# Fileset [manuscript_NS2023PRAppl.pdf](https://mdr.nims.go.jp/filesets/d246c1a6-638e-4da6-97a1-3e45bd396003/download) ## Creator [N. Sekiguchi](https://orcid.org/0000-0003-1533-1411), [M. Fushimi](https://orcid.org/0000-0001-6197-1407), A. Yoshimura, C. Shinei, [M. Miyakawa](https://orcid.org/0000-0002-0838-8156), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), [T. Teraji](https://orcid.org/0000-0002-7731-0547), [H. Abe](https://orcid.org/0000-0001-9659-8382), [S. Onoda](https://orcid.org/0000-0003-1425-0708), [T. Ohshima](https://orcid.org/0000-0002-7850-3164), M. Hatano, [M. Sekino](https://orcid.org/0000-0002-3387-1932), [T. Iwasaki](https://orcid.org/0000-0001-6319-7718) ## Rights © 2024 American Physical Societ[In Copyright](http://rightsstatements.org/vocab/InC/1.0/) ## Other metadata [Diamond quantum magnetometer with dc sensitivity of sub-10 pT

Hz - 1 / 2

toward measurement of biomagnetic field](https://mdr.nims.go.jp/datasets/c8e5b032-ca83-4a74-822d-a52f32b09bc9) ## Fulltext Diamond quantum magnetometer with dc sensitivity of < 10 pT Hz−1/2 towardmeasurement of biomagnetic fieldN. Sekiguchi,1, ∗ M. Fushimi,2 A. Yoshimura,1 C. Shinei,3 M. Miyakawa,4 T. Taniguchi,4T. Teraji,3 H. Abe,5 S. Onoda,5 T. Ohshima,5 M. Hatano,1 M. Sekino,2 and T. Iwasaki11Department of Electrical and Electronic Engineering,Tokyo Institute of Technology, Meguro, Tokyo 152-8550, Japan2Department of Bioengineering, The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan3Research Center for Electronic and Optical Materials,National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan4Research Center for Materials Nanoarchitectonics,National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan5Takasaki Institute for Advanced Quantum Science,National Institutes for Quantum Science and Technology, Takasaki, Gunma 370-1292, Japan(Dated: April 20, 2024)We present a sensitive diamond quantum sensor with a magnetic field sensitivity of 9.4 ±0.1 pT/√Hz in a near-dc frequency range of 5 to 100 Hz. This sensor is based on the continuous-wave optically detected magnetic resonance of an ensemble of nitrogen–vacancy centers along the[111] direction in a diamond (111) single crystal. The long T ∗2 ∼ 2 µs in our diamond and the re-duced intensity noise in laser-induced fluorescence result in remarkable sensitivity among diamondquantum sensors. Based on an Allan deviation analysis, we demonstrate that a sub-picotesla field of0.3 pT is detectable by interrogating the magnetic field for a few thousand seconds. The sensor headis compatible with various practical applications and allows a minimum measurement distance ofabout 1 mm from the sensing region. The proposed sensor facilitates the practical application of di-amond quantum sensors. The sensitivity presented is realized without a magnetic flux concentrator,so that a sensitivity of tens of fT/√Hz can be achivable by using a flux concentrator.I. INTRODUCTIONThe biomedical applications of quantum sensors havebeen studied for over a decade [1]. The realization ofmagnetoencephalography (MEG) under ambient condi-tions is a major goal (conventional MEG requires a mag-netically shielded room). In addition to clinical diagno-sis [2–5], ambient-condition MEG can be used for dailydiagnosis, brain-machine interfaces [6, 7], and fundamen-tal research on brain function [8–12]. A quantum mag-netometer that uses nitrogen–vacancy (NV) centers indiamond is a candidate for realizing ambient-conditionMEG given that it can be operated with high sensitiv-ity at room temperature in an ambient magnetic field[13–22]. A spatial resolution on the millimeter scale orbelow, far better than the centimeter-scale resolution ofconventional MEG [4], is expected for a diamond quan-tum magnetometer [23].Magnetometry based on continuous-wave optically de-tected magnetic resonance (CW-ODMR) is the mostwidely used method for measuring magnetic fields usingNV centers [13–20]. In this method, a microwave (MW)field continuously drives the magnetic resonance of theNV center spin and the spin state is continuously readout as the intensity of the laser-induced fluorescence fromthe NV center. Compared with other methods based onpulsed MWs and/or light [16, 21, 22, 24], the CW-ODMR∗ sekiguchi.n.ac@m.titech.ac.jpmethod