Diamond magnetometer enhanced by ferrite flux concentrators

Ilja Fescenko,1, ∗ Andrey Jarmola,2, 3 Igor Savukov,4 Pauli Kehayias,1, 5 Janis Smits,1, 6

Joshua Damron,1 Nathaniel Ristoff,1 Nazanin Mosavian,1 and Victor M. Acosta1, †

1Center for High Technology Materials and Department of Physicsand Astronomy, University of New Mexico, Albuquerque, NM, USA

2ODMR Technologies Inc., El Cerrito, CA, USA3Department of Physics, University of California, Berkeley, CA, USA

4Los Alamos National Laboratory, Los Alamos, NM, USA5Sandia National Laboratory, Albuquerque, NM, USA

6Laser Center of the University of Latvia, Riga, Latvia

Magnetometers based on nitrogen-vacancy (NV) centers in diamond are promising room-temperature, solid-state sensors. However, their reported sensitivity to magnetic fields at low fre-quencies (.1 kHz) is presently &10 pT s1/2, precluding potential applications in medical imaging,geoscience, and navigation. Here we show that high-permeability magnetic flux concentrators, whichcollect magnetic flux from a larger area and concentrate it into the diamond sensor, can be used toimprove the sensitivity of diamond magnetometers. By inserting an NV-doped diamond membranebetween two ferrite cones in a bowtie configuration, we realize a ∼ 250-fold increase of the magneticfield amplitude within the diamond. We demonstrate a sensitivity of ∼ 0.9 pT s1/2 to magneticfields in the frequency range between 10 and 1000 Hz, using a dual-resonance modulation techniqueto suppress the effect of thermal shifts of the NV spin levels. This is accomplished using 200 mWof laser power and 20 mW of microwave power. This work introduces a new dimension for diamondquantum sensors by using micro-structured magnetic materials to manipulate magnetic fields.

I. Introduction

Quantum sensors based on nitrogen-vacancy (NV) cen-ters in diamond have emerged as a powerful platformfor detecting magnetic fields across a range of lengthscales [1]. At the few-nanometer scale, single NV cen-ters have been used to detect magnetic phenomena incondensed-matter [2, 3] and biological [4, 5] samples.At the scale of a few hundred nanometers, diamondmagnetic microscopes have been used to image biomag-netism in various systems, including magnetically-labeledbiomolecules [6] and cells [7, 8] and intrinsically-magneticbiocrystals [9, 10]. At the micrometer scale, diamondmagnetometers have detected the magnetic fields pro-duced by neurons [11], integrated circuits [12, 13], andthe nuclear magnetic resonance of fluids [14, 15].

Diamond magnetometers with larger active volumesare expected to offer the highest sensitivity [16]. How-ever, in order to be competitive with existing technolo-gies, they must overcome several technical drawbacks,including high laser-power requirements and poor sensi-tivity at low frequencies. The most sensitive diamondmagnetometer reported to date featured a projected sen-sitivity of ∼0.9 pT s1/2 using 400 mW of laser power [17].However this magnetometer used a Hahn-echo pulse se-quence which limited the bandwidth to a narrow rangearound 20 kHz. For broadband, low-frequency operation,the highest sensitivity reported to date is ∼15 pT s1/2 inthe 80–2000 Hz range, using & 3 W of laser power [11]. Adiamond magnetometer based on infrared absorption de-

∗ iliafes@gmail.com† vmacosta@unm.edu

tection realized a sensitivity of ∼30 pT s1/2 at 10–500 Hz,using 0.5 W of laser power [18].

To understand the interplay between sensitivity andlaser power, we consider a diamond magnetometer basedon continuous-wave, fluorescence-detected magnetic res-onance (FDMR) spectroscopy. Here, the sensitivity isfundamentally limited by photoelectron shot noise as:

ηpsn ≈Γ

γnvC√ξPopt/Eph

, (1)

where γnv = 28 GHz/T is the NV gyromagnetic ratio,Γ is the FDMR full-width-at-half-maximum linewidth,and C is the FDMR amplitude’s fractional contrast. Thefactor ξPopt/Eph constitutes the photoelectron detectionrate, where Popt is the optical excitation power, ξ is thefraction of excitation photons converted to fluorescencephotoelectrons, and Eph = 3.7×10−19 J is the excitationphoton energy (532 nm). To set an optimistic bound onηpsn, we insert the best reported values (ξ = 0.08 [17],Γ/C = 1 MHz/0.04 [11]) into Eq. (1) to obtain ηpsn ≈2 pT s1/2 W1/2. Even in this ideal case (Appendix XIII),∼4 W of optical power is needed to realize a sensitivity of1 pT s1/2, and further improvements become impractical.

The need for such a high laser power presents chal-lenges for thermal management and has implications forthe overall sensor size, weight and cost. Applicationswhich call for sub-picotesla sensitivity, such as magne-toencephalography (MEG) [19] and long-range magneticanomaly detection [20, 21], may require alternative ap-proaches to improve sensitivity. Avenues currently be-ing pursued often focus on reducing the ratio Γ/C [16].Approaches to reduce Γ include lowering 13C spin den-sity and mitigating strain and electric-field inhomogene-ity [22, 23], increasing the nitrogen-to-NV− conversion

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|B (r

)| / |

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Gap, (µm)

Enha

ncem

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0 3000 6000 90000

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Relative permeability, μr

MN

60fe

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300

b)

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= 0°

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θ = 43 µm= 0°

δ

δ

μr = 6500

= 43 µmδ

Bextθ( = 0°)

Enha

ncem

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0 45 90 135 180-300

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Rel

ativ

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ticfie

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c)

Fit: є cosθ Bz

(r0) / |Bext|

Bx (r0) / |Bext|

x

z

FIG. 1. Simulations of magnetic flux concentrators. (a) Model geometry. Two identical solid cones, configured in abowtie geometry, are placed in an external magnetic field, Bext. (b) Simulated x-z plane cut of the relative magnetic fieldamplitude, |B(r)|/|Bext|, for cones with relative permeability µr = 6500 and a tip gap of δ = 43 µm, upon application ofBext at θ = 0. Arrows indicate the direction and magnitude of B(r). The point at the geometric center is labeled r0. (c)Vector components of the relative magnetic field amplitude at r0 as a function of θ, for cones with µr = 6500 and δ = 43 µm.The relative axial magnetic field amplitude is fit to the function Bz(r0)/|Bext| = ε cos θ, where in this case ε = 280. (d)Enhancement factor as a function of δ for cones with µr = 6500. (e) Enhancement factor as a function of µr for δ = 43 µm.

yield [24–26], and designing techniques to decouple NVcenters from paramagnetic spins [23, 27]. Methods toincrease C include using preferentially-aligned NV cen-ters [28, 29], detecting infrared absorption [18, 30], anddetecting signatures of photo-ionization [31–33].

In this Manuscript, we report a complementary ap-proach to improve the sensitivity of diamond magnetome-ters. Our approach uses microstructured magnetic fluxconcentrators to amplify the external magnetic field am-plitude by a factor of ∼250 within the diamond sensor.Using a dual-resonance magnetometry technique to sup-press the effect of thermal shifts of the NV spin levels,we realize a sensitivity of ∼0.9 pT s1/2 in the 10–1000 Hzrange, using a laser power of 200 mW. We show that,with further improvements, a magnetic noise floor of∼0.02 pT s1/2 at 1000 Hz is possible before ferrite ther-mal magnetization noise limits the sensitivity.

II. Experimental design

Magnetic flux concentrators have previously been usedto improve the sensitivity of magnetometers based on theHall effect [34], magnetoresistance [35], magnetic tunneljunctions [36], superconducting quantum interference de-vices [37], and alkali spin precession [38]. Typically, themagnetometer is positioned in the gap between a pair

of ferromagnetic structures which collect magnetic fluxfrom a larger area and concentrate it into the gap. Thefractional increase in magnetic field amplitude due to theflux concentrators, ε, is a function of their geometry, gapwidth, and relative permeability (µr). Ideally, the con-centrators are formed from a soft magnetic material withlow remanence, high µr, low relative loss factor [38], andconstant susceptibility over a broad range of magneticfield amplitudes and frequencies. The improvement insensitivity is generally accompanied by a reduction inspatial resolution, as the total magnetometer size is larger(Appendix II). Diamond sensors usually have sub-mm di-mensions, whereas the flux concentrators used here havedimensions of ∼10 mm. Thus our device is best suitedfor applications that require a spatial resolution &10 mm,such as MEG and magnetic anomaly detection.

The optimal flux concentrator geometry depends on anumber of factors, which include the sensor dimensionsand target application. Here, we consider a pair of iden-tical cones (height: 10 mm, base diameter: 10 mm), with∼370–µm diameter flat tips, arranged in a bowtie con-figuration, Fig. 1(a). A static magnetic field, Bext, isapplied at an angle θ from the cone symmetry axis (z)and the resulting magnetic field, B(r), is simulated us-ing finite-element magnetostatic methods. Figure 1(b)shows a plane-cut of the relative magnetic field ampli-

3

0.96

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1.00 2840 2860 2880

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(nor

m.) Microwave frequency (MHz)

Fit (є=254)

-40 -20 0 20 40

2700

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2900

3000

3100

External field, Bext (µT)

FDM

R fr

eqne

ncy

(MH

z)

f+f-

f+f-

Laser (532 nm, 200 mW)

Fluorescence (650-800 nm)

Dichroicmirror

Lens

mu-metal shield

Diamond microwave loop

Balancedphotodetector

Beamsplitter

Ferrite cones

Helmholtz coils

a) b)

c)

300 µm

microwave loop

MN60 ferrite

[100]diamond

Bext

Bgap

FIG. 2. Experimental setup and enhancement measurement. (a) Schematic of the experimental setup. Inset:photograph of the diamond membrane in the gap between ferrite cones. (b) Fluorescence-detected magnetic resonance (FDMR)spectrum obtained at Bext = 2.62 µT. Two peaks are present, with central frequencies f± extracted from Lorentzian fits. (c)Measured FDMR frequencies as a function of Bext. Error bars are smaller than the plot markers. The gray solid lines are a fitusing the NV spin Hamiltonian (Appendix I), assuming Bgap = εBext, with ε = 254.

tude, |B(r)|/|Bext|, for cones with µr = 6500 and atip gap of δ = 43 µm, upon application of Bext atθ = 0. Throughout the gap (Appendix II), B(r) isaligned along z with a uniform relative magnetic field|B(r)|/|Bext| ≈ 280.

