Abstract
Negatively charged boron vacancies (V_{B}^{−}) in hexagonal boron nitride (hBN) are a rapidly developing qubit platform in twodimensional materials for solidstate quantum applications. However, their spin coherence time (T_{2}) is very short, limited to a few microseconds owing to the inherently dense nuclear spin bath of the hBN host. As the coherence time is one of the most fundamental properties of spin qubits, the short T_{2} time of V_{B}^{−} could significantly limit its potential as a promising spin qubit candidate. In this study, we theoretically proposed two materials engineering methods, which can substantially extend the T_{2} time of the V_{B}^{−} spin by four times more than its intrinsic T_{2}. We performed quantum manybody computations by combining density functional theory and cluster correlation expansion and showed that replacing all the boron atoms in hBN with the ^{10}B isotope leads to the coherence enhancement of the V_{B}^{−} spin by a factor of three. In addition, the T_{2} time of the V_{B}^{−} can be enhanced by a factor of 1.3 by inducing a curvature around V_{B}^{−}. Herein, we elucidate that the curvatureinduced inhomogeneous strain creates spatially varying quadrupole nuclear interactions, which effectively suppress the nuclear spin flipflop dynamics in the bath. Importantly, we find that the combination of isotopic enrichment and strain engineering can maximize the T_{2} time of V_{B}^{−}, yielding 207.2 μs and 161.9 μs for single and multilayer h^{10}BN, respectively. Furthermore, our results can be applied to any spin qubit in hBN, strengthening their potential as material platforms to realize highprecision quantum sensors, quantum spin registers, and atomically thin quantum magnets.
Introduction
Optically addressable spin defects in wide bandgap materials are promising solidstate qubit platforms that enable cuttingedge quantum applications such as quantum computation^{1}, quantum sensing^{2}, and quantum network^{3,4}. Spin defects in diamond^{5}, which show long spin coherence times and many other attractive features such as hightemperature quantum functionality^{6} and spintophoton interfaces^{7}, is one of the leading qubit systems for quantum applications. Progresses made in the research on diamond have inspired several pioneering studies, wherein quantum spin defects have been developed in silicon carbide and nitrides, broadening the palette of spin qubits in materials^{8,9,10,11}. Notably, the spin qubits in nondiamond hosts provide unique opportunities for building advanced platforms for quantum systems by taking advantage of the wellestablished material technologies developed for their hosts^{12,13}. Recently, the search for qubit systems in twodimensional (2D) van der Waals (vdW) materials has gained significant attention owing to their potential superiority in light extraction, heterostructuring, defect positioning, strain engineering, and nanophotonic integration^{14,15,16,17}.
Among the 2D vdW materials, quantum spin defects in hexagonal boron nitride (hBN) are gaining prominence for the development of optically active spin qubits. Owing to its wide bandgap of ~6 eV, hBN hosts a variety of color centers from near infrared to ultraviolet^{15,16,18}. Some of the color centers have been found to emit bright single photons even at room temperature, and these findings have sparked worldwide research efforts in this direction^{19,20,21,22,23,24,25}. Subsequently, the search for optically addressable spin qubits in hBN has become the focus of materials research, and several breakthroughs have been made recently. Notably, negatively charged boron vacancies (V_{B}^{−}) were discovered as optically addressable spin qubits in hBN^{26,27,28}. Since the first report by Gottscholl et al.^{26}, several important achievements have been made^{29}, including the realization of coherent Rabi oscillations^{30}, deterministic defect generations^{31,32,33}, nanoscale quantum sensing^{34}, and coupling to nanophotonic structures^{35}. In addition, carbonrelated defects^{36,37,38} and boron vacancy complexes have also been recently identified as optically accessible spin qubits in hBN. Overall, significant advances have been made in the development of defectbased spin qubits in hBN and their use in quantum applications. Nonetheless, further research is required to realize robust spin qubits in hBN.
One of the most compelling issues for hBN spin qubits is their short spin coherence time (T_{2}) due to the dense nuclear spin bath in the hBN lattice^{26,39}. For V_{B}^{−}, the Hahnecho T_{2} time has been measured to be several microseconds^{30}. Notably, several wellknown schemes, such as isotopic purification^{40}, dynamical decoupling (DD)^{41}, and clock transitions (CT)^{42,43}, are available to extend the spin coherence times in materials. However, in hBN, a conventional isotopic purification is impossible because all the naturally occurring boron (^{10}B and ^{11}B) and nitrogen (^{14}N and ^{15}N) isotopes exhibit nonzero nuclear spins. In addition, the coherence protection schemes such as DD and CT are often limited by the intrinsic Hahnecho T_{2} time. When the T_{2} time is extremely short like in hBN, pulse requirements in the DD and CT schemes become considerably challenging^{11}. Considering the fundamental role of the coherence time in determining the retention time of the quantum information, the short T_{2} time of the hBN spin qubits is one of the most pressing problems that needs to be solved for advancing these platforms in hBN.