has a simpler experimental setup and is easierto apply to actual measurements. Millimeter-scale mag-netocardiography [23] has been realized using the CW-ODMR method. However, measurement of an encephalo-magnetic field requires the detectable level of at least onthe order of pT [2, 5, 8, 10, 11] with a feasible interro-gation time. The frequency of a clinically relevant en-cephalomagnetic field ranges from nearly dc to ∼ 100 Hz[3, 4, 12]. Reported sensitivities, without using a mag-netic flux concentrator (MFC), have not reached the re-quired level in this frequency range. For example, sen-sitivities of around 20 to 30 pT/√Hz [16, 18] have beendemonstrated. In addition, a sensitivity of 15 pT√Hzin a higher frequency range (80 Hz to 3.6 kHz) has beenreported [15]. The amplification of a target field by usinga MFC is a way to achieve the superior sensitivity, whileit requires calibration to the target field and costs the in-trinsic spatial resolution of a diamond quantum magne-tometer. Field sensitivities of approximately 1 pT/√Hzwith a MFC have been reported with the sensing volumeof & 103 mm3 [19, 20]. A short standoff distance fromfield generating sources in the brain is also required giventhat the decay of an encephalomagnetic field is inverselyproportional to the square of the distance [3]. There-fore, for biomedical applications, the sensitivity in thenear-dc frequency range of a diamond quantum magne-tometer that can closely approach the target object mustbe improved.Here, we develop a CW-ODMR-based diamond mag-netometer for practical applications (e.g., MEG of a liv-ing animal). The sensor head of the magnetometer wasmailto:sekiguchi.n.ac@m.titech.ac.jp2designed to approach the target object to a distance ofabout 1 mm with a sensing volume of 0.03 mm3. Bycarefully tuning the experimental conditions and using ahigh-quality diamond, we achieved a record-breaking sen-sitivity of 9.4±0.1 pT/√Hz in a near-dc frequency rangeof 5 to 100 Hz. Based on the Allan deviation, the mini-mum detectable field was found to be 8.5 and 0.3 pT formeasurement periods of 1 second and several thousandseconds, respectively.II. EXPERIMENTAL SETUPA. Sensor headIn this work, we synthesized a single-crystalline dia-mond using a high-pressure–high-temperature (HPHT)method with a 12C isotopically enriched carbon source.The reduced concentration of 13C was about 500 ppm.The amount of titanium in the metal solvent in theHPHT synthesis was adjusted to control the initial con-centration of neutral substituted nitrogen (N0s ) in the di-amond crystal [25]. The initial [N0s ] was estimated tobe 5.6 ppm using electron spin resonance. The origin ofnitrogen in diamond crystals seems to be impurities in-troduced from the source material, solvent, or pressuretransmitting medium during the growth process. Sincethis nitrogen is of natural origin, the isotope abundanceof N0s is the same as the natural abundance (14N, 99.6%;15N, 0.4%). After this HPHT synthesis, a piece of thecrystal was cut out parallel to the (111) crystal plane.The dimensions of this diamond sample were approxi-mately 1 mm × 0.7 mm in area and 0.4 mm in thick-ness. Negatively charged NV (NV−) centers were thenproduced using electron beam irradiation followed by an-nealing at 1000 C◦ for 2 hours in vacuum. The energyand total fluence of the irradiation were 2.0 MeV and5 × 1017 cm−2, respectively. The concentrations of theproduced NV− and residual N0s were estimated to be 1.2and 2.3 ppm, respectively, using electron spin resonance[26]. A full width at half maximum of 0.19 MHz forthe CW-ODMR peak was experimentally measured in-dependent of this work. This linewidth indicates a longdephasing time of T ∗2 ∼ 2 µs.The conceptual design of our sensor head is shown inFig. 1(a). This sensor head was designed to closely ap-proach the head of a living animal and measure the en-cephalomagnetic field along the z axis by an ensemble ofNV centers oriented to the surface-normal [111] directionparallel to the z axis. The sensor head components de-scribed in this section (see below) were integrated usingplastic and aluminum holders. Hence, the sensor headcan be freely moved as a unit and easily positioned closeto the target object.The diamond containing NV centers was attached bya high-thermal-conductivity glue to a polycrystalline di-amond plate (10 × 10 × 0.5 mm3) in order to dissipatethe heat due to laser illumination. The other side of theLaser(532 nm)PDflMagnetLPFPolycrystallinediamond plateNV center diamondHemisphericallensMW guideMW amp.RF (2.16 MHz)MWMW circuitzxy[111](a)Linearlypolarized(b)Laser(532 nm)T-Z amp.