Figure 1(c) shows the vector components of the rela-tive magnetic field at the center of the bowtie geometry(r = r0) as a function of θ. The relative axial mag-netic field is well described by Bz(r0)/|Bext| ≈ ε cos θ,where ε is the enhancement factor (in this simulationε = 280). On the other hand, the relative transversemagnetic field, Bx(r0)/|Bext|, is less than 0.1 for all val-ues of θ. Thus, the structure acts as a filter for the axialcomponent of external magnetic fields, producing a uni-form field throughout the gap of:

Bgap ≈ ε |Bext| cos θ z. (2)

For the remainder of the manuscript, we consider onlyexternal magnetic fields applied along z (θ = 0) and de-scribe Bgap according to Eq. (2).

Fig. 1(d) shows simulation results of the enhance-ment factor as a function of gap length for cones withµr = 6500. For δ in the 20–100 µm range, ε varies from560 to 120, indicating that large enhancement factors arepossible for typical diamond membrane thicknesses. Fig-ure 1(e) is a plot of the simulated ε as a function of µrfor δ = 43 µm. For µr & 500 the enhancement factor isrelatively constant at ε ≈ 280. This indicates that a widerange of magnetic materials can be used for flux concen-tration and minor variations in µr (due, for example, totemperature variation) have a negligible impact on Bgap.

We elected to use MN60 ferrite (µr ≈ 6500) as the ex-perimental concentrator material, owing to its low ther-

mal magnetic noise [38, 39]. The ferrite cones were micro-machined to have approximately the same dimensionsas simulated in Fig. 1. Figure 2(a) depicts the experi-mental setup. An NV-doped diamond membrane with[100] faces is positioned in the gap between the ferritecones. The membrane was formed from a commercially-available, type Ib diamond grown by high-pressure high-temperature (HPHT) synthesis. The diamond had beenirradiated with 2–MeV electrons at a dose of∼1019 cm−2.It was subsequently annealed in a vacuum furnace at800–1100° C [9] and mechanically polished and cut intoa membrane of dimensions ∼300× 300× 43 µm3.

Approximately 200 mW of light from a 532 nm laseris focused by a 0.79 NA lens to a ∼40 µm diameterbeam that traverses the diamond membrane parallel toits faces. The same lens is used to collect NV fluores-cence, which is then refocused onto one of the channelsof a balanced photodetector, producing ∼1.2 mA of pho-tocurrent. A small portion of laser light is picked off fromthe excitation path and directed to the other photode-tector channel for balanced detection. Microwaves aredelivered by a two-turn copper loop wound around oneof the ferrite cones. The ferrite cones provide a & 2-foldenhancement in the microwave magnetic field amplitudewithin the diamond (Appendix IX). All measurementswere performed using . 20 mW of microwave power.

The ferrite-diamond assembly is positioned at the cen-ter of a pair of Helmholtz coils (radius: 38 mm), whichproduce a homogenous magnetic field parallel to thecone axis of amplitude Bext. The coils’ current responsewas calibrated using three different magnetometers (Ap-pendix X). A 1.5-mm-thick cylindrical mu-metal shield(diameter: 150 mm, height: 150 mm) surrounds the

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Diamond sensor

(Fig. 2a) Lock-inamplifier

signal

cos(2 fmod t)πreference signal:

b)frequency-modulated

microwaves in

photodetector

phase shift

3.5

FIG. 3. Dual-resonance magnetometry concept. (a) Microwave frequency modulation used for dual-resonance magne-tometry. (b) Schematic of the lock-in technique. Both microwave signals depicted in (a) are combined and delivered throughthe microwave loop. NV fluorescence is continuously excited and its time-varying intensity is recorded by the balanced pho-todetector. This signal is then fed to a lock-in amplifier and demodulated by the reference signal. (c) Lock-in signal as afunction of Bext for both single-resonance and dual-resonance modulation protocols. The microwave frequencies were centeredabout the f± values measured by FDMR spectroscopy at Bext = 1.73 µT. In all cases, fmod = 15 kHz and the lock-in uses a 12dB/octave low-pass filter with a 100 µs time constant. For the f− scan, the lock-in reference signal had a π phase shift relativeto the modulation function. The right vertical axis converts the lock-in signal to the amplitude of photocurrent oscillations atfmod, which is used to estimate the photoelectron-shot-noise-limited sensitivity, Appendix XIII.

Helmholtz coils, providing a shielding factor of ∼ 100.To measure the enhancement factor, we recorded the

NV FDMR spectrum as a function of Bext. Figure 2(b)shows a typical FDMR spectrum acquired at Bext =2.62 µT. Two peaks are present, with central frequenciesf±. These frequencies correspond to NV electron-spintransitions between the ms = 0 and ms = ±1 magneticsublevels (Appendix I). For magnetic field amplitudeswithin the diamond in the range 0.5 mT . εBext . 5 mT,the transition frequencies may be approximated as:

f± ≈ D(∆T )± γnv εBext/√

3, (3)

where, in our experiments (Appendix IV), D(∆T ) ≈2862 MHz + χ∆T is the axial zero-field splitting param-eter which shifts with changes in temperature, ∆T , asχ ≈ −0.1 MHz/K [40]. The 1/

√3 factor in Eq. (3) comes

from projecting Bgap onto the four NV axes which areall aligned at 55° with respect to the cone axis.

Figure 2(c) plots the fitted f± values as a function ofBext. These data were obtained by scanning Bext backand forth between ±50 µT two times. For a given Bext,the extracted f± are nearly identical regardless of scanhistory, indicating negligible hysteresis (Appendix XII).The data were fit according to the NV spin Hamiltonian(Appendix I), which reveals an experimental enhance-ment factor of ε = 254± 19. The uncertainty in ε is pri-marily due to uncertainty in the Bext current calibration(Appendix X). The experimental enhancement factor is∼10% smaller than the one simulated in Fig. 1(b). Thiscould be explained by a ∼4 µm increase in δ due to adhe-sive between the diamond and ferrite tips (Appendix III).

Having established that the ferrite cones provide a

∼250-fold field enhancement, we now turn to methods ofusing the device for sensitive magnetometry. A commonapproach in diamond magnetometry [41, 42] is to mod-ulate the microwave frequency about one of the FDMRresonances and demodulate the resulting fluorescence sig-nal using a lock-in amplifier (Appendix VI). We call thismethod “single-resonance” magnetometry, as each reso-nance frequency is measured independently. For exam-ple, to measure f+, the microwave frequency is varied asF(t) ≈ f+ +fd cos (2πfmodt), where fd is the modulationdepth and fmod is the modulation frequency. The lock-in amplifier demodulates the photodetector signal usinga reference signal proportional to cos (2πfmodt). The re-sulting lock-in output is proportional to variations in f+.

However, a single FDMR resonance can shift due tochanges in temperature in addition to magnetic field, seeEq. (3). To isolate the shifts due only to changes in mag-netic field, the difference frequency (f+ − f−) must bedetermined. Previous works accomplished this by mea-suring both resonances either sequentially [43] or simulta-neously by multiplexing modulation frequencies [44, 45].The magnetic field was then inferred by measuring f+and f− independently and calculating the difference.

Here, we use an alternative “dual-resonance” ap-proach, which extracts the magnetic field amplitude di-rectly from a single lock-in measurement (Appendix VI).Two microwave signal frequencies, centered about f±, aremodulated to provide time-varying frequencies, F±(t) ≈f± ± cos (2πfmodt). In other words, each tone is mod-ulated with the same modulation frequency and depth,but with a relative π phase shift, Fig. 3(a). The photode-tector signal is then demodulated by the lock-in ampli-

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Single resonance, f-

MN60 ferrite magnetization-noise limit

Single resonance, f+Dual resonance

Single resonance, f-

Microwaves off0.01

FIG. 4. Sub-picotesla diamond magnetometry. (a) Time-domain lock-in signals for single-resonance (f−) and dual-resonance modulation. Throughout, fmod = 15 kHz and the lock-in uses a 12 dB/octave low-pass filter with a 100 µs timeconstant. The adjacent plot is a zoom of the dual-resonance signal where the 580 pTrms test field at 135 Hz can be seen. Thetest-field frequency for f+ and f− single-resonance experiments were 125 and 130 Hz, respectively, with the same 580 pTrmsamplitude. (b) Magnetic noise spectra of single-resonance (two shades of gray) and dual-resonance (blue) signals. A referencespectrum obtained with microwaves turned off (green) shows noise from the un-modulated photodetector signal. Each spectrumwas obtained by dividing a 100–s data set into one hundred 1–s segments, taking the absolute value of the Fourier Transform ofeach segment, and then averaging the Fourier Transforms together. Spectra were normalized such that the test field amplitudesmatched the calibrated 580 pTrms values (Appendix VIII). The dashed red line is the projected value of ηpsn for dual-resonancemagnetometry (Appendix XIII). The dashed magenta line is the calculated thermal magnetization noise produced by the ferritecones (Appendix XIV). (c) Frequency dependence of the test field amplitude measured by dual-resonance magnetometry.

fier using a reference signal proportional to cos (2πfmodt),Fig. 3(b). In this way, the lock-in output is proportionalto (f+−f−) and is unaffected by thermal shifts ofD(∆T ).Furthermore, the dual-resonance lock-in signal’s responseto magnetic fields is larger than in the single-resonancecase. Figure 3(c) shows the experimental lock-in signalas a function of Bext for dual-resonance modulation andboth of the f± single-resonance modulation protocols.The slope for dual-resonance modulation is ∼1.3 timeslarger than that of single-resonance modulation. This isclose to the expected increase of 4/3 (Appendix VII).

