#  $T_{2}$-limited dc Quantum Magnetometry via Flux Modulation

Yijin Xie  Caijin Xie  Yunbin Zhu  Ke Jing  Yu Tong  CAS Key Laboratory of
Microscale Magnetic Resonance and School of Physical Sciences, University of
Science and Technology of China, Hefei 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei 230026, China  Xi Qin  CAS Key Laboratory of
Microscale Magnetic Resonance and School of Physical Sciences, University of
Science and Technology of China, Hefei 230026, China  Hefei National
Laboratory, Hefei 230088, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Haosen Guan  CAS Key Laboratory of Microscale
Magnetic Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Chang-Kui Duan  Ya Wang  Xing Rong
<EMAIL_ADDRESS>Jiangfeng Du<EMAIL_ADDRESS>CAS Key Laboratory of
Microscale Magnetic Resonance and School of Physical Sciences, University of
Science and Technology of China, Hefei 230026, China  Hefei National
Laboratory, Hefei 230088, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China

###### Abstract

High-sensitivity magnetometry is of critical importance to the fields of
biomagnetism and geomagnetism. However, the magnetometry for the low-frequency
signal detection meets the challenge of sensitivity improvement, due to
multiple types of low-frequency noise sources. In particular, for the solid-
state spin quantum magnetometry, the sensitivity of low frequency magnetic
field has been limited by short $T_{2}^{*}$. Here, we demonstrate a
$T_{2}$-limited dc quantum magnetometry based on the nitrogen-vacancy centers
in diamond. The magnetometry, combining the flux modulation and the spin-echo
protocol, promotes the sensitivity from being limited by $T_{2}^{*}$ to
$T_{2}$ of orders of magnitude longer. The sensitivity of the dc magnetometry
of 32 $\rm pT/Hz^{1/2}$ has been achieved, overwhelmingly improved by 100
folds over the Ramsey-type method result of 4.6 $\rm nT/Hz^{1/2}$. Further
enhancement of the sensitivity have been systematically analyzed, although
challenging but plenty of room is achievable. Our result sheds light on
realization of room temperature dc quantum magnetomerty with femtotesla-
sensitivity in the future.

Sensing weak magnetic fields plays an important role in many frontier science
and technology, such as biological signal measurement Boto2018 ; Cohen1967 ,
geomagnetic survey Lenz2006 , magnetic imaging applications Glenn2017 , and
exotic interactions examination Jiao2021 . Quantum sensors have been proposed
with high sensitivity and other advantages Degen2017 ; Budker2007 ;
Dolabdjian2017 ; Dolabdjian2017 . In particular, a type of solid-state spin
system, the electron spins of Nitrogen-Vacancy (NV) centers in diamond, has
been proposed to be utilized as a quantum magnetometry with potential
sensitivity up to femtotesla at room temperature Taylor2008 . Since then,
great efforts have been paid to enhance its sensitivity for detecting static
or low-frequency magnetic fields, including cavity-enhanced infrared
absorption magnetometry Jensen2014 ; Chatzidrosos2017 , the total internal
reflection Clevenson2015 , flux concentration Xie2021 ; Fescenko2020 , and
continuously excited Ramsey measurements combined with lock-in detection
Zhang2021c . The NV-based magnetometry has been improved to be sub-picotesla
sensitivity Xie2021 ; Fescenko2020  with flux concentration and picotesla
sensitivityBarry2017  without flux concentration.

There is still a great gap between the current low-frequency sensitivity and
the expected femotesla-sensitivity for NV-based magnetometry. The main
obstacle is the short $T_{2}^{*}$ of the ensemble NV center system, which
essentially limits sensitivities of the continuous wave type and Ramsey-type
NV-based magnetometers. Recently, intense efforts have been paid to increase
$T_{2}^{*}$ by improving the material of the diamond or by developing advanced
quantum control technologies Barry2020 ; Bauch2018 ; Wolf2015 ; Bauch2018 ;
Zhang2021c . Schemes have been proposed to break the $T_{2}^{*}$ limit,
including the ancilla-assisted frequency up-conversion Liu2019a  and fast
rotation of diamonds Wood2018a . By dynamically engineering quantum states of
NV centers, these works turn the static magnetic field into a pseudo ac one.
However, significant sensitivity advancement still remains elusive.

