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Akirabha Chanuntranont, Daiki Saito, Kazuki Otani, Tomoki Ota, Yuki Ueda, Masato Tsugawa, Shuntaro Usui, Yuto Miyake, [Tokuyuki Teraji](https://orcid.org/0000-0002-7731-0547), Shinobu Onoda, Takahiro Shinada, Hiroshi Kawarada, Takashi Tanii

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This is the pre-peer reviewed version of the following article:Chanuntranont, A., Saito, D., Otani, K., Ota, T., Ueda, Y., Tsugawa, M., Usui, S., Miyake, Y., Teraji, T., Onoda, S., Shinada, T., Kawarada, H. and Tanii, T. (2025), Real-Time Nuclear Magnetic Resonance Detection Using Maximum Likelihood Estimation with Single-Shallow-Nitrogen-Vacancy Centers in Quantum Heterodyne Measurements. Phys. Status Solidi A, 222: 2400307, which has been published in final form at https://doi.org/10.1002/pssa.202400307. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Real‐Time Nuclear Magnetic Resonance Detection Using Maximum Likelihood Estimation with Single‐Shallow‐Nitrogen‐Vacancy Centers in Quantum Heterodyne Measurements](https://mdr.nims.go.jp/datasets/bb52529d-7987-4ce0-a21b-b97471f5c069)

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Real-Time Nuclear Magnetic Resonance Detection UsingMaximum Likelihood Estimation with Single-Shallow-Nitrogen-Vacancy Centers in Quantum HeterodyneMeasurementsAkirabha Chanuntranont* Daiki Saito Kazuki Otani Tomoki Ota Yuki Ueda Masato Tsugawa Shuntaro Usui Yuto Miyake Tokuyuki Teraji Shinobu Onoda Takahiro Shinada Hiroshi Kawarada Takashi TaniiA. Chanuntranont, D. Saito, K. Otani, T. Ota, Y. Ueda, M. Tsugawa, S. Usui, Y. Miyake, Prof. H. Kawarada, Prof. T. TaniiSchool of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, JapanEmail Address: akirabha@akane.waseda.jpT. TerajiNational Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 304-0044, JapanS. OnodaNational Institutes for Quantum Science and Technology, 1233 Watanuki, Takasaki, Gunma 370-1292, JapanT. ShinadaCenter for Innovative Integrated Electronic Systems, Tohoku University, 468-1 Aramaki-aza-aoba, Aoba, Sendai, Miyagi 980-8572, JapanKeywords: quantum NMR, nanopillar, maximum likelihood, NV center, diamondSingle NV centers in diamond are highly promising quantum NMR sensors. However, their sensing ability is greatly limitedby their low SNR. Not only does low SNR make quantum NMR with single NV centers vulnerable to noise, but it also necessitateslonger measurement durations so that a sufficient number of observations can be acquired to combat noise. In this paper, we identifytwo sources of low SNR in qdyne measurements and provide solutions to overcome them. The first source is the high TIR rate at theair-diamond interface, which is solved by fabricating nanopillars on the diamond surface. The second source is the use of FFT as thesignal processing method. Instead, we suggest the use of MLE, which requires much fewer data points for peak detection. Our solu-tions yield a 3.5× improvement in photon count rate and shortens necessary measurement durations by several orders of magnitude.1 IntroductionSensors for nanoscale magnetometry are desired across many practices. Applications of such sensors in-clude materials characterization, electronics defect detection, and structural analysis of biomolecules.Over the past decade, many studies have proposed nitrogen vacancy (NV) centers in diamond as promis-ing sensors for quantum magnetometry. Diamond NV centers have a magnetic field sensitivity of 1 nT/√Hz,six orders of magnitude more sensitive than conventional nuclear magnetic resonance (NMR) sensors,and a magnetic moment sensitivity of 10−3 µB, among the highest of any known sensor [1]. Furthermore,recent advances in quantum sensing sequences, such as qdyne, have allowed the frequency resolution ofdiamond NV centers as quantum NMR sensors to scale with total measurement time as T−3/2, arbitrar-ily beyond their T2 coherence times [2, 3, 4]. Some other advantages of diamond NV centers include theirrelatively long quantum coherence times at ambient conditions, their susceptibility to optical control andreadout, and their bio-compatibility [5, 6]. These properties give diamond NV centers great potential asboth high performance quantum sensors in laboratories and for field deployment in commercial settings.However, despite various demonstrations of their potential, diamond NV centers are still not readyfor regular use as