III. Results

We next show that the combination of flux concentrationand dual-resonance modulation enables diamond mag-

netometry with sub-pT s1/2 sensitivity over a broad fre-quency range. A 1.73 µT bias field and 580 pTrms oscil-lating test field in the 125–135 Hz range were appliedvia the Helmholtz coils. The lock-in signal was con-tinuously recorded for 100 s using either dual-resonanceor single-resonance modulation. Figure 4(a) shows themagnetometer signals as a function of time. For single-resonance modulation, the signals undergo low-frequencydrifts, likely due to thermal shifts of D(∆T ). These driftsare largely absent for dual-resonance modulation.

Figure 4(b) shows the magnetic noise spectrum forthe different modulation techniques. In addition to thecalibrated test field signals, numerous peaks appear forboth single and dual-resonance modulation. We attributethese peaks to ambient magnetic noise that is not suffi-ciently attenuated by the single-layer mu-metal shield. In

6

1 10 100 1000

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Frequency (Hz)

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netic

nois

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T s1/

2 ) test fields

Twinleaf VMR270 pT s1/2

Fluxgate22 pT s1/2

SENSYS

NV-ferrite0.9 pT s1/2

FIG. 5. Magnetometer comparison. Magnetic noise spec-tra of a commercial magnetoresistive magnetometer (TwinleafVMR), fluxgate magnetometer (SENSYS FGM-100) and ourdual-resonance NV-ferrite magnetometer reproduced fromFig. 4(b). Each magnetometer was placed in a similar locationwithin the experimental apparatus and subject to the samebias and test field amplitudes. The test field frequency was130 Hz for both commercial sensors and 135 Hz for NV-ferrite.The manufacturer-specified sensitivities are 300 pT/

√Hz and

10 pT/√

Hz for the VMR and fluxgate, respectively.

regions without peaks, the noise floor for single-resonancemagnetometry is ∼1.5 pT s1/2 for frequencies &300 Hz,but it exhibits nearly 1/f behavior for lower frequencies.On the other hand, the noise floor for dual-resonancemagnetometry is ∼0.9 pT s1/2 for frequencies &100 Hzand remains at this level, to within a factor of two, forfrequencies down to ∼10 Hz. The remaining noise be-low 10 Hz may be due to thermal variation in the gaplength, δ (Appendix XVI). For reference, a spectrum ob-tained with the microwaves turned off is also shown. Itfeatures a constant noise floor of ∼0.8 pT s1/2 through-out the 1–1000 Hz frequency range. This level is con-sistent with the projected photoelectron shot-noise limit,ηpsn = 0.72 pT s1/2, which was calculated based on theaverage photocurrent and lock-in slope (Appendix XIII).

The frequency response of the magnetometer was de-termined by recording magnetic spectra at different test-field frequencies, while holding the amplitude of the driv-ing current constant. Figure 4(c) plots the test-field am-plitude, recorded by dual-resonance diamond magnetom-etry, as a function of frequency. The amplitude decays byless than a factor of two over the 1–1000 Hz range. Theobserved decay is due to a combination of the lock-in am-plifier’s low-pass filter and a frequency-dependent mag-netic field attenuation due to metal components withinthe Helmholtz coils (Appendix VIII).

Finally, we compared the performance of our magne-tometer with two commercial vector sensors: a magne-toresistive magnetometer and a fluxgate magnetometer.Figure 5 shows the magnetic noise spectra obtained undercomparable experimental conditions. Evidently, the NV-ferrite magnetometer outperforms the commercial mag-netometers throughout the frequency range.

IV. Discussion and conclusion

The demonstration of broadband, sub-picotesla diamondmagnetometry is a significant step towards applicationsin precision navigation, geoscience, and medical imaging.Since only 200 mW of laser power and 20 mW of mi-crowave power were used, the device holds promise forfuture miniaturization and parallelization efforts. More-over, our magnetometer operates at microtesla ambientfields, which raises the intriguing possibility of operatingin Earth’s magnetic field without an additional bias field.

Our implementation used a commercially-available,type Ib HPHT diamond processed using standardelectron-irradiation and annealing treatments [24]. Thismaterial exhibits relatively broad FDMR resonances(Γ ≈ 9 MHz), which leads to a photoelectron-shot-noise-limited sensitivity of ηpsn = 0.72 pT s1/2 even after the∼250-fold flux-concentrator field enhancement. State-of-the-art synthetic diamonds have recently been fabricatedthat feature several orders of magnitude narrower reso-nances [23, 46]. The excitation photon-to-photoelectronconversion efficiency in our experiments (ξ ≈ 10−2) couldalso be improved by at least an order of magnitude withoptimized collection optics [17]. With these additions,ηpsn could be further improved by several orders of mag-nitude, Eq. (1). However, at this level, thermal magne-tization noise intrinsic to the flux concentrators becomesrelevant.

Thermal magnetic noise originating from dissipativematerials can be estimated using fluctuation-dissipationmethods [38, 47]. The noise has contributions due tothermal eddy currents and magnetic domain fluctuations.As discussed in Appendix XIV, we find that thermal eddycurrents in the ferrite cones produce an effective whitemagnetic noise of ∼7 × 10−5 pT s1/2. This negligibly-low noise level is a consequence of our choice of low-conductivity ferrite. On the other hand, thermal mag-netization noise results in a larger, frequency-dependentmagnetic noise. At 1 Hz, this noise is 0.5 pT s1/2, and itscales with frequency as f−1/2, reaching ∼0.02 pT s1/2 at1 kHz. This noise, shown in Fig. 4(b), is not a limitingfactor in our experiments, but it may have implicationsfor future optimization efforts. If a material with a lowerrelative loss factor could be identified, it would result inlower thermal magnetization noise (Appendix XV).

In summary, we have demonstrated a diamond mag-netometer with a sensitivity of ∼0.9 pT s1/2 over the10–1000 Hz frequency range. The magnetometer oper-ates at ambient temperature and uses 0.2 W of laserpower. These improved sensor properties are enabledby the use of ferrite flux concentrators to amplify mag-netic fields within the diamond sensor. Our results maybe immediately relevant to applications in precision nav-igation, geoscience, and medical imaging. More broadly,the use of micro-structured magnetic materials to manip-ulate magnetic fields offers a new dimension for diamondquantum sensors, with potential applications in magneticmicroscopy [6–13] and tests of fundamental physics [48].

7

Acknowledgments

The authors acknowledge advice and support from A.Laraoui, Z. Sun, D. Budker, P. Schwindt, A. Mounce,M. S. Ziabari, B. Richards, Y. Silani, F. Hubert, andM. D. Aiello. This work was funded by NIH grants1R01EB025703-01 and 1R21EB027405-01, NSF grantDMR1809800, and a Beckman Young Investigator award.

Competing interests I. Fescenko, A. Jarmola, andV. M. Acosta are co-inventors on a pending patent appli-cation. A. Jarmola is a co-founder of ODMR Technolo-gies and has financial interests in the firm. The remainingauthors declare no competing financial interests.

Author contributions V. M. Acosta and I. Savukovconceived the idea for this study in consultation with I.Fescenko and A. Jarmola. I. Fescenko carried out sim-ulations, performed experiments, and analyzed the datawith guidance from V. M. Acosta. P. Kehayias, J. Smits,J. Damron, N. Ristoff, N. Mosavian, and A. Jarmola con-tributed to experimental design and data analysis. Allauthors discussed results and helped write the paper.

8

Appendix I.NV electron spin Hamiltonian

Neglecting hyperfine coupling (which is not resolvedin our experiments), the NV ground-state electron spinHamiltonian can be written as [49]:

H

h= DS2

z′ + E(S2x′ − S2

y′) + γnvBBB ·SSS, (AI-1)

where h is Planck’s constant, γnv = 28.03 GHz/T is theNV gyromagnetic ratio, and E ≈ 3 MHz is the transversezero-field splitting parameter. The axial zero-field split-ting parameter, D ≈ 2862 MHz, is temperature depen-dent, as discussed in Appendix IV. SSS = (Sx′ , Sy′ , Sz′) aredimensionless electron spin operators, and the zzz′ direc-tion is parallel to the NV symmetry axis. For a magneticfield of amplitude Bgap applied normal to a diamond with[100] faces, the Hamiltonian for NV centers aligned alongany of the four possible axes is the same. In matrix form,it is:

H

h=

D +

γnvBgap√3

γnvBgap√3

E

γnvBgap√3

0γnvBgap√

3

EγnvBgap√

3D − γnvBgap√

3

,

(AI-2)The eigenstates and eigenfrequencies can be found bydiagonalizing the Hamiltonian. The two microwave tran-sition frequencies observed in our experiments, f±, cor-respond to the frequency differences between the eigen-state with largely ms = 0 character and the eigenstateswith largely ms = ±1 character. We used this Hamil-tonian to fit the f± versus Bext data in Fig. 2(c). Weassumed Bgap = εBext and used solutions to Eq. AI-2 tofit for ε = 254. The values of E and D were determinedseparately from low-field FDMR data and were not fitparameters.

Note that Eq. (3) in the main text, which approxi-mates f± as being linearly dependent on Bext, is merelya convenient approximation. As can be seen in Fig. 2(c),the exact values of f± are generally nonlinear functionsof Bext. This is especially pronounced near zero field,ε |Bext| . E/γnv ≈ 0.1 mT, where f± undergo an avoidedcrossing, and also at high field, where mixing due totransverse fields produces nonlinear dependence. How-ever, for magnetic fields 0.5 mT . εBext . 5 mT,the transition frequencies f± are approximately linear inBext.