λ/2 λ/2BBBB BBPDref IrefIflIdiffPDflPBS PBSPBS-VrbVrbLNPBS L MMDiamondλ/2FIG. 1. Experimental setup (not to scale). (a) Sensor headdesign and MW circuit diagram. PDfl: fluorescence photodi-ode; LPF: long-pass filter. (b) Optical setup. λ/2: halfwaveplate; PBS: polarizing beam splitter; NPBS: non-polarizingbeam splitter; L: lens; M: mirror; PDref : reference photodi-ode; BB: beam block.polycrystalline diamond plate had a current flow guidefor MWs. The MW guide was made of thin copper film;the distance between the lower side of the MW guideand the excited NV centers was 0.8 mm. A bias mag-netic field of 0.9 mT along the z axis was applied by aring samarium-cobalt magnet.We used a hemispherical lens with a high refractiveindex of 2.0 to enhance the collection efficiency of thelaser-induced fluorescence from the NV center ensemble[17]. The fluorescence collection efficiency from the di-amond surface facing the lens [top surface in Fig. 1(a)]was assumed to be as high as about 56% based on a pre-viously reported numerical calculation [17] for a similarsetup. The fluorescence that was not emitted from thissurface was considered to be emitted mainly from theside faces due to the high refractive index (2.4) of the di-amond [27]. Some of the fluorescence from the side facesof the diamond was collected by the lens since the lensdiameter (4 mm) was larger than the size of the diamond.3Fluorescence was also collected by an elliptically shapedreflective inner surface of an aluminum block. Stray greenlight and part of the fluorescence from neutrally chargedNV (NV0) centers were filtered out by a long-pass filterwith a cut-on wavelength of 633 nm. The transmittedfluorescence was detected by a reverse-biased photodiode(PDfl).B. CW-ODMR measurement setupThe NV ensemble was excited by a green laser at532 nm from a side face of the diamond, as shown inFig. 1(a). Figure 1(b) shows the optical setup. A laserbeam with a diameter of about 3 mm was focused by alens with a focal length of 300 mm. The beam diameterat the diamond was estimated to be 70 µm. The exci-tation volume was estimated to be 4 × 10−3 mm3. Thelaser light was linearly polarized along the y axis, whichis perpendicular to the chosen NV orientation. The flu-orescence photocurrent Ifl = 6.6 mA was observed at anincident light power of 0.39 W, which corresponds to adetected fluorescence power of about 13 mW.The noise in the fluorescence due to the intensity fluc-tuation of the incident laser was reduced using a bal-anced detection technique. The reference light, whichwas picked up by a non-polarizing beam splitter, was de-tected by a reverse-biased photodiode (PDref). In thiswork, we connected the anode of PDfl to the cathode ofPDref to obtain the difference between their photocur-rents, Ifl and Iref , respectively. The difference photocur-rent Idiff was amplified by a lab-built transimpedanceamplifier with a gain of 10 kV/A. The power of the refer-ence light was finely adjusted using a halfwave plate anda polarizing beam splitter to achieve a high reductionrate for the intensity noise. The polarization fluctuationof the laser was converted into an intensity fluctuationby a polarization beam splitter just after the laser. Thebeam diameter at PDref was expanded by a lens to bal-ance the nonlinear response of the photodiode with thatof PDfl, since the nonlinear response depends on spot size[28].The magnetic resonance between the ground states |0〉and | − 1〉 was driven by applying an MW current tothe MW guide. To enhance the amplitude of a CW-ODMR peak, we simultaneously drove the three transi-tions associated with the hyperfine spin state by three-tone MWs [15], which was generated by mixing radio-frequency (RF) waves at 2.16 MHz with MWs and sum-ming the mixed waves with bypassed MWs. In this work,the enhancement factor for the peak amplitude was about2.5.We adopted lock-in detection, achieved by sinusoidallymodulating the MW frequency, to avoid