Selective addressing of solid-state spins at the nanoscale via magnetic resonance frequency encoding

The nitrogen-vacancy (NV) centre in diamond is a leading platform for nanoscale sensing and imaging, as well as quantum information processing in the solid state. To date, individual control of two NV electronic spins at the nanoscale has been demonstrated. However, a key challenge is to scale up such control to arrays of NV spins. Here we apply nanoscale magnetic resonance frequency encoding to realize site-selective addressing and coherent control of a four-site array of NV spins. Sites in the array are separated by 100 nm, with each site containing multiple NVs separated by ~15 nm. Microcoils fabricated on the diamond chip provide electrically tuneable magnetic-field gradients ~0.1 G/nm. Tailored application of gradient fields and resonant microwaves allow site-selective NV spin manipulation and sensing applications, including Rabi oscillations, imaging, and nuclear magnetic resonance (NMR) spectroscopy with nanoscale resolution. Microcoil-based magnetic resonance of solid-state spins provides a practical platform for quantum-assisted sensing, quantum information processing, and the study of nanoscale spin networks.


Introduction
In recent years, nitrogen vacancy (NV) colour centres in diamond have been successfully applied to a wide range of problems in quantum information, sensing, and metrology in both the physical and life sciences 1 . For example, single NV centres have been used for a loophole-free Bell test of quantum realism 2 , probing nanoscale phenomena in condensed matter systems [3][4][5][6][7] , and NMR spectroscopy and imaging of nanoscale ensembles of nuclear spins [8][9][10] including single proteins 11 and individual proton spins 12 . Large ensembles of NV centres have provided magnetic imaging with combined micron-scale resolution and millimetre field-of-view, e.g., for mapping paleomagnetism in primitive meteorites 13 and ancient Earth rocks 14 , genetic studies of magnetotactic bacteria 15,16 , and identifying biomarkers in tumour cells 17 . However, it remains a challenge to realize the intermediate regime of mesoscopic arrays of NV spins with selective nanoscale addressing and coherent control of NVs at each site in the array. Such a capability could be a platform technology for applications such as high-spatial-dynamic-range magnetic imaging [18][19][20] and quantum-assisted sensing [21][22][23] , as well as scalable quantum information processing 24,25 and simulation 26,27 .
In the present work, we experimentally demonstrate selective coherent manipulation of an array of four NV spin sites, equally spaced by ~100 nm, using a frequency encoding technique inspired by magnetic resonance imaging (MRI). In frequency encoding, the positions of spins at different locations in a sample are mapped onto their resonance frequency using tuneable magnetic-field gradients; and then frequency-tailored pulse sequences address and control spins at specific target positions. This technique is widely employed in conventional biomedical MRI for image slice-selection with millimetre-scale resolution 28 , and was recently used in trapped-ion experiments 29 to control coherently ions separated by a few microns. Creating strong and spatially homogeneous magnetic-field gradients that can be switched rapidly compared to the spin coherence lifetime is the primary technical challenge for multi-site spin control. For our NV-diamond experiment, we achieve a tuneable gradient strength ~0.1 G/nm over a 1.2 × 8 μm 2 area by use of a micrometre-scale electromagnetic coil (microcoil) fabricated onto the diamond by e-beam lithography (Fig. 1). The field gradient is spatially uniform to 5%, can be modulated at ~1 MHz, and its application requires no active cooling of the sample. Gradients of comparable magnitudes over such a large area and with comparable switching rates are difficult to achieve using ferromagnets (Supplementary Figs. 1,2) 30 . We apply these electrically tuneable magnetic-field gradients to a series of demonstrations on the array of NV spins, including site-selective electron spin resonance (ESR) spectroscopy, Rabi oscillations, Fourier imaging 20 , and nuclear magnetic resonance (NMR) spectroscopy, all with spatial resolution ≈30 nm.

