1)gs,xx < 0 and Ags,yy ≡ A(1)gs,yy < 0, lead-ing to |A(i)gs,+| > |A(i)gs,−|. A similar symmetry argumentcan be applied to ES, yielding |A(i)es,+| > |A(i)es,−|, as Ap-pendix A describes. Therefore, the flip-flop interaction isdominant.Under GSLAC or excited-state LAC (ESLAC) con-ditions, the flip-flop interaction induces the transition|mS = 0,mI⟩ ↔ |mS = −1,mI + 1⟩. Optical excitationfurther drives transitions such as |mS = −1,mI + 1⟩ →|mS = 0,mI + 1⟩, initializing the electron spin while si-multaneously experiencing the flip-flop interaction. Thisprocess increases the total angular momentum of thesystem, resulting in nuclear spin polarization toward in-creasing mI . In contrast, the flip-flip interaction inducesthe transition |mS = 0,mI + 1⟩ ↔ |mS = −1,mI⟩, lead-ing to polarization toward decreasing mI , causing depo-larization.The flip-flop and flip-flip transitions are maximized un-der the following conditions, respectively,(flip-flop:) Ev,0,mI= Ev,−1,mI+1, (5)(flip-flip:) Ev,0,mI+1 = Ev,−1,mI, (6)where mI = {+1/2,−1/2,−3/2}, and Ev,mS ,mI≡⟨mS ,mI | Ĥv |mS ,mI⟩ represents the diagonal elementsof the Hamiltonian in Eq. (1). The magnetic fields thatsatisfy conditions Eqs. (5) and (6) are as follows:(flip-flop:) B+−,v,mI=Dv + (mI + 1)Av,zzγe + γn, (7)(flip-flip:) B++,v,mI=Dv +mIAv,zzγe + γn. (8)These equations tell that the magnetic fields that maxi-mize the flip-flop and flip-flip transitions differ. By usingzero-field splitting and hyperfine interactions (listed inTable I, introduced later), we can calculate the specificfield conditions. Then, we derive the range of the GSLACby the upper and lower limits of these magnetic fields asfollows:BGSLAC,low < Bz < BGSLAC,up,BGSLAC,low ≡ B+−,gs,+1/2 −|Ags,yy|γe + γn= 115.8 mT,BGSLAC,up ≡ B++,gs,−3/2 +|Ags,yy|γe + γn= 131.5 mT. (9)4III. SIMULATION BASED ON LINDBLADEQUATIONOne of the purposes of this study is to investigatehow far the experimental results can be explained by themodel based on the three 15N nuclear spins adjacent tothe V−B in h10B15N. In this subsection, we introduce theLindblad equation that considers the spin Hamiltonianand optical transitions described in the previous subsec-tion and explain how to calculate the nuclear spin polar-ization and ODMR spectra.The density matrix ρ̂ in the present model can be ex-pressed as follows:ρ̂ =∑m,nρmn |m⟩ ⟨n| ,|m⟩ , |n⟩ =∣∣∣level,mS ,m(i)I〉, (10)where∣∣∣level,mS ,m(i)I〉≡∣∣∣level,mS ,m(1)I ,m(2)I ,m(3)I〉(i = 1, 2, and 3). For level = GS or ES, the basis states∣∣∣level,mS ,m(i)I〉are labeled by mS = {+1, 0,−1} andm(i)I = {+1/2,−1/2}. The basis states∣∣∣MS,mS ,m(i)I〉are labeled by mS = {0} and m(i)I = {+1/2,−1/2}. ρ̂ isa 56×56 matrix (56 = (2×3+1×1)×23), and it followsthe Lindblad equation described below.∂ρ̂∂t= − iℏ[Ĥ, ρ̂] + D̂(ρ̂). (11)The Hamiltonian Ĥ in Eq. (11) is expressed as follows:Ĥ = Ĥgs ⊕ Ĥes ⊕ Ĥms,Ĥ(gs,es) = D(gs,es)Ŝz2+ γeB(Ŝz cos θ + Ŝx sin θ)+ E⊥(Sx2 − Sy2)−3∑i=1γnB(Î(i)z cos θ + Î(i)x sin θ) + Ĥhf,(gs,es),Ĥhf,(gs,es) =3∑i=1{A(gs,es),zzŜz Î(i)z +A(i)(gs,es),+(Ŝ+Î(i)− + h.c.)+A(i)(gs,es),−(Ŝ+Î(i)+ + h.c.)},Ĥms = −3∑i=1γnB(Î(i)z cos θ + Î(i)x sin θ)}. (12)Ĥgs, Ĥes, and Ĥms are the spin Hamiltonians for GS, ES,and MS, respectively. Ĥgs and Ĥes are similar to Eq. (1),but they additionally consider the tilt of the bias mag-netic field θ and strain E⊥. B denotes the magnitude ofthe magnetic field, and Bz = B cos θ. The strain param-eter E⊥ = 52 MHz originates from lattice distortions [32]and the surrounding charge distribution [33]. Ĥhf,(gs,es)represents the hyperfine interaction terms for GS and ES.Unless otherwise specified, the simulation in this paperuse the hyperfine parameters shown in Table I. All hy-perfine parameters, except for Ags,zz, are based on thefirst-principles calculations [23], scaled by a factor of ni-trogen isotope gyromagnetic ratios γ15Nn /γ14Nn = −1.4.Ags,zz = −64 MHz is derived from the experimental dataas described below.TABLE I. Hyperfine parameters used in our simulation [23].Parameter ValueAgs,xx −64 MHzAgs,yy −125 MHzAgs,zz −64 MHzA(1)es,xx −5.1 MHzA(2,3)es,xx −64 MHzA(1)es,yy −4.9 MHzA(2,3)es,yy −73 MHzA(i=1,2,3)es,zz −60 MHzA(1)es,xy 0 MHzA(2,3)es,xy 8.3 MHzA(i=1,2,3)es,yx −A(i=1,2,3)es,xyTABLE II. Transition or relaxation rates used in our sim-ulation. We neglect the electron spin and nuclear spin T1relaxations in ES.k Process Rate Γk1 Laser excitation: ΓL 2.14 µs−12 Radiative decay: Γ0 0.10 µs−13 Non-radiative decay: γe0 2.0 ns−14 Non-radiative decay: γe1 0.74 ns−15 Non-radiative decay: γg0 38 µs−16 Non-radiative decay: γg1 5.6 µs−17, 8 electron spin T1 in GS: 1/3T1,gs 1/(12 µs)/39 electron spin T2 in GS: 1/T2,gs 1/(180 ns)10 electron spin T2 in ES: 1/T2,es 1/(2 ns)11, 12 nuclear spin T1 in GS: 1/2T1,n 1/(2 ms)/213 nuclear spin T2 in GS and ES: 1/T2,n 1/(200 µs)The second term on the right-hand side of Eq. (11) isexpressed as follows:D̂(ρ̂) =∑kΓk(L̂kρ̂L̂†k − 12{L̂†kL̂k, ρ̂}). (13)Here, L̂k are jump operators describing non-Hermitianprocesses, and Γk are the transition or relaxation rates.The optical transitions from |m⟩ to |n⟩ are included inEq. (11) with a jump operator of L̂k = |n⟩ ⟨m|. T1 re-laxations are included in Eq. (11) with a jump operatorof L̂k = Ŝ+/√2, Ŝ−/√2, Î+, or Î−. The phase relax-ation in the superposition of |m⟩ and |n⟩ (e.g., T2 relax-ation) is included in Eq. (11) with a jump operator ofL̂k = |m⟩ ⟨m| − |n⟩ ⟨n|. As shown in Table II, we con-sider a total of 13 non-Hermitian processes (k = 1, 2, · · · ,and 13) in Eq. (11). The processes corresponding to5k = 1, 2, · · · , and 6 represent the optical transitions de-picted in Fig. 1(b). The processes for k = 7, 8, · · · , and13 describe the T1 and T2 relaxations of the V−B electronspin and 15N nuclear spins.Unless otherwise noted, we perform the simulation byinserting the parameters listed in Table II into Eq. (13).All optical-transition parameters (k = 1, 2, · · · , and 6)are determined by fitting the time-resolved PL (see Ap-pendix C). The T1 relaxation rates of the V−B electronspin in the GS (k = 7 and 8) are experimentally deter-mined, and the T2 relaxation rate in the ES (k = 10)is adopted from Ref. [27]. The other relaxation rates(k = 9, 11, 12, and 13) are adjusted to match the simu-lation results with the experimental results.Under the above conditions, we solve the steady-statesolution of the Lindblad equation for the following twocases:(A) Without microwave irradiation.