large residualnoise at low frequencies. The amplified difference pho-tocurrent was fed into a lock-in amplifier and demod-ulated with the modulation frequency as the reference.The 3-dB cutoff frequency of the low-pass filter in the5 0 5Detuning (MHz)3020100102030Demod. photocurrent (µA) (a)25 0 25Detuning (kHz)1050510Demod. photocurrent (µA) (b)FIG. 2. Lock-in CW-ODMR spectrum (a) over hyperfinemanifold and (b) in near-resonant region of central peak.Measured demodulated photocurrent is shown by filled cir-cles. The solid line in (a) represents the fitted curve obtainedfrom the summation of five derivative Lorentzian functions.The linear function shown by the dashed line in (b) was fittedto the near-resonant data to obtain a zero-crossing slope.lock-in amplifier was 149.4 Hz, which corresponds to anoise-equivalent-power bandwidth of 168.8 Hz. The de-modulated output was recorded on a computer via ananalog-to-digital converter.The sensor head and optical setup were inside a roomthat was shielded from magnetic fields by three permalloylayers to reduce environmental field fluctuations. Thetotal shielding factor of this room was about 2 × 10−4at 1 Hz and about 1 × 10−5 at 10 Hz. An additionalpermalloy box was placed around the sensor head. Thefront face of the shield box remained open to introducethe incident laser and the target object.III. RESULTSA. CW-ODMR measurementFigure 2(a) shows a CW-ODMR spectrum of the en-semble of [111]-oriented NV centers. The vertical axis inthe figure is the demodulated signal Ĩ in the photocur-rent, which was calculated using the gains at the tran-simpedance and lock-in amplifiers. The horizontal axis isthe detuning δ from the resonance frequency of the cen-tral peak at which the three hyperfine spin states weresimultaneously driven. The fluorescence photocurrent Iflat a far-detuned MW frequency was Ifl = 6.6 mA. Inthis measurement, the frequency and depth of the modu-lation were 6.2 kHz and 160 kHz, respectively. We foundthat a modulation frequency of 3 to 7 kHz yielded a low-noise output. The modulation frequency was finely tunedwithin this range on each day of the experiment becausethe frequencies of some noise peaks in the intensity noiseslightly shifted over time. The modulation frequency ofhigher than several kilohertz caused a decrease in the4amplitude of the CW-ODMR peaks, probably becausethe dynamics of the population in the ground states wasslower than the modulation at several kilohertz for therelatively low laser power of < 1 W and large laser spotsize of 70 µm in diameter. The MW and RF wave powerwas tuned to yield a maximum zero-crossing slope atthe central peak. The black solid curve is fitted to themeasured data using the summation of five derivativeLorentzian functions. The corresponding full width athalf maximum of the derivative Lorentzian function wasabout 0.48 MHz . This linewidth is greater than the in-homogeneous broadening of the CW-ODMR peak due tothe inhomogeneity in the bias magnetic field, which wasestimated to be approximately 0.2 MHz.To determine the zero-crossing slope, we measured aCW-ODMR spectrum at the near-resonance region of thecentral peak, as shown in Fig. 2(b). The demodulatedphotocurrent linearly depends on the detuning in thisregion. The zero-crossing slope dĨ/dδ was measured to be324 pA/Hz by fitting the data with a linear function, asshown by the black dashed line. This slope correspondsto the photocurrent response to magnetic field variationas (dĨ/dδ) × γe = 9.06 A/T, where γe = 28.0 GHz/T isthe gyromagnetic ratio for an NV center.B. Reduction in intensity noiseThe reduction rate for the intensity noise was esti-mated at Ifl = 25 mA. We measured the standard de-viations of Ĩ with and without the reference light to be3.0 nA and 130 nA (see Supplemental Material [29] forfurther details), respectively. Here, the MW source wasswitched off to isolate the sensor from the noise associ-ated with the environmental magnetic field. The relativeintensity noise (RIN) in the incident light was roughlyestimated to be RIN = 10log10(130 nA2168.8 Hz×25 mA2)=−130 dBc/Hz at a modulation frequency of 6.2 kHz. Thephoton shot noise with and without the reference lightwas calculated to be 1.6 nA and 1.2 nA, respectively.The details of the photon shot noise