Frequency-encoding of NV spin sites in diamond
The frequency-encoding system for site-selective addressing of NV spins in diamond consists of fabricated arrays of NV centres and micron-scale coils, integrated into a home-built scanning confocal microscope as illustrated in Fig. 1a. NV centres are located at a depth of ≈20 nm from the surface of a [100]-cut diamond chip (4 × 4 × 0.5 mm 3 ) and are arranged in a two-dimensional array ( Supplementary Fig. 3). A uniform, static magnetic field of B0 = 128 G, created by an external Helmholtz coil pair, is applied along the diamond [111] direction, which corresponds to one of the four NV crystallographic orientations (Fig. 1a inset). A microwave antenna for coherent spin-state control and a magnetic-field gradient microcoil are fabricated directly on the diamond surface. A strong field gradient, aligned nominally with B0, is created by sending electric currents through the microcoil in an anti-Helmholtz configuration. In each optically-addressable confocal volume, four NV sites spaced by 100 nm are created by mask implantation (Fig. 1b) 31 . Figure  1c shows the energy level diagram of the negatively charged NV centre. The ground-state spin triplet is optically initialized into spin state |0〉, coherently addressed with microwaves, and read out by spin-state-dependent fluorescence. On applying a magnetic field gradient, the NV spins in different sites acquire a position-dependent Zeeman splitting between the | ± 1〉 states with well-resolved resonance frequencies. NV spins in each site can thus be selectively manipulated by microwaves with distinct, site-specific frequencies.
As shown by scanning electron microscope (SEM) image in Fig. 1d, the spacing between the gradient microcoil wires is 2.5 μm. In each NV site, there are approximately three NV centres with the same crystallographic orientation, as determined by the rate of fluorescence counts in a typical confocal volume (Fig. 1e), which is consistent with the estimated NV concentration of 5 × 10 11 cm -2 . See Methods and Supplementary Figs. 4, 5 for more details of the mask implantation and microcoil fabrication. Superresolution optical imaging with a stimulated-emission depletion (STED) microscope confirms the formation of four NV sites with a spacing of about 100 nm within a confocal volume (Fig. 1e, inset). More details of the STED imaging apparatus is found in the Methods section.

Demonstration of site-selective NV spin addressing and control
As a benchmark demonstration of site-selective NV addressing via frequency encoding, we performed optically-detected electron spin resonance (ESR) measurements with a DC electric current of 250 mA sent through the microcoil (Fig. 2a). Four ESR peaks, corresponding to the four NV sites in the array, are clearly observed (Fig. 2b). We fit the data to a sum of four Lorentzian curves, and determined the splitting between adjacent resonances to be ∆f = 29(3) MHz. From the SEM and STED images we find the mean separation between NV sites to be Δx = 96(7) nm, and thus the field gradient to be dB/dx = 2π × Δf/γΔx = 0.11(1) G nm -1 . Note that the observed ~30% variation in ESR peak linewidth is consistent with a simple model of inhomogeneous line broadening in the presence of the applied magnetic field gradient due to multiple (typically three) NV centres being randomly distributed in position within each site.
Next we demonstrated site-selective coherent Rabi driving of NV spins via the pulse sequence illustrated in Fig. 2c. After initializing all NV spins into the |0〉 state with a 5 µs long green laser pulse, frequency encoding is instantiated with a DC field gradient, and a microwave pulse tuned to the ESR frequency of a target NV site is applied for a duration τMW. Finally, another 5 µs laser pulse is applied to read out the NV spin states via a fluorescence measurement. NVs in the target site exhibit Rabi oscillations as the duration of the microwave pulse is varied. As the microwave frequency is adjusted to match the ESR frequency at each of the four sites, we obtain the data shown in Fig. 2d. The fidelity of such site-selective control of the NV spins via frequency encoding is estimated to be >97.4% (see Methods). By fitting each Rabi oscillation data set with a sinusoid, we determine the Rabi frequencies at all four NV sites, with results that are consistent with a numerical simulation of the microwave field produced by the antenna (Fig. 2e, see also Supplementary Discussion 1).