In this study, by taking the V_{B}^{−} spin as a representative spin qubit system in hBN, we developed two unconventional methods to enhance the T_{2} time of spin qubits significantly in hBN via isotopic enrichment and strain engineering. We combined density functional theory (DFT) and cluster correlation expansion (CCE) to compute the theoretical T_{2} time of V_{B}^{−} in hBN single layer and multiple layers to be 45.9 μs and 26.6 μs, respectively, in the presence of an intrinsic nuclear spin bath in natural hBN. Subsequently, we showed that the T_{2} times were increased to 143.4 and 81.1 μs in the single and multilayer h^{10}BN, respectively, which is enriched with the ^{10}B isotope. This result is somewhat counterintuitive as the ^{10}B isotope has a larger nuclear spin (I = 3) than that of the ^{11}B isotope (I = 3/2). We found that this T_{2} enhancement by a factor of three resulted from the smaller gyromagnetic ratio of ^{10}B than that of ^{11}B, despite the much larger nuclear spin of ^{10}B. In addition, we demonstrated that the T_{2} time could be further increased to 207.2 μs and 161.9 μs for the single and multilayer h^{10}BN, respectively, by inducing inhomogeneous strain around the V_{B}^{−}. We considered a Gaussianshaped bubble around V_{B}^{−} as a representative inhomogeneous strain and showed that the T_{2} increased in proportion to the Gaussian width and height, demonstrating the impact of the local inhomogeneous strain on the T_{2} time. We found that a spatial inhomogeneity in the nuclear spin quadrupole interaction effect, induced by the inhomogeneous strain, plays a crucial role in suppressing the nuclear spin flipflop dynamics, thus enhancing the V_{B}^{−} spin coherence. Our results pave the way for effective protection of the spin coherence in hBN, and this method should be applicable to V_{B}^{−} as well as to any potential spin qubits in hBN^{44,45}. Improved spin coherence of V_{B}^{−} would also open possibilities of further coherence extension using DD or CT, thereby advancing spin qubits in hBN as promising platforms for quantum information science and technology.
Results
Spin decoherence of V_{B} ^{−} in natural hBN
We considered a central spin model to compute the decoherence dynamics of a V_{B}^{−} spin interacting with the nuclear spin bath in the hBN lattice. Figure 1a shows a schematic of the spin model, in which a V_{B}^{−} defect is created in the middle of a large hBN supercell. The ground state of V_{B}^{−} is a spintriplet state, and the spin density is highly localized at the vacancy site (see inset of Fig. 1a). In our model, we assumed a localized S = 1 at the vacancy site and treated its +1 and 0 spin sublevels as qubit states. The nuclear spin bath, which strongly interacts with the V_{B}^{−} spin via hyperfine interaction, is derived from the spinbearing boron and nitrogen nuclei in hBN. Notably, all the naturally occurring boron and nitrogen isotopes exhibit nonzero nuclear spins: 19.9% of ^{10}B with I = 3, 80.1% of ^{11}B with I = 3/2, 99.6% of ^{14}N with I = 1, and 0.4% of ^{15}N with I = ½. In our model, we randomly distributed the nuclear spins in the hBN lattice according to their natural abundance. To compute the homogeneous dephasing time (T_{2}) of the V_{B}^{−} spin, we considered the Hahnecho pulse sequence^{46} and used the CCE method to expand the manybody nuclearspin correlation effects systematically on the V_{B}^{−} spin decoherence^{47}. Notably, the CCE method combined with DFT calculations enables the quantum manybody computation of T_{2} without the assumption of any adjustable theoretical parameter. Owing to its predictive power, the CCE method was successfully applied to a wide range of solidstate qubit systems, yielding excellent agreement with the experimental results^{11,48,49,50,51,52} (further details on the theoretical methods and the system’s spin Hamiltonian are provided in the Method and Supplementary Information sections; see Supplementary Note 1 and 2, and Supplementary Fig. 1).
Figure 1b presents the computed spin coherence of V_{B}^{−} in the bulk of natural hBN. Evidently, the spin coherence rapidly decays during the free evolution time, within tens of microseconds. By fitting the coherence with a stretched exponential function (\({{{\mathrm{exp}}}}\left( {t_{{{{\mathrm{free}}}}}/T_2} \right)^n\)), we computed the T_{2} time and stretching exponent (n) of the decay to be 26.64 μs and 2.64, respectively. In addition, we determined T_{2} and n of the V_{B}^{−} spin in the singlelayer hBN to be 45.85 μs and 2.27, respectively, owing to the reduced number of nuclear spins surrounding V_{B}^{−} (see Supplementary Fig. 2). Notably, the computed T_{2} time corresponds to the upper limit of T_{2} set by the intrinsic nuclear spin bath of hBN. The experimentally measured T_{2} time can be smaller than the theoretical T_{2}^{30}, if other decoherence sources such as other paramagnetic defects are present in hBN.
To examine the nuclear bath dynamics generating the intracrystalline magnetic noise, we analyzed the impact of the spin Hamiltonian terms on the V_{B}^{−} spin decoherence. Importantly, we find that the nuclear quadrupole interaction^{53} plays an important role in determining the bath dynamics in hBN. Figure 1b shows the V_{B}^{−} decoherence computed with a partial spin Hamiltonian, in which the quadrupole Hamiltonian terms are excluded from the model. We observe that the spin coherence decays much faster in the “withoutthequadrupole” model than in the “withthequadrupole” model. Without the quadrupole interaction effect in the model, the T_{2} time of V_{B}^{−} is reduced to 17.92 μs and 34.98 μs for the multi and singlelayer hBN, respectively^{39}. Notably, the nuclear spin flipflop transitions, driven by the magnetic dipolar coupling between the nuclear spins, are the dominant sources of intracrystalline magnetic noise in a nuclear spin bath^{48,52,54}. Our results show that the nuclear quadrupole interaction in hBN plays a significant role in suppressing such nuclear flipflop transitions in the hBN.