Appendix II.Flux concentrator simulations

Our flux concentrator model and simulations are de-scribed in Sec. II and Fig. 1 of the main text. Here wedescribe supplementary results demonstrating the fieldhomogeneity in the gap, the enhancement factor as the

FIG. A6. Enhancement factor and field homogeneity.(a) The model geometry. See Fig. 1(a) for additional dimen-sions. (b) Enhancement factor, ε, as a function of the gaplength, δ. (c) Enhancement factor as a function of the axialdisplacement z. The gap is shaded in light gray, while the fer-rite concentrators are shaded in dark gray. (d) Enhancementfactor as a function of the transverse displacement x.

gap length approaches zero, and the approximate pointspread function. Figure A6(a) describes the geometryused for the simulations. Figure A6(b) shows the en-hancement factor as a function of δ, with the rangeextending to δ ≈ 0. The largest enhancement factorsare observed for small gaps, approaching ε = 5000 forδ = 0. We chose a gap of δ ≈ 43 µm in our exper-iments as a compromise that offers moderate enhance-ment (ε ≈ 250) while still providing substantial opticalaccess and straightforward fabrication and construction.

To visualize the homogeneity of the magnetic fieldwithin the gap, we plot line cuts of the relative field am-plitude along the axial and transverse directions. Fig-ure A6(c) shows the relative magnetic field along thecone symmetry axis. Figure A6(d) shows the relativefield along a transverse line passing through r0. Bothplots predict a high degree of magnetic field homogeneity;residual variations of the relative field are . 1% through-out the region filled by the diamond membrane.

Future NV-flux concentrator devices may involve theuse of sensor arrays to perform imaging. While a detailedanalysis of the design space for imaging applications isbeyond the scope of this work, we performed simulationsto estimate the point spread function of our device. Asmall (1–mm diameter) current loop was positioned tohave an axial displacement of 1 mm below the base of thebottom cone. The magnetic field amplitude in the gap,Bgap, was simulated as a function of the current loop’slateral displacement, x. Figure A7 shows the resultingmagnetic field profile. While it does not a have simpleGaussian shape, it can be approximated as having a full-width-at-half-maximum (FWHM) resolution of ∼11 mm.

9

-40 -20 0 20 400

10

20

30

40

Lateral displacement, x (mm )

Fiel

din

gap,

B gap(a

rb. )

FWHM ≈ 11 mm

current loop1 mm

FIG. A7. Flux concentrator point spread function.The value of Bgap due to a small current loop located belowthe device is recorded as a function of lateral displacement.The resulting field profile has a FWHM linewidth of ∼11 mm.Inset: geometry for scanning.

Appendix III.Experimental setup: cones

The ferrite cones were ordered from Precision Ferrites& Ceramics, Inc. The diamond membrane was gluedon the tip of one of the ferrite cones with LOCTITEAA3494 UV-curing adhesive. The second cone with themicrowave loop was mounted inside a metallic holder andmicro-positioned to contact the exposed face of the dia-mond membrane by use of a Thorlabs MicroBlock Com-pact Flexure Stage MBT616D. When in the desired po-sition, the holder was glued to the support of the bottomcone by superglue, and then detached from the micro-positioning stage.

Appendix IV.Experimental setup: optics

To excite NV fluorescence, a Lighthouse PhotonicsSprout-G laser is used to form a collimated beam of 532nm light. The beam is focused with a Thorlabs asphericcondenser lens ACL25416U-B (NA=0.79) onto the edgeof the diamond membrane. Fluorescence is collected bythe same lens and is spectrally filtered by a SemrockFF560-FDi01-25x36 dichroic mirror. A second lens re-images the fluorescence onto a photodetector. For mag-netometry experiments, including all data in the figuresin the main text, we used a Thorlabs PDB210A balancedphotodetector. For beam characterization (Fig. A8), weused a CMOS image sensor, and for observing Rabi os-cillations (Fig. A13), we used a Thorlabs PDA8A high-speed photodetector. Figure A8(a) shows an image of thefluorescence spot from the entrance edge of the diamondmembrane. The FWHM spot diameter of ∼40 µm wasselected to match the diamond membrane thickness. Itwas adjusted by tailoring additional telescoping lenses inthe excitation path.

With this optical system, we obtained a excitation

FIG. A8. Beam profile and absorption length. (a)Image of the fluorescence spot at the entrance edge of the di-amond membrane. The FWHM spot diameter is ∼ 40 µm.The dashed lines indicate the approximate edges of the dia-mond. (b) Fluorescence intensity produced by a ∼1 mm di-ameter laser beam entering the edge of a diamond membrane.The inset shows a fluorescence image of the top face. Redmarkers depict the normalized fluorescence intensity alongthe cut shown by the dashed line in the inset. The black solidline is an exponential fit, revealing a 1/e absorption length of0.6 mm.

photon-to-photoelectron conversion efficiency of ξ ≈0.01. The primary factors limiting ξ are due to the lim-ited optical access afforded by the ferrite cones, loss of flu-orescence exiting orthogonal faces of the diamond mem-brane, and incomplete absorption of the excitation beamwithin the diamond. To characterize the latter, we useda separate apparatus to image the fluorescence from thetop face of a larger membrane, Fig. A8(b). This largermembrane was the starting piece from which we cut thesmaller membrane used in magnetometry experiments.We found that the 1/e absorption length of this materialis 0.6 mm. Thus we expect that only ∼40% of the laserlight was absorbed in the ∼300 µm-long diamond mem-brane used in magnetometry experiments. This approx-imation neglects the effects of laser light that is reflectedat the air-diamond interfaces.

The large absorbed optical power results in significantheating of the diamond membrane. The experimentally-measured axial zero-field splitting parameter D ≈2862 MHz, Fig. 2(c), indicates a local diamond temper-ature of ∼385 K [40]. While the elevated temperatureleads to a large shift in D, it does not significantly di-

10

Diamond

sensor

(Fig.2a)

Laser

Amplifier

Power

combiner

lock-inDemodu-

lator

LPF

Function

generator

Current

supply

Vtest

to HCs

to MW loop

Balanced PD

LabVIEW DAQ card

φmod = π

φmod = 0

+

-

fmod

Vpd

clock sync.

Vout

MW generators

a) b)

c)

VCO

VCO

f+ + f-2

fd cos(2π fmod t)

f+ - f-2

to MW loopMixer

T feedback

B feedback

f- f+

fd

f+ + f-2

f+ - f-2

FDMR

MW

FIG. A9. Electronics. (a) Schematic of the electronics portion of the experimental apparatus. Vpd is the photodetector signal,Vout is the lock-in amplifier’s in-phase output signal, Vtest is the test signal waveform, and fmod is the modulation frequency.(b) Alternative electronic scheme for dual-resonance microwave signal generation and feedback. A voltage-controlled oscillator(VCO) produces a carrier frequency fcar = (f+ + f−)/2 ≈ D(∆T ) that is mixed with the signal from a second VCO withfrequency fdiff = (f+ − f−)/2, creating two sidebands at the FDMR frequencies. The sidedand frequencies are modulatedby adding a reference signal fd cos (2πfmodt) to the second VCO. This arrangement allows for rapid feedback to correct fortemperature and magnetic field drifts by adjusting the bias voltage to the VCOs. (c) Microwave signal spectrum resulting fromthe alternative electronics scheme in (b). A typical FDMR spectrum is shown in red for reference. DAQ: data acquisition card;HC: Helmholtz coils; LPF: low-pass filter; MW: microwave; PD: photodetector.

minish the contrast or broaden the FDMR resonances.Future devices may employ active cooling or optimizedheat sinks to reduce the diamond temperature.

Appendix V.Experimental setup: electronics

Figure A9(a) shows a schematic of the electronic de-vices used in our experimental setup. Microwaves aresupplied by two Stanford Research SG384 signal gener-ators. The clocks of the generators are synchronized bypassing the 10 MHz frequency reference output of onegenerator to the frequency reference input of the other.Both generators are configured to modulate the mi-crowave frequency with a modulation frequency fmod =15 kHz and depth fd = 3.3 MHz. In dual-resonancemodulation, the signal generators are configured suchthat their modulation functions, F±, have a relative πphase shift (see Sec. II). The signals from both generatorsare combined with a Mini-Circuits ZAPD-30-S+ 2-waypower combiner, amplified by a Mini-Circuits amplifierZHL-16W-43-S+, and finally delivered to a two-turn mi-crowave loop made from polyurethane-enameled copperwire (38 AWG). Prior to performing dual-resonance mag-netometry, the microwave powers for each f± resonancewere independently adjusted to give approximately thesame lock-in slope Fig. 3(c).

The photodetector output signal, Vpd, is fed to a SignalRecovery 7280 lock-in amplifier using 50 Ω termination.The lock-in multiplies Vpd by a reference signal, propor-tional to cos (2πfmodt), output from one of the signalgenerators. The demodulated signal is processed by thelock-in’s low pass filter, which was set to 12 dB/octavewith a 100 µs time-constant. The lock-in amplifier’s in-phase component, Vout, is digitized at 50 kS/s by a Na-tional Instrument USB-6361 data acquisition unit.

External fields, Bext, are produced by a pair ofHelmholtz coils (radius: 38 mm) driven by a TwinleafCSUA-50 current source. To create oscillating test sig-nals, a Teledyne LeCroy WaveStation 2012 function gen-erator provides a sinusoidal waveform, Vtest, to the mod-ulation input of the current source. The same functiongenerator was used to slowly sweep the magnetic fieldfor the lock-in signals shown in Fig. 3c (in this case, nooscillating test signals were applied).

While our tabletop prototype uses scientific-grade mi-crowave generators, a simpler system could be used to de-liver the requisite dual-resonance microwave waveforms.Figures A9(b) shows an alternative scheme which usesonly voltage-controlled oscillators and a mixer. Thisscheme has the benefit of allowing for rapid feedback tocompensate for thermal and magnetic field drifts, whichwould enable a higher dynamic range [45].