Demonstration of site-selective NV imaging and NMR spectroscopy
To illustrate the utility of the frequency encoding technique, we performed siteselective 1D imaging of the array of NV centres. The experimental protocol (Fig. 3a) integrates our previously-demonstrated NV Fourier imaging technique 20 with frequency encoding to resolve the array's sub-diffraction-limit spatial structure with site-selection capability. In analogy with conventional MRI, an alternating (AC) magnetic field gradient, synchronized with a Hahn echo NV pulse sequence, phase-encodes spatial information about the NV sites in wavenumber or ' -space' onto the NV spins' phase, while also isolating the NV spins from local magnetic field variations that induce dephasing. In particular, the array of NV spins with real-space positions (i = A,…, D) is exposed to an AC gradient of magnitude ( / ) and thus acquires a position-dependent phase = 2 , where = (2 ) −1 ( / ) defines the ℎ point in Fourier or -space.
Here, /2 = 2.8 MHz/G is the NV gyromagnetic ratio and is the total NV spin precession time in the Hahn echo sequence. The optically-detected NV signal for a point in -space is proportional to the sum across all NV sites of the cosine of the acquired NV spin phase at each site: s( )~∑ cos (2 ). By incrementally stepping through a range of field gradient amplitudes with fixed, one measures the NV signal as a function of to yield a -space image. In the following discussion we drop subscript j for simplicity. The real-space image is then reconstructed by a Fourier transformation of the -space image: gives the relative positions of the NV sites in the array.
Note that the resolution of the real-space image is ( ) −1 , where = (2 ) −1 | / | is the maximum value used in the measurement. In the presence of an additional DC frequency-encoding field gradient and frequency-tuned microwave pulses, only NV centres at a specific target site in the array (e.g., ) are subject to the phase-encoding protocol and hence contribute to both the Fourier image (e.g., s( , )~cos (2 )) and the real space image (e.g., ( )).
We demonstrated this nanoscale NV imaging protocol using a phase-encoding magnetic field gradient of sinusoidal form ( / ) = sin(2 / ), where is the gradient magnitude for the ℎ point in -space. Example 1D -space and real-space NV images are shown in Figs. 3b and 3c, respectively, for 512 equally-spaced -space points and the NV spin free precession time fixed at τ = 0.9 µs. The maximum k-space value is kmax = 0.021(1) nm -1 , which is induced by an AC gradient magnitude of = 0.0068 (3) G nm -1 corresponding to a microcoil current of I = 25 mA. kmax implies a 1D real-space resolution of δx = (2kmax) -1 = 24(2) nm, which is much less than the 100 nm separation between sites in the array of NV spins, but is insufficient to resolve individual NVs within one site. Note that site-selective Hahn-echo measurements using the DC field gradient, but without phase-encoding, confirm that the coherence times ( 2 ) for NV centres in all array sites are ≥2 μs and thus decoherence is insignificant during the imaging protocol (see Figs. 4a and 4b and Supplementary Fig. 6).
The This enhanced gradient field could allow selective addressing with >95% fidelity of a micron-scale array of dipolar-coupled NV centres, each spaced by ≈10 nm (Supplementary Discussions 4 and 5). Such a network of strongly-interacting spins with high spatial dynamic range has many potential applications, including in quantum sensing and imaging 32 , quantum information processing 33,34 , studies of quantum spin transport 35 , and as quantum simulators for exotic quantum and topological phases (e.g., spin liquids and supersolids 36 , quantum spin Hall effect 37 , and topological insulators 38 ). We also emphasize the simplicity and flexibility of the gradient microcoil design, which facilitates integration with other systems such as microfluidics and micro-electro-mechanical systems (MEMS). Furthermore, the frequency-encoding technique should be integratable with a wide range of NV sensing protocols, including for DC and AC magnetic fields, electric fields, and temperature. These advantages open new directions for applications, including wide-field NMR imaging of nanoscale nuclear spin diffusion and effusion in cellular or microfluidic environments via q-space detection 39 , and spatially-selective nanoscale imaging of strongly correlated spins in condensed matter systems 36 .

Creation of 2D NV centre arrays with mask implantation
The diamond sample used in this experiment is an electronic-grade, single-crystal, and 99.999% 12 C high-purity chemical vapour deposition (CVD) [100]-cut chip (4 × 4 × 0.5 mm 3 ) obtained from Element 6 Corporation. Registration markers are fabricated on the diamond substrate by e-beam lithography and reactive ion etching. All subsequent fabrication steps use the same spatial coordinates defined by these markers. A polymethyl methacrylate (PMMA) ion implantation mask is used to spatially control NV centre formation in a three-level hierarchical structure of 2D NV arrays ( Supplementary Fig. 3). 15 N + ions are implanted with a dose of 1 × 10 13 cm -2 at an implantation energy of 14 keV. The conversion efficiency from nitrogen ions to NV centres after high-temperature vacuum annealing (1200 °C, 4 hours) is approximately 6%, which is determined by comparing the measured NV fluorescence signal from a confocal spot with that from a single NV centre.
From simulations using the stopping and range of ions in matter (SRIM) program 40 , the NV centres are estimated to be 21(7) nm below the diamond surface. Typical NV spin coherence times are T2 * ≈ 580 ns and T2 ≈ 4.5 µs.