To understand the role of quadrupole interaction in the pairwise nuclear spin transitions, we show the energy levels of two ^{10}B nuclear spins, defined as \(\left {m_{I_1},m_{I_2}}\rangle \right.\), where \(m_I\) = 3, 2, 1, 0, −1, −2, −3, in Fig. 1c–e. Figure 1c shows the energy level splitting due to the Zeeman interaction (H_{Z}) and hyperfine field imposed by the V_{B}^{−} spin (H_{Hyper}). The Zeeman splitting yields spin manifolds, which are separated from each other in energy by 13.7 MHz in the presence of an external magnetic field of 3 T. Each manifold contains certain number of twospin states such as \(\left\{ {\left {{{{\mathrm{  3,  1}}}}}\rangle \right.{{{\mathrm{,}}}}\left {{{{\mathrm{  2,  2}}}}}\rangle \right.{{{\mathrm{,}}}}\left {{{{\mathrm{  1,  3}}}}} \right.} \rangle\right\}\), which have the same Zeeman energy, as shown in Fig. 1d. These states in a manifold are split by a small energy of the order of hundreds of Hz because of the hyperfine field (denoted as Δ_{HF} in Fig. 1d). In the “withoutthequadrupole” decoherence model, the magnetic dipolar interaction drives the flipflop transitions between the states, confined in each manifold, with a transition rate of hundreds of Hz (denoted as Ω_{FF} in Fig. 1d). However, the transitions are strongly suppressed between the states belonging to different manifolds because of a large Zeeman energy mismatch. Remarkably, the energy levels in a manifold exhibit significant splitting in the presence of the quadrupole interaction, as shown in Fig. 1e. We found that the quadrupoleinduced splitting (Δ_{HF+Q} in Fig. 1e) ranges from tens of kHz to a few MHz, owing to the large nuclear quadrupole interaction in hBN (see Table 1). Thus, the flipflop transitions within a Zeeman manifold are significantly suppressed, compared to those evaluated using the “withoutthequadrupole” model, because of the large quadrupoledriven energy mismatch. Our analysis reveals the importance of the quadrupole interaction in determining the nuclear bath dynamics, which in turn governs the spin decoherence of the spin qubits in hBN.
Isotopic enrichment of hBN to increase T_{2}
Figure 2 depicts the computed T_{2} of V_{B}^{−} in hBN as a function of the composition ratio of ^{10}B and ^{14}N in hBN. Surprisingly, we find that the T_{2} time substantially increases as the ratio of ^{10}B and ^{14}N increases toward 100% in the lattice, despite their larger nuclear spins than those of ^{11}B and ^{15}N. When hBN is 100% enriched with ^{10}B and ^{14}N (i.e., h^{10}B^{14}N), the T_{2} time is increased to 143.39 μs and 81.11 μs in the single and multilayer h^{10}B^{14}N, respectively. These values are three times larger than those in natural hBN. In contrast, the T_{2} times in single and multilayer h^{11}BN are 45.85 μs and 26.64 μs, respectively, which are unanticipated, because a smaller nuclear spin (e.g., I = 3/2 of ^{11}B vs. I = 3 of ^{10}B) results in a smaller number of flipflop transition channels in the nuclear spin–spin interaction as well as lower intracrystalline noise in h^{11}B^{15}N than in h^{10}B^{14}N. However, our results show that the opposite is true—the intracrystalline magnetic noise is significantly reduced in h^{10}B^{14}N.
To elucidate the microscopic origin of the enhanced T_{2} in h^{10}B^{14}N, we consider a hypothetical central spin model, wherein the gyromagnetic ratio (γ_{B}) and the nuclear spin number (I_{B}) at the boron sites are variable. Figure 3 shows T_{2} as a function of γ_{B} and I_{B}. Considering the actual I_{B} and γ_{B} of ^{10}B and ^{11}B (2.875 \({{{\mathrm{rad}}}}\;{\mathrm{G}}^{{{{\mathrm{  1}}}}}\;{{{\mathrm{ms}}}}^{{{{\mathrm{  1}}}}}\) and 8.585 rad G^{−1} ms^{−1 }^{55}, respectively), we consider the range of γ_{B} to be from 1 to 10 \({{{\mathrm{rad}}}}\;{\mathrm{G}}^{{{{\mathrm{  1}}}}}\;{{{\mathrm{ms}}}}^{  1}\), and that of I_{B} from ½ to 3 in Fig. 3. We find that for a given γ_{B}, the T_{2} time decreases as I_{B} increases, indicating that the intracrystalline magnetic noise increases as the total number of nuclear spin flipflop channels in the bath (proportional to (2I_{B})^{2}) increases. In addition, T_{2} increases quadratically as γ_{B} decreases. For instance, for I_{B} = 3/2, the T_{2} changes from 24 μs to 695 μs when γ_{B} changes from 10 to 1 \({{{\mathrm{rad}}}}\;{\mathrm{G}}^{{{{\mathrm{  1}}}}}\;{{{\mathrm{ms}}}}^{  1}\). The γ_{B}dependent change in T_{2} observed mainly because the magnetic dipolar coupling strength is proportional to (γ_{B})^{2}. Thus, decreasing γ_{B} reduces the flipflop transition rate quadratically in the nuclear spin bath. In h^{10}BN, the total number of flipflop transitions is increased by four times, but the flipflop transition rate is substantially decreased by approximately nine times compared to those in h^{11}BN. In the case of ^{14}N isotope, the same analysis yields four times larger number of flipflop transition channels and approximately two times smaller flipflop transition rates in hB^{14}N compared to those in hB^{15}N. Therefore, based on the result shown in Fig. 3, we can conclude that in hBN, the spin coherence of V_{B}^{−} is much more sensitive to the change in γ_{B} than that in I_{B}, and the increased T_{2} in h^{10}B^{14}N stems from the smaller γ_{B} of the ^{10}B and ^{14}N nuclear spins despite their large nuclear spin numbers.
