5.5.1 Shallow NV centers
The NV centers used in this work for dark-spin readout are created by implanting
14N (at 3· 1011cm−2density with 6 keV energy) into electronic grade diamond
(Element 6, [100] cut) and annealing the diamond (2 hours at 800◦C in vacuum) via previously reported procedures [37,62]. Waveguiding nanopillars [4] are used to isolate individual NV centers and enhance NV collection rates (200-400 kCPS for single NV centers). We note that the nominal depth of implanted N at 6 keV is 10± 3 nm [104], though in reality NV centers can lie deeper in the crystal because of channeling during implantation [97]. The coherence properties of these shallow NV centers exhibited considerable variation, with T∗2values
5.5.2 Scaniing magnetic gradients
Magnetic tips are optimized from previous iterations of scanning gradients [37]. For all tips used in this work, nickel films are thermally evaporated onto
laser-pulled quartz rods, with radii of curvature of 35 nm. In order to minimize off-axis fields that quench NV spin contrast, and are inevitable during tip scanning, the gradient is maximized while minimizing the total value of the tip field. This is accomplished by evaporating a layer of nickel equal to the tip radius of curvature directionally onto its end, resulting in the majority of the tip field coming from as small a volume as possible. Tips are brought within a radius of curvature of dark spins of interest for the highest possible gradients. We note that we observed significant reductions in MRI performance when using films in thickness below 30 nm (due to weak magnetization) or above 40 nm (due to off-axis fields).
5.5.3 Quantifying magnetic field gradients using Ramsey interfer- ometry
We measure the gradient of the magnetic field from our scanning tip by using a Ramsey interferometry sequence applied to a single NV center for a fixed evolution time τ 5.2.2a). Without the presence of the tip, the spin evolves coherently between the mS= 0 and mS = 1 states with a period of 1/τ, as long as
the detuning from the NV resonance is less than the MW Rabi frequency (31 MHz). In the presence of the magnetic tip (Fig. 5.2.2a) and with fixed τNV, the
tip’s magnetic field maps these Ramsey oscillations in frequency space to real space. The spatial period of the fringes therefore measures the local magnetic field gradient (of the field component projected along the NV axis) in the scan range.
5.5.4 Pulsing normalization and references
In order to quantify the NV DEER signal from proximal dark spins, all presented dark spin measurements are normalized by (or are referenced to) the NV sensor running a sequence identical to the DEER sequence, except that the dark-spin
addressing π-pulses are removed. For instance, for DEER spin-echo pulsing the normalized dark-spin signal is found from the fluorescence rate of that
measurement divided by the fluorescence rate from a spin-echo measurement on the NV with the same evolution time.
This normalization ensures that any observed dark-spin signal is not the result of (a) reduction of NV fluorescence collection, (b) reduction of NV contrast, (c) reduction of NV coherence, or (d) detuning of NV addressing MWs, all of which in principle can occur when scanning the magnetic tip. Such issues would cause a reduction in the magnitude of the normalized DEER signal we observe, but do not easily allow for any false positive signals. We note that we rule out possible cross-talk between the DEER RFs and the NV MWs (which could potentially lead to fictitious normalized signals). This is done by ensuring that when the RF is detuned from any dark-spin resonances, the fluorescence from the DEER signal is equal to the fluorescence from a NV spin-echo (as can be seen in the normalized fluorescence going to 1 in DEER ESR measurements like those in Fig. 5.3.1a, 5.3.1c).
During pulsing experiments, measurements are also referenced to the fluorescence of the NV spin states (mS = 0 and typically mS = 1), to provide
context for the strength of the DEER signal. In particular, to claim coupling of individual dark spins via DEER, it is necessary to prove that the DEER signal goes beyond the NV mixed-state. By measuring the NV reference states by either using an empty pulse sequence or using a single π-pulse, the fluorescence of the DEER can be directly compared to the fluorescence of a mixed NV spin state.
For optimizing signal-to-noise and increasing clarity, we find it useful to vary the phase of the NV readout π/2 pulse by 180 degrees, so as to probe coherence and DEER originating from both the mS = 0 and mS = 1 states and to ensure
that pulse errors are insignificant.
All normalization and references are interwoven in experiments. For pulse evolution measurements (e.g. Fig. 5.3.2a) each signal/reference is measured once at each time interval before incrementing to a new one. For scanning (or frequency sweeping ESR), a fixed pulse sequence runs in the background,
asynchronously from piezoelectric scanners (MW frequency scanning), where the pulse sequence repeats many thousands of times within each pixel (frequency point). By minimizing the time between signal and references, we are able to reject low-frequency common-mode noise such as fluorescence rate drifts and ensure proper normalization.
5.5.5 Precision and accuracy determinations
Unless otherwise specified, error bars on extracted dark spin distances refer to the precision of the measurement. These estimates use the fact that the dominant noise source in NV-based measurements is photon shot noise, which is
uncorrelated. Consequently, the noise on different pixels/data points can be well approximated to be independent from one another (particularly when
fluorescent rates are normalized to avoid slow-drifts in count rates). Thus, the error induced on distance estimates from these random variables can be found by either using the covariance matrix from regression fitting or by calculating how the χ2from a model varies as a function of distance parameters.
Estimating the extent of systematic errors is more difficult. In general, piezoelectric nonlinearities, piezoelectric hysteresis, and errors in
point-spread-function determination can all lead to inaccuracies which are unaccounted when estimating measurement precision. In order to provide an upper bound for the contributions of these issues to measurement accuracy, for the single dark spin imaging, we can compare the observed dark-spin coupling rate to the NV (oscillation period in Fig. 5.3.2a) to the calculated magnetic field from an individual Bohr magneton residing at the extracted displacement from the NV (Fig. 5.3.2d), taking into effect the directions of the dark-spin
quantization axis and the NV axis. We find the extracted dark-spin coupling rate is 72 kHz and the measured coupling rate is 85 kHz. Because the coupling rate is a strong function of distance (B∝ r−3), the agreement between these two coupling rates indicates that the extracted distance is accurate to within 6%.
5.5.6 Dark-spin deconvolution
Deconvolution of the dark-spin imaging in Fig. 5.3.1d is performed using a Weiner deconvolution implemented in MATLAB as has been used and described previously [37].