(B) With microwave irradiation.We now explain the two cases of (A) and (B) in turn.For case (A), we numerically obtain the steady-stateρ̂sat with the Lindblad equation by converting Eq. (11)into a Liouville space form [see Appendix D]. We canobtain the polarization of the three adjacent 15N nuclearspins using the following formula:Psim,true =∑mImIρgs,mI(3/2)∑mIρgs,mI,ρgs,mI=∑∑m(i)I =mI∑mS〈GS,mS ,m(i)I∣∣∣ ρ̂sat ∣∣∣GS,mS ,m(i)I〉,(14)where Psim,true is a true (genuine) nuclear spin polariza-tion in the simulation. The present approach is similarto that proposed previously [25, 27].In experiments, the polarization is usually estimatedusing the fitting of ODMR spectra with the sum of fourLorentzians [25, 26]. We refer to this method as “4-dip fitting” and discuss experimental polarization valuesPexp,4dip and Pexp,4dip,area defined in Eqs. (17) and (19),respectively, later. This estimation assumes that the con-trast or area of a decomposed spectral component is pro-portional to the nuclear spin population. However, as wewill discuss in detail in Sec. VD, consideration shouldbe given that the estimation accuracy is degraded if theODMR spectra are distorted by factors other than thenuclear spin population.Now, we calculate the ODMR spectra in case (B) andcompare them either directly or with the results of 4-dipfitting to make a fairer comparison with the experimentalresults. We consider microwave irradiation by rewritingEq. (12) as follows:Ĥ(gs,es) → Ĥ(gs,es) +Bmw cos (2πfmwt)(γeŜx + γn3∑i=1Î(i)x ),Ĥms → Ĥms +3∑i=1γnBmw cos (2πfmwt)Î(i)x , (15)where Bmw and fmw are the microwave amplitude andfrequency, respectively. Using the ρ̂sat obtained in case(A) as the initial condition, we calculate the time evo-lution under microwave irradiation using Eq. (11). Wenumerically obtain the state at t = 1 µs when the sys-tem has sufficiently relaxed and regard this as the steadystate under microwave irradiation. The PL intensity Isimunder microwave irradiation is expressed as follows:Isim = Γ0ρes,ρes =∑mS ,m(i)I〈ES,mS ,m(i)I∣∣∣ ρ̂(t = 1µs)∣∣∣ES,mS ,m(i)I〉.(16)By calculating the PL intensity for each microwave fre-quency, we obtain ODMR spectra. By applying 4-dip fit-ting to these simulated spectra to emulate experimentalpolarization estimation, we obtain Psim,4dip. In Sec. VD,we evaluate the accuracy of the 4-dip fitting by how wellPsim,4dip agrees with Psim,true [Eq. (14)].We performed numerical simulation on a workstationequipped with dual intel CPUs (56 cores in total). Areasonable calculation speed was obtained using the Ju-lia packages DifferentialEquations.jl and Distributed.jl tocalculate time evolution.IV. EXPERIMENTAL METHODWe measure the ODMR of V−B in h10B15N synthesizedusing a metathesis reaction [26]. The hBN flake, approx-imately 70 nm thick, was stamped onto an 800 nm widegold wire on a sapphire substrate, which was fabricatedusing electron beam lithography (Elionix, ELS-F125).We used a helium ion microscope (Carl Zeiss Microscopy,Orion Plus HIM) to irradiate a (300 nm)2-sized spot withhelium ions (acceleration voltage of 30 keV, irradiationdose of 1015 /cm2) to create a V−B ensemble [13, 34].Using a home-built confocal microscope [35] equippedwith optical filters optimized for the PL wavelength ofV−B [36], we illuminated the V−B spot with a green laser(wavelength of 532 nm). The PL in the wavelength rangefrom 750 to 1000 nm was detected using a single-photoncounting module (Excelitas Technologies, SPCM-AQRH-16-FC-ND). Microwaves for manipulating the V−B elec-tron spin were supplied to the gold wire via coaxial cablesand coplanar waveguides from an analog signal genera-tor (Keysight, E8257D). The microwave magnetic fieldwas spatially concentrated around the thin gold wire630 60 90 120 150010002000300040005000600070008000120 12466006800700072000.940.960.981.004100 4300 45000.960.981.005300 5500 57000.960.981.007000 7200 74000.940.960.981.00MW frequency (MHz)(a) (b)(c)(d)MW frequency (MHz)Bz (mT)MW frequency (MHz)(b)(c) (d)unpolarizedESLAC>GSLACPL ratioPL ratioPL ratioPL ratioMagnetic field (mT)FIG. 2. ODMR spectra at an optical power of 0.73 mW. (a) Magnetic field dependence of the spectra is shown on a colormap. The vertical axis shows the microwave frequency, the horizontal axis indicates the magnetic field Bz, and the colorbar represents the PL ratio, which is the ratio of the PL intensity with and without microwaves. The map shows magneticresonances for both mS = 0 ↔ +1 and mS = 0 ↔ −1. (b)(c)(d) ODMR spectra corresponding to the mS = 0 ↔ +1 transitionat (b) Bz = 30.7 mT, (c) Bz = 75.0 mT, and (d) Bz = 132.8 mT. The vertical axis shows the PL ratio, and the horizontal axisshows the microwave frequency. The black solid lines represent the fitting results of four Lorentzians (“4-dip fitting”). Thedecomposed Lorentzians are shown as solid blue, green, or red lines in (b), (c), and (d), respectively.