calculation are de-scribed in Sec. III C. We obtained the following reductionrate for the fluorescence intensity noise:√3.0 nA2 − 1.6 nA2130 nA2 − 1.2 nA2 = 1.9× 10−2.We found that this “red–green” balance detection exhib-ited a similar reduction rate to that for the “green–green”balanced detection with the reference and incident lights.C. Photocurrent dependence of noiseWe analyzed the noise components (photon shot noise,fluorescence intensity noise, and electrical noise) of thedetectors and circuits by measuring their dependence on0100200n I,far (pA/Hz)(a)0200400600Slope (pA/Hz)(b)0 5 10 15 20 25 30Fluorescence photocurrent Ifl (mA)810121416n B,far (pT/Hz)(c)FIG. 3. Fluorescence photocurrent dependence of (a) floor ofnoise spectral density of Ĩ, (b) zero-crossing slope dĨ/dδ, and(c) estimated floor of equivalent magnetic field noise spectraldensity.the fluorescence photocurrent Ifl. The demodulated pho-tocurrent Ĩ was recorded for 5 s and Fourier-transformedto provide a single-sided noise amplitude spectral densitynĨ . To evaluate the noise nĨ,far without influence fromenvironmental magnetic field noise, the analysis was per-formed with an MW carrier frequency of 2.4 GHz, whichwas far-detuned from the resonance. We observed no ex-cess noise due to the application of the far-detuned MWs.The noise density nĨ,far was almost flat up to the cutofffrequency of the lock-in amplifier. The average 〈nĨ,far〉 ofthe noise density within the 100-Hz bandwidth was takenas a measure of the intrinsic noise of our diamond sensorat a given Ifl. The dependence on Ifl of 〈nĨ,far〉 is shownin Fig. 3(a). Here, we varied Ifl by varying the incidentlaser power using a halfwave plate and a polarizing beamsplitter just before the non-polarizing beam splitter. Inthe figure, the measured 〈nĨ,far〉 is represented by opencircles. The relative uncertainty in the data, shown as er-ror bars, was independently evaluated to be 5%. 〈nĨ,far〉at Ifl = 0 represents the electrical noise density 〈nĨ,elec〉and was measured to be 20 pA/√Hz by blocking the laser5beam before the non-polarizing beam splitter.We fitted the noise model in Eq. (1) to the data.〈nĨ,far〉 =√〈nĨ,elec〉2 + p1Ifl + p2I2fl. (1)The second and third terms represent the photon shotnoise 〈nĨ,psn〉 and fluorescence intensity noise 〈nĨ,int〉, re-spectively. This noise model well describes the data, asshown by the black solid curve in Fig. 3(a). The fit-ted parameters were p1 = (5.0± 0.6)× 10−19 A/Hz andp2 = (5.0 ± 0.5) × 10−17 /Hz. This agreement impliesthat the reduction rate for the intensity noise was almostconstant for different Ifl, since the noise model assumedp2 to be independent of Ifl. The obtained p2 and theestimated RIN of −130 dBc/Hz give another estimationof the reduction rate of √p2 × 10−RIN/20 = 2.2 × 10−2,which is consistent with the value estimated in Sec. III B.The black dashed curve is the sum of 〈nĨ,elec〉 and thecalculated shot noise given by√〈nĨ,elec〉2 + 2× 2qeIfl, (2)where qe = 1.6× 10−19 C is the elementary charge. Thefactor of 2 for the shot noise term was introduced be-cause the shot noise at the two photodiodes was assumedto be independent. The measured shot noise coefficientp1 = (5.0 ± 0.6) × 10−19 A/Hz is close to the calculatedvalue of 2× 2qe = 6.4× 10−19 A/Hz. The intensity noise〈nĨ,int〉 =√p2I2fl is equivalent to 〈nĨ,psn〉 at the fluores-cence photocurrent Ifl,eqv = p1/p2 = 10 ± 1.6 mA. Thephoton shot noise surpassed the laser intensity noise ata low fluorescence photocurrent (< Ifl,eqv).The sensor noise nB in the magnetic field measurementdepends on demodulated photocurrent noise nĨ and zero-crossing slope dĨ/dδ as nB = nĨ/(γedĨ/dδ). The fluores-cence dependence of the slope was measured, as shownin Fig. 3(b). The error bars are the estimated standarddeviations of the slope; they are much smaller than themarker size. Here, the modulation parameters and pow-ers of the MWs and RF waves were fixed over all mea-surements; they were tuned at Ifl = 7.2 mA to maximizethe slope. Note that the optimal parameters and powerdepend on the incident laser power [30, 31]. Neverthe-less, we confirmed that tuning these values resulted in animprovement in the slope of about 3% at Ifl = 29.3 mA.We thus assumed that the relative uncertainty of the datawas several percent in this