Effects of inhomogeneous lattice strain on spin coherence
Next, we investigate the impact of inhomogeneous lattice strain on the T_{2} of V_{B}^{−} spin in hBN. We consider a curved hBN to create an inhomogeneous strain around V_{B}^{−}. Reportedly, various feasible methods have been developed to study the curvatureinduced phenomena in hBN by creating bubbles^{56,57}, pillars^{33}, folds^{58}, and wrinkles^{20}. As a representative model of the local curvature and inhomogeneous strain, we consider a Gaussian lattice deformation as illustrated in Fig. 4a. The lattice deformation can be described in terms of the full width at half maximum (FWHM) and height of the Gaussian function, and the V_{B}^{−} defect is created on top of the Gaussian deformation. To crosscheck the dependence of our results on the shape of lattice deformation, we also considered two other types of lattice deformations, which are Lorentzianshape and pillarshape deformations (see Supplementary Fig. 3). We employed DFT to compute the spin Hamiltonian parameters in the deformed hBN lattice. The computed effects were subsequently included into the spin Hamiltonian to examine the effect of the inhomogeneous lattice strain on the spin coherence of V_{B}^{−}.
Figure 4b shows the computed T_{2} time of the V_{B}^{−} in natural hBN (no isotopic modification) as a function of the FWHM and height of the Gaussian lattice deformation. Evidently, the T_{2} time increases significantly with the increasing function height and FWHM. At a Gaussian height of 3.5 Å, the T_{2} in the single and multilayer hBN is 58.7 μs and 34.2 μs, respectively, as shown in Fig. 4b, c. These values are 12.9 μs and 7.5 μs greater than those in natural and flat hBN. In Supplementary Fig. 4, we computed the V_{B}^{−} spin decoherence in the presence of the Lorentzianshape and pillarshape lattice deformations. We found that the V_{B}^{−} spin coherence is consistently enhanced in these situations: when the deformation height is 3.5 Å, the T_{2} time in the singlelayer hBN reaches 54.6 μs and 72.2 μs in the presence of the Lorentzian and the pillarshape deformations, respectively. This result shows that the T_{2} enhancement due to inhomogeneous strain is valid regardless of lattice deformation shapes.
To understand the curvatureinduced enhancement of the T_{2} time, we analyze the impact of the lattice deformation on the V_{B}^{−} spin decoherence in terms of two factors: modification of the nuclear spin–spin distances and spatial inhomogeneity induced in the nuclear spin quadrupole interaction in the deformed hBN lattice. In Supplementary Fig. 6, we consider a model that does not include the nuclear spin quadrupole interaction, and compute T_{2} as a function of FWHM and function height same as before (Fig. 4). We find that the T_{2} time shows negligible changes, as shown in Supplementary Fig. 6, implying that the enhanced V_{B}^{−} spin coherence, obtained using the Gaussian lattice deformation, is not derived from the mere modification of the nuclear spin–spin distances; instead, it is mediated by the modified nuclear quadrupole interaction produced by the inhomogeneous strain.
To visualize the role of nuclear quadrupole interaction in determining the straindriven enhancement of T_{2}, we compute the firstorder quadrupoleinteractioninduced energy shift (ζ) of a nuclear spin in hBN. As described in Supplementary Note 3, the energy shift is calculated as: Q_{C} \(\times \;\zeta \; \times \left( {3m_I^2  I\left( {I{{{\mathrm{ + 1}}}}} \right)} \right)\), where Q_{C} is the coefficient of quadrupole interaction (6), and ζ = V_{zz} − (V_{xx} + V_{yy})/2, where V_{ii} (i = x, y, z) is the electric field gradient computed using DFT. Figure 5a–c shows ζ as a function of position with respect to the V_{B}^{−} site, indicating that a difference of ζ between two different points, shown in Fig. 5a–c, is proportional to the difference between the quadrupoleinduced energy shifts of the two nuclear spins at these two different points. As shown in Fig. 5a–c, the inhomogeneity of ζ becomes broader and larger as the Gaussian lattice deformation increases. In Supplementary Fig. 7, we also show the quadrupole coupling constant (C_{q} = eQV_{zz}/ħ) and the asymmetry parameter (η_{as} = (V_{xx} − V_{yy})/V_{zz}) in the presence of the lattice deformation. In terms of the ^{10}B–^{10}B twospin energy levels presented in Fig. 1e, a larger difference of ζ between two ^{10}B spins results in a larger level splitting in their twospin energy levels.