(∼ 800 nm), allowing us to drive the V−B electron spinwithout using a microwave amplifier. This configurationprovides uniform microwave field strength over a widefrequency range from 1 MHz to 8 GHz. A magnetic fieldparallel to the quantization axis of the V−B was appliedin the range from 10 to 150 mT using an electromag-net (GMW, 5203). The alignment of the magnetic fieldwas performed by moving a manual stage carrying theelectromagnet.V. RESULTS AND DISCUSSIONA. Magnetic field dependence of ODMR spectraWe measured the ODMR spectra for mS = 0 ↔ +1and mS = 0 ↔ −1 at Bz = 10–150 mT. We presentthe result obtained at the optical power of 0.73 mW asfunctions of the microwave frequency and Bz as a colormap in Fig. 2(a). The transitions between mS = 0 ↔ ±1decrease the PL ratio, which is the ratio of the PL in-tensity with and without microwaves. The upper (lower)branch corresponds to the resonance of mS = 0 ↔ +1(mS = 0 ↔ −1). Each resonance frequency shift is equiv-alent to the product of the magnetic field and the gyro-magnetic ratio γe due to the Zeeman splitting. Note thatthe ODMR signal of the ES is small enough to be ignoredat the microwave and optical power in this experiment(< 0.2% at Bz = 0 mT).We focus on the ODMR spectra at several character-istic magnetic fields, as indicated by the arrows (b), (c),and (d) in Fig. 2(a). Figure 2(b) shows the spectrumat Bz = 30.7 mT, away from the LACs. The spec-trum consists of four dips. They correspond to the elec-tron spin resonances under mI = +3/2,+1/2,−1/2, and−3/2. The separation between the dips equals the hy-perfine interaction strength of |Ags,zz| = 64 MHz, whichis consistent with the previous work [24, 26, 27]. Thesymmetric shape of the spectrum tells that the nitrogennuclear spins are not polarized.On the other hand, the ODMR spectra are not sym-metric at specific magnetic fields. Figure 2(c) displaysthe spectrum at Bz = 75.0 mT on the ESLAC; The dipsfor mI > 0 are enhanced, suggesting that the nuclearspins are polarized in the mI > 0 direction. Figure 2(d)shows the spectrum at Bz = 132.8 mT, slightly abovethe range of the GSLAC (Bz > BGSLAC,up = 131.5 mT)[see Eq. (9)]. Again, as in Fig. 2(c), the asymmetric spec-trum suggests the nuclear spin polarization. Note thatthe spectra under the GSLAC [inset in Fig. 2(a)] are morecomplex and varied, making it difficult to estimate thepolarization straightforwardly. We will discuss this pointin more detail later in Fig. 5.To estimate the nuclear spin polarization, we use the4-dip fitting. The fitting by four Lorentzians (solid lines)7reproduces the experimental ODMR spectra (markers)well, as presented in Figs. 2(b), (c), and (d). The fit-ting yields the contrast CmIand line width νmIof eachdip. The area of each dip AmIcan be estimated fromthe contrast and line width. In the conventional method,the nuclear spin polarization is calculated by consideringthat the area of each dip corresponds to the occupancy ofeach mI . However, the correspondence between the diparea and the occupancy does not always hold becausethe line width varies in a complex manner due to com-petition between microwave driving and other inhomo-geneous broadening. Therefore, in this study, we simplyestimate nuclear spin polarization by assuming that thecontrast of each dip corresponds to the occupancy of eachmI using the following equation:Pexp,4dip =∑mImICmI(3/2)∑mICmI. (17)We obtain Pexp,4dip = 0.1±0.1%, 14±1%, and 26±3% forFigs. 2(b), (c), and (d), respectively. The accuracy of theestimation method based on the 4-dip fitting, includingthe area-based one, is discussed later in Sec. VD.B. Magnetic field dependence of polarizationWe discuss the magnetic field dependence of the po-larization of the adjacent 15N nuclear spins, estimatedby analyzing the experimental ODMR spectra. First,we focus on the result obtained at the optical power of0.73 mW shown in the middle panel of Fig. 3. The mark-ers present the magnetic field dependence of the polar-ization Pexp,4dip [Eq. (17)] obtained for the ODMR spec-tra shown in Fig. 2(a) using the 4-dip fitting. Pexp,4dipincreases around ESLAC (∼ 75 mT) and GSLAC (∼120 mT) except for the middle of GSLAC (gray area).We can successfully reproduce a similar behavior us-ing the simulation discussed in Sec. III; The magneticfield dependence of Psim,true [Eq. (14)] is superposed bythe black solid line in the same panel. Pexp,4dip andPsim,true behave similarly in the magnetic fields. It isevident that the flip-flop interaction contributes to theincrease in polarization. Furthermore, we obtain a steepdecrease in polarization Psim,true at the middle of GSLAC(Bz = 122.0 mT). The flip-flip interaction for the groundstate is most pronounced at this magnetic field.Next, we show the optical power dependence. Thetop and bottom panels of Fig. 3 are the results obtainedat the optical powers of 0.18 mW and 1.55 mW, re-spectively. Similar to the behavior in the middle panel(0.73 mW), the polarization Pexp,4dip changes around ES-LAC and GSLAC in both cases. Also, Fig. 3 indicatesthat the higher the optical power, the larger the polar-ization change appears near the ESLAC.We remark that the 4-dip fitting analysis is notstraightforward near GSLAC, shown in gray in Fig. 3,because mixing electron and nuclear spins results in com-plex ODMR spectra. Although our simulation depicted: Simulation: Simulation: Simulation: Experiment (4-dip): Experiment (4-dip): Experiment (4-dip)0.18 mW0.73 mW1.55 mWGSLACPolarizationPolarizationPolarization0 30 60 90 120 1500.00.10.20.30.40 30 60 90 