Three-dimensional solid-state qubit arrays with long-lived spin coherence

Three-dimensional arrays of silicon transistors increase the density of bits. Solid-state qubits are much larger so could benefit even more from using the third dimension given that useful fault-tolerant quantum computing will require at least 100,000 physical qubits and perhaps one billion. Here we use laser writing to create 3D arrays of nitrogen-vacancy centre (NVC) qubits in diamond. This would allow 5 million qubits inside a commercially available 4.5x4.5x0.5 mm diamond based on five nuclear qubits per NVC and allowing $(10 \mu m)^3$ per NVC to leave room for our laser-written electrical control. The spin coherence times we measure are an order of magnitude longer than previous laser-written qubits and at least as long as non-laser-written NVC. As well as NVC quantum computing, quantum communication and nanoscale sensing could benefit from the same platform. Our approach could also be extended to other qubits in diamond and silicon carbide.


I. INTRODUCTION
Demonstrated qubit fidelities for a single nitrogenvacancy center (NVC) and its nearby nuclear spins [1] are above the required thresholds for fault-tolerant quantum computing [2]. In addition, two NVCs in different cryostats have been optically entangled [3,4]. However, useful faulttolerant quantum computing will require 10 6 -10 9 physical qubits [2]. It will not be practical to build this with one cryostat per NVC so it is important to consider how many NVCs can be used inside of one diamond as inspired by previous experiments [1,[5][6][7] and theoretical proposals [8][9][10][11]. We focus on optical entanglement rather than the possibility of having NVCs so close to each other that they directly interact magnetically [12] because no scalable fabrication technique has been found for the latter approach. For communications [4,13], having an array of NVCs will provide many spin-photon interfaces within one cryostat [14], increasing data rates and allowing multiplexing. Sensing with two-dimensional (2D) arrays of * gavin.morley@warwick.ac.uk NVCs will combine the high resolution of single NVC sensing [15][16][17] with the simultaneous imaging achieved with wide-field microscopy [18]. Stacking two of these 2D arrays will then increase the sensitivity by subtracting the background noise measured by the array that is further from the sample of interest.
For all these technologies, we envision a fiber array or a spatial light modulator (SLM) that simultaneously sends and receives optical photons to and from a 2D array of NVCs through a lens or a microlens array. A solid immersion lens (SIL) array would be used to collect more of the fluorescence [3,4,19,20] or an optical cavity array [21] would be used to improve the optical performance of NVCs in the middle of a diamond that is thin, for example, 4 µm [22,23]. For sensing, the NVCs need to be approximately 10 nm from the surface with no SIL or cavity, so delta doping with nitrogen would be used to control the depth of NVCs with a precision of 4 nm [24], followed by plasma etching to leave them at the surface.
The creation of 2D NVC arrays has been demonstrated previously [24][25][26][27][28][29][30][31][32][33][34][35], but not three-dimensional (3D) arrays. With ion implantation through a mask, high-precision placement of 10 nm [26] has been shown with electron spin T 2 times of up to 50 µs. Longer T 2 times of up to 530 µs were achieved by using isotopically pure 12 C diamond, but with less precise placement and again requiring ion implantation through a mask [27]. For quantum computing, it is important to use diamond with natural isotopic abundance because the 1.1% 13 C nuclear spins provide a valuable register of five or more qubits that can be used to store quantum information for longer than the electron spin [5][6][7]36]. Ion implantation has been shown to lead to cryogenic optical transitions that are too broad for quantum computing applications [37]. In contrast, laser writing similar to that presented here has produced transform-limited cryogenic optical transitions including photoluminescence excitation with linewidths below 14 MHz [25]. Localized electron irradiation into a diamond provided a 2D array with a T 2 of up to 1.3 ms due to the use of 12 C diamond and no need for a mask [24]. Delta doping was used to reach a depth precision of 4 nm with an in-plane precision of 450 nm. Laser-written optical waveguides have been made in diamond and optically detected magnetic resonance (ODMR) measured on the NVC ensembles inside [38,39]. 2D arrays of NVC ensembles have been created on the diamond surface with short-pulse laser nanoablation [40]. Earlier work from our collaboration has shown that 2D arrays of NVCs (some single, some double, some triple) can be laser written with no mask, but the T 2 time measured was typically only 30-80 µs [25]. These short times may have been due to the creation of too much damage by overly energetic laser write pulses. Recent work from our collaboration describes preferential orientation and near 100% yield for 5 × 5 2D arrays of laser-written NVCs in diamond, but the T 2 times are 170 µs or less [41]. This reduced spin coherence may be due to the high concentration of nitrogen currently needed for the in situ annealing technique used or it may be that the in situ annealing does not heal the damage as well as traditional annealing in a furnace. The literature on 2D NVC arrays does not report the dependence of T 2 on NVC depth. The depth dependence of NVC T 2 times has been studied for shallow NVCs, finding that T 2 < 100 µs [42], which is good for nanoscale sensing but too short for quantum computing.
In this paper, we present the first 3D arrays of NVCs and find that their spin coherence times are consistently as long as naturally occurring NVCs over a range of depths. These coherence times are the result of (a) plasma etching to remove the subsurface polish damage, (b) a global thermal anneal, (c) high purity diamond, and (d) precisely calibrated laser-write-pulse energy to avoid creating unnecessary amounts of damage.