Nuclear spin flipflop transitions are well understood in terms of a pseudospin model^{48,52}, which can model a flipflop transition between any two ^{10}B–^{10}B nuclear spin states having the same Zeeman energy. The two states of a pseudospin are separated by an energy gap (Δ), and the transition between them is mediated by the magnetic dipolar interaction with a flipflop transition rate (Ω) (see Supplementary Note 4). Figure 5d, e shows the histogram of (log(Ω/Δ)) for all the possible pseudospins in the Gaussiandeformed hBN. Notably, a small log(Ω/Δ) indicates a high suppression in the flipflop transition. Furthermore, a large Gaussian deformation around the V_{B}^{−} defect produces a large number of pseudospins with small log(Ω/Δ) values appearing in the histogram. Our results demonstrate that the energy gap between numerous ^{10}B–^{10}B pseudospins in hBN increases as the Gaussian lattice deformation becomes larger, which in turn produces a larger inhomogeneous strain around V_{B}^{−}. This results in a considerable suppression of the nuclear flipflop transitions in hBN, leading to the enhancement of the T_{2} time of V_{B}^{−}.
Maximizing the spin coherence time
The two methods suggested in this study can be combined to maximize the T_{2} time. Figure 6a, b presents the computed T_{2} as a function of the Gaussian height and FWHM for the V_{B}^{−} in the Gaussiandeformed h^{10}B^{14}N. Evidently, the maximum achievable T_{2} is 207.17 μs and 161.92 μs in the single (Fig. 6c) and multilayer hBN (Fig. 6d), respectively. These values are approximately six times larger than that in natural and flat hBN. Supplementary Fig. 5 shows the V_{B}^{−} spin decoherence in the Lorentziandeformed singlelayer h^{10}B^{14}N and in the presence of pillarshape lattice deformation as well. We found that T_{2} time reaches 188.9 μs and 308.4 μs for the Lorentzian and the pillarshape deformation, respectively, in the singlelayer h^{10}B^{14}N. We remark that the combined effect of the smaller gyromagnetic ratio of ^{10}B and the inhomogeneous quadrupole interaction induced by the inhomogeneous strain leads to the enhancement of T_{2} of the V_{B}^{−} spin.
Discussion
In summary, we proposed two theoretically effective methods to extend the spin coherence time of the V_{B}^{−} spin qubit in hBN. The spin coherence time is otherwise limited to a few microseconds because of the strong intracrystalline magnetic noise caused by the dense nuclear spin bath in the hBN lattice. To predict the T_{2} time of the spin qubit accurately, we performed firstprinciples calculations by combining the DFT and CCE theory. In a natural hBN host, the theoretical upper limit of the V_{B}^{−} T_{2} time is 45.85 μs and 26.64 μs in single and multilayer hBN, respectively. Next, we demonstrated that the T_{2} time can be increased to 143.39 μs and 81.11 μs in single and multilayer h^{10}BN, respectively, in which all the boron atoms in the lattice are replaced by the ^{10}B isotope. It is evident that such an isotopic enrichment technique has been already developed and applied to hBN^{59,60}. By analyzing the magnetic dipolar interaction between the boron nuclear spins, we showed that the smaller gyromagnetic ratio of ^{10}B plays a key role in suppressing the nuclear flipflop dynamics, despite its large nuclear spin compared to that of ^{11}B. Then, we showed that the T_{2} time of V_{B}^{−} can be further increased by introducing an inhomogeneous strain around the V_{B}^{−} defect. Applying a Gaussiantype lattice deformation (as a representative model of the inhomogeneous strain), we identified that the inhomogeneous strain contributes to the suppression of the nuclear spin flipflop dynamics in the bath by producing spatially varying nuclear quadrupole interactions in hBN. We found that applying both the isotopic enrichment and inhomogeneous strain could increase the T_{2} time of V_{B}^{−} by six times that in a pristine hBN bulk, and consequently, the T_{2} can reach up to 207.17 μs and 161.92 μs for the single and multilayer hBN, respectively.
Our study on V_{B}^{−} not only provides a fundamental understanding of the decoherence of V_{B}^{−} spins in hBN but paves the way to engineer their T_{2} times. With an increased T_{2} time, the V_{B}^{−} spin coherence could be further enhanced by combining other active quantum control schemes such as DD or CTs. In addition, the essential physics developed in this study can be applied to any localized spin in any 2D vdW material. Thus, the proposed approach provides a universal tool for engineering the coherence of potential spin qubits in 2D vdW materials. Combining these engineering schemes with the unique characteristics of 2D vdW materials would be a promising strategy for the development of robust lowdimensional quantum systems.