Fabrication of gradient microcoil and microwave antenna
The magnetic field gradient microcoil and microwave antenna are fabricated on the diamond chip near the NV arrays ( Supplementary Fig. 4)

STED microscopy
Superresolution NV optical images (e.g., Fig. 1e) are recorded on a homebuilt CW-STED microscope 41 Stimulated Emission Depletion (STED) microscopy is based on applying a strong, spatially structured optical depletion field to switch off peripheral fluorescent emitters through stimulated emission. In our system, the depletion field is a doughnut-shaped optical beam at 750 nm, which is applied to NV centres in the field of view at the same time as a 532 nm Gaussian-shaped excitation laser beam. The doughnut-shaped depletion beam rapidly drives to the ground electronic state all NV centres that are off the dark spot in the centre of the beam axis, while limiting unwanted NV re-excitation. Thus, only NVs on the beam axis produce NV fluorescence, which is imaged as the microscope is scanned.
The doughnut beam is created by a vortex phase plate (RPC Photonics), which imprints a is routinely achieved by applying a 300 mW depletion doughnut beam while the excitation beam power is kept as low as 100 µW. With the same experimental parameters, a four site array of NV centres can be STED imaged across a 600 nm × 250 nm field-of-view with 100 × 100 pixels, at a speed of 10 ms/pixel, realizing a per pixel NV contrast-to-noise ratio of ~5. With such a STED imaging procedure, four individual NV sites, each separated by 96 (7) nm are distinguished. 2D Gaussian filtering (4 × 1.5 pixels) and thresholding are applied to improve the image rendering, as in Fig. 1e.

Characterization of the magnetic field gradient
To characterize the performance of the microcoil, the magnetic field gradient is measured as a function of electric current through the microcoil by optically detected NV ESR. An external Helmhotz coil pair is used to apply a uniform magnetic field of B0 =128 along the  9b). As the current is increased, the resonance bands become broader because the four NV sites split. A higher resolution ESR scan of the highlighted region in Supplementary Fig.   9a quantifies the splitting of the resonance band for on-axis NV centres ( Supplementary   Fig. 9c). At around I = 200 mA, the band begins to clearly split into four peaks, corresponding to four proximal NV sites, and the ESR contrast decreases to ~2%. Consistency with simulation is again found (Supplementary Fig. 9d). By fitting the ESR spectrum with a curve comprised of four Lorentzian lineshapes for a given current value, the resonance frequency of each NV site is extracted fi (i = 1,…,4). Since the separation between NV sites is known to be Δx = 96 (7) nm from the SEM and STED images, the field gradient at each current value can be obtained using dB/dx = 2π × Δf/γΔx, where Δf is the measured frequency splitting between adjacent resonance peaks. Repeating this analysis for all current values, the field gradient per unit current is found to be dB/dx/I = 0.45(2) G nm -1 A -1 ( Supplementary Fig. 10). In particular, at I = 250 mA, the measured field gradient is dB/dx = 0.11(1) G nm -1 . A numerical simulation of the field gradient is performed in three steps.
First, the magnetic field spatial distribution produced by the microcoil at a fixed current is simulated using COMSOL. Next, the magnetic field spatial distribution for the entire current range is determined under the assumption that the field is linearly proportional to the current. Finally, the ESR resonance peaks for all NV centres are calculated by diagonalising the ground state Hamiltonian with the obtained field distribution as an input. The resulting analysis yields a field gradient per unit current of dB/dx/I = 0.48 G nm -1 A -1 , which is in reasonable agreement with the measured value ( Supplementary Fig. 10).

Estimation of Rabi driving fidelity
From the site-selective Rabi driving measurements presented in Fig. 2, the fidelity of coherent NV spin driving in the presence of a DC gradient and with a Rabi frequency of Ω can be estimated by evaluating the off-resonant excitation ("crosstalk") error: err ≈ dip + 1 + 2 + off 43 . The first term, dip~( /Ω) 2~1 0 −9 , is the error induced by dipolar-coupling (κ = 0.3 kHz) between nearest neighbour NV centres; the second term,      The open and filled circles indicate measurements at low temperature (< 77 K) and roomtemperature, respectively. Ferromagnets (red) can produce the strongest gradients (>10 G nm -1 ) at short length scales (<100 nm) [1][2][3]; however, switching the gradients requires slow mechanical motion of the magnet. Conventional Helmholtz and other 'volume' coils (green), as used in MRI [4][5][6], can create gradients ~10 -3 G nm -1 (= 100 T m -1 ) over 100 mm. Micro-electromagnetic coils (blue) offer intermediate performance [7,8]. Our device (back dot) provides gradients >0.1 G nm -1 that can be switched at ~1 MHz. The flexibility of microcoil design is useful for developing hybrid devices with microfluidic channels, Hall sensors, and microelectromechanical (MEMS) systems [9,10]. These advantages make microcoils relevant for many applications, ranging from atomic manipulation [11,12], to ferrofluid actuation [13], to manipulation of magnetotactic bacteria [14,15] and DNA [16,17]. Step 1: Registration markers (cross mark) for alignment are fabricated on a diamond substrate with e-beam lithography and reactive ion etching. All subsequent fabrication uses the spatial coordinates defined by these markers.