In this work, a dc quantum magnetometry with $T_{2}$-limited based on the
ensemble of NV centers in diamond is demonstrated. A Flux Concentration and
Modulation (FCM) technique has been utilized to transfer the dc magnetic
signal to an ac one. Then the ac signal can be detected with the spin-echo
sequence Hahn1950  applied on the NV magnetometry. Our work has achieved a
$\sim$140 folds sensitivity enhancement from 4.6 $\rm nT/Hz^{1/2}$ to 32 $\rm
pT/Hz^{1/2}$. The low-frequency sensitivity limit has been successfully
boosted from $T_{2}^{*}$ to $T_{2}$. We also systematically analyzed
limitations for enhancement of the sensitivity with the current magnetometry,
and our result paves a way towards future realization of room temperature dc
quantum magnetomerty with femotesla-sensitivity.

Fig. 1: The system of the NV centers in diamond and the basic idea of our
scheme. (a) Diagram of diamond containing NV centers and other impurities. 13C
and Ns represent for isotope of carbon atom and isolated substitutional
nitrogen atom in diamond lattice, respectively. (b) Experimental results of
Ramsey experiment and spin-echo experiment. The data of Ramsey experiment is
fitted by exponential damping and sum of three cosine function with different
angular frequency. $T_{2}^{*}=1.13\pm 0.03~{}\rm\mu s$ is obtained. The
oscillation in $T_{2}^{*}$ data is induced by the hyperfine interaction
between NV and 14N. The data of spin-echo signal is fitted by exponential
damping function and get $T_{2}=102.0\pm 0.5~{}\rm\mu s$. (c) The schematic of
the magnetic field modulation. The raw external signal is in the low-frequency
domain originally, while the low-frequency noise and spin bath noise are
dominant. The modulation method transfers the raw signal into high-frequency
domain. The noise in the high-frequency domain presents less intensity
compared to the low-frequency domain.

The NV center consists of a substitutional nitrogen adjacent to a lattice
vacancy. The energy levels of NV center is shown in Fig. 1(a). The ground
state of NV center is a spin triplet state ${}^{3}A_{2}$, which contains
sublevels of $m_{s}=0$ and $m_{s}=\pm 1$ Barry2020 . The excited state of NV
center includes some discrete electronic states and continuous vibronic states
with higher energy. A 532 nm laser can be used to excite the NV center. Then
the NV center decays to the ground state with fluorescence emitting from 600
to 850 nm Barry2020 . The intensity of fluorescence is dependent on the spin
state of NV center in ground state. Thus the readout of NV center can be
realized by the 532 nm laser excitation and fluorescence acquisition.

There are many types of impurities in the diamond, such as 13C and Ns shown in
Fig. 1(a). The inevitable interactions between these impurities and the
electron spin of NV center are the main contributions to the short dephasing
time $T_{2}^{*}$. In our experiment, the $T_{2}^{*}$ is $1.13\pm 0.03~{}\rm\mu
s$ (Fig. 1(b)). This dephasing effect can be greatly suppressed by dynamical
decoupling technique such as spin-echo sequence, and the coherence time of the
NV centers can be prolonged to $T_{2}$. The prolonged coherence time $T_{2}$
with spin-echo sequence is $102.0\pm 0.5~{}\rm\mu s$ (Fig. 1(b)).
Unfortunately, the dynamical decoupling technique also cancels the effect of
dc magnetic field on NV centers. Therefore, the prolonged coherence time via
dynamical decoupling technique cannot ensure the improvement of the dc
magnetic sensitivity of the NV magnetometry. In this work, we utilized the FCM
method to transfer the low-frequency signal into the high-frequency domain,
while the Ramsey-type protocol can be replaced by the spin-echo protocol
accordingly, as demonstrated in Fig. 1(c). Our method will result in an
extending limit to $T_{2}$ for the low-frequency magnetic field detection.