Methods
Quantumbath approach for decoherence and CCE
According to the quantumbath theory, the spin qubit decoherence occurs because of the entanglement between the qubit and its environment^{61}. In our spin model, the V_{B}^{−} spin serves as a qubit, and the main environmental degrees of freedom are the nuclear spins residing in the hBN lattice. The nuclear spins are coupled to each other via magnetic dipolar coupling, and the nuclear spin bath interacts with the qubit through hyperfine interactions. The full spin Hamiltonian for the entire quantum manybody system is described in Supplementary Note 1. To compute the homogeneous dephasing time—T_{2}—of the V_{B}^{−} ensembles, the Hahnecho pulse sequence, which features a π pulse between two free evolution time τ, is considered. The decoherence of the qubit is obtained by computing the offdiagonal elements of the reduced density matrix of the qubit after tracing out the bath degrees of freedom at the end of the \(2\tau\) free evolution time, as: \({{{\mathcal{L}}}}\left( {2\tau } \right) = \frac{{{{{\mathrm{Tr}}}}\left[ {\rho \left( {2\tau } \right){{{\mathrm{S}}}}_ + } \right]}}{{{{{\mathrm{Tr}}}}\left[ {\rho \left( 0 \right){{{\mathrm{S}}}}_ + } \right]}}\), where S_{+} is the electron spin raising operator, and ρ is the density operator of the entire system. To compute the coherence function, we employed the CCE method^{47,62}, which systematically expands the coherence function. We find that the CCE expansion converges at the CCE2 level of the theory, implying that the nuclear spin–spin pairwise correlation effect is the dominant source of decoherence. Extensive numerical convergence tests were performed in terms of the CCE order, bath size, and coupling radius, and the corresponding results are summarized in Supplementary Note 2 and Supplementary Fig. 1.
DFT calculations
We performed DFT calculations using planewave basis functions with an energy cutoff of 85 Ry, as implemented in the Quantum Espresso (QE) code^{63} and the GIPAW module^{64,65}, to compute the nuclear quadrupole interaction and the hyperfine interaction in hBN containing V_{B}^{−} with a Gaussian lattice deformation. We used the Perdew–Burke–Ernzerhof exchangecorrelation functional along with the projectoraugmented wave (PAW) pseudopotentials^{66,67}. To simulate an isolated V_{B}^{−} defect in the hBN, we used large supercells containing either single or multilayer hBN. For the singlelayer hBN calculations, we built a 336atom supercell starting from a 4atom orthorhombic unit cell. The hBN layer in the supercell was separated from its periodic images in the outofplane direction by a 10Åthick vacuum space. For the multilayer hBN, a 672atom supercell was used.
To crosscheck our results obtained with the orthorhombic supercell, we computed the spin Hamiltonian parameters and the V_{B}^{−} spin decoherence by using a 450atom hexagonal supercell for hBN. We show the results in Supplementary Tables 1, 2, 3, and Supplementary Figs. 8, 9 in Supplementary Information. We found that the computed EFG and hyperfine parameters obtained with the hexagonal supercell are consistent with the orthorhombic supercell results except for a few atoms near the V_{B}^{−} defect. In particular, the EFG parameters show a noticeable difference for a few N atoms near the V_{B}^{−} due to a slight difference in the atomic relaxation depending on the supercell shape (see Supplementary Fig. 8 and Supplementary Table 2). We found, however, that this difference did not affect the main conclusion of this study. In Supplementary Fig. 9, we compare the V_{B}^{−} spin decoherence in multilayer hBN, computed with the orthorhombic supercell and with the hexagonal supercell. We found that the two results show only a minor difference in T_{2}: 25.43 μs and 26.64 μs for the hexagonal cell and the orthorhombic cell, respectively. We remark that the subμs variation found in the T_{2} time depending on the supercell shape does not affect the main conclusion of our study as our prediction on the T_{2} enhancement in isotopically enriched hBN and in strainengineered hBN is on the order of tens of μs.
Data availability
The data that support the findings of this study are available upon reasonable request to the corresponding author.
Code availability
The codes that were used in this study are available upon reasonable request to the corresponding author.
References
Bradley, C. E. et al. A tenqubit solidstate spin register with quantum memory up to one minute. Phys. Rev. X 9, 031045 (2019).
Barry, J. F. et al. Sensitivity optimization for NVdiamond magnetometry. Rev. Mod. Phys. 92, 015004 (2020).
Pompili, M. et al. Realization of a multinode quantum network of remote solidstate qubits. Science 372, 259–264 (2021).
Humphreys, P. C. et al. Deterministic delivery of remote entanglement on a quantum network. Nature 558, 268–273 (2018).
Doherty, M. W. et al. The nitrogenvacancy colour centre in diamond. Phys. Rep. 528, 1–45 (2013).
Liu, G. Q., Feng, X., Wang, N., Li, Q. & Liu, R. B. Coherent quantum control of nitrogenvacancy center spins near 1000 kelvin. Nat. Comm. 10, 1344 (2019).
Togan, E. et al. Quantum entanglement between an optical photon and a solidstate spin qubit. Nature 466, 730–734 (2010).
Weber, J. R. et al. Quantum computing with defects. PNAS 107, 8513–8518 (2010).
Atatüre, M., Englund, K., Vamivakas, N., Lee, S. Y. & Wrachtrup, J. Material platforms for spinbased photonic quantum technologies. Nat. Rev. Mater. 3, 38–51 (2018).