Supplementary
Step 2: A polymethyl methacrylate (PMMA) ion implantation mask is used to spatially control NV centre formation in a threelevel hierarchical structure. In this work 15 N + (10 13 cm -2 , 14 keV) and 12 C + (10 12 cm -2 , 20 keV) co-implantation is implemented to enhance conversion efficiency from 15

Microwave inhomogeneity
In Fig. 2d, a variation in Rabi frequency is observed across the four NV sites, which can be explained by the non-uniform microwave field distribution created by the antenna. The antenna + NV array system is in the microwave near-field; hence the spatial variation of the microwave field is determined by the dimensions of surrounding conductors rather than the wavelength of the radiation. Since the electric field is zero (E = 0) in a conductor, no This boundary effect alters the magnitude as well as the direction of the microwave field in the near field, which leads to a significant change in the NV Rabi frequency across the dimensions of the four-site array. Note that such microwave field gradients have also been used for selective addressing and motional-internal state coupling of atomic systems [19,20].

Transverse magnetic field induced by gradient microcoil
The gradient field produced by the microcoil is nominally aligned with the uniform static field B0 in order to perform NV frequency encoding and Fourier imaging. There is also a small transverse component to the gradient field ⊥ (x), orthogonal to the direction of B0, which is exploited for site-selective NMR spectroscopy of 15 N nuclear spins ( Fig. 4 and Supplementary Fig. 6). Thus, for the experimental protocol shown in Fig. 4a, the envelope of the NV Hahn echo signal in the presence of a DC magnetic field gradient is modulated as ( ) ∝ sin 2 ( /2) sin 2 ( /2), where A ≈ 2 × 3 MHz is the NV-15 N hyperfine coupling strength and ω B ≈ 14γ n ⊥ is the 15 N nuclear spin Larmor frequency. Here γ n = 2π × 0.43 kHz G -1 is the gyromagnetic ratio of the bare 15 N nuclear spin, and the factor of 14 is an effective enhancement of this gyromagnetic ratio due to virtual transitions arising from higher order NV-15 N hyperfine interactions [18]. From the observed modulation frequency B = 2π × 0.36(2) MHz, the transverse field at NV site C is determined to be ⊥ = 59(4) G, which agrees well with a COMSOL simulation of the gradient field produced by the microcoil (Supplementary Fig. 6c). Similar agreement between measurement and simulation is found for the other NV sites in the array.

Extension of selective addressing to multiple NV sites and spatial dimensions
In future work, straightforward methods can be used to extend selective control to two or more NV sites as well as a second spatial dimension. For example, multiple NV sites could be addressed simultaneously, in the presence of a frequency-encoding DC magnetic field gradient, by applying a microwave signal with spectral components at the NV ESR frequencies of all target sites. To selectively address a two-dimensional (2D) array of NV sites (e.g., a square lattice), one cannot simply employ simultaneous frequency-encoding gradients in both dimensions because a degeneracy of the resonance frequency occurs.
Instead, one can maintain the spin polarization of a target row of NV sites, while dephasing the other rows; and then performing 1D frequency encoding along the target row to select the desired site. This procedure can be realized, for instance, by (i) putting all NV spins in the 2D array onto the equator of the Bloch sphere with no DC gradient applied and a uniformly resonant microwave π/2 pulse; (ii) bringing only the target x = x0 row back to the initial polarized state with a frequency-encoded 3π/2 microwave pulse in the presence of a DC field gradient; (iii) waiting for T2 * dephasing of the other (non-target) NV rows; and then (iv) performing 1D frequency encoding in the y direction to select the target NV site (x0, y0).