measurement.We found that the slope saturated as the incident laserpower increased. In general, the slope is expected to in-crease as Ifl increases, since the amplitude of the lock-in CW-ODMR peak is proportional to the contrast ofthe CW-ODMR peak multiplied by Ifl. Although the in-creased laser power causes optical broadening of the peak[30, 31] in principle, the observed linewidth was mainlygoverned by microwave power broadening. Therefore, thesaturation of the slope could be explained by a chargestate conversion of NV centers, which led to a decrease inthe contrast. Indeed, [N0s ] was only about twice as large5 20 40 60 80 100Frequency (Hz)4102050n B (pT/Hz)nB, resnB, farPSN limitFIG. 4. Single-sided noise amplitude spectral density in mag-netic field measurement. The blue and orange curves are thesensor noise measured with resonant and far-detuned MWs,respectively. Calculated photon-shot-noise-limited sensitivityof 6.9 pT/√Hz is indicated by the dashed line. PSN: photonshot noise.as [NV−] in our diamond [32–34]. A detailed investiga-tion of this saturation is beyond the scope of this work.The magnetic field noise density 〈nB,far〉 expected fromthe measured 〈nĨ,far〉 and dĨ/dδ did not monotonicallydecrease as Ifl increased, as shown in Fig. 3(c), becauseof the saturation of the slope. The error bars indicatethe uncertainties computed from a relative uncertaintyof 5% in the slope and the covariance matrix used in thecurve fitting to 〈nĨ,far〉 with Eq. (1). The photocurrentdependence of 〈nB,far〉 suggests that good sensitivity toa magnetic field can be achieved at Ifl from 5 to 20 mA.D. Magnetic field noise spectral density andsensitivityWe measured single-sided noise amplitude spectraldensity nB,res in a magnetic field measurement where theMWs were resonant with the central CW-ODMR peak(δ = 0). The noise spectrum nB,res was computed us-ing the discrete Fourier transform from a measured timetrace of Ĩ for 5 s with sampling frequency Fs = 400 Hz.Figure 4 shows the measured nB,res, which was averagedover 19 time measurements, at Ifl = 6.4 mA (blue solidcurve). The optimal power of the reference light was es-timated from the CW-ODMR peak contrast of 3% anda reference light power that had been optimized withthe far-detuned MWs. The zero-crossing slope dĨ/dδwas 332 ± 0.7 pA/Hz. Note that the displayed nB,reswas digitally filtered by narrow-band notch filters forthe harmonics of a 50-Hz power line and a band-passfilter with 3-dB cutoff frequencies of 5 and 100 Hz, whichcorresponds to the bandwidth of the target object (e.g.,brain of a living animal). The noise-equivalent powerbandwidth fNEP of the digital filtering was numerically6calculated to be 91.9 Hz. In this numerical calculation,white noise with a standard deviation of σ was numeri-cally computed with sampling frequency Fs and digitallyfiltered. The standard deviation σ′ of the filtered noise,given by σ′ = σ√fNEP/(Fs/2) [35], was numerically cal-culated to yield fNEP [29].The achieved noise density indicated a very low floor inthe single-sided spectrum in the near-dc range. The low-est noise density floor, about 9 pT/√Hz, was measurednear 40 Hz and from 70 to 90 Hz. The sudden drop innB,res at 90 Hz was due to the digital band-pass filter. Alow noise density of 15 pT/√Hz was obtained even near5 Hz, even though magnetic field noise generally deterio-rates at lower frequency [15, 16, 18, 19, 23, 24]. The noisespectral density nB,far measured with the far-detunedMWs at Ifl = 6.4 mA is shown by the orange trace inFig. 4. We found that nB,far reached the photon-shot-noise-limited sensitivity of 6.9 pT/√Hz (black dashedline). This nB,far value is better than the value of 〈nB,far〉of around 10 pT/√Hz at Ifl = 7.2 mA estimated inSec. III C. We attributed this improvement to the in-crease in the slope from 292 pA/Hz (Ifl = 7.2 mA) to332 pA/Hz (Ifl = 6.4 mA) achieved by fine tuning theoptical path and measurement parameters.The noise spectrum nB,res shows many peaks that arepossibly due to environmental magnetic field noise andvibration of the sensor head. To clarify the source ofthese peaks, we provided support to the sensor head,which may shift the frequency of peaks associated withmechanical vibration of the head. We observed that thenoise peaks around 25–30 Hz