Zhang, G., Cheng, Y., Chou, J. P. & Gali, A. Material platforms for defect qubits and singlephoton emitters. Appl. Phys. Rev. 7, 031308 (2020).
Wolfowicz, G. et al. Quantum guidelines for solidstate spin defects. Nat. Rev. Mater. 6, 906–925 (2021).
Anderson, C. P. et al. Electrical and optical control of single spins integrated in scalable semiconductor devices. Science 366, 1225–1230 (2019).
Widmann, M. et al. Electrical charge state manipulation of single silicon vacancies in a silicon carbide quantum optoelectronic device. Nano Lett. 19, 7173–7180 (2019).
Liu, X. & Hersam, M. C. 2D materials for quantum information science. Nat. Rev. Mater. 4, 669–684 (2019).
toth, M. & Aharonovich, I. Single photon sources in atomically thin materials. Annu. Rev. Phys. Chem. 70, 123–142 (2019).
Azzam, S. I., Parto, K. & Moody, G. Prospects and challenges of quantum emitters in 2D materials. Appl. Phys. Lett. 118, 240502 (2021).
Liu, Y. et al. Van der Waals heterostructures and devices. Nat. Rev. Mater. 1, 16042 (2016).
Tran, T. T., Bray, K., Ford, M. J., Toth, M. & Aharonovich, I. Quantum emission from hexagonal boron nitride monolayers. Nat. Nanotechnol. 11, 37–41 (2016).
Gross, G. et al. Tunable and highpurity room temperature singlephoton emission from atomic defects in hexagonal boron nitride. Nat. Comm. 8, 705 (2017).
Yim, D., Yu, M., Noh, G., Lee, J. & Seo, H. Polarization and localization of singlephoton emitters in hexagonal boron nitride wrinkles. ACS Appl. Mater. Inter. 12, 36362–36369 (2020).
Noh, G. et al. Stark tuning of singlephoton emitters in hexagonal boron nitride. Nano Lett. 18, 4710–4715 (2018).
Bourrellier, R. et al. Bright UV single photon emission at point defects in hBN. Nano Lett. 16, 4317–4321 (2016).
Caldwell, J. D. et al. Photonics with hexagonal boron nitride. Nat. Rev. Mater. 4, 552–567 (2019).
Exarhos, A. L., Hopper, D. A., Grote, R. R., Alkauskas, A. & Bassett, L. C. Optical signatures of quantum emitters in suspended hexagonal boron nitride. ACS Nano 11, 3328–3336 (2017).
Dietrich, A. et al. Observation of Fourier transform limited lines in hexagonal boron nitride. Phys. Rev. B 98, 081414 (2018).
Gottscholl, A. et al. Initialization and readout of intrinsic spin defects in a van der Waals crystal at room temperature. Nat. Mater. 19, 540–545 (2020).
Abdi, M., Chou, J. P., Gali, A. & Plenio, M. B. Color centers in hexagonal boron nitride monolayers: a group theory and ab initio analysis. ACS Photonics 5, 1967–1976 (2018).
Ivády, V. et al. Ab initio theory of the negatively charged boron vacancy qubit in hexagonal boron nitride. npj Comp. Mater. 6, 41 (2020).
Gao, X. et al. Highcontrast plasmonicenhanced shallow spin defects in hexagonal boron nitride for quantum sensing. Nano Lett. 21, 7708–7714 (2021).
Gottscholl, A. et al. Room temperature coherent control of spin defects in hexagonal boron nitride. Sci. Adv. 7, eabf3630 (2021).
Kianinia, M., White, S., Fröch, J. E., Bradac, C. & Aharonovich, I. Generation of spin defects in hexagonal boron nitride. ACS Photonics 7, 2147–2152 (2020).
Gao, X. et al. Femtosecond laser writing of spin defects in hexagonal boron nitride. ACS Photonics 8, 994–1000 (2021).
Li, C. et al. Scalable and deterministic fabrication of quantum emitter arrays from hexagonal boron nitride. Nano Lett. 21, 3626–3632 (2021).
Gottscholl, A. et al. Spin defects in hBN as promising temperature, pressure and magnetic field quantum sensors. Nat. Comm. 12, 4480 (2021).
Fröch, J. E. et al. Coupling hexagonal boron nitride quantum emitters to photonic crystal cavities. ACS Nano 14, 7085–7091 (2020).
Chejanovsky, N. et al. Singlespin resonance in a van der Waals embedded paramagnetic defect. Nat. Mater. 20, 1079–1084 (2021).
Auburger, P. & Gali, A. Towards ab initio identification of paramagnetic substitutional carbon defects in hexagonal boron nitride acting as quantum bits. Phys. Rev. B 104, 075410 (2021).
Stern, H. L. et al. Roomtemperature optically detected magnetic resonance of single defects in hexagonal boron nitride. Nat. Comm. 13, 618 (2022).
Ye, M., Seo, H. & Galli, G. Spin coherence in twodimensional materials. npj Comp. Mater. 5, 44 (2019).
Anderson, C. P. et al. Fivesecond coherence of a single spin with singleshot readout in silicon carbide. Sci. Adv. 8, eabm5912 (2022).