120 1500.00.10.20.30.40 30 60 90 120 1500.00.10.20.30.4Magnetic field (mT)FIG. 3. Magnetic field dependence of the polarization ofthe adjacent 15N nuclear spins. The top, middle, and bottompanels are obtained at optical powers of 0.18, 0.73, and 1.55mW, respectively. Markers indicate the polarization Pexp,4dipestimated using the 4-dip fitting of ODMR spectra for mS =0 ↔ +1 and Eq. (17). The black solid lines are the truenuclear polarization Psim,true [Eq. (14)] in the steady stateunder continuous optical excitation in our simulation. Thegray area indicates the range of GSLAC (BGSLAC,low < Bz <BGSLAC,up) [Eq. (9)]. We use θ = 0.6◦ in the simulation.by the solid black lines qualitatively reproduces the trendunder GSLAC, even this observation is theoretically non-trivial. A more quantitative discussion and analysis willbe made later in Secs. VC and VD.So far, our simulation reproduces the observed behav-ior qualitatively well. At the end of this subsection,we point out an experimental result that cannot be re-produced. Figure 4(a) shows the estimated polarizationPexp,4dip for mS = 0 ↔ +1 and mS = 0 ↔ −1 un-der a laser power of 0.73 mW. The polarizations ob-tained in these two transitions agree very well, exceptnear Bz ∼ 91 mT; A prominent peak appears at Bz ∼ 91mT only for the mS = 0 ↔ +1 transition. Such a be-havior does not exist in the simulated Psim,4dip shownby a solid line. To examine this phenomenon more,we present the ODMR spectra for mS = 0 ↔ −1 andmS = 0 ↔ +1 at Bz ∼ 91 mT in Figs. 4(b) and (c), re-spectively. The ODMR contrasts corresponding to eachnuclear spin state clearly differ between the two spectra.We believe this is due to something we did not consider8PL ratio800 1000 12000.920.940.960.981.005800 6000 62000.940.960.981.000.73 mW( )( ) Polarization(a)(b) (c)MW frequency (MHz)(c)(b)GSLAC0 30 60 90 120 1500.00.10.20.30.4Magnetic field (mT)FIG. 4. (a) Comparison of Pexp,4dip as a function of Bzbetween mS = 0 ↔ +1 (green) and mS = 0 ↔ −1 (navy).The optical power is fixed at 0.73 mW. The black solid linepresents the corresponding Psim,true. Pexp,4dip (mS = 0 ↔+1) and Psim,true are the same as those shown in the middlepanel of Fig. 3. (b) ODMR spectrum and the result of the4-dip fitting for mS = 0 ↔ −1 (navy) at Bz = 91.0 mT.(c) ODMR spectrum and the result of the 4-dip fitting formS = 0 ↔ +1 (green) at Bz = 91.2 mT.in our simulation model, such as nuclear spins other thanthe nearest-neighbor nitrogen atoms [37] or fast and com-plex optical transitions [38, 39]. We leave the identifica-tion of this cause as a future work.C. ODMR spectra at GSLACWe discuss the ODMR spectra at GSLAC in detail andinvestigate their consistency with our simulation model.First, we show the experimental ODMR spectra Iexp(red markers) and simulated spectra Isim (black solidlines) at various magnetic fields Bz between 115.3 and127.0 mT in the left panel of Fig. 5. As the magneticfield approaches 121 mT, the experimental spectra be-come broad so that the four dips are almost indistin-guishable. Correspondingly, the simulation results indi-cate that the spectra shape becomes complex, and manydips appear around 121 mT.The above behavior is due to mixing the electron andnuclear spins in V−B at GSLAC. The number of dips in-creases because the complex mixing increases the numberof transitions between eigenstates. Theoretically, strongmixing can occur under conditions where Eq. (9) is sat-isfied. In the present case, the magnetic field conditionswith strong mixing are estimated to beBz = 120.2, 121.1,and 122.0 mT, which are consistent with experiments andsimulations. Such mixing is also observed in the LAC ofdiamond nitrogen-vacancy centers [40, 41].Next, we discuss the differences between the experi-mental and simulated spectra. The simulated spectraare sharper than the experimental ones due to hyper-fine interactions not fully considered in our simulations.Our model described in Sec. III considers line broadeningdue to microwave or laser irradiation [42], T2 relaxation,and T1 relaxation (see Table II). On the other hand, itdoes not count inhomogeneous linewidths due to hyper-fine interactions with boron nuclear spins, which is themain reason for the experimental linewidth broadening ofV−B [37]. Accurately accounting for multiple boron spinswith large nuclear spins is difficult because it dramati-cally increases the size of the Hamiltonian and, thus, thecomputational cost.We, therefore, reproduce the experimental spectra byempirically incorporating inhomogeneous linewidths. Weregard them as resulting from a static magnetic field dis-tribution by assuming that all but the nearest-neighbornitrogen nuclear spins are stationary during individualODMR measurements. Thus, the spectra are expressedby the following convolution:Iconv(B) ∝∑B′=B+nδBP (B′ −B)Isim(B′), (18)where P (B) =(∆B/2)2B2 + (∆B/2)2is the shape of inhomo-geneous broadening, ∆B = 50 (MHz)/γe = 1.78 mT isthe effective field width of the broadening, δB = 0.1 mT,and n = {−25,−24, · · · , 25}. We appropriately choosethe range of n so that the simulation reproduces the ex-perimental spectral shape, while a slight change in n doesnot affect the result. We perform the normalization toensure that the area of the spectrum remains conservedafter the convolution.The right panel of Fig. 5 compares the experimentalspectra Iexp (red markers) and the simulated spectra con-sidering the inhomogeneous broadening Iconv (blue solidlines). They agree satisfactorily. It is significant to notethat the agreement implies that even in GSLAC, theODMR spectra can be explained only by the dynamicsof