11

Appendix VI.Dual-resonance magnetometry

We perform our magnetometry experiments with alock-in amplifier in order to reduce technical noise, par-ticularly at low frequencies. Such noise could arise froma variety of sources, but a common source in NV mag-netometry experiments is due to intensity fluctuations ofthe laser that are not fully canceled by balanced pho-todetection. The lock-in method allows us to tune ourphotodetector signal to a narrow frequency band, wheresuch technical noise is minimal. In our experiments,this is accomplished by modulating the microwave fre-quency at a modulation frequency fmod = 15 kHz anddepth fd = 3.3 MHz. The resulting photodetector sig-nal, Vpd, has components at fmod and higher harmonics,in addition to the DC level. The lock-in amplifier iso-lates the component at fmod, in a phase-sensitive man-ner, by multiplying Vpd by a reference signal proportionalto cos (2πfmodt). The product signal is passed through alow-pass filter, and the in-phase component, Vout, servesas the magnetometer signal.

The lock-in signal, Vout, can be converted to magneticfield units by one of two methods. In the first case, onecan sweep the magnetic field and measure the depen-dence of Vout on Bext, as in Fig. 3(c) of the main text.The slope can be used to infer the conversion of Vout tomagnetic field units. This method works well providedthat the slope never changes. In practice, the slope canchange due to drifts of the laser or microwave powers. Italso can’t account for any dependence of Vout on mag-netic field frequency, as the slope is measured at DC.Thus, we always apply a calibrated oscillating test fieldand re-normalize our magnetometer conversion based onthe observed amplitude. Typically the difference in con-version factors using the two methods is small (. 10%).

We now turn to describing the principle of dual-resonance magnetometry. In single-resonance magne-tometry, the microwave frequency is modulated aboutone of the FDMR resonances (for example, f+) and de-modulated at the same frequency. The in-phase lock-inoutput Vout is proportional to small deviations in f+.This allows one to infer both the magnitude and signof changes in f+. If the relative phase between the mi-crowave modulation function, F+, and the reference sig-nal were shifted by π radians, the magnitude of Voutwould be the same but the sign would reverse.

In dual-resonance magnetometry, we exploit this fea-ture of phase-sensitive detection. The microwave mod-ulation function for one resonance has a π phase shiftwith respect to the modulation function of the secondresonance. The reference signal has the phase of the firstmodulation function. In this way, if both f+ and f−shift by equal amounts in the same direction [due to achange in D(∆T )], their contributions to the lock-in sig-nal cancel and Vout = 0. If f+ and f− shift by equalamounts but in opposite directions (due to a change inBext), their contributions to the lock-in signal add to-gether and Vout changes in proportion to their shift. In

ms = -1 ms = +1

ms = 0

Dual resonance: P(ms = 0) = 1/3

ms = -1 ms = +1

ms = 0

Single resonance: P(ms = 0) = 1/2

FIG. A10. Single and dual-resonance spin popula-tions. NV spin level populations, represented by the num-ber of magenta circles, are shown under single-resonance anddual-resonance microwave excitation.

other words, the lock-in output is unaffected by ther-mal shifts of the NV spin levels (which shift f+ and f−by equal amounts in the same direction), but it remainsproportional to changes in magnetic field (which shift f+and f− by approximately equal amounts in opposite di-rections).

Note that dual-resonance modulation could also beused to make an NV thermometer which is unaffected bychanges in magnetic field. This would be accomplishedby applying the same modulation phase to both F± sig-nals and monitoring the in-phase lock-in signal.

Appendix VII.Sensitivity enhancement in dual-resonance

magnetometry

The dual-resonance magnetometry approach was pri-marily used because it is unaffected by thermal shifts ofthe NV spin levels. This enabled better low-frequencyperformance. However the dual-resonance approach alsohas a fundamental advantage in sensitivity for all fre-quencies. Compared to the single-resonance approach,it offers a ∼4/3-fold improvement in photoelectron-shot-noise-limited sensitivity. This improvement comes aboutdue to a ∼4/3-fold increase in the FDMR contrast.

To understand where the factor of 4/3 arises, considerthe limiting case when the microwave excitation rate ismuch larger than the optical excitation rate. In thisregime, a resonant microwave field drives the spin levelsit interacts with into a fully mixed state, Fig. A10. Forsingle-resonance excitation, when the microwave field ison resonance, the probability that NV centers will be inthe ms = 0 level is P0 = 1/2. For dual-resonance ex-citation, both microwave transitions share the ms = 0level and thus P0 = 1/3 when both microwave fields areon resonance. Defining the fluorescence intensity of anNV center in the ms = 0 level as I0 and the fluorescenceintensity of an NV center in either of the ms = ±1 levelsas I1, the FDMR contrast is given by:

C =I0 − [P0I0 + (1− P0)I1]

I0. (AVII-1)

12

For the single-resonance case, the contrast is Cs =12I0−I1I0

. In the dual-resonance case, the contrast is

Cd = 23I0−I1I0

. The ratio is therefore Cd/Cs = 4/3.Since the photoelectron-shot-noise-limited sensitivity isproportional to 1/C [Eq. (1)], this corresponds to a 4/3reduction in the magnetic noise floor.

To derive the factor of 4/3 we assumed that the mi-crowave excitation rate was larger than the optical ex-citation rate. In experiments, we use 20 mW of mi-crowave power. This corresponds to a microwave Rabifrequency of ∼0.7 MHz (Appendix IX) or a spin flip rateof ∼1.4×106 s−1. The optical intensity used in our exper-iments was Iopt ≈ 0.2 W/(40 µm)2 = 12.5 kW/cm2 (Ap-pendix IV). The NV absorption cross section at 532 nmis σnv ≈ 3 × 10−17 cm2 [24], so this corresponds to anoptical excitation rate of Ioptσnv/Eph ≈ 106 s−1. Thus,in our experiments, the microwave excitation rate is com-parable to, or slightly larger than, the optical excitationrate. The improvement in dual-resonance sensitivity wasthus not exactly 4/3, but it was close (∼1.3). Anotherassumption that we implicitly made is that the FDMRlinewidth is the same under single-resonance and dual-resonance excitation. This assumption is reasonably ac-curate in our experiments, see Fig. 3(c) of the main text.

Appendix VIII.Magnetometer frequency response

Figure 4(c) of the main text shows the amplitude oftest fields, recorded by dual-resonance diamond magne-tometry, as a function of their frequency. A moderatedecay (∼ 40%) of the signal amplitude was observed overthe 1–1000 Hz range. In order to determine the causesof this signal decay, we performed a series of frequency-response measurements under different conditions.

Figure A12 shows the results of these experiments.In all cases, we use fmod = 15 kHz and the lock-inuses a 12 dB/octave low-pass filter with a time constantτli = 100 µs. We first isolated the lock-in amplifier’sfrequency response by applying a sinusoidal voltage, os-cillating at fmod = 15 kHz, with an amplitude mod-ulation of constant depth and variable modulation fre-quency. The resulting lock-in response is well describedby a second-order Bessel filter with a cutoff frequency of1/(2πτli). While this filter is largely responsible for themagnetometer decay at frequencies &1 kHz, it can onlyaccount for a small fraction of the decay observed overthe 1–1000 Hz range.

Next, we removed the ferrite cones from the assem-bly and performed dual-resonance magnetometry. Theobserved frequency response is similar to that observedwith the ferrite cones in place. The decay is slightly lesspronounced, but evidently the ferrite cones do not ac-count for the observed decay.

Finally, we removed the metal mounting hardware usedin the apparatus that were located within the Helmholtzcoils, Fig. A11. We again performed dual-resonance di-amond magnetometry without the ferrite cones in place.

Brass crewwBrass screw

Cumount

Al post

FIG. A11. Photo of apparatus with shield removed.Metal mounting components that were removed to generatethe data in Fig. A12 are labeled. The brass screw was usedfor mounting to a translation stage during initial alignment(Appendix III). Other unlabeled metal parts, such as brassnuts, were not found to contribute to the frequency-dependentmagnetic field attenuation.

FIG. A12. Frequency response of different magne-tometer configurations. The blue trace is the normalizedmagnetometer frequency response, reproduced from Fig. 4(c).The red trace is the same NV magnetometer setup exceptwithout the ferrite cones. The brown trace is the NV mag-netometer without ferrite cones and with metal components(Fig. A11) removed from the interior of the Helmholtz coils.The black trace is the lock-in filter response as measured byamplitude-modulated voltage inputs.

In this case, we observe a frequency response which isnearly identical to the lock-in amplifier’s frequency re-sponse.

We therefore conclude that metal components withinthe Helmholtz coils are responsible for most of the decayin the 1–1000 Hz range observed in Fig. 4(c). The lock-in amplifier’s low-pass filter contributes as well, but toa lesser degree. The ferrite cones may also contributea small amount to the observed decay, but future workwould be needed to isolate their response independently.

The frequency dependence of our magnetometer leavesan ambiguity as to how best to normalize the magneticnoise spectra in Fig. 4(b). As seen in Fig. 4(c), when we

13

apply a test current which is expected to produce an am-plitude of 580 pTrms, it produces the correct amplitudeat 1 Hz, but at 125–135 Hz it produces an amplitude of∼540 pTrms. Since 125–135 Hz is the frequency range ofthe test fields applied in Fig. 4(b), we therefore had todecide whether to normalize the noise spectra so that thetest-field peaks appeared at 580 pT s1/2 or ∼540 pT s1/2.Conservatively, we chose the former. We multiplied eachspectrum by 580/540 = 1.07, which raised the test fieldpeaks to 580 pT s1/2 and also raised the noise floor by 7%.If we had instead chosen to normalize the test-field peaksto 540 pT s1/2, our noise floor estimates would improveby ∼7% to ∼0.84 pT s1/2.