Maximum gradient strength and switching bandwidth
Though the frequency encoding demonstrations reported in the present work are performed with a maximum current of 250 mA, the gradient microcoil can be operated at a much higher current (and hence larger gradient strength) before it breaks down due to ohmic heating. The maximum current that the microcoil can support is determined by measuring the device temperature rise as a function of electric current ( Supplementary Fig. 8). With water cooling, the temperature rises by nearly 40 K with I = 1.4 A, with a corresponding gradient of 0.6 G nm -1 . The switching bandwidth of the gradient pulse is determined by measuring the rising NV Rabi signal contrast as a function of delay between the initiation of the gradient pulse and π/2 pulse. Using impulse response function theory, the 3 dB switching bandwidth is f3dB = (ln(0.9)-ln(0.1))/(2πTr) ~ 0.9 MHz, where Tr = 400 ns is the measured 10%-90% rise time in the NV Rabi signal contrast.

Relation between spatial dynamic range and switching bandwidth
To characterize the maximum number of individually addressable NV sites, the spatial dynamic range of site-selective control, DR = L/D is introduced. Here L is the length scale across which the strong gradient is applied, and D is the separation between individually addressable NV sites. For a microcoil, L is determined by the separation between the two wires and D is inversely proportional to the gradient strength. A magnetic field gradient of dB/dx = 0.1 G nm -1 and 1 G nm -1 corresponds to D ≈ 100 nm and 10 nm, respectively, assuming a Zeeman shift of about 30 MHz is needed for selective addressing of neighbouring NV sites with high fidelity. Since the gradient strength decreases quadratically as the coil separation increases (dB/dx ~ I/L 2 ) for a fixed value of gradient strength or D, the electric current I should be increased by I ~ L 2 ~ DR 2 . However the maximum achievable current I is also limited by the switching bandwidth, i.e., I ~ BW -1 .
BW can be affected by at least 3 factors: a, self-inductance of the micro-coil; b, peripheral circuits, such as in the printed circuit board (PCB) that interfaces the current source and micro-coil; and c, finite speed of the pulsed current source, which is a current amplifier in our case. COMSOL simulation indicates that the micro-coil has a very small selfinductance ~ 1 nH, and hence the corresponding bandwidth limit is on the order of GHz.
Finite bandwith due to peripheral circuits can be, in principle, eliminated by integrating the current source close to the micro-coil. So the current amplifier is expected to be the primary switching bandwidth limitation. Furthermore, assuming that the output from the current amplifier is slew rate (SR) limited (which is usually the case when high current output or large voltage is needed), BW can be expressed as: BW = SR/(2πVp), where Vp stands for peak output voltage [21]. Since current is proportional to voltage, then BW and current are related as I ~ BW -1 . Thus we conclude that the spatial dynamic range and switching bandwidth are related via DR~BW -1/2 ( Supplementary Fig. 2).

Outlook for fabrication of high NV density arrays
High-precision nitrogen ion implantation is a key technical challenge for preparing either single NV centre arrays or strongly-coupled dense NV spin baths. Controlling the separation and number of NV centres with high precision remains an active area of research.
It is thus of importance to evaluate the minimum limits of NV site area (A), NV-NV separation within a single site (d), and spacing between sites (D), at the present state of technology. The NV site area is constrained by the mask diameter and lateral implantation straggle. Using a polymethyl methacrylate (PMMA) mask, apertures with diameter ≈ 30-60 nm can be created. Further reduction to 10 nm by reducing the mask thickness or even to ~1 nm by combining with atomic layer deposition of alumina may be possible; however, the lateral implantation straggle (7.0 nm at 14 keV implantation energy) will eventually limit the NV site area to Amin ≈ (10 nm) 2 . Nevertheless, this is small enough for quantum information processing architectures based on spin chains with error correction methods [22 ,23]. The NV-NV separation within a single NV site is determined by the NV density.
With an 15 N + implantation dosage of 10 13 cm -2 and N-to-NV conversion efficiency of 6 %, the diamond sample used in the present work has d = 15 nm. By increasing the implantation dosage and irradiating with neutrons or electrons, a minimum separation of dmin ≈ 4 nm (corresponding to an NV-NV dipolar interaction strength ≈ 0.6 MHz) may be achievable, although the sample at this density might suffer from low fluorescence contrast due to NV -/NV 0 charge state conversion and a shorter T2 * due to remaining paramagnetic impurities.
The spacing between NV sites, on the other hand, is limited by the NV site area, and thus cannot be smaller than Dmin ≈10 nm. If two NV centres are randomly picked from each of two NV sites with area A and spacing D, the mean and standard deviation of separation between them are given by ~D and ~A 1/2 , respectively.