Fig. 2: Schematic of experimental setup. (a) Principle of FCM technique. Bold
arrow, light arrow and dashed arrow represent for the magnetic field in FCs,
in interval between FCs and in modulation chip, respectively. Thickness of
arrow indicates the intensity of magnetic field. $h$ represents for the gap
between modulation chip and FCs. With $h$ decreasing to $h^{\prime}$, the
intensity of magnetic field in modulation chip increases, and the intensity of
magnetic field in interval between FCs decreases conversely. (b) Experimental
setups. CRS represents for home-built control and readout system. The CRS
controls output of laser and microwave. The experiment is conducted by spin-
echo sequence with $\pi/2-\pi-\pi/2$ microwave pulses. Waveform generator
provides sinusoid signal for the excitation of piezoelectric bender. The
output of the waveform generator is triggered by CRS. Dashed box on the right
side is 3D rendering of the probe. LPF is low-pass filter. PD is photo diode.
Fluorescence emitted from diamond is collected by a compound parabolic
concentrator and finally detected by a PD. The photocurrent is recorded by CRS
for analysis. Meanings of lines with different colors are shown below.

We utilized the mechanical motion scheme Tian2013 ; Edelstein2006 ; Guedes2008
to realize the FCM, as shown in Fig. 2(a). The mechanical motion scheme is
implemented by a piezoelectric bender integrated with a modulation chip made
of high permeability permalloy. The diamond sample is clamped by two Flux
Concentrators (FCs). The modulation chip is placed on the top of the diamond
sample and FCs. The FCs result in magnification of magnetic field on the
diamond. The reduction of the gap $h$ leads to the decrease of magnetic field
intensity in the diamond Tian2013 . Thus, with the periodically driven
modulation chip, the magnetic field intensity in the diamond will become time-
dependent. The external dc magnetic field can be modulated to an ac magnetic
field, which has the same frequency as the driven signal, and it can be then
detected by the NV magnetometry with spin-echo pulse sequence. This scheme
provides a potential improvement in sensitivity by a factor of
$\sqrt{T_{2}/T_{2}^{*}}$.

In our experiment, the resonant frequency of the piezoelectric bender is about
10.8 kHz with the load of modulation chip (See Sec. I in Supplementary
Material). The maximum vibration amplitude of the piezoelectric bender is 3.6
$\rm\mu m$. We set the vibration amplitude to about 3 $\rm\mu m$ to avoid the
overload of the bender. The amplitude corresponds to the modulation efficiency
of 9.6% (See Sec. I in Supplementary Material).

Fig. 2(b) shows the schematic of this work. The optical system is composed of
a 532 nm laser, a battery of lenses, and an Acoustic Optical Modulator (AOM)
to generate initialization and readout laser pulses. The microwave system
consists of a microwave (MW) source, an In-phase/Quadrature (IQ) mixer, a
Positive-Intrinsic-Negative diode (PIN), and a power amplifier. Microwave from
the MW source is multiplied with a radio-frequency (RF) signal fed into the IQ
mixer. Then the output is gated by the PIN to generate the spin-echo sequence
and amplified by a power amplifier. The MW signal is finally sent to a double
split-ring microwave resonator Bayat2014  to drive the NV centers. The Control
and Readout System (CRS) shown in the figure stands for a home-built system
Qin2020 . Both of the AOM and the PIN timing is controlled by the Transistor-
Transistor Logic (TTL) signals from the CRS. RF signals utilized to multiply
with the MW signal are generated from the Arbitrary Waveform Generator (AWG)
integrated into CRS. Another waveform generator provides a sinusoid signal for
the excitation of a piezoelectric bender and it is also triggered by the CRS.
Inside the probe, a couple of FCs is used to amplify the external magnetic
field on the diamond. The length of FC is 4 cm and the width is 8 cm. The
amplified magnetic field is modulated by a vibratory modulation chip glued
onto the piezoelectric bender. The fluorescence is collected through a
compound parabolic concentrator contacting one side of the diamond. After
being separated from the excitation laser by a Long-Pass Filter (LPF), the
fluorescence is transferred to the photodiode (PD). The signal from the PD are
amplified by a current amplifier and finally sampled by the CRS. Another PD,
which is not shown in this figure, is utilized to cancel the fluctuation of
the laser’s intensity by monitoring the variation of the intensity of the
laser as the reference signal Clevenson2015 ; Barry2017 .