the three adjacent 15N nuclear spins. Slight spectraldifferences between the experiment and simulation wouldbe caused by variations in experimental microwave poweror by polarization or mixing of nuclear spins other thanthe nearest-neighbor nitrogen spins.D. Comparison based on the 4-dip fittingWe are interested in how much quantitative informa-tion about DNP can be obtained from the experimentalODMR spectra. Therefore, this subsection compares ex-periments and simulations using the 4-dip fitting. We9PL (a. u.)MW frequency (MHz)115.3 mT118.4 mT119.3 mT120.2 mT121.1 mT122.0 mT122.9 mT123.8 mT127.0 mT115.3 mT118.4 mT119.3 mT120.2 mT121.1 mT122.0 mT122.9 mT123.8 mT127.0 mT6500 6800 7100 74000.20.40.60.81.06500 6800 7100 74000.60.70.80.91.0FIG. 5. ODMR spectra near GSLAC at various magnetic fields Bz between 115.3 to 127.0 mT under the optical power of0.18 mW. (Left panel) The experimental spectra Iexp (red markers) and the simulated spectra Isim (black solid lines). (Rightpanel) The red markers show Iexp (the data are the same as those in the left panel), and the simulated spectra consideringinhomogeneous broadening Iconv are demonstrated by blue solid lines. In this simulation, we use θ = 0.6◦, ΓL = 2.14 MHz,and Bmw = 0.25 mT.treat the simulated spectra with inhomogeneous broad-ening as if they were experimental data and emulate thepolarization estimation in the same way as in Eq. (17).We define the resulting polarization as Psim,4dip, as brieflymentioned at the end of Sec. III.We start to investigate the validity of our model bycomparing this Psim,4dip and Pexp,4dip [Eq. (17)]. Fig-ure 6(a) shows Psim,4dip (blue triangles) and Pexp,4dip (redcircles). Clearly, Pexp,4dip and Psim,4dip are in good agree-ment; their behavior in magnetic fields is quantitativelysimilar. This agreement indicates that our simulation re-produces essential features of the experimental ODMRspectra nicely. We have already seen such examples inthe right panel of Fig. 5.How about the nuclear polarization? The true polar-ization Psim,true [Eq. (14)] obtained in the simulation issuperposed in Fig. 6(a) by a solid black line. We noticethat Psim,true and Psim,4dip disagree. In particular, thereis a difference of a few times around Bz=100–120 mT.Note that both values are purely derived from the model,not experiments. This discrepancy implies that the accu-racy of the polarization estimation by Eq. (17) using the4-dip fitting is unsatisfactory, especially around GSLAC.Now, it is interesting to examine the conventional es-timation method using the 4-dip fitting. Usually, thefollowing formula, which relies on the area of the spec-trum instead of the contrast, is used to estimate polar-ization [25–27].Pexp,4dip,area =∑mImIAmI(3/2)∑mIAmI, (19)where AmIis the area of the spectral component cor-responding to the mI state. We apply Eq. (19) to theexperimental data and the simulation. The resulting val-ues are denoted as Pexp,4dip,area and Psim,4dip,area, respec-tively. Figure 6(b) compares Psim,4dip,area (blue trianglemarkers) and Pexp,4dip,area (red circles) with Psim,true su-perposed in a solid black line. As previously, Pexp,4dip,areaand Psim,4dip,area agree well. However, they significantlydeviate from the true value Psim,true, especially aroundGSLAC. This observation proves that the conventionalarea-based polarization estimation using the 4-dip fittingdoes not work quantitatively.Interestingly, the results using contrast (Psim,4dip) arecloser to the true polarization Psim,true than the conven-tional area-based estimation (Psim,4dip,area). It may sug-gest the usefulness of the contrast-based fitting methodfor estimating polarization.The quantitative inaccuracy of the 4-dip fitting in bothexperiments and simulation is due to the mixing between10(a)(b)PolarizationPolarizationMagnetic field (mT)0 30 60 90 120 1500.10.00.10.20.30.40 30 60 90 120 1500.10.00.10.20.30.4FIG. 6. Comparison of the 15N nuclear spin polarization ob-tained by several methods. (a) Estimated polarization basedon the spectral contrast. The blue triangles show Psim,4dip,and the red circles present Pexp,4dip [Eq. (17)]. (b) Estimatedpolarization based on the spectral area. The blue trianglesshow Psim,4dip,area, and the red circles display Pexp,4dip,area[Eq. (19)]. The black solid lines in both (a) and (b) arePsim,true [Eq. (14)]. Psim,true and Pexp,4dip are the same asthose shown in the top panel of Fig. 3.the V−B electron spin and the 15N nuclear spins. Asthe mixing becomes intense, the number of dips in Isimincreases, invalidating the assumptions of the 4-dip fit-ting, namely, ρgs,mI∝ CmIor ρgs,mI∝ AmI. For in-stance, the differences between Psim,true and Psim,4dip, aswell as Psim,true and Psim,4dip,area, are most pronouncedat around GSLAC. The left panel of Fig. 5 shows thatIsim contains numerous additional dips beyond the mainfour dips, indicating that the effects of mixing cannotbe ignored. In contrast, at Bz = 127.0 mT, the edgeof GSLAC, Psim,true closely matches both Psim,4dip andPsim,4dip,area and Isim exhibits a more evident four-dipstructure with reduced mixing effects. The correlationbetween the number of dips in the simulated ODMRspectrum Isim and the differences between Psim,true andPsim,4dip is consistent with the conclusion that mixingdegrades the quantitative reliability of the 4-dip fitting.E. Maximum polarizationBased on our simulation model, we discuss maximumpolarization and the factors limiting it. Our experimentsand simulations [Figs. 3, 4, and 6] suggest that the po-larization of 15N nuclear spins is at most approximately20 %. A similar maximum polarization (∼30 %) was re-PolarizationPolarization1.89 MHz3.77 MHz7.54 MHz15.1 MHz= 0 MHz(a)(b)(a)0 30 60 90 120 1500.00.20.40.60.81.010−2 100 1020.00.20.40.60.81.0(at MHz)(at )at mT (b) mT Magnetic field (mT)(μs)μsFIG. 