Appendix IX.Ferrite microwave field enhancement

A feature of our magnetometer is that it uses a sim-ple, non-resonant coil for microwave excitation and onlyrequires 20 mW of microwave power. This is partiallyenabled by an enhancement of the microwave magneticfield provided by the ferrite cones. Figure A13 showsRabi oscillations of the same diamond-coil configurationwith and without ferrite cones. The Rabi frequency withferrite is a & 2-times larger, indicating an equivalent & 2-fold enhancement in the microwave magnetic field.

0 100 200 300 400 500

0.95

1.00

1.05

Time (ns)

Fluo

resc

ence(n

orm

.)

no ferrite, fRabi=7 MHz

with ferrite, fRabi=16 MHz

Laser

MW

readout

500 ns

FIG. A13. Rabi frequency with and without ferrite.(top) Protocol used to observe continuous-wave Rabi oscilla-tions. (bottom) Rabi oscillations observed with and withoutferrite cones (the setup was identical otherwise). Black solidcurves are fits to an exponentially-damped sinusoidal functionrevealing fRabi = 16 MHz with ferrite and fRabi = 7 MHzwithout ferrite. A microwave power of ∼10 W was used forboth traces in order to clearly visualize the Rabi oscillations.

Appendix X.Calibration of Helmholtz coils

Our magnetometer signal’s accuracy relies on a care-ful calibration of the conversion between the current ap-plied to the Helmholtz coils and Bext. Here, we call

this conversion factor Mcal. Theoretically, we estimatedMcal = 165 µT/A based on the known coil geometryand number of turns. We verified this estimate experi-mentally by applying currents to the Helmholtz coils andmeasuring the resulting magnetic field using three differ-ent magnetometers.

FG, shield, 174.6 µT/AFG, no shield, 166.0 µT/AVMR, shield,150.6 µT/A

0.00 0.05 0.100

5

10

15

20

Current (A)

Exte

rnal

field

,Bex

t(µT)

a)

f+f-

Fit with Mcal=175 µT/A

0 2 4 6 8 102830284028502860287028802890

Current (A)

Freq

uenc

y(M

Hz)

b)

FIG. A14. Helmholtz coils current calibration. (a)Helmholtz coils current calibration performed with two com-mercial vector magnetometers. FG: SENSYS FGM3D/100fluxgate magnetometer; VMR: Twinleaf VMR magnetometer.(b) NV FDMR frequencies versus current in the Helmholtzcoils. Solid lines are a fit using Eq. AI-2, where Mcal =175 µT/A is the fit parameter.

First, two commercial vector magnetometers (TwinleafVMR and SENSYS fluxgate, see Fig. 5) were used tocalibrate the Helmholtz coils. Each magnetometer wasplaced in the center of the coils at approximately thesame location as the NV-ferrite structure would be. Thecurrent in the Helmholtz coils was varied and the axialmagnetic field component was recorded. Figure A14(a)shows the resulting calibration curves. The data were fitto linear functions, revealing Mcal (listed in the legend).For the fluxgate magnetometer, Mcal is approximatelythe same as the theoretical estimate when the top of themagnetic shields was removed. When the shield remainedin place, the calibration factor was ∼ 5% larger. TheVMR magnetometer reported a lower magnetic field thanother methods. In both cases we relied on conversionconstants between voltage and magnetic field units asprovided by the manufacturers.

Note that when the current was turned off, we still ob-served a small residual axial field of Bext = −0.2 µT usingboth magnetometers. This is due to the finite attenuation

14

provided by the shields. When the shields were removed,the axial component of the lab field was approximately−20 µT. Since the shields provide a ∼100-fold attenua-tion, this leads to a small residual field of −0.2 µT.

Next, we used NV magnetometry, with the ferritecones removed from the setup (Appendix VIII), to mea-sure the FDMR frequencies as a function of the currentin the coils. Figure A14(b) shows the observed f± valuesalongside a fit according to the NV spin Hamiltonian,Eq. AI-2, with Mcal = 175 µT/A as a fitting parameter.

The value of Mcal used throughout the main text wasthe average of all three values reported by the magne-tometers with the shields on. It is Mcal = 167±14 µT/A,where the uncertainty is the standard deviation. If wehad removed the VMR magnetometer from the analy-sis, we would have obtained Mcal ≈ 175 µT/A. Thiswould decrease the reported sensitivity by ∼5% to ∼0.95 pT s1/2.

Appendix XI.Sensitivity without ferrite

We used the same dual-resonance magnetometry tech-nique described in the main text to record the dia-mond magnetometry signal with the ferrite cones re-moved. Fig. A15 shows the resulting magnetic noisespectrum alongside the spectrum with ferrite [repro-duced from Fig. 4(b)]. The noise floor without ferrite is∼ 300 pT s1/2. This is slightly larger than the expected254-fold increase, most likely due to a suboptimal choiceof microwave power.

FIG. A15. Sensitivity with and without ferrite cones.Magnetic noise spectra for dual-resonance magnetometry with(blue) and without (red) the ferrite cones.

Appendix XII.Flux concentrator hysteresis

The data in Fig. 2(c) of the main text were obtained bysweeping Bext from zero to +50 µT, then from +50 µTto −50 µT, and finally from −50 µT back to zero. Tocheck whether hysteresis results in any remanent fields

over the course of these measurements, we separated thef± data into three segments: 0– + 50 µT, +50–− 50 µT,and −50–0 µT. We fit the three data sets separatelyaccording to Eq. AI-2 with a residual magnetic field offsetof Bext as the only fitting parameter. The resulting offsetmagnetic fields were found to be 8.8 nT, -9.2 nT, and9.8 nT, respectively. This variation lies within the fituncertainty, so we take 10 nT as an upper bound. Notethat this corresponds to a remanent field within the gapof . 2.5 µT.

Appendix XIII.Photoelectron shot-noise limit

The photoelectron-shot-noise-limited sensitivity of ourmagnetometer is given by:

ηpsn =

√q Idc

dIac/dBext, (AXIII-1)

where Idc = 2.3 mA is the sum of the average pho-tocurrent in both channels of the balanced photodetector,dIac/dBext = 33 Arms/T is the lock-in slope expressed interms of the AC photocurrent rms amplitude [Fig. 3(c)],and q = 1.6 × 10−19 C is the electron charge. ThusEq. (AXIII-1) evaluates to ηpsn = 0.58 pT s1/2. Thisnoise can be thought of as the standard deviation of thetime-domain magnetometer data obtained in 1 secondintervals. In the frequency domain it corresponds to thestandard deviation of the real part of the Fourier Trans-form expressed in pT s1/2. In our experiments, we reportthe absolute value of the Fourier Transform. In order torepresent ηpsn in this way, it must be multiplied by 1.25 to

reveal a magnetic noise floor of ηpsn = 0.72 pT s1/2. Thisconversion was checked by simulating Poissonian noiseand observing the noise floor in the absolute value of theFourier Transform.

A similar value ηpsn ≈ 0.75 pT s1/2 was obtained byinserting experimental values from FDMR spectra intoEq. 1 in the main text. In this case, we used ξ = 0.01,Popt = 200 mW, Γ = 9 MHz, and C = 0.04. The effect offlux concentrators is incorporated by multiplying γnv byε. Note that the expression in Eq. 1 refers to the sensitiv-ity to the magnetic field component along the NV axis.Since we use this measurement to infer the total field am-plitude (which is directed at 55° with respect to the NVaxes), the right hand side of Eq. 1 must be multiplied by

1/ cos 55° =√

3.In Sec. I of the main text, we claimed that the lowest

value of Γ/C [11] was 1 MHz/0.04. To be accurate, thereported contrast in this paper was 0.05 and the linewidthwas 1 MHz. However this experiment measured the pro-jection of the field onto NV axes that were aligned at 35°with respect to the field (the field was aligned normal toa [110]-cut diamond face). Incorporating the projectionfactor (cos 35° = 0.82) in Eq. 1 has the same effect asscaling down the ratio Γ/C by the same proportion. Wethus reported the ratio as Γ/C ≈ 1 MHz/0.04.

15

Finally, we would like to clarify some issues with theoptimistic estimation of ηpsn made in Sec. I of the maintext. There, we combined the highest-reported valueof ξ with the lowest reported value of Γ/C. In realitysuch a combination may be difficult to achieve as thereare competing factors. For example, realizing high ξ re-quires high optical depth. This is challenging to realizewhen Γ is small, because the latter implies a low NVdensity. In principle this could still be accomplishedwith a multipass configuration or by using a large dia-mond. However, as one moves to lower Γ, the optimalexcitation intensity also decreases (since the optical ex-citation rate should not exceed Γ). This means that, fora fixed power, the beam area must increase, which fur-ther constrains the geometry and favors larger diamonddimensions. A lower excitation intensity also results in asmaller magnetometer bandwidth, since the optical repo-larization rate is lower. Finally, realizing a high value ofξ requires getting waveguides and/or lenses very close tothe diamond. Realizing such a high optical access mayinterfere with other magnetometer components (concen-trators, microwave loop, heat sinks, etc.). Most of thesetechnical challenges are not insurmountable, but theyneed to be addressed. Our flux concentrator solutionoffers a complementary path that may alleviate some ofthese engineering constraints.

Appendix XIV.Ferrite thermal magnetic noise

Thermal magnetic noise originating from dissipativematerials can be estimated using fluctuation-dissipationmethods [38, 47]. The noise is inferred by calculatingthe power loss (P ) incurred in the material due to a hy-pothetical oscillating magnetic field (angular frequency:ω) produced by a small current loop (area: A, current:I) situated at the location of the magnetometer. Themagnetic noise detected by the sensor is then given by:

δBgap =

√8kTP

AIω, (AXIV-1)

where k is the Boltzmann constant. The power loss hasseparate contributions due to thermal eddy currents andmagnetic domain fluctuations:

Peddy =

∫V

1

2σE2dV, Physt =

∫V

1

2ωµ′′H2dV.