Fig. 3:  Sensitivities of different methods. (a)-(c) The signal $S$ as the
function of the applied magnetic fields for different methods. $S$ was
obtained by the reference readout and the differential technique (See Sec. I
in Supplementary Material). $\Delta B$ in abscissa represents for relative
applied magnetic field. Four repetitive field-sweep spectra were tested. Each
point on the figure was the average of four measurements. Error bars denote
the standard deviation of four measurements. The maximum slope max$|\partial
S/\partial B|$ was obtained from the fitting. (d)-(f) Magnetic field amplitude
spectrum density of magnetometer based on different methods. The 0.1 Hz
magnetic field signal was applied by coils for calibration. Green, blue and
purple dashed lines represent for sensitivities of Ramsey protocol, Ramsey
protocol with FC, and FCM method, respectively.

To demonstrate the enhancement of the sensitivities of NV-magnetometry, we
carried out experiments to evaluate the sensitivities for three types of NV
magnetometry. The first one is the Ramsey-type NV-magnetometry. The second one
is Ramsey-type NV-magnetometry with FCs. Because of the magnification of the
external magnetic field via FCs, the sensitivity of the second type of
magnetometer is expected to be improved compared with first one without FCs.
Both the sensitivities of the first and second types of magnetometry are
limited by $T_{2}^{*}$. The third type is NV-magnetometry with FCM. The
sensitivity of this type are expected to be further improved compared with the
second one, and it is essentially limited by $T_{2}$. All the experiments were
carried out with the same diamond in a magnetic shield.

We first measured the sensitivity of Ramsey-type NV-magnetometry. A coil was
utilized to generate the external magnetic field, which is calibrated by a
Tunnel MagnetoResistance (TMR) senor (See Sec. I in Supplementary Material). A
stepped magnetic field generated by the coil was used to test the
max$|\partial S/\partial B|$, as shown in the Fig. 3(a). The max$|\partial
S/\partial B|$ of $0.959\pm 0.004$ mT-1 was obtained. Then the time domain
signal was measured by repeating the Ramsey sequence under resonance and non-
resonance conditions, respectively. The non-resonance condition means that the
microwave frequency was set to be far from the resonance frequency of the NV
centers. The time domain signal with about one hour continuously acquisition
was then converted to amplitude spectral density (ASD) Xie2021  to display the
noise floor in frequency domain, as shown in the Fig. 3(d). The sampling rate
for the acquisition was 1.15 Hz with one point accumulated by 4000 times. The
ASD was computed using the Welch’s method with a 1380-point Blackman-Harris
window with 50% overlap. The ASD measured under resonance condition was higher
than the non-resonance condition. The thermal fluctuation caused by laser and
MW is the the main reason to separate the two ASD spectra Fescenko2020 . To
avoid this impact, previous works Jensen2014 ; Zheng2019 ; Zheng2020  using
the ASDs measured under non-resonance conditions to evaluate the sensitivity
of the NV-magnetometry. Due to the data rate limit of the acquisition system
in the current setup, we met the challenge of increasing the sampling rate, so
the estimated frequency range is from $\sim$0.003 Hz to $\sim$0.6 Hz in ASD.
To estimate the magnetometry performance, we followed the previous works and
used the non-resonant noise floor as the low-frequency sensitivity Jensen2014
; Zheng2019 ; Zheng2020 , which was obtained as $4.6\pm 1.3$ nT/Hz1/2 for
Ramsey protocol in Fig. 3(d).

The experimental results of sensitivity of the Ramsey-type NV-magnetometry
with FCs is shown in Fig. 3(b and e). The magnification of the external
magnetic field via FCs is measured to be 85. As shown in the Fig. 3(b), the
max$|\partial S/\partial B|$ with FCs was obtained as $68.3\pm 1.3$ mT-1,
which is about two orders of magnitude of that without FCs. The noise floors
were also obtained as shown in the Fig. 3(e). The sensitivity of Ramsey-type
NV-magnetometry with FCs is $76\pm 20$ pT/Hz1/2.

The experimental results for sensitivity of NV-magnetometry with FCM are shown
in Fig. 3(c and f). The vibration phase and the position of the modulation
chip have been optimized (See Sec. I of Supplementary Material). As shown in
the Fig. 3(c), the max$|\partial S/\partial B|$ of FCM method was $289.5\pm
7.6$ mT-1. Due to the prolongation of $\tau$ with the FCM method, each data
point was accumulated by 2000 times in order to maintain the same sampling
rate of $\sim$1.15 Hz compared to the Ramsey experiment. The sensitivities of
$32\pm 8$ pT/Hz1/2 was obtained for FCM method.