quantum sensors. In quantum NMR applications, a few major hurdles that have tobe overcome include the poor charge stability of shallow NV centers, the limited number of detectablenuclear species due to insufficient T2 times, and the poor signal to noise ratio (SNR) of the measure-ment itself. The first two problems directly affect the ability of the NV center to act as a coherent sen-sor and to detect various nuclear species. The third problem affects the total measurement duration andthe frequency resolution of the obtained NMR spectrum, often necessitating measurement times in the1order of hours or days for biologically relevant samples [6, 2, 3, 4]. Overcoming the problem of low SNRwould not only shorten the required measurement duration and improve frequency resolution of quan-tum NMR with NV centers, bringing these sensors closer to commercial readiness, but may also speedup research in the field as a whole. Thus, improving the SNR of diamond NV centers is the focus of thiscurrent work.In this paper, we identify and overcome two sources of low SNR in diamond NV centers for qdyne-based quantum NMR. Particularly, this paper is focused on qdyne measurements with single shallow NVcenters, defined as being 5 nm below the diamond surface. The depth of 5 nm was chosen with the goalof single protein structural analysis in mind. NV centers at this depth were shown to have favourablesensing characteristics for protein analysis, such as having a sensing volume that is approximately thesize of an individual protein [6]. This paper is organized as follows. In Section 2, we identify the firstsource of low SNR as the high rate of total internal reflection (TIR) in diamond leading to low photonoutput. We present the solution to the TIR problem as the fabrication of nanopillars on the diamondsurface, resulting in a measured 3.5× increase in photon counts. This result was recently published in[7], but we expand on the discussion here. In Section 3 we identify the second source of low SNR as asub-optimal choice of signal processing method, namely the fast Fourier transform (FFT). We proposeinstead the use of maximum likelihood estimation (MLE) and demonstrate significant improvements inpeak clarity and orders of magnitude reduction in required measurement duration. In Section 4 we pro-vide a conclusion of our results and directions for future research. Section 5 contains the experimentalsetup and methods used to obtain our data.2 Photonic Signal Enhancement with NanopillarsFigure 1: Optical excitation and readout of theNV center state.Most quantum NMR sensing schemes with diamond NV cen-ters are based on optical excitation and readout [8]. Qdyneis a type of an optically based quantum NMR scheme. Op-tical readout schemes rely on counting single photons emit-ted from the NV center, thus the photon count rate is a keymeasure of signal strength [9]. As shown in Figure 1, opti-cal excitation and readout of the NV center state is achievedthrough the top facet of the diamond. The refractive indexof diamond is approximately 2.42 at the air-diamond inter-face, which corresponds to a narrow critical angle of 24◦,beyond which emitted photons are trapped within the di-amond due to TIR. The high TIR rate at the air-diamondinterface is the physical cause of low SNR in NMR measure-ments with single diamond NV centers.Measurements from a single shallow NV center in a dia-mond sample with a smooth top facet yield a photon countrate of 90 kC/s. Of these counts, approximately 50 kC/s isattributed to background fluorescence. Thus, the NV cen-ter contributes only 40 kC/s, which is below the backgroundrate and results in low SNR. In practice, we find that NVcenter contributions of at least 100 kC/s, or twice the back-ground rate, are required to obtain sufficient SNR for qdynemeasurements within tractable time. Various photonic solutions, such as the solid immersion lens [10],the reflecting mirror [11], and the inverted nanocone [12] have been proposed to increase the photoncount rate. While these solutions take advantage of diamond’s optical properties to increase the pho-ton count rate, they do so without regard for quantum NMR requirements such as NV center depth andcharge stability.A solution that holds promise for quantum NMR applications is the nanopillar. Unlike other pho-2Figure 2: FFT spectrum of observed data for qdyne measurements (a) without a nanopillar and (b) with a