(AXIV-2)Here σ is the electrical conductivity, µ′′ is the imaginarypart of the permeability (µ = µ′− iµ′′), E and H are theamplitudes of the induced electric and magnetic fields,and the integration is carried out over the volume V ofthe dissipative material. In the small excitation limit,E and H scale linearly with the driving dipole moment(AI), so the magnetic noise in Eq. (AXIV-1) is indepen-dent of the size and driving current in the loop.

We numerically calculated magnetic noise contribu-tions due to Peddy and Physt for our flux concentrator

geometry [Figs. 1(a-b)]. We used MN60 material param-eters [38] (σ = 0.2 Ω−1m−1, µ′ = 6500µ0, µ′′ = 26µ0,where µ0 is the vacuum permeability) and a cone gap ofδ = 47 µm, which resulted in the experimental enhance-ment factor ε = 254. We find that thermal eddy currentsproduce white magnetic noise at the level of δBgap ≈0.02 pT s1/2. Since we are interested in our sensitivityin relation to the external field [38], noise produced lo-cally by the ferrite cones translates to an equivalent ex-ternal field noise of δBext = δBgap/ε ≈ 7× 10−5 pT s1/2.This negligibly-low noise level is a consequence of ourchoice of low-conductivity ferrite materials. On the otherhand, thermal magnetization noise results in a larger,frequency-dependent magnetic noise. At 1 Hz, the ef-fective noise is 0.5 pT s1/2, and it scales with frequencyas f−1/2. The thermal magnetization noise is annotatedin Fig. 4(b). It is not a limiting factor in our presentexperiments, but it may have implications for future op-timization efforts. If a material with a lower relativeloss factor (µ′′/µ′2) could be identified, it would result inlower thermal magnetization noise (Appendix XV).

Appendix XV.Thermal magnetic noise for various materials

We also used Eqs. (AXIV-1) and (AXIV-2) to estimatethe magnetic noise produced by cones of the same dimen-sions as in Fig. 1(a), but made from different magneticmaterials. Specifically, we considered low-carbon steel1018 [50], MnZn ferrite MN80 [51], and mu-metal [38].The results of these estimates are listed in Tab. A1 alongwith the material parameters used for the analysis. Inall cases, the hysteresis noise is dominant for frequencies. 10 Hz.

1

2

5

0.51 2 5

Steel 1018MN80 ferriteMN60 ferritemu-metal

-1/2

FIG. A16. Magnetization noise vs. relative loss factor.Calculated hysteresis magnetic noise (at 1 Hz) as a functionof the square root of the relative loss factor,

√µ′′/µ′ in four

magnetic materials.

To minimize hysteresis noise, one must limit the rel-ative loss factor (µ′′/µ′2). We found that the hysteresis

noise scales proportional to√µ′′/µ′2, Fig. A16. Another

16

Material µ′/µ0µ′/µ0µ′/µ0 µ′′/µ0µ′′/µ0µ′′/µ0 σσσ (S/m) Enhancement, ε δBhystδBhystδBhyst (1 Hz), pT s1/2 δBeddyδBeddyδBeddy, pT s1/2

Steel 1018 250 5 5.18×106 223 6.8 0.4

MnZn MN80 2030 6.1 0.2 251 0.85 0.00007

MnZn MN60 6500 26 0.2 254 0.54 0.00007

mu-metal 30000 1200 1.6×106 255 0.8 0.2

TABLE A1. Thermal magnetic noise for different cone materials. Magnetic noise arising from Hysteresis and Johnsonnoise were numerically calculated by the method described in Refs. [38, 47] using finite-element methods. The values of µ′,µ′′, and σ are taken from references: low-carbon steel 1018 [50], MnZn ferrite MN80 [51], MnZn ferrite MN60 [38], and mu-metal [38]. Note that µ′ and µ′′ are in general frequency dependent. Here we take the values for the lowest reported frequencyand assume that the response is relatively flat below 1 kHz. The enhancement ε is determined from magnetostatic simulationsas in Fig. 1 of the main text. The effective external magnetic noises δBext = δBgap/ε are defined by Eqs. AXIV-2-AXIV-1.

δBext is reported at 1 Hz. It scales with frequency as f−1/2.

design consideration is the geometry of the flux concen-trators, but such an optimization is beyond the scopeof this work. If the Johnson noise matters, as in theconductive mu-metal, it could be further decreased bypassivating the skin effect with a lamination.

Finally, we estimated the magnetic noise producedby the mu-metal magnetic shield used in our experi-ments. Here, we used an analytical expression for afinite closed cylinder [47] and inserted mu-metal pa-rameters from Tab. A1 along with the shield dimen-sions (height: 150 mm, diameter: 150 mm, thickness:1.5 mm). The calculated Johnson noise for our shield isδBeddy = 0.02 pT s1/2 and the hysteresis noise at 1 Hz is

δBhyst = 0.007 pT s1/2. These are much lower than theobserved noise floors and can safely be neglected. Notethat the noise from the shields is enhanced by the fluxconcentrators. This effect was incorporated in the calcu-lations, but we still arrived at negligibly-low values.

Appendix XVI.Sensitivity to variation of the gap length

An important systematic effect in our device couldarise from temporal variations in the cone gap length.According to the data in Fig. 1(d), a small change in gaplength in the vicinity of δ ≈ 43 µm produces a changein the enhancement factor given by dε/dδ ≈ 6/µm. Thisvariation extrapolates to a variation in the magnetometer

reading given by:

dBext

dδ=dε

dδ

Bext

ε. (AXVI-1)

For ε ≈ 254 and Bext = 2 µT, Eq. (AXVI-1) predictsthat a change in δ of just 20 pm produces an error inestimation of Bext of 1 pT.

Fortunately the gap length remains relatively stablein our construction such that this effect may only be aproblem at low frequencies. If the material in the gapexpands and contracts due to changes in temperature,this produces a thermal dependence of the magnetometeroutput given by:

dBext

dT=dBext

dδ

dδ

dT. (AXVI-2)

If the material in the gap has a thermal expansioncoefficient α, then the temperature dependence of thegap length is dδ/dT = αδ. For diamond, α ≈ 0.7 ×10−6K−1. Using this value, and inserting Eq. (AXVI-1)into Eq. (AXVI-2), we find a magnetometer temperaturedependence of dBext/dT ≈ 1.5 pT/K. This temperaturedependence is more than 6 orders of magnitude smallerthan the thermal dependence in single-resonance magne-tometry (Sec. II). Nevertheless, to reach this limit, caremust be taken to mechanically stabilize the gap usingan approach which does not significantly increase dδ/dT .For example, using mechanical clamping and/or very thinadhesive layers would be beneficial.

[1] C. L. Degen, F. Reinhard, and P. Cappellaro, “Quantumsensing,” Reviews of Modern Physics 89, 035002 (2017).

[2] F. Casola, T. van der Sar, and A. Yacoby, “Probingcondensed matter physics with magnetometry based onnitrogen-vacancy centres in diamond,” Nature ReviewsMaterials 3, 17088 (2018).

[3] V. M. Acosta, L. S. Bouchard, D. Budker, R. Folman,T. Lenz, P. Maletinsky, D. Rohner, Y. Schlussel, andL. Thiel, “Color centers in diamond as novel probesof superconductivity,” Journal of Superconductivity andNovel Magnetism 32, 85 (2019).

[4] R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen,

17

“Nitrogen-vacancy centers in diamond: nanoscale sen-sors for physics and biology,” Annual Review of PhysicalChemistry 65, 83 (2014).

[5] Y. Wu, F. Jelezko, M. B. Plenio, and T. Weil, “Dia-mond quantum devices in biology,” Angewandte ChemieInternational Edition 55, 6586 (2016).

[6] S. Lourette, L. Bougas, M. Kayci, S. Xu, andD. Budker, “Noncovalent force spectroscopy using wide-field optical and diamond-based magnetic imaging,”arXiv:1908.10052 (2019).

[7] S. Steinert, F. Ziem, L. T. Hall, A. Zappe, M. Schweik-ert, N. Gotz, A. Aird, G. Balasubramanian, L. Hollen-berg, and J. Wrachtrup, “Magnetic spin imaging underambient conditions with sub-cellular resolution,” NatureCommunications 4, 1607 (2013).

[8] D. R. Glenn, K. Lee, H. Park, R. Weissleder, A. Ya-coby, M. D. Lukin, H. Lee, R. L. Walsworth, and C. B.Connolly, “Single-cell magnetic imaging using a quantumdiamond microscope,” Nature Methods 12, 736 (2015).

[9] I. Fescenko, A. Laraoui, J. Smits, N. Mosavian, P. Ke-hayias, J. Seto, L. Bougas, A. Jarmola, and V. M.Acosta, “Diamond magnetic microscopy of malarialhemozoin nanocrystals,” Physical Review Applied 11,034029 (2019).

[10] J. M. McCoey, M. Matsuoka, R. W. de Gille, L. T. Hall,J. A. Shaw, J.-P. Tetienne, D. Kisailus, L. C. L. Hollen-berg, and D. A. Simpson, “Quantum magnetic imagingof iron biomineralisation in teeth of the chiton Acantho-pleura hirtosa,” arXiv:1902.09637 (2019).

[11] J. F. Barry, M. J. Turner, J. M. Schloss, D. R. Glenn,Y. Song, M. D. Lukin, H. Park, and R. L. Walsworth,“Optical magnetic detection of single-neuron action po-tentials using quantum defects in diamond,” Proceedingsof the National Academy of Sciences 113, 14133 (2016).

[12] A. Nowodzinski, M. Chipaux, L. Toraille, V. Jacques, J.-F. Roch, and T. Debuisschert, “Nitrogen-vacancy cen-ters in diamond for current imaging at the redistributivelayer level of integrated circuits,” Microelectronics Reli-ability 55, 1549 (2015).