A test signal with the frequency of 0.1 Hz was applied in all sensitivity
tests (See Sec. I of Supplementary Material). In the measurement of Ramsey-
type NV-magnetometry without FCs, there is no observation of such a signal. In
the measurements of Ramsey-type NV-magnetometry with FCs and NV-magnetometry
with FCM, there are clear observations of the test signal. It is clear that
the signal to noise ratio of the test magnetic field by NV-magnetometry with
FCM is much better than that from the Ramsey-type NV-magnetometry with FCs. We
have successfully demonstrated the improvement of the sensitivity of dc NV-
magnetometry by FCM.

To further investigate the potential of these methods, we establish a model
according to previous works Barry2020 ; Degen2017 ; Xie2021  (See Sec. II in
Supplementary Material) for the sensitivity evaluation as follows,

$\eta\approx A\frac{n_{f}}{G\alpha
E_{{}_{F}}\gamma_{e}e^{-\left(\tau/T_{\rm{coh}}\right)^{p}}C\sqrt{N}}\frac{\sqrt{t_{m}+\tau}}{\tau}\rm,$
(1)

where $A$ is a coefficient for different pulse sequences. $A=1$ ($A=\pi/2$) is
corresponding to the Ramsey protocol (spin-echo sequence) Barry2020 . $n_{f}$
is the ratio between overall noise of system and the shot noise. For the shot-
noise limited sensitivity, we have $n_{f}=1$. $G$ is the magnification of FC.
$\alpha$ is the angle factor used for describing the misalignment between
magnetic field and the NV symmetry axis Xie2021 . $E_{{}_{F}}$ is modulation
efficiency of the FCM method (See Sec. I of Supplementary Material).
$\gamma_{e}$ is the gyromagnetic ratio of electron. $\tau$ is the evolution
time of the NV center. $T_{\rm coh}$ is coherence time of the NV centers. $p$
is stretched exponential parameter depended on the origins of the dephasing
Barry2020 . $C$ is the measurement contrast Taylor2008 . $N$ is the average
number of photons detected per measurement. $t_{m}$ is the additional time in
the pulse sequence (See Sec. II of Supplementary Material).

The parameters in the equation (4) can be experimentally measured as shown in
table I in Supplementary Material. For the Ramsey-type NV-magnetometry without
FCs, $G=1$ and $\tau=0.7~{}\rm\mu s$. The predicted sensitivity according to
the equation (4) is 3.3 nT/Hz1/2, which is consistent with the experimental
result in Fig. 3(d). For the method of Ramsey-type NV-magnetometry by FC, the
predicted sensitivity by equation (4) is 67 pT/Hz1/2 with the parameters
$G=85$ and $\tau=0.7~{}\rm\mu s$. The predicted sensitivity is also consistent
with the experimental result. For the FCM method, $\tau$ has been set to
$92.7~{}\rm\mu s$ to match the modulation frequency. The spin-echo sequence
prolonged the coherence time, which was $102~{}\rm\mu s$ in Fig. 1(b). The
experimental modulation efficiency of the FCM, $E_{{}_{F}}$, was 9.6%. The
contrast C is 0.0045. With $T_{\rm coh}=T_{2}$, the predicted sensitivity of
NV-magnetometry by FCM is 39 pT/Hz1/2, which agrees with the experimental
result. The slight differences between the evaluated sensitivities and the
experimental values may come from the error of the parameters used in the
evaluation. For example, the coefficient $A$ was not exactly $\pi/2$, since
the modulated magnetic field was not a perfect sine. Nevertheless, the
predicted sensitivities are consistent with the experimental results.
Therefore, our model is suitable for predicting the performance of NV-
magnetometry and we would like to utilize such model to evaluate the potential
of sensitivity NV-magnetometry with FCM with current state-of-art
technologies.