nanopillar. (c)Dependence of FFT peak amplitude on photon count rate.tonic structures, nanopillars do not rely on diamond’s optical properties to improve photon count rates.Instead, nanopillars improve photon count rates by exploiting wave-guiding characteristics, coupling pho-ton emission modes to the nanopillar axis so that photons are mainly directed through the top facet ofthe nanopillar [13]. This mode coupling approach admits greater flexibility in NV center depth, which isa key determinant of sensing properties. While originally optimized for applications with single NV cen-ters at a depth of 20 nm below the top facet [9], we present nanopillars with new dimensions designed forNV centers 5 nm below the top facet for biological sensing applications.A grid search described in [7] was performed to determine suitable nanopillar dimensions. A nanopil-lar with a top facet of 93 nm, base of 261 nm and length of 260 nm was fabricated using the method out-lined in Section 5. This nanopillar was found to increase the photon count rate to 180 kC/s, while slightlyreducing the background counts to 40 kC/s. Thus, the NV center in the nanopillar was found to con-tribute 140 kC/s, which is both 3.5× greater than the background rate and the rate of the NV centerwithout nanopillars. Hence, an improvement of 3.5× in photon counts was achieved and our conditionthat the photon counts be at least twice that of the background is satisfied. A detailed analysis of NVcenter characteristics and nano-NMR results with the nanopillars are available in [7], but it suffices tosay that the NV center maintained its NMR sensing capabilities. Notably, the height of this nanopillaris shorter than those used by other groups, whose nanopillars usually exceed 1 µm in length [9]. The fo-cus on short nanopillars is driven by an interest to find the minimal dimensions that satisfy our sensingrequirements and not use any more material than is necessary. Although these dimensions may not beglobally optimal, they are sufficient for our purpose.Photon counts alone do not constitute our final signal. In qdyne measurements, the photon counts3must be processed to extract frequency data. Usually, this is achieved through the FFT, where resonantfrequencies appear as dominant peaks in the FFT spectrum [8, 3, 2, 4]. To see how the improvement inphoton counts translates to SNR improvement in the FFT spectrum, the count data obtained from a60 s 2MHz coil measurement described in Section 5 was analyzed. The FFT spectra for measurementsperformed without and with the nanopillar are shown in Figure 2 (a) and (b) respectively. It can be seenthat the measurement with the nanopillar yields an FFT peak with twice the amplitude of the mesure-ment without the nanopillar. Both peaks are correctly situated around the target 2MHz frequency, withthe small descrepancy most likely being owed to slight differences between the clock of the analog signalgenerator and the data timing generator. The level of noise, measured as the floor level fluctuation, issimilar in both spectra. Thus, an SNR improvement of 2× was achieved in the FFT spectrum.Simulations based on the measured values were performed to examine the relationship between FFTpeak amplitude and the photon count rate. The results of these simulations are presented in Figure 2(c), which shows the average FFT peak amplitude for various photon count rates. The correspondinglocations of the experimentally obtained spectra are also indicated by grey circles. It can be seen thatthe peak amplitude increases linearly with the photon count rate, which is expected from the linear na-ture of the FFT. Meanwhile, the floor level fluctuation remains approximately constant, if not increasingvery slowly with the photon count rate. Thus, increasing the photon count rate is beneficial for quantumNMR measurements and linearly increases the SNR.3 Statistical Signal Enhancement with Maximum Likelihood EstimationThe most common way to process a qdyne signal is the FFT. The definition of the FFT for time seriesobservations x0, ...xn−1 is shown in Equation 1,Xk =n−1∑m=0xme−i2πkm/n (1)where k ∈ 0, 1, ..., n− 1 and Xk are the relative contributions of each frequency component. The popu-larity of the FFT is owed largely to its unbiased nature, as it is a pure transform from the time domainto the frequency domain. In this sense, the FFT may be regarded as a naive method of signal process-ing. Equation 