7. (a) Bz dependence of Psim,true for several differ-ent flip-flip interaction strengths (Ags,−). T2,n is fixed at200 µs. (b) T2,n dependence of Psim,true at Bz = 132 mT,corresponding to (a). The present situation corresponds toAgs,− = 15.1 MHz, calculated from the values in Table I. Weset θ = 0◦ and ΓL = 2.14 MHz in this simulation. The blackhorizontal dashed line in (a) and (b) represents the polar-ization Pe between the electron spin sublevels mS = 0 andmS = −1 when Ags,− = 0 MHz.ported in the previous studies [25, 27]. These values arefar from the high polarization (∼100 %) obtained in theDNP of a nitrogen nuclear spin of the NV center [43]. Un-derstanding the factors that limit maximum polarizationis one of the most fundamental issues in DNP studies.As described in Sec. II, the flip-flip interactionA(i)− Ŝ+Î(i)+ + h.c. reduces the polarization [24, 25, 27].According to Eq. (4), the flip-flop and flip-flip interac-tions in the GS can be described by two independentparameters: Ags,+ ≡ |A(i)gs,+| = |Ags,xx + Ags,yy|/4 andAgs,− ≡ |A(i)gs,−| = |Ags,xx − Ags,yy|/4, respectively. Tocheck the impact of the flip-flip term on the flip-flopterm, the magnetic field dependence of the polarizationPsim,true simulated for several different Ags,− is shownin Fig. 7(a) while maintaining Ags,+. Except for thenarrow magnetic field region (Bz ∼ 121.4 mT), the po-larization increases as Ags,− decreases. If the flip-flipinteraction were quenched (Ags,− = 0 MHz), the maxi-mum polarization could exceed 65 %, but for the present11flip-flip interaction strength, which is calculated to beAgs,− = 15.1 MHz from the values in Table I, the maxi-mum is below 30 %. Thus, the flip-flip interaction is themain factor that limits the maximum polarization.We continue to discuss in more detail to find otherfactors that suppress the maximum polarization. Whenconsidering only the flip-flop interactions, the maximumpolarization achievable in the DNP coincides with theelectron spin polarization. On the other hand, as indi-cated by the horizontal black dashed line in Fig. 7(a),even in the absence of the flip-flip interaction (Ags,− =0 MHz), the polarization does not match with the elec-tron spin polarization Pe between the electron spin sub-levels mS = 0 and mS = −1, as can be calculated asfollows:Pe =ρgs,mS=0 − ρgs,mS=−1ρgs,mS=0 + ρgs,mS=−1,ρgs,mS=∑m(i)I〈GS,mS ,m(i)I∣∣∣ ρ̂sat ∣∣∣GS,mS ,m(i)I〉. (20)Usually, the discrepancy between maximum nuclear spinpolarization and electron spin polarization can occur ow-ing to the fast nuclear spin relaxation. In the simula-tions, we set the nuclear spin decoherence and relaxationas 1/T2,n = 1/200 µs−1 and 1/T1,n = 1/2 ms−1, respec-tively (Table II). They are much smaller than the inverseof the flip-flop or flip-flip interaction strengths, so suchtrivial suppression should be weak.Figure 7(b) presents the T2,n dependence of the polar-ization at 132 mT, simulated for several different Ags,−corresponding to Fig. 7(a). Without the flip-flip inter-action (Ags,− = 0 MHz), the polarization is the largestat T2,n ∼ 1 µs, and the maximum value is nearly iden-tical to the electron spin polarization Pe. This observa-tion implies the existence of a polarization suppressionthat depends on nuclear spin coherence. As expected,the polarization suppression occurs as T2,n decreases be-low ∼ 0.1 µs. Interestingly, the polarization is againsuppressed as T2,n exceeds ∼ 10 µs, which is counter-intuitive. This suggests that polarization suppression forlong T2,n is not trivial simply due to spin relaxation.Figure 7(b) indicates that the coherence-dependentsuppression appears similarly for finite Ags,−. The maxi-mum polarization is suppressed as the flip-flip interactionbecomes stronger. This suppression includes the influ-ence of reduced electron spin polarization in the pres-ence of the flip-flip interaction (data not shown). Thestronger suppression for short T2,n and large Ags,− is dueto lowering the effective polarization speed under the co-existence of flip-flop and flip-flip interactions. The above-mentioned nontrivial suppression at long T2,n also varieswith the flip-flip interaction. As the flip-flip interactionincreases, the dependence on T2,n weakens and convergesto a constant value. Thus, the polarization in the presentinteraction Ags,− = 15.1 MHz remains almost constantat slightly above 20% for T2,n ≳ 1 µs. This behavior in-fers that the above nontrivial suppression for longer T2,nis irrelevant in the present case.F. Role of dark statesFinally, we examine the suppression that occurs whenT2,n is very long. Generally, in coherent quantum sys-tems, it is known that transitions can be suppressed bydestructive interference of the entanglement states, called“dark states” [44]. The dark states tend to arise in highlysymmetric systems [45]. This requirement is satisfied inhBN, where the three nitrogen nuclear spins adjacent toV−B are equivalent.To describe the dark state in V−B , we represent the or-thogonal bases for the three nuclear spins by the followinglinear combinations:|ϕ1⟩ =1√3(|++−⟩+ ei2π3 |+−+⟩+ ei4π3 |−++⟩),|ϕ2⟩ =1√3(|++−⟩+ ei4π3 |+−+⟩+ ei2π3 |−++⟩),|ϕ3⟩ =1√3(|++−⟩+ |+−+⟩+ |−++⟩) ,|± ± ±⟩ ≡∣∣∣m(1)I = ±1/2,m(2)I = ±1/2,m(3)I = ±1/2〉.(21)Using these orthogonal bases, we can express the flip-flopon the mS = 0 in GS as follows,Ags,+3∑i=1Ŝ−Î(i)+ |mS = 0⟩ |ϕ1⟩ = 0,Ags,+3∑i=1Ŝ−Î(i)+ |mS = 0⟩ |ϕ2⟩ = 0,Ags,+3∑i=1Ŝ−Î(i)+ |mS = 0⟩ |ϕ3⟩ =√3 |mS = −1⟩ |+++⟩ .