According to detailed evaluations in the Sec. II of Supplementary Material,
the sensitivity can be further improved in the following steps by cutting-edge
technologies. (i) When the width of the diamond is reduced, the magnification
of the magnetic field by FCs can be increased. (ii) Through polishing
modulation chip and FCs together with increasing vibration amplitude of
piezoelectric bender, $E_{{}_{F}}$ can be increased to above fifty percent.
This high modulation efficiency is achievable since a recent experiment
reported a modulation efficiency up to 68.7% with a similar method Du2019 .
(iii) The properties of the diamond can be further optimized. The coherence
time can be improved with better diamonds together with advanced quantum
control technologiesBauch2018 ; Wolf2015 . According to the detailed work on
the diamond samples Barry2020 , $T_{2}$ is expected to be $\sim 700\ \rm\mu s$
with 12C enriched diamond, and the NV concentration of about 0.019 ppm is
reachable. If the total internal reflection method Clevenson2015  is utilized,
a reliable $N$ of $2.6\times 10^{10}$ with the above NV concentration is
expected. (iv) The contrast $C$ in experiment is decreased by the
inhomogeneity of the magnetic field in diamond. The inhomogeneity is mainly
due to the FCs’ remanence as analyzed in the Supplementary Material. When such
remanence of FCs is reduced via the demagnetization procedure, $C$ can be
promoted to $\sim 1.2\times 10^{-2}$. With above improvements, an optimized
shot-noise-limited sensitivity is expected to be $\sim 3~{}\rm fT/Hz^{1/2}$.
Besides, the reduction of $n_{f}$ is essential on the way to the shot-noise-
limited sensitivity, since it is about 20 in the current experiments. It could
be helpful to improve the laser stability and reduce the noise of data
acquisition system for decreasing $n_{f}$ Fescenko2020 ; Schloss2018 ;
Chatzidrosos2017 . The further reduction of $n_{f}$ can lead to an achievable
sensitivity of femtotesla level in the future study.

Our work demonstrates a dc quantum magnetometry with $T_{2}$-limited based on
diamonds via flux modulation. Compared to the recent progress in dc NV-
magnetometry via rotating the diamond, our method have a unique advantage as
follows. Since FCM can be realized by Micro-ElectroMechanical Systems (MEMS),
the power consumption and the size of the sensor can be further reduced. Very
recently, a CMOS-integrated NV-Magnetometry has been successfully demonstrated
Kim2019 . We further anticipate that a CMOS-integrated NV-Magnetometry with
FCM by MEMS will provide a highly integrated low frequency magnetometry with
high sensitivity. Our work provides a clear route towards femtotesla
magnetometry at room temperature, which will play an important role in
biological signal detection, such as magnetocardiography.

## Acknowledgments

We thank Wenzhe Zhang for the helpful discussion. This work was supported by
the Chinese Academy of Sciences (Grants No. XDC07000000, No. GJJSTD20200001,
No. QYZDY-SSW-SLH004, No. QYZDB-SSW-SLH005), Innovation Program for Quantum
Science and Technology (Grant No. 2021ZD0302200), the National Key R$\&$D
Program of China (Grant No. 2018YFA0306600), the National Natural Science
Foundation of China (Grant No. 81788101), Anhui Initiative in Quantum
Information Technologies (Grant No. AHY050000), Hefei Comprehensive National
Science Center, and the Fundamental Research Funds for the Central
Universities. X. R thank the Youth Innovation Promotion Association of Chinese
Academy of Sciences for the support.

Y. X., C. X. and Y. Z. contributed equally to this work.

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Supplementary Material

###### Contents

  1. I Materials and methods
    1. I.1 Sample preparation
    2. I.2 Probe with flux concentration and modulation
    3. I.3 Position optimization of the modulation chip
    4. I.4 Vibration phase optimization of the piezoelectric bender
    5. I.5 Experimental setup
    6. I.6 Experimental method
  2. II Further sensitivity improvement
    1. II.1 Magnetic field sensitivity
    2. II.2 Improvement of the magnification ($G$)
    3. II.3 Improvement of the modulation efficiency ($E_{{}_{F}}$)
    4. II.4 Improvement of the coherence time ($T_{\rm coh}$)
    5. II.5 Improvement of the average number of photons detected per measurement ($N$)
    6. II.6 Improvement of the measurement contrast ($C$)
    7. II.7 The potentially achievable sensitivity

##  Appendix I Materials and methods

###  I.1 Sample preparation

The diamond sample used in this work was a single crystal chip (Element Six,
DNV B1), grown by using chemical vapor deposition (CVD). Its typical initial
nitrogen concentration and NV concentration were 0.8 ppm and 0.3 ppm,
respectively. The diamond sample was cut into a piece with the size of 1.0 mm
$\times$ 1.0 mm $\times$ 0.4 mm, and the 1.0 mm $\times$ 1.0 mm facet is
perpendicular to $[100]$ crystal axis.