II. FABRICATION WITH LASER WRITING
Our 3D arrays are created by laser writing over 2000 NVCs into a diamond with a natural isotopic abundance of 13 C. Figure 1(a) is a schematic of the writing process and Fig. 1(b) is an overview of all of the arrays written. Figure 2 shows one of the 3D arrays in the form of four 2D images. We measure the spin-echo coherence time for 22 of the single centers and find that 15 of them have T 2 > 500 µs at room temperature. All our measurements are at room temperature where the electron spin coherence time is limited by the natural isotopic abundance of 13 C. We use dynamic decoupling to probe the electron spin coherence without 13 C limitations, finding a coherence time limited by the electron spin T 1 as has been reported for naturally occurring NVCs.
To create the arrays, an electronic grade (EL grade) diamond is bought from Element Six, and plasma etching is used to remove 20 µm of subsurface polishing damage [17,43]. Arrays of ensembles of vacancies are generated in the diamond lattice by single 250-fs pulses from a 790-nm laser focused tightly beneath the surface of the diamond using a high numerical aperture (NA) oil objective. The light-matter interaction is highly nonlinear, limiting any material modification to the center of the focal volume and giving an inherent three-dimensional resolution to the fabrication [44]. The refractive index mismatch at the oil-diamond interface causes refraction leading to a depth-dependent spherical aberration of the laser focus, which can limit three-dimensional fabrication resolution. Adaptive optics using a liquid crystal SLM is used to correct for the aberration [45], ensuring that the fabrication is the same at each depth. The full width at half maxima of the intensity distribution of the laser focus inside the diamond are theoretically estimated to be 350 nm radially and 1.7 µm longitudinally. However, the expected dimensions over which the light-matter interaction is appreciable are likely to be much lower [46]. The diamond sample is mounted on a three-axis precision translation stage and moved relative to the laser focus to fabricate arrays of vacancy ensembles. In order to find the fine range of pulse energies to use in these experiments, an initial calibration study is carried out on the same fabrication run inside a nominally identical diamond by writing arrays of points across a coarse range of pulse energies. This sample is subsequently characterized using a scanning confocal microscope to find the pulse energy that produces just visible vacancy ensembles using an air objective. Previous work has shown that writepulse energies slightly lower than this are optimal for NVC creation [25]. Twenty-two 3D arrays labeled A to V are laser written with different energies from 14 to 19 nJ and with different pitches from 2 to 5 µm, with each 3D array having 21 × 20 2D arrays stacked with up to six depths for a total of over 44 000 writing sites. Once set up, laser writing of a 2000-point array typically takes 5 min. Following laser writing and characterization, the sample is buried in diamond grit (from Element Six) and annealed for three hours at 1000°C in a tube furnace in a nitrogen atmosphere to convert the introduced vacancies into NVCs [25].