(22)Since |ϕ1⟩ and |ϕ2⟩ are transparent to the flip-flop in-teraction, they are dark states. With these dark stateskept coherent, the transition to the fully polarized state|+++⟩ is suppressed, reducing the achievable polariza-tion. This mechanism is consistent with the polarizationsuppression for long T2,n observed in Fig. 7(b). In otherwords, when these states are phase relaxed, i.e., someof them transition to |ϕ3⟩, it can increase the polariza-tion [46].We note that the influence of the dark state or entan-glement may appear in other defects in hBN, as it is anuclear-spin-rich crystal with high symmetry. It wouldbe more pronounced in isotopically enriched hBN withincreased symmetry. Despite these arguments, our simu-lations suggest that the impact of the dark state hardlyappears experimentally in the present case, which is at-tributed to the strong flip-flip interaction due to the sym-metry of V−B . However, this does not negate the entan-glement effect in other defects in hBN [47–56]. Therefore,focusing on the influence of the nuclear spin coherence inhBN remains essential.12VI. CONCLUSIONWe have established that the DNP and ODMR spectrain h10B15N can be explained by a model that considersthe electron spin of V−B and three adjacent nitrogen nu-clear spins. Our model based on the Lindblad equationreproduced the ODMR spectra in a wide range of mag-netic fields, including GSLAC. We found that the polar-ization estimations based on the 4-dip fitting, includingthe conventional area-based method, are not highly accu-rate. Developing a rigorous process to estimate the truepolarization from ODMR spectra quantitatively is an im-portant future direction. Our model also revealed thatDNP in V−B is limited by the flip-flip interactions and lessaffected by nuclear spin coherence. We discussed the rel-evance of the symmetry-induced polarization suppressionmechanisms, which may be observed in other defects inhBN, a nuclear-spin-rich highly-symmetric crystal. Ourquantitative understanding of the fundamental behaviorof the most representative quantum defects in hBN willcontribute to the theoretical developments and applica-tions of the dynamics of electron and nuclear spins ofdefects in two-dimensional materials.ACKNOWLEDGMENTSWe thank Mr. Tomohiko Iijima (AIST) for the usage ofAIST SCR HIM for helium ion irradiations, Dr. Toshi-hiko Kanayama (AIST) for helpful discussions since theintroduction of HIM at AIST in 2009, and Prof. KoheiM. Itoh (Keio University) for letting us use the confocalmicroscope system. This work was partially supported byJST, CREST Grant No. JPMJCR23I2, Japan; Grants-in-Aid for Scientific Research (Grants No. JP24K21194,No. JP23K25800, and No. JP22K03524); “Advanced Re-search Infrastructure for Materials and Nanotechnologyin Japan (ARIM)” (Proposal No. JPMXP1224UT1056and No. JPMXP1224NM0055) of the Ministry of Ed-ucation, Culture, Sports, Science and Technology ofJapan (MEXT); “World Premier International ResearchCenter Initiative on Materials Nanoarchitectonics (WPI-MANA)” supported by MEXT; the Mitsubishi Founda-tion (Grant No. 202310021); and the Cooperative Re-search Project of RIEC, Tohoku University. Y.N. andS.N. acknowledge the financial support from FoPM, theWINGS Program, The University of Tokyo, and theJSPS Young Researcher Fellowship (No. JP24KJ0692and No. JP24KJ0657).Appendix A: Symmetry of hyperfine interactionsWe explain the necessary conditions for the hyperfineinteractions imposed by the symmetry of V−B . Figure 8(a)shows a schematic of the hyperfine interactions when V−Bis in the GS. According to first-principles calculations,the symmetry of the system in the GS is D3h [23]. TheVB(b)(a)GS: VBES: VB-- VB--FIG. 8. Configuration of the hyperfine interactions betweenthe V−B electron spin and the three adjacent 15N nuclear spinsin (a) GS and (b) ES. (b) shows that, according to first-principles calculations [23], two of the hyperfine interactions(red, i = 2 and 3) are equivalent, while the remaining one(blue, i = 1) is significantly smaller in the case of ES.symmetry concerning a rotation by ϕ ≡ 2π/3 in the x-yplane requires the following condition:A(2)xx = Axxcos2ϕ+Ayysin2ϕ,A(2)yy = Axxsin2ϕ+Ayycos2ϕ,A(2)xy = (Axx −Ayy)sinϕcosϕ,A(2)yx = A(2)xy , (A1)where A(1)xx ≡ Axx and A(1)yy ≡ Ayy. The mirror symme-try with respect to the x-z plane imposes the followingconditions:A(1)xy = A(1)yx = 0,A(3)xx = A(2)xx ,A(3)yy = A(2)yy ,A(3)xy = −A(2)xy ,A(3)yx = A(3)xy . (A2)We get Eq. (4) from Eqs. (3), (A1), and (A2).On the other hand, first-principles calculations indi-cate that in the ES, the system quickly relaxes to a statewith C2v symmetry, as shown in Fig. 8(b) [23]. From asimilar discussion as in the GS case, the following relationis obtained:A(1)xy = A(1)yx = 0,A(2)xx = A′xxcos2ϕ+A′yysin2ϕ,A(2)yy = A′xxsin2ϕ+A′yycos2ϕ,A(2)xy = (A′xx −A′yy)sinϕcosϕ,A(2)yx = A(2)xy ,A(3)xx = A(2)xx ,A(3)yy = A(2)yy ,A(3)xy = −A(2)xy ,A(3)yx = A(3)xy , (A3)130 30 60 90 120 150PL (a.u.)0.70.80.91.01.1ExperimentMagnetic field (mT)FIG. 9. Magnetic field orientation dependence of PL inten-sity.where A(1)xx ≡ Axx ̸= A′xx and A(1)yy ≡ Ayy ̸= A′yy.Eqs. (3) and (A3) lead to the following conditions:A(i=1)es,+ =Aes,xx +Aes,yy4,A(i=2,3)es,+ =A′es,xx +A′es,yy4,A(i=1)es,− =Aes,xx −Aes,yy4,A(i=2,3)es,− =A′es,xx −A′es,yy4e2iϕ(i),ϕ(2) = −2π/3, ϕ(3) = −ϕ(2). (A4)Under Axx, Ayy, A′xx, A′yy < 0, it immediately followsthat |A(i)es,+| > |A(i)es,−|.Appendix B: Effects of magnetic field misalignmentMagnetic field misalignment alters the magnetic fielddependence of PL intensity. Figure 9 shows