BarGill, N., Pham, L. M., Jarmola, A., Budker, D. & Walsworth, R. L. Solidstate electronic spin coherence time approaching one second. Nat. Comm. 4, 1743 (2013).
Wolfowica, G. et al. Atomic clock transitions in siliconbased spin qubits. Nat. Nanotechnol. 8, 561–564 (2013).
Miao, K. C. et al. Universal coherence protection in a solidstate spin qubit. Science 369, 1493–1497 (2020).
Bhang, J., Ma, H., Yim, D., Galli, G. & Seo, H. Firstprinciples predictions of outofplane group IV and V dimers as highsymmetry, highspin defects in hexagonal boron nitride. ACS Appl. Mater. Interfaces 13, 45768–45777 (2021).
Smart, T. J., Li, K., Xu, J. & Ping, Y. Intersystem crossing and excitondefect coupling of spin defects in hexagonal boron nitride. npj Comp. Mater. 7, 59 (2021).
Schweiger, A. & Jeschke, G. Principles of Pulse Electron Paramagnetic Resonance (Oxford Univ. Press, 2001).
Yang, W. & Liu, R. B. Quantum manybody theory of qubit decoherence in a finitesize spin bath. Phys. Rev. B 78, 085315 (2008).
Seo, H. et al. Quantum decoherence dynamics of divacancy spins in silicon carbide. Nat. Comm. 7, 12935 (2016).
Morley, G. W. et al. Quantum control of hybrid nuclearelectronic qubits. Nat. Mater. 12, 103–107 (2013).
Witzel, W. M., Carroll, M. S., Morello, A., Cywiński., Ł. & Sarma, S. D. Electron spin decoherence in isotopeenriched silicon. Phys. Rev. Lett. 105, 187602 (2010).
Kanai, S. et al. Generalized scaling of spin qubit coherence in over 12,000 host materials. PNAS 119, e2121808119 (2022).
Zhao, N., Ho, S. W. & Liu, R. B. Decoherence and dynamical decoupling control of nitrogen vacancy center electron spins in nuclear spin baths. Phys. Rev. B 85, 115303 (2012).
Slichter, C. P. Principles of Magnetic Resonance (Springer, 1990).
Maze, J. R., Taylor, J. M. & Lukin, M. D. Electron spin decoherence of single nitrogenvacancy defects in diamond. Phys. Rev. B 78, 094303 (2008).
Winter, M. The periodic table of the elements. https://www.webelements.com (2021).
He, L. et al. Isolating hydrogen in hexagonal boron nitride bubbles by a plasma treatment. Nat. Comm. 10, 2815 (2019).
Liu, W. et al. An ultrastable and robust singlephoton emitter in hexagonal boron nitride. Phys. E Low. Dimens. Syst. Nanostruct. 124, 114251 (2020).
Wang, G. et al. Bending of multilayer van der Waals materials. Phys. Rev. Lett. 123, 116101 (2019).
Vuong, T. Q. P. et al. Isotope engineering of van der Waals interactions in hexagonal boron nitride. Nat. Mater. 17, 152–158 (2018).
He, M. et al. Phonon engineering of boron nitride via isotopic enrichment. J. Mater. Res. 36, 4394–4403 (2021).
Breuer, H. P. & Petruccione, F. The Theory of Open Quantum Systems (Oxford Univ. Press, 2007).
Witzel, W. M. & Sarma, S. D. Quantum theory for electron spin decoherence induced by nuclear spin dynamics in semiconductor quantum computer architectures: spectral diffusion of localized electron spins in the nuclear solidstate environment. Phys. Rev. B 74, 035322 (2006).
Giannozzi, P. et al. QUANTUM ESPRESSO: a modular and opensource software project for quantum simulations of materials. J. Phys. Condens. Matter 21, 395502 (2009).
Yates, J. R., Pickard, C. J. & Mauri, F. Calculation of NMR chemical shifts for extended systems using ultrasoft pseudopotentials. Phys. Rev. B 76, 024401 (2007).
Bahramy, M. S., Sluiter, M. H. F. & Kawazoe, Y. Pseudopotential hyperfine calculations through perturbative corelevel polarization. Phys. Rev. B 76, 035124 (2007).
Blöchl, P. E. Projector augmentedwave method. Phys. Rev. B 50, 17953 (1994).
Corso, A. D. Pseudopotentials periodic table: from H to Pu. Comput. Mater. Sci. 95, 337–350 (2014).
Lovchinsky, I. et al. Magnetic resonance spectroscopy of an atomically thin material using a singlespin qubit. Science 355, 503–507 (2017).
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIT) (Nos. 2018R1C1B6008980, 2018R1A4A1024157, 2019M3E4A1078666 and 2021R1A4A1032085) and by the National Supercomputing Center with supercomputing resources including technical support (KSC2021CRE0033). This work was supported by the Ajou University research fund.
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J.L. performed the theoretical calculations. H.P. and J.L. developed the CCE code. H.S. devised and supervised the project. All authors contributed to the data analysis and production of the manuscript.
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Lee, J., Park, H. & Seo, H. Firstprinciples theory of extending the spin qubit coherence time in hexagonal boron nitride. npj 2D Mater Appl 6, 60 (2022). https://doi.org/10.1038/s41699022003362
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DOI: https://doi.org/10.1038/s41699022003362
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