1 does not perform any sort of inference and produces reliable spectra for all time-seriesdata, regardless of their original generating functions.While the FFT is a reliable and generally applicable tool, its naive nature means it performs poorlyat identifying frequencies of interest within noisy signals. The typical method to increase the chance ofdetection is to acquire more data, which lengthens the duration of the measurement. In biological sens-ing applications, qdyne measurements usually run in the order of several hours to acquire enough datapoints for the FFT to successfully identify a peak. Not only is it tedious to wait several hours for rawmeasurement data, but it is also difficult to control environmental factors over the course of long mea-surements. Moreover, a lack of quick litmus tests for signal availability often lead to measurements beingperformed on areas that do not contain particles of interest, which results in wasted time. Thus, a signalprocessing method with the ability to detect signals with few data points is highly desired.Statistical inference methods that allow the input of prior knowledge are better solutions for noisysignals with known generating functions. Previously, Bayesian inference via Markov-chain Monte Carlo(MCMC) simulations was proposed as an alternative to the FFT in qdyne measurements [4]. While BayesianMCMC inference is a powerful statistical inference tool in theory, it is still largely intractable for mostreal data. For example, we found that running 3,000 steps of Bayesian MCMC inference for qdyne datawith 200,000 data points on four Nvidia A10 GPUs required over 100 hours and, at 64-bit precision, oc-cupied over 70GB of memory. In this case, both the number of MCMC samples and processed data pointsare still far smaller than those used in actual analysis. For example, a typical qdyne measurement hasseveral billions of data points, for which at least 100,000 steps of MCMC inference are desired. Clearly,Bayesian MCMC inference is still not ready for practical use cases.4Instead, we propose the use of MLE as an excellent alternative for processing qdyne signals. The ba-sic concept of MLE is to measure the likelihood that the data was generated from a certain function andchoose function parameters that maximize that likelihood. Both the definition of the likelihood functionand the input of known parameters serve as the provision of prior knowledge. To derive the likelihoodfunction, a model of the measurement data must be defined. The raw signal S from the NV center in aqdyne measurement with a single target frequency is given in Equation 2,S = A sin(2πνt⃗+ ϕ0)(2)where A is the alternating magnetic field strength, ν is the Larmor frequency of the target nuclear spin,t⃗ is the elapsed probing time and ϕ0 is the initial phase of the NV center. The qdyne probing protocolrelies on the XY8-k sequence, which acts as a filter function F⃗xy8k defined as a square wave in Equation3,F⃗xy8k ={10, 11, ..., 1τ/dtxy8k ,−10,−11, ...,−1τ/dtxy8k}(τ8k/dtxy8k)(2τ/dtxy8k)(3)where dtxy8k is the discretized time step of the XY8-k sequence, τ is the Rabi period of the NV center, kis the number of repetitions of the XY8 sequence. The magnetic resonance signal B detected by qdyne ismasked by this filter function as in Equation 4.B =∫ t0S · F⃗xy8kdtxy8k (4)The signal B determines the quantum state Q of the NV center to be read out. In our measurements,the readout pulse of the XY8-k sequence is a 3π/2 pulse, which corresponds to a cosine readout func-tion. This function is defined in Equation 5,Q = cos (2πγNVB) (5)where γNV is the gyromagnetic ratio of the NV center. As mentioned in Section 2, this readout is de-tected as photon counts. The photon emissions from an NV center during a qdyne measurement may bemodelled as a non-homogenous Poisson process (NHPP). The time-varying arrival rate λ of the detectedphotons are approximated as in Equation 6,λ =Pmax − Pmin2Q+Pmax + Pmin2+ λBG (6)where λBG is the arrival rate of background photons and Pmax, Pmin are the average photon counts instates ms = 0 and ms = −1 respectively. The NHPP representation of detected photon counts P isgiven in Equation 7.P ∼ P (λ) (7)Finally, the log-likelihood function of Equation 7 is shown in Equation 8,ln p(Y⃗ |θ⃗) = −N∑k=1λk +N∑k=1Yk lnλk −N∑k=1ln (Yk!) (8)where Y⃗ is a vector of observed photon counts, θ⃗ is a vector of function parameters, and λk is the