[13] A. Horsley, P. Appel, J. Wolters, J. Achard, A. Tal-laire, P. Maletinsky, and P. Treutlein, “Microwave devicecharacterization using a widefield diamond microscope,”Physical Review Applied 10, 044039 (2018).

[14] D. R. Glenn, D. B. Bucher, J. Lee, M. D. Lukin, H. Park,and R. L. Walsworth, “High-resolution magnetic reso-nance spectroscopy using a solid-state spin sensor,” Na-ture 555, 351 (2018).

[15] J. Smits, J. T. Damron, P. Kehayias, A. F. McDowell,N. Mosavian, I. Fescenko, N. Ristoff, A. Laraoui, A. Jar-mola, and V. M. Acosta, “Two-dimensional nuclear mag-netic resonance spectroscopy with a microfluidic diamondquantum sensor,” Science Advances 5, eaaw7895 (2019).

[16] J. F. Barry, J. M. Schloss, E. Bauch, M. J. Turner,C. A. Hart, L. M. Pham, and R. L. Walsworth, “Sen-sitivity optimization for NV-diamond magnetometry,”arXiv:1903.08176 (2019).

[17] T. Wolf, P. Neumann, K. Nakamura, H. Sumiya,T. Ohshima, J. Isoya, and J. Wrachtrup, “Subpicotesladiamond magnetometry,” Physical Review X 5, 041001(2015).

[18] G. Chatzidrosos, A. Wickenbrock, L. Bougas, N. Leefer,T. Wu, K. Jensen, Y. Dumeige, and D. Budker, “Minia-ture cavity-enhanced diamond magnetometer,” PhysicalReview Applied 8, 044019 (2017).

[19] E. Boto, R. Bowtell, P. Krger, T. M. Fromhold, P. G.Morris, S. S. Meyer, G. R. Barnes, and M. J. Brookes,“On the potential of a new generation of magnetometersfor MEG: A beamformer simulation study,” PLoS ONE11, e0157655 (2016).

[20] J. Lenz and S. Edelstein, “Magnetic sensors and theirapplications,” IEEE Sensors Journal 6, 631 (2006).

[21] P. Frontera, S. Alessandrini, and J. Stetson, “Shipboardcalibration of a diamond nitrogen vacancy magnetic fieldsensor,” in 2018 IEEE/ION Position, Location and Nav-igation Symposium (PLANS) (2018) pp. 497–504.

[22] K. Fang, V. M. Acosta, C. Santori, Z. Huang, K. M.Itoh, H. Watanabe, S. Shikata, and R. G. Beausoleil,“High-sensitivity magnetometry based on quantum beatsin diamond nitrogen-vacancy centers,” Physical ReviewLetters 110, 130802 (2013).

[23] E. Bauch, C. A. Hart, J. M. Schloss, M. J. Turner, J. F.Barry, P. Kehayias, S. Singh, and R. L. Walsworth, “Ul-tralong dephasing times in solid-state spin ensembles viaquantum control,” Physical Review X 8, 031025 (2018).

[24] V. M. Acosta, E. Bauch, M. P. Ledbetter, C. Santori,K.-M. C. Fu, P. E. Barclay, R. G. Beausoleil, H. Linget,J. F. Roch, F. Treussart, S. Chemerisov, W. Gawlik, andD. Budker, “Diamonds with a high density of nitrogen-vacancy centers for magnetometry applications,” Physi-cal Review B 80, 115202 (2009).

[25] S. Chakravarthi, C. Moore, A. Opsvig, E. Hunt,A. Ivanov, I. Christen, S. Dunham, and K.-M. C. Fu,“A window into NV center kinetics via repeated anneal-ing and spatial tracking of thousands of individual NVcenters,” arXiv:1907.07793 (2019).

[26] T. R. Eichhorn, C. A. McLellan, and A. C. B. Jayich,“Optimizing the formation of depth-confined nitrogen va-cancy center spin ensembles in diamond for quantumsensing,” arXiv:1901.11519 (2019).

[27] G. de Lange, T. van der Sar, M. Blok, Z.-H. Wang, V. Do-brovitski, and R. Hanson, “Controlling the quantum dy-namics of a mesoscopic spin bath in diamond,” ScientificReports 2, 382 (2012).

[28] H. Ozawa, Y. Hatano, T. Iwasaki, Y. Harada, andM. Hatano, “Formation of perfectly aligned high-densityNV centers in (111) CVD-grown diamonds for magneticfield imaging of magnetic particles,” Japanese Journal ofApplied Physics 58, SIIB26 (2019).

[29] C. Osterkamp, M. Mangold, J. Lang, P. Balasubrama-nian, T. Teraji, B. Naydenov, and F. Jelezko, “Engineer-ing preferentially-aligned nitrogen-vacancy centre ensem-bles in CVD grown diamond,” Scientific Reports 9, 5786(2019).

[30] K. Jensen, N. Leefer, A. Jarmola, Y. Dumeige, V. M.Acosta, P. Kehayias, B. Patton, and D. Budker, “Cavity-enhanced room-temperature magnetometry using ab-sorption by nitrogen-vacancy centers in diamond,” Phys-ical Review Letters 112, 160802 (2014).

[31] B. J. Shields, Q. P. Unterreithmeier, N. P. de Leon,H. Park, and M. D. Lukin, “Efficient readout of a sin-gle spin state in diamond via spin-to-charge conversion,”Physical Review Letters 114, 136402 (2015).

[32] E. Bourgeois, A. Jarmola, P. Siyushev, M. Gulka,J. Hruby, F. Jelezko, D. Budker, and M. Nesladek,“Photoelectric detection of electron spin resonance ofnitrogen-vacancy centres in diamond,” Nature Commu-nications 6, 8577 (2015).

[33] D. A. Hopper, R. R. Grote, A. L. Exarhos, and L. C.

18

Bassett, “Near-infrared-assisted charge control and spinreadout of the nitrogen-vacancy center in diamond,”Physical Review B 94, 241201 (2016).

[34] P. Leroy, C. Coillot, A. F. Roux, and G. M. Chanteur,“High magnetic field amplification for improving the sen-sitivity of Hall sensors,” IEEE Sensors Journal 6, 707(2006).

[35] M. J. Caruso, T. Bratland, C. H. Smith, and R. Schnei-der, “A new perspective on magnetic field sensing,” Sen-sors (Peterborough, NH) 15, 34 (1998).

[36] R. C. Chaves, P. P. Freitas, B. Ocker, and W. Maass,“MgO based picotesla field sensors,” Journal of AppliedPhysics 103, 07E931 (2008).

[37] S. Bondarenko, A. Shablo, P. Pavlov, and S. Perepelkin,“Ferromagnetic concentrator of a magnetic field for theplanar HTSC SQUID,” Physica C: Superconductivity372-376, 158 (2002).

[38] W. C. Griffith, R. Jimenez-Martinez, V. Shah,S. Knappe, and J. Kitching, “Miniature atomic mag-netometer integrated with flux concentrators,” AppliedPhysics Letters 94, 023502 (2009).

[39] Y. J. Kim and I. Savukov, “Ultra-sensitive magnetic mi-croscopy with an optically pumped magnetometer,” Sci-entific Reports 6, 24773 (2016).

[40] D. M. Toyli, D. J. Christle, A. Alkauskas, B. B. Buckley,C. G. Van de Walle, and D. D. Awschalom, “Measure-ment and control of single nitrogen-vacancy center spinsabove 600 K,” Physical Review X 2, 031001 (2012).

[41] R. S. Schoenfeld and W. Harneit, “Real time magneticfield sensing and imaging using a single spin in diamond,”Physical Review Letters 106, 030802 (2011).

[42] C. S. Shin, C. E. Avalos, M. C. Butler, D. R. Trease,S. J. Seltzer, J. Peter Mustonen, D. J. Kennedy, V. M.Acosta, D. Budker, A. Pines, and V. S. Bajaj, “Room-temperature operation of a radiofrequency diamond mag-

netometer near the shot-noise limit,” Journal of AppliedPhysics 112, 124519 (2012).

[43] H. Clevenson, M. E. Trusheim, C. Teale, T. Schroder,D. Braje, and D. Englund, “Broadband magnetometryand temperature sensing with a light-trapping diamondwaveguide,” Nature Physics 11, 393 (2015).

[44] J. M. Schloss, J. F. Barry, M. J. Turner, and R. L.Walsworth, “Simultaneous broadband vector magnetom-etry using solid-state spins,” Physical Review Applied10, 034044 (2018).

[45] H. Clevenson, L. M. Pham, C. Teale, K. Johnson, D. En-glund, and D. Braje, “Robust high-dynamic-range vec-tor magnetometry with nitrogen-vacancy centers in dia-mond,” Applied Physics Letters 112, 252406 (2018).

[46] H. Zheng, J. Xu, G. Z. Iwata, T. Lenz, J. Michl,B. Yavkin, K. Nakamura, H. Sumiya, T. Ohshima,J. Isoya, J. Wrachtrup, A. Wickenbrock, and D. Bud-ker, “Zero-field magnetometry based on nitrogen-vacancyensembles in diamond,” Physical Review Applied 11,064068 (2019).

[47] S. K. Lee and M. V. Romalis, “Calculation of magneticfield noise from high-permeability magnetic shields andconducting objects with simple geometry,” Journal ofApplied Physics 103 (2008).

[48] P.-H. Chu, Y. J. Kim, and I. Savukov, “Search forexotic spin-dependent interactions with a spin-exchangerelaxation-free magnetometer,” Physical Review D 94,036002 (2016).

[49] S. Rochester and D. Budker, “”Nitrogen-Vacancy centersin diamond”,” Budker Group Physics Tutorials (2013).

[50] N. Bowler, “Frequency-dependence of relative perme-ability in steel,” AIP Conference Proceedings 820, 1269(2006).

[51] T. W. Kornack, S. J. Smullin, S.-K. Lee, and M. V.Romalis, “A low-noise ferrite magnetic shield,” AppliedPhysics Letters 90, 223501 (2007).