the Bz de-pendence of PL intensity. Black markers represent exper-imental data, and the solid lines in different colors corre-spond to PL intensity I = Γ0ρes simulated by the steady-state solutions of the Lindblad equation without consid-ering microwaves. Each color represents a misalignmentangle θ = 0◦, 0.5◦, 1.5◦, and 3.0◦. We note that the ex-perimental and simulated behaviors qualitatively agree.However, the changes in the Bz dependence of PL inten-sity for θ = 0.5◦–1.5◦ are not significant, making it diffi-cult to estimate the angle θ accurately. The PL intensityreduction near the LAC is attributed to the mixing ofthe V−B electron spin ground-state mS levels [30, 57, 58].Appendix C: Time-resolved PLWe experimentally estimate the optical transition pa-rameters of V−B shown in Fig. 1(b) by fitting the time-resolved PL using a five-level model of V−B [30, 59]. TheTime (ns)Count (a.u.)experiment (0.73 mW)fit  (0.73 mW)experiment (0.18 mW)fit  (0.18 mW)experiment (0.73 mW)fit  (0.73 mW)experiment (0.18 mW)fit  (0.18 mW)(a) (b)0 300 600 900 12000.20.40.60.81.00 300 600 900 12000.20.40.60.81.0FIG. 10. Time-resolved PL of V−B at Bz = 35 mT andoptical powers of 0.18 (light blue) and 0.73 mW (pink). Thevertical axis represents photon counts, and the horizontal axisshows the time elapsed after the readout laser is turned on.(a) Time-resolved PL after polarization into mS = 0 by laserirradiation. (b) Time-resolved PL after applying an adiabaticinversion pulse [60, 61] corresponding to the mS = 0 ↔ +1transition.estimated optical transition parameters are used in oursimulations based on the Lindblad equation.Figure 10 shows experimental data of the time-resolvedPL obtained at Bz = 35 mT. The optical powers areset at 0.18 and 0.73 mW, indicated by light blue andpink markers, respectively. Figure 10(a) shows the time-resolved PL after polarizing to mS = 0 via initializa-tion laser irradiation, and Fig. 10(b) shows the time-resolved PL after applying an adiabatic inversion pulsecorresponding to the mS = 0 ↔ +1 transition [60, 61].The vertical axis represents photon counts, and the hori-zontal axis represents the time elapsed since the readoutlaser is turned on.We fit the experimental data using the time-resolvedPL described by the rate equations. The occupationvector for each level in Fig. 1(b) is given by n =[n0,gs, n1,gs, n0,es, n1,es, nms]T, where n0,gs (n0,es) repre-sents the occupation of the mS = 0 state in the GS (ES),n1,gs (n1,es) represents the occupation of the mS = ±1state in the GS (ES), and nms corresponds to that inthe MS. The time-resolved PL intensity at a time t,Γ0(n0,es + n1,es), follows the rate equation below:n(t) = eAtn(t = 0),A =−ΓL 0 Γ0 0 γg00 −ΓL 0 Γ0 γg1ΓL 0 −(γe0 + Γ0) 0 00 ΓL 0 −(γe1 + Γ0) 00 0 γe0 γe1 −(γg0 + γg1) .(C1)We set the initial condition of the occupationvector in the case of Figs. 10(a) and (b) asn(t = 0) = [p, 1− p, 0, 0, 0]Tand n(t = 0) =[p− q, 1− p+ q, 0, 0, 0]T, respectively. The fitting pa-rameters are {ΓL,Γ0, γe0, γe1, γg0 , γg1 , p, q}. The time-resolved PL obtained from this fitting is shown by the14solid lines in Fig. 10. The values estimated from thisfitting are compiled in Table II.Appendix D: Lindblad equation in Liouville spaceWe describe the method for calculating the Lindbladequation to obtain Psim,true, as mentioned in Sec. III. Wevectorize the density matrix ρ̂ as follows:ρ̂ =ρ11 ρ12 · · · ρ1nρ21 ρ22 · · · ρ2n....... . ....ρn1 ρn2 · · · ρnn → |ρ⟩⟩ =ρ11ρ12...ρ1nρ21...ρnn. (D1)We convert Eq. (11) into the following Liouville spaceform [62]:∂∂t|ρ⟩⟩ = L̂ |ρ⟩⟩,L̂ = − iℏĤ + D̂ ,Ĥ = Ĥ ⊗ I56 + I56 ⊗ ĤT ,D̂ =∑kΓk{L̂k ⊗ L̂†Tk − 12(L̂†kL̂k)⊗ I56 −12I56 ⊗ (L̂†kL̂k)T}.(D2)I56 is the 56× 56 identity matrix. Given that L̂ is time-independent, we calculated the steady-state solution ρ̂satof Eq. (D2) by numerically solving the null space of the3136 × 3136 matrix L̂ . Importantly, unless the relax-ation of the 15N nuclear spin is included in the Lindbladequation, the non-zero steady-state solution of Eq. (D2)is not uniquely determined.[1] J. Weber, W. Koehl, J. Varley, A. 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Gyamfi, Fundamentals of quantum mechanics inLiouville space, European Journal of Physics 41, 063002(2020).https://doi.org/10.1038/s41563-020-00850-yhttps://doi.org/10.1038/s41563-020-00850-yhttps://doi.org/10.1038/s41563-021-00979-4https://doi.org/10.1038/s41563-021-00979-4https://doi.org/10.1038/s41467-023-38672-6https://doi.org/10.1021/acsanm.3c01047https://doi.org/10.1021/acsanm.3c01047https://doi.org/10.1038/s41467-024-51129-8https://doi.org/10.1038/s41467-024-51129-8https://doi.org/10.1021/acs.nanolett.4c01333https://doi.org/10.1038/s41563-024-01887-zhttps://doi.org/10.1038/s41563-024-01887-zhttps://arxiv.org/abs/2407.13148https://arxiv.org/abs/2407.13148https://arxiv.org/abs/2407.13148https://arxiv.org/abs/2409.01601https://arxiv.org/abs/2409.01601https://arxiv.org/abs/2409.01601https://arxiv.org/abs/2409.20186https://arxiv.org/abs/2409.20186https://arxiv.org/abs/2409.20186 Systematic investigation of dynamic nuclear polarization with boron vacancy in hexagonal boron nitride Abstract Introduction  DNP in hexagonal boron nitride Simulation based on Lindblad equation Experimental method Results and discussion Magnetic field dependence of ODMR spectra Magnetic field dependence of polarization ODMR spectra at GSLAC Comparison based on the 4-dip fitting  Maximum polarization  Role of dark states Conclusion Acknowledgments Symmetry of hyperfine interactions Effects of magnetic field misalignment Time-resolved PL Lindblad equation in Liouville space References