arrivalrate predicted by a model with θ⃗ parameters at time k. The log-likelihood is used instead of the directlikelihood function for two reasons. First, the likelihood shrinks exponentially with the number of ob-servations, often beyond the accuracy of 64-bit floating point numbers and resulting in zero likelihood.Second, the log-likelihood is a summation, while the likelihood function is a product, so computing thelog-likelihood is more efficient. The MLE model presented in Equation 2 to Equation 8 clearly containsmore information about the target signal than the FFT defined in Equation 1. Also, conveniently, de-spite the arity of the total measurement model, the log-likelihood function is only dependent on the ar-rival rate λ. Thus, it is possible to reuse the same log-likelihood function for all NHPP photon countingprocesses while only swapping out the λ model.5Figure 3: Evolution of the (a) FFT spectrum and (b) MLE likelihood curve for qdyne measurement data at varyingelapsed times.6To see whether the prior information improves performance, this model was used to analyze the samecoil measurement data from the nanopillar presented in Section 2. The data was truncated at variousmeasurement times in order to see the evolution of the peak shapes as the number of observations is in-creased. This evolution is shown in Figure 3 (a) and (b) for FFT and MLE analyses respectively. Thelikelihood curve in the MLE analysis shows the likelihood that the observed data originated from a givenfrequency. Visually, the likelihood curve is more stable and has a clearer peak than the FFT spectrumacross all measurement durations. These characteristics are most likely due to the model’s ability to cap-ture noise variances well, leading to stable likelihood curves over time. The likelihood peak is also cor-rectly positioned around 2.0MHz across all measurement durations, even after only 0.3 s or 12,758 obser-vations. In contrast, the FFT spectrum does not show a clear peak until around 150 s, or approximately6 million observations. The sensitivity of the FFT lags behind MLE because FFTs perform a pure trans-form of every data point from the time to frequency domain, while MLE measures the similarity betweenthe observed data and a pre-defined model. Using MLE to process the data requires several orders ofmagnitude fewer data points than FFT analysis, which translates to significantly shorter measurementdurations.On the other hand, the FFT spectrum is more information dense than the MLE likelihood curve.The FFT spectrum provides the contribution of all constituent frequencies in the signal, while the likeli-hood curve only searches for one frequency. In principle, it is possible to search for the presence of mul-tiple frequencies using MLE by making a more detailed model, but this comes at the cost of computa-tional efficiency. Similarly, the resolution of the FFT spectrum is significantly higher than that of theMLE likelihood curve. Again, it is possible in principle to improve the resolution of MLE by improvingthe model, which is an area of active development.Meanwhile, it should also be noted that the frequency axes of the FFT spectra in Figure 3 are de-modulated frequencies, while those of the MLE curves are raw frequencies. Undemodulated FFT spectrafrom qdyne measurements yield frequencies in the kHz range due to the nature of the heterodyne tech-nique [2]. Thus, the FFT frequencies need to be shifted back into the MHz regime by adding some off-set. This technique contributes to the high resolution of FFT, as seen in Figure 3, but the demodulationrequires an estimate of the offset, which is often done heuristically and can introduce error. In contrast,MLE analysis accepts frequency inputs as-is due to the use of a complete measurement model. Thus, wecan be sure that the likelihood curve corresponds to our frequency axis exactly. Also, as shown in Figure3, the ability of input frequencies directly into the estimation model allows MLE to search a wider rangethan FFT.Figure 4: MLE likelihood curve of randomly generated count data.A concern with MLE is whether itis prone to hallucinations. To explorethis question, we used our model toperform MLE on simulated noise datawith no underlying signal. A resultfrom this analysis is shown in Figure 4.As can be seen, the likelihood curve inFigure 4 is different from those in Fig-ure 3 (b), with a likelihood dip insteadof a peak. The