disappeared, and thereforeascribed these peaks to sensor head vibration. Becausethe peaks at 17, 35, 46, 49, and 70 Hz appeared at thesame frequencies even with sensor-head support, thesepeaks were ascribed to environmental noise. We couldnot identify the source of the peaks at around 55–61 Hz,since these peaks were not present in a noise spectrumobtained on another day without sensor-head support.The sensitivity of our sensor was evaluated from theaverage power of the measured noise, which was digitallyfiltered. The sensitivity η is defined as η = δB√T , whereδB is the minimum detectable magnetic field for mea-surement time T . In this definition, the noise spectrumis assumed to be frequency-independent (white noise).The standard deviation of the measured noise, whichwas evaluated from its time traces to be 128 ± 2 pT(see Supplemental Material [29] for details), is consid-ered to represent δB for measurement time T = F−1s =2.5 ms. Since the bandwidth of the digital band-passfilter was narrower than the measurement bandwidthFs/2 and the lock-in amplifier’s bandwidth, fNEP forthe digital filtering was substituted for the measurementbandwidth; that is, the sensitivity was equivalent toη = δB/√2fNEP [15, 35]. We achieved a sensitivity ofη = 9.4± 0.1 pT/√Hz.101100101102103Averaging time (s)101100101102Allan deviation (pT)8.5 pT0.3 pTFIG. 5. Allan deviation as function of averaging time. Opencircles show calculated overlapping Allan deviation from acontinuous measurement for 200 minutes. Estimated uncer-tainties of the Allan deviations are indicated by the error bars,which are much smaller than the marker circle size.E. Allan deviationThe Allan deviation of the noise for a measurement ofabout 200 minutes was computed to evaluate the stabil-ity of our sensor. We continuously tuned the MW carrierfrequency to the resonance using the demodulated pho-tocurrent Ĩ output, which was low-pass-filtered with acutoff frequency of 10 Hz. The bandwidth of the feed-back response was approximately 2 Hz. The measurednoise was recorded every minute on a computer. Thenotch and band-pass digital filters used in the sensitivityanalysis (see Sec. III D) were not used in this analysis.We then computed the overlapping Allan deviation, asshown in Fig. 5. The open circles indicate Allan devia-tions for a given averaging time. The error bars representthe standard deviations of the Allan deviations; they aremuch smaller than the marker size.We found that the 1-second interrogation yielded anAllan deviation of 8.5 pT, which is consistent with theevaluated sensitivity (η = 9.4 pT/√Hz). The Allan de-viation showed a bump around the 10-second averagingtime. This bump may arise from a periodic fluctua-tion of several tens of seconds. We found that the noisespectrum for this measurement showed a broad peak ataround 10–50 mHz. Both this broad peak and the bumpin the Allan deviation were found to be reproducible.However, the cause of the long-term fluctuation was notidentified. Aside from the bump, the Allan deviationdecreased as the averaging time increased and appearedto saturate at a value of 0.3 pT after a few thousandseconds. The zero-crossing slopes before and after theAllan deviation measurement were found to be almostthe same. We thus conclude that our sensor remainedstable and could measure a magnetic field with a sensi-tivity of η = 9.4 ± 0.1 pT/√Hz at 5–100 Hz for at least200 minutes.7IV. DISCUSSIONThe demonstrated sensitivity of 9.4 ± 0.1 pT/√Hz isthe best reported value for diamond quantum sensorsbased on the CW-ODMR of an ensemble of NV centers[15, 16, 18, 35] without a MFC in the frequency range of5 to 100 Hz. The previous best sensitivities were around20 to 30 pT/√Hz in the low frequency range [16, 18]and 15 pT/√Hz in the relatively high frequency rangeof 80 Hz–3.6 kHz [15]. Moreover, in our study, the noisefloor of nB stayed below 20 pT/√Hz even at 5 Hz, asshown in Fig. 4. Given that the noise environment isgenerally cleaner at higher frequency, the very low noisefloor of about 9 pT/√Hz will continue into the kilohertzrange if we use a higher cutoff frequency for the lock-inamplifier’s low-pass filter. The Allan deviation analysisshowed that our diamond magnetometer can interrogatea magnetic field for a long time with remarkable