Single nuclear spin detection and management in a van der Waals materials

Pattern preparation

The hBN skinny flakes had been tape-exfoliated from a monocrystalline hBN crystal and transferred onto Si/SiO₂ substrates. Then we irradiated the hBN flakes with 2.5 keV ¹³CO₂ (99.0% ¹³C, Sigma-Aldrich) ions with a dose density of 10¹² cm⁻² utilizing a home-built ion implanter. The pattern is then annealed at 1,000 °C at 10⁻⁵ Torr for two h to activate the carbon-related defects. For ODMR measurements, we transferred the hBN flakes to a coplanar waveguide utilizing the usual dry switch technique with propylene carbonate stamps. The waveguide is made from 200-nm-thick silver with a 4-nm-thick Al₂O₃ layer on prime.

Sensitivity of a single spin defect

The ODMR distinction varies between defects and may attain as excessive as 200% (Supplementary Fig. 24). Among the many greater than 100 spin defects investigated, roughly 25% exhibit a distinction increased than 10%. A single hBN spin defect in our pattern has a typical sensitivity of (5,mu {rm{T}}/sqrt{{rm{Hz}}}) for DC magnetic-field sensing, calculated utilizing ((8{rm{pi }}/3sqrt{3})(1/{gamma }_{{rm{e}}})(Delta nu /Csqrt{I})) (ref. ²⁹), during which Δν is linewidth (20 MHz), C is the distinction (30%) and I is the photon rely fee (170 kcts s⁻¹). Moreover, though most group II and III defects exhibit steady behaviours below a weak laser excitation (≤15 μW), the steadiness of group I defects varies significantly.

Estimation of nuclear spin polarization

We estimate the polarization of the ¹³C nuclear spin by evaluating the imbalance between III-2 and III-4 within the ODMR spectrum. The ODMR is taken after the SWAP gate to switch the electron polarization to the ¹³C nuclear spin. By utilizing the fitted relative populations of the hyperfine foundation states, the polarization could be calculated by the equation

Spin readout effectivity

The effectivity of a single-shot spin readout is a crucial issue to estimate how effectively we will decide the digital spin state of a spin defect, which is very depending on the defect properties. The readout effectivity is outlined by the signal-to-noise ratio from a single readout pulse and could be expressed as³⁸

during which α₀ and α₁ are the imply numbers of detected photons for a single measurement of the brighter state and darker state, respectively. We estimate the effectivity based mostly on the pulsed ODMR measurements. The pulsed ODMR distinction is 17.5% after we set the readout length at 5 μs below a P_MW = 60 mW microwave drive. The distinction reaches 28% when P_MW = 2 W. For every readout laser pulse, we acquire roughly 0.9 photons from the darker state below the 15-μW laser pumping. These yield the effectivity of 0.08 and 0.12 for P_MW = 60 mW and P_MW = 2 W, respectively. See particulars in Supplementary Info Part IV.

Gate constancy

For Rabi oscillations restricted by a pure dephasing course of, we will write the π-gate constancy as F_π = 0.5(1 + exp(−1/Q_π)), during which Q_π = T_Rabi/T_π is the standard issue of a π gate³⁹. We extract the coherence time and π-gate time of nuclear spin Rabi by becoming the ends in Fig. 3 to the perform C(τ) = asin(πτ/T_π + b)exp(−τ/T_Rabi) + d, during which C(τ) is the sign distinction of Rabi. In consequence, we acquire a F_π,n = 99.75% π-gate constancy, with T_π,n = 0.60 μs and T_Rabi,n = 117 μs. Equally, we additionally estimate the digital spin π-gate constancy to be 96.2%, utilizing the identical defect and transition within the nuclear spin management experiments.

DFT calculations

We use Quantum Espresso⁴⁰, an open-source plane-wave software program, to carry out the DFT calculations. Each the Perdew–Burke–Ernzerhof useful and the Heyd–Scuseria–Ernzerhof hybrid useful (the issue of 0.32 for Fock alternate)^41,42 are used for the exchange-correlation interplay. We use the optimized norm-conserving Vanderbilt (ONCV) pseudopotential^43,44 for the calculations of excitation vitality and the GIPAW pseudopotential⁴⁵ for the calculation of hyperfine interplay parameters and ZFS. We set the kinetic vitality cut-off to be 55 Ry, which is ample for converging the related properties. Geometry optimizations are carried out with a power threshold of 0.001 Ry Bohr⁻¹. We choose the 6 × 6 × 1 or increased supercell measurement of hBN for the calculations of hyperfine parameters and excitation energies. For these calculations, we pattern a ok-point mesh of three × 3 × 1 for the calculation of excitation energies⁴⁶ and the Γ level for the hyperfine parameters and ZFS^28,46. We calculate the zero-phonon line by the constraint occupation DFT technique⁴⁷, the hyperfine parameters utilizing the QE-GIPAW code⁴⁸, the ZFS through the use of the ZFS code⁴⁶ and we cross-compare outcomes between the ZFS code and the PyZFS code⁴⁹. The important thing outcomes are summarized in Supplementary Tables 2 and 3.