likelihood dip signifiesthat MLE correctly found no similar-ity between the data and the proposedmodel. It is interesting that the like-lihood curves in Figure 4 and Figure3 (b) both show activity in the same range, that is from approximately 1.8MHz to 2.2MHz, while therest of the curve is flat. We believe that the existence of this active region is owed to the frequency fil-tering properties of the XY8-k sequence, as this active region coincides with the detectable frequenciesfor the given XY8-k parameters. It makes sense that the likelihood of observations originating from fre-quencies beyond this filter should be equal, as the model cannot differentiate between those frequencies.7Within this sensitive area, however, peaks will occur when an underlying frequency agrees well with ob-servations and valleys otherwise. From our tests, it appears that the model is robust to hallucinationswhen the number of observations exceeds 200,000 samples. Nevertheless, more work is needed to developsystematic measures and guards for hallucinations.4 ConclusionIn this work, two sources of low SNR in qdyne measurements with single shallow NV centers in diamondwere identified and overcome. The first source is the high TIR rate of diamond, which is overcome throughthe fabrication of nanopillars. The nanopillars presented in work increased the photon counts 3.5× andresulted in 2× improvement in FFT SNR. The second source is the choice of signal processing method,for which we suggest the use of MLE. Adopting MLE vastly improves the signal clarity and can poten-tially reduce necessary measurement durations by orders of magnitude.The implementation of these two approaches can vastly improve and speed up development of quan-tum NMR sensing with diamond NV centers. In addition, shortening required measurement durationsand improving robustness against noise makes NV centers more attractive for field deployment. How-ever, there are still outstanding issues with both approaches. The variety of nuclear species detectablewith NV centers in nanopillars are still limited, likely due to insufficient T2 times. In MLE, systematicmeasures and guards for hallucinations must be developed. Until then, we suggest the use of MLE pri-marily as a litmus test for available particles and frequencies when performing nano-NMR measurements,while using FFT for the final analysis. Solutions to these problems are currently being investigated aspart of ongoing research.5 Experimental SectionDiamond Substrate and NV Centers :All experiments were performed using a type-Ib high-pressure high-temperature (100) single crystallinediamond substrate with a size of 5mm×5mm×1mm. On the top facet of this substrate, a 20µm thick99.95% 12C enriched homoepitaxial diamond film was grown using plasma-assisted chemical vapor depo-sition as described in [14]. The growth conditions were 140 Torr gas pressure, 1.2 kW microwave power,120 to 180W/cm3 microwave power density, 1% methane concentration, 0% oxygen concentration, 200sccm total flow rate and 800±10 ◦C substrate temperature. Single 15N ion implantation was performedin this diamond film at an acceleration energy of 2.5 keV and a fluence of 1.5 × 1011 cm−2 as describedin [15]. Single NV centers were formed through thermal annealing at 1000 ◦C in 10% H2 forming gas.The NV centers were located between 2 nm and 11 nm in depth, with approcimately 50% being locatedwithin 4 nm of the surface [15].Nanopillar Fabrication:Nanopillars were fabricated on a diamond substrate already containing shallow NV centers following aprocess described in [7]. The top facet of the diamond substrate was spin coated with PMGI (Microchem,PMGI SF 6S) and ZEP-520A (ZEON, ZEP-520A) at 4000 rpm for 50 s with a slope of 10 s to form adouble-layer resist film. The thickness of the PMGI and ZEP-520A layers were measured in a scanningelectron microscope (SEM) to be 220 nm and 200 nm respectively. The resist film was baked for 5min at225 ◦C to harden. Electron beam (EB) lithography was performed to pattern regular arrays of nanoholesonto the ZEP-520A layer of the film. The pattern was stabilized in low light through immersion in ZED-N50 for 3min 30 s followed by a rinse in ZEP for 30 s. Wet etching via immersion in a 2.38% NMD-3 so-lution for 35 s was performed to transfer the pattern onto the PMGI layer. Ti cylinder masks approxi-mately 130 nm thick were deposited into the nanohole pattern via EB vapor