III. EXPERIMENTS
We study this sample with our home-built scanning confocal microscope using ODMR with 532-nm excitation and single-photon counting modules for the broadband red fluorescence. The electron spins are coherently controlled with microwaves at 2.1-3 GHz. A 25-mT magnetic field is applied along the [111] direction for the spin coherence measurements, suppressing the periodic spin-echo collapse and revival [47]. Each spin coherence measurement takes 2-12 h, depending on the desired signal-to-noise ratio. The equipment is controlled using open-source QUDI software [48] with some of our modifications to enable automated data collection over many NVCs [49]. Figure 2 contains images from our scanning confocal microscope of one of the 3D arrays (array M) before and after annealing. Before annealing, laser-written spots are visible due to the fluorescence of neutrally charged vacancies V 0 , which are lattice sites in the diamond with missing carbon atoms. This is shown in the top row of Fig. 2. The characteristic V 0 fluorescence spectrum confirms the identity of these vacancies as shown in Fig. 5(a) of Appendix B. None of these preanneal spots were visible with our air objective, but these images were collected with our oil objective (the numerical aperture equals 1.4). We anneal the diamond at 1000°C for 3 h in a nitrogen environment [25] and repeat the imaging, as shown in the bottom row of Fig. 2. The spots that can be seen are NVCs as confirmed by the characteristic fluorescence spectrum shown in Fig. 5(b) of Appendix B. We analyze the precision of the position for 167 single NVCs by three-dimensional fitting of the point spread function as shown in Appendix G. This reveals that the NVCs are in the desired locations to within ±220 nm in the transverse (X-Y) plane and ±250 nm in the vertical (Z) direction. The high precision in the Z direction is due to the nonlinearity of the writing. The precision is probably limited by the concentration of nitrogen in this material, which is in the range from 1 to 5 parts per billion (ppb) corresponding to an average spacing between nitrogen atoms of 180 to 100 nm. This implies that the vacancies generally bond to one of their nearest nitrogen dopants.
To check if the NVCs are single centers, we use automated Hanbury Brown-Twiss (HBT) experiments on over 600 sites to measure the photon arrival autocorrelation function g 2 (τ ), as shown in Fig. 3. We classify a site as a single emitter where g 2 (0) < 0.5, a double for 0.5 ≤ g 2 (0) < 0.66, and a triple for 0.66 ≤ g 2 (0) < 0.75. Lower boundary lines could have been chosen to account for the fact that NVCs can switch between different charge states and the different emission dipoles that result from different NVC orientations. For array M, 87% of the NVCs are single centers while 11% are doubles and 2% are triples. To reach higher yields, the sites with no NVCs should be repeatedly rewritten with another laser pulse and reannealed until no sites are empty. One of our previous papers reported single NVC creation with laser writing [25]: higher pulse energies were used, which led to a higher yield of single NVCs, but there were a similar number of the unwanted double and triple NVCs, which would lead to a lower yield of single NVCs in a repeat-until-success strategy. We greatly reduce this problem here. Appendix E contains confocal imaging and HBT statistics for some of the other arrays. Of the 2050 sites written in array M, 8% develop into a single NVC, 0.6% produce a double NVC, and <0.1% produce a triple NVC. This is consistent with Poissonian statistics (0.0064 = 0.08 2 and 0.0005 = 0.08 3 ), which may suggest that NVC creation is limited by the nitrogen density rather than by the highly nonlinear laser writing. We may expect that if there were an abundance of NVCs, the nonlinear nature of the laser writing would lead to more doubles and triples for a given yield of single NVCs. Figure 4 shows measurements of the spin coherence from 22 of the single NVCs in the 3-µm pitch arrays M and I, which are written with 17.5 and 17 nJ pulses, respectively. The longest room-temperature spin-echo coherence times without 12 C enrichment we find in the literature are T 2 = 687 µs [3] and T 2 = 650 µs [50], which are slightly below (but within the error of) our five longest times. long times we measure demonstrate that our laser-writing technique does not introduce excess damage or impurities to the environment of the NVCs. Our new calibration step to ensure we use the optimum write-pulse energy may be needed to achieve this as that is a key difference from our previous work [25]. This allows the use of lower writepulse energies, which only create NVCs without leaving extra damage behind in the lattice. Previous NVC optical entanglement work has applied electric fields to Stark shift the optical fluorescence frequency so that the two NVCs have indistinguishable emissions [3,4]. Figure 4(c) shows an NVC between two electrically conducting wires that we laser write in three dimensions at the same time as the arrays. It is known that these laser-written wires in diamond are graphitic and that they conduct with a dc resistivity of around 0.1 cm at room temperature [51].

IV. DISCUSSION AND CONCLUSION
Further extensions of the work presented here could include studying the optical coherence [25] as a function of depth, and studying the spin coherence over a greater range of depths. Looking at shallower NVCs would reveal the effects of the surface and provide clearer statistics on the NVCs that are 6-µm deep. The depth range presented here is ideal for solid immersion lenses, but approximately 2µm deep would be used for membrane experiments inside an optical cavity [23]. Testing the Stark shift that can be achieved at low temperatures with the graphitic wires is another important future experiment, as this would allow the optical emission of two NVCs to be brought to the same wavelength as required for optical entanglement [4]. To evaluate the usefulness of quantum sensing using gradiometry with two 2D arrays, it will be necessary to measure the crosstalk in the excitation and detection. Having a larger spacing between the 2D arrays may help with this.
In conclusion, this work demonstrates creation of arrays of long-lived spin qubits in a solid-state system with alignment to control structures in three dimensions inside diamond, making this a compelling approach for engineering larger quantum systems. Finding that the spin coherence times at room temperature are as long as naturally occurring NVCs shows that laser writing is suitable for quantum computer designs calling for an array of many NVCs in one diamond. The data for this paper is available online [52].  18, 17, 16, 15, 14, 16, 16.5, and 17 nJ. The labeling of the arrays is shown in Fig. 1(b).