Simulation of ODMR spectrum

The continual-wave ODMR spectra proven in Fig. 5 are simulated utilizing the MATLAB toolbox EasySpin⁵⁰ based mostly on knowledge from the ab initio calculations. EasySpin additionally takes the nuclear Zeeman and quadrupole interplay under consideration. Subsequently, the continuous-wave ODMR linewidth could be decided in accordance with the hyperfine couplings with essentially the most plentiful nuclear-spin-active isotopes: ¹³C, ¹¹B and ¹⁴N. In our simulation, we think about a ¹³C nuclear spin, ten nearest ¹¹B nuclear spins and two proximate ¹⁴N nuclear spins. The opposite nuclei, positioned additional away, couple extra weakly to the electron, scaling with ∝ 1/r³ (during which r is the space from the central carbon website) and, thereby, have a negligible impact on the ODMR linewidth.

Spin-pair mannequin

The S = 1 transitions are persistently noticed alongside the S = 1/2 transitions throughout the identical emitters. To clarify the coexistence of each spin transitions in ODMR, we use a weakly coupled spin-pair mannequin^32,33 and carry out numerical simulations to research the underlying mechanism.

Prolonged Information Fig. 1a offers a simplified illustration of the spin-pair mannequin, consisting of two unbiased defects (defects A and B), separated by ≥1 nm, forming a defect advanced. This advanced hosts two unpaired electrons that set up totally different inside cost states relying on their spatial occupancy. When each electrons are localized on the identical defect (defect A), they kind a closed-shell spin singlet GS with a metastable spin triplet state (S = 1), which could be accessed by means of laser excitation and intersystem crossing transitions (left panel in Prolonged Information Fig. 1b). This state corresponds to a strongly coupled spin-pair cost state and explains the S = 1 transitions.

Alternatively, laser excitation can induce cost hopping that transfers one electron from defect A to defect B, forming a weakly coupled defect pair. On this configuration, every defect hosts a single electron (S = 1/2), as illustrated in the best panel of Prolonged Information Fig. 1b. This spin-dependent cost hopping yields a corresponding spin-dependent PL sign³².

The precise GS, which can be the optically lively state, is decided by the lowest-energy cost configuration and depends upon the particular defect species and the native Fermi vitality degree. We thought of two attainable energy-level configurations (Supplementary Info Part XII), with Prolonged Information Fig. 1c depicting the almost definitely state of affairs. In essentially the most steady cost state, each electrons occupy the identical defect website, yielding a S = 0 GS and a metastable S = 1 state. Laser excitation can then generate a metastable spin-pair cost state by selling transitions from the S = 1 to the S = 1/2 manifold. The pronounced asymmetry within the Rabi oscillations, mirrored by an growing distinction baseline in each S = 1/2 and S = 1 transitions, strongly helps the metastable nature of those spin manifolds.

Within the presence of a ¹³C nuclear spin, the defect electron spins can couple to the nuclear spin by means of hyperfine interactions, which differ throughout totally different cost states. Within the weakly coupled spin-pair state, the nuclear spin is primarily coupled to a single electron spin at defect A. That is per our experimental observations, during which hyperfine coupling constants A_zz of 130 and 300 MHz had been measured for group II and group III defects, respectively.

Given their distinct hyperfine options, group II and III defects are prone to have extra well-defined and deterministic constructions. In contrast, group I defects, missing hyperfine splitting, could embody a variety of chemical configurations: both much like group II and III defects (however involving ¹²C as an alternative of ¹³C) or very totally different chemical constructions. This structural variability could account for the broader vary of stability noticed in group I defects, whereas group II and III defects are likely to exhibit extra constant and steady behaviour below experimental circumstances.

Simulation of spin photodynamics

On the idea of the attainable energy-level fashions, we numerically simulate the spin photodynamics utilizing the Lindblad grasp equation:

during which ρ(t) is the time-dependent density matrix, Γ_ok represents transition charges and L_ok are the related Lindblad operators. This simulation permits us to foretell PL alerts in each continuous-wave ODMR and pulsed Rabi experiments (see particulars in Supplementary Info).

Within the continuous-wave ODMR simulation, we use the total Hamiltonian, which incorporates the optical manifold, spin-pair states and spin-triplet states. For mannequin 1, during which the GS is a spin singlet (S = 0), the Hamiltonian is written as:

during which H_eg, H_m,S1 and H_pair describe the optical manifold, the spin S = 1 metastable state and the spin-pair state, respectively (see detailed expressions in Supplementary Info). For every subspace, we think about a ¹³C nuclear spin coupled by means of hyperfine interplay. The total Hamiltonian is a direct sum of particular person spin manifolds, that means that there isn’t a coherent interplay between them. As a substitute, they’re related via incoherent transitions described by the Lindblad operators L_ok.