deposition. The diamondwas immersed in 20mL of remover PG and gently perturbed until the double-layer resist film was re-moved. Inductively coupled plasma (ICP) reactive ion etching with O2 gas at 30 SCCM and 1Pa pres-sure, 700W ICP power and 250W bias power was performed to etch the diamond substrate around the8Figure 5: Fabricated nanopillar array.Ti masks and form nanopillars. The Ti mask was removed through hot mixed acid cleaning (HNO3 :H2SO4 = 1 : 3) at 200 ◦C for 30min. The diamond surface was terminated with oxygen through O2annealing at 465 ◦C for 8 h to improve the charge stability of the NV centers. A SEM image of the fab-ricated nanopillar array is shown in Figure 5. It can be seen that this method yields large quantities ofhighly uniform, evenly spaced nanopillars.CFM Setup:A home-built scanning confocal fluorescence microscope (CFM) was used to perform quantum NMR ex-periments. A schematic of the CFM is shown in Figure 6. Optical control signals were generated by agreen 532 nm laser source (Changchun New Industries Optoelectronics Technology, MGL-III-532 nm 300mW-1%) and pulsed by an acousto-optic modulator (Gooch & Housego, AOMO 3350-120). An air ob-jective lens (Olympus, MPLAPON 50×) was mounted on a piezo stage (PI, NanoCube P-611.3S) forscanning. The diamond was mounted on the sample holder. Microwave (MW) pulses from an analogradio frequency (RF) signal generator (Keysight, E4428C) were pulsed by an RF circuit composed ofa phase shifter, two switches and a combiner (Mini-Circuits, ZX10Q-2-25-S+, ZASWA-2-50DR+ andZX10-2-442-S+) and were amplified by a power amplifier (Mini-Circuits, ZHL-16W-43-S+). A data tim-ing generator (Textronics, DTG5274) controlled RF switching. The MW pulses were supplied to the NVcenter via a Cu nanowire placed on the diamond surface. A 30mT static magnetic field was supplied byan Nd magnet. Emissions from the NV center were sent through a long-pass filter (≥650 nm) and de-tected by a single-photon counting module (Laser Components, COUNT-100C). The photon countingprotocol was implemented in a field-programmable gate array (FPGA) board (Digilent, Cora Z7-10) andsent to a desktop computer as binary data. The various components of the CFM were managed usingthe Qudi software package [16].Qdyne Measurements :All qdyne measurements were performed on the same single NV center approximately 5 nm below thediamond surface. The target of the measurements was a 2MHz alternating magnetic field supplied by acoil wound around the diamond sample. The coil had a loop diameter of 38mm, wire diameter of 0.40mmand was hand-wound with 50 turns. A qdyne measurement was first performed on the diamond samplebefore nanopillar fabrication and again after fabrication. Each measurement was performed for 600 s atambient conditions.FFT, MLE and Simulation Implementation:FFT analysis was performed in Python with the Scipy library [17]. MLE analysis was implemented inPython according to Equation 2 through Equation 8 with the JAX library [18]. This library was cho-9REFERENCESFigure 6: CFM schematic.sen to take advantage of GPU acceleration. The MLE calculation was sharded across two GPUs (Nvidia3060) operating in 64-bit precision. Likelihood curves were produced by calculating the log-likelihood ofall frequencies from 1.5MHz to 2.5MHz with a step of 1 kHz. The measurement model in MLE was alsoused to simulate qdyne measurement data. These simulations were also sharded across the two GPUswith 64-bit precision.AcknowledgementsThis work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI (No.JP22H01921, JP18H03766, JP23H00169, 20H02187 and 20H05661), MEXT Q-LEAP (JPMXS0118068379),JST CREST (JPMJCR1773), JST Moonshot R&D (JPMJMS2062), MIC R&D for construction of aglobal quantum cryptography network (JPMI00316), Advanced Research Infrastructure for Materialsand Nanotechnology in Japan (ARIM) and Design & Engineering by Joint Inverse Innovation for Ma-terials Architecture (DEJI2MA) of the Ministry of Education, Culture, Sports, Science and Technology(MEXT).Conflict of InterestThe authors bear no conflict of interest in this work.References[1] M. Chen, C. Meng, Q. Zhang, C.-K. Duan, F. Shi, J. Du, National Science Review 2017, 5.[2] S. Schmitt, T. Gefen, F. M. Stürner, T. Unden, G. Wolff, C. Müller, J. Scheuer, B. Naydenov,M. Markham, S. Pezzagna, J. Meijer, I. Schwarz, M. Plenio, A. Retzker, L. 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