sensitiv-ity. Therefore, our sensor is capable of detecting a repet-itive biomagnetic field, for example, a stimulus-evokedfield, with a strength on the order of 1 pT by accumulat-ing the signals.CW-ODMR-based magnetometry has advantages overpulsed-MW-based magnetometry for practical applica-tions; it has a simpler experimental setup and looser re-quirements for the inhomogeneities of the bias magneticfield and MWs. Additionally, the use of a single orien-tation of NV center axis in our magnetometer leads to alower requirement for the bias field alignment comparedwith that for multiple orientations [15, 18, 24]. The sen-sor head design, which can approach the target objectto a distance of 1 mm, relies on a simple setup and re-duced requirements for the bias field. The simplified ge-ometry between the single orientation of NV centers andthe magnetic field to be measured also facilitates variouspractical applications. We note that a better sensitivityof around 2 pT/√Hz in the low-frequency range (from10 Hz), achieved using the Ramsey method, has beenrecently reported [24]; however, our sensor is more suit-able for practical applications such as biomagnetic fieldmeasurement because of its simplified setup and shortmeasurement distance.We attributed a major part of the sensitivity im-provement in this work to the long dephasing time ofT ∗2 ∼ 2 µs, achieved by decreasing the concentrationof 13C to about 500 ppm and using a relatively lowinitial nitrogen concentration of 5.6 ppm. The narrowlinewidth of a CW-ODMR peak due to the long T ∗2 re-sulted in a high response signal to a magnetic field ofγe(dĨ/dδ) = 9.3 A/T, even with the use of only a singlecrystallographic orientation of NV centers. The photon-shot-noise-limited sensitivity is comparable to previouslyreported values [15, 18]. In addition, the approximatelyfive-fold improvement in the intensity noise reduction inour balanced detection over the balanced detection re-ported in a previous study [18] contributed to the goodsensitivity.A reduction in the RIN of a laser can enhance sen-sitivity. The RIN of our laser (Coherent Verdi G5)was measured to be about −130 dBc/Hz at a modula-tion frequency of 6.2 kHz; the typical estimated RIN forstate-of-the-art solid-state lasers at the same frequency is−140 dBc/Hz [36, 37]. Therefore, a 10-dB improvementin n2Ĩ,intis feasible. This would result in a photon-shot-noise-limited sensitivity at up to Ifl ∼ 100 mA.It is expected that the zero-crossing slope can be in-creased by extending the dephasing time T ∗2 for the dia-mond. For example, a very long dephasing time of 8.5 µswith [NV−] = 0.4 ppm has been reported [16]. This longdephasing time will offer a four-fold improvement if thesame fluorescence intensity is available since the shot-noise-limited sensitivity is proportional to the linewidthof a CW-ODMR peak [15, 30]. Although the lower [NV−]emits weaker fluorescence, a photocurrent of up to 10 mAcan be obtained by increasing the incident laser power.In addition, the fluorescence collection efficiency can beboosted to approximately unity by using a total internalreflection lens and a light pipe [24, 38].The sensitivity can be further enhanced by using anMFC to concentrate the magnetic flux to be measuredonto the diamond [19, 20, 39, 40]. MFCs have achievedan enhanced field sensitivity of approximately 1 pT/√Hzin the near-dc range with the high concentration factorof up to 1347 [19, 20] . Using a high MFC concentra-tion factor, the sensitivity of our sensor could be furtherimproved to tens of fT/√Hz. However, the concentra-tion factor is strongly dependent on the MFC geometry[41], so precise design is required to effectively concen-trate a magnetic field in practical applications includ-ing biomagnetic measurements. Moreover, fluctuationsin the concentration factor caused by temperature vari-ations result in magnetic-field noise [19]. In biomagneticmeasurements involving living animals whose body tem-perature tends to vary, this noise would be expected toobstruct the measurement and reduce the sensor stabil-ity.V. CONCLUSIONSWe demonstrated a sensitive diamond magnetometerwith a magnetic field sensitivity of 9.4± 0.1 pT/√Hz ina near-dc frequency range of 5 to 100 Hz. 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