APPENDIX B: PHOTOLUMINESCENCE SPECTROSCOPY
Photoluminescence (PL) spectra under excitation with a 532-nm laser are measured using an Andor Shamrock spectrograph with an iDus series silicon CCD through a multimode fiber.
Before the sample is annealed, PL spectra are collected such as that shown in Fig. 5(a), clearly showing the characteristic GR1 emission spectrum associated with neutral vacancies in diamond. No sites are visible by confocal microscopy with our oil objective in array D (16 nJ) before annealing. Following the anneal, PL spectra are collected from sites including M9-4 (labeled NVC2 above), which clearly show the characteristic spectrum of NVC − as in Fig. 5(b). The background emission from a nearby region of diamond is collected for an equal duration and 064005-8 FIG. 11. Examples of T 2 spin-echo coherence time measurements from arrays M and I along with a histogram of the echo decay coherence times subtracted to eliminate artifacts of spectrometer response, second-order Raman emission, and fluorescence from the immersion oil.

APPENDIX C: NVC COLOR CENTER YIELD
A narrow range of pulse energies are used to create three-dimensional arrays, each of which has a pitch of 2, 3, or 5 µm in the x, y, and z directions. The proportion of sites that developed NVCs are shown in Fig. 6. While the yield of NVCs in the 2-µm pitch arrays is high with a high proportion of single NVCs, the close packing of sites in depth make it more difficult to image individual sites in the z direction. The 3-µm pitch array develops a good yield of well-spaced NVCs, which can be reliably characterized. Despite a higher pulse energy, the 5-µm pitch array develops a lower yield of NVCs. This suggests that a smaller pitch leads to more damage, which is reasonable given that the sites are written on top of each other in the z direction.

APPENDIX D: BIREFRINGENCE AND THE INFLUENCE OF STRAIN
Array I has the region of highest birefringence, in the upper right quadrant of the images in Fig. 7, however throughout the arrays, n < 80 × 10 −6 , which is comparable to low birefringence optical diamond and is relatively low for electronic grade diamond. The laser writing sites that do and do not develop fluorescing NVCs are not strongly correlated with the birefringence indicated by brightness through crossed polarizers in Fig. 7(c). The highest brightness corresponds to a birefringence averaged through the plate of approximately n = 80 × 10 −6 .

APPENDIX E: HANBURY BROWN-TWISS MEASUREMENTS
Fluorescence from the NVCs being measured is split by a nonpolarizing beam splitter and the autocorrelation of the photon arrival events at two Excelitas near-infrared (NIR) enhanced single photon counting modules (SPCMs) is accumulated using a time tagger. The data are normalized by the accumulated autocorrelation for large τ and fitted using the photon arrival time autocorrelation model for a three-level system g (2) with a modification to allow the depth [d = 1 − g (2) (0)] of the dip at τ = 0 to vary as a fit parameter. τ 2 , τ 3 , and c relate to the lifetimes of states and the internal dynamics of the NVCs. The fitted g (2) (0) with measurement uncertainty inferred from the standard deviation of g (2) (τ ) at large τ is binned into g (2) (0) < 0.5 for single NVCs, 0.5 ≤ g (2) (0) < 0.66 for double NVCs, and 0.66 ≤ g (2) (0) < 0.75 for triple NVCs. Examples are shown 064005-10 in Fig. 8. In cases where insufficient light is detected at automatically visited sites, some sites are inspected manually and where pairs of nearby NVCs confound the automated focusing, these are classified as optically resolved double sites as are sites where more than one optically distinct NVC can be seen in confocal imaging of the grid site. The full HBT statistics are shown in Fig. 8(d).

APPENDIX F: SPIN COHERENCE MEASUREMENTS
Arrays M and I are focused upon for measurements of the spin coherence: confocal images of their layers are shown in Figs. 9 and 10, respectively, which show both the distribution of single and double centers and the location of the NVCs where the spin-echo coherence time (T 2 ) is measured. There is no obvious clustering of the multiple NVC sites or sites with long T 2 times. Some of the longest spin-echo coherence times measured as well as a histogram are shown in Fig. 11 and Table I gives all of the fitted coherence times.

APPENDIX G: POSITION MEASUREMENTS
In array M, automated measurements of position are made relative to a reference single NVC using confocal microscopy using a PI Mars P-563 closed-loop nanopositioning stage. By fitting oversampled confocal scans of single NVCs with Gaussian profiles to approximate the point spread function (PSF) of the optical system, the positions of single centers relative to a repeatedly measured reference point are inferred to better than the diffraction limit in three dimensions. Repeated measurements show that this technique has an accuracy of around 20 nm over a 20-min period and remains self-consistent in the presence of slow drifts in sample position as shown in Fig. 12(d). The site positions are fitted to a grid of the design pitch and the position displacements reported are the measured displacements from the fitted orientation after rotation to compensate for sample tilt and skew as shown in Fig. 12.