###  I.2 Probe with flux concentration and modulation

As shown in Fig. S1, the probe with Flux Concentration and Modulation (FCM)
contains a commercial piezoelectric bender (Harbin Core Tomorrow, NAC2223), a
modulation chip and two Flux Concentrators (FCs). The piezoelectric bender is
a cuboid piezo-bimorph, whose size is 21.0 mm $\times$ 7.8 mm $\times$ 1.8 mm.
Both end of the piezoelectric bender are glued onto an aluminium alloy holder.
The modulation chip is a cuboid made of 1J85 alloy, whose size is 4 mm
$\times$ 2 mm $\times$ 1 mm. The modulation chip is glued onto the center of
piezoelectric bender. The size of the FCs is shown in Fig. S1. The FCs are
specially shaped sheet metals made of 1J85 alloy. The thickness of FC is 1 mm.

The resonance frequency and the corresponding vibration amplitude of the
piezoelectric bender were 10.795 kHz and 3.6 $\mu$m with the excitation
voltage of 75 V, as shown in Fig. S1. In sensitivity tests, to protect the
piezoelectric bender, the vibration amplitude was set to about 3.0 $\mu$m with
the excitation voltage of 30 V. The vibration amplitude was tested by a high
speed laser doppler vibrometer (Sunny Optical, LV-S01).

Fig. S1: Schematic of probe with FCM. The red double headed arrow indicates
the vibration direction of piezoelectric bender. Black box on left side shows
frequency-response curve of the piezoelectric bender. The frequency-response
curve was obtained with the load of modulation chip. Black box on right side
shows the size of FCs.

###  I.3 Position optimization of the modulation chip

Fig. S2: Position optimization of the modulation chip. (a) Schematic of the
modulation chip and the coordinate system. $h$ is the the minimum distance
between modulation chip’s lower surface and FCs’ upper surface. $h=0$ means
the two surfaces are in contact. (b) The x-y position optimization of the
modulation chip. An optimal x-y position is indicated by the arrow. (c) $B$ as
the function of $h$ measured at the optimal x-y position.

The position of the modulation chip should be optimized to achieve high
modulation efficiency. The modulation efficiency $E_{{}_{F}}$, which is an
important parameter for the sensitivity of FCM method, can be defined as pan12

$E_{{}_{F}}=\frac{B_{\rm max}-B_{\rm min}}{2B_{\rm init}}\times 100\%.$ (2)

$B_{\rm max}$ and $B_{\rm min}$ are maximum and minimum intensity of magnetic
field in diamond clamped by FCs during the flux modulation. $B_{\rm init}$ is
the intensity of magnetic field in diamond between FCs with modulation chip at
the equilibrium position.

According to the equation (2), the optimized x-y position of the modulation
chip can be achieved by changing $h$ and measuring $B$ at different x-y
positions, denoted as B-h curve. The schematic of the FCM method and the
coordinate system is shown in Fig. S2(a). A magnetic field of about 12.3
$\mu$T is applied during the optimization process. The magnetic field $B$ in
diamond is obtained by measuring the resonance frequency extracted from the CW
spectra. The optimization of the modulation chip’s x-y position is shown in
Fig. S2(b). The irregular shape in Fig. S2(b) is due to the imperfect surfaces
of the modulation chip and the FCs, discussied in Sec. II.3. Fig. S2(c) shows
the B-h curve measured at the optimal x-y position shown in Fig. S2(b). $B$
decreases with $h$ decreasing. Since the vibration amplitude of piezoelectric
bender is set to 3 $\mu$m, the equilibrium position of modulation chip is
chosen as $h=3~{}\mu$m to achieve maximal modulation efficiency. $B_{\rm
max}=$ 10.95 Gauss, $B_{\rm min}=$ 8.95 Gauss and $B_{\rm init}=$ 10.47 Gauss
with above configuration are indicated on the Fig. S2(c). Thus
$E_{{}_{F}}=9.6$% is obtained by using (2).

###  I.4 Vibration phase optimization of the piezoelectric bender

Fig. S3:  Vibration phase optimization of the piezoelectric bender. (a) Spin-
echo sequence and modulated magnetic field. (b) The frequency of the fringe as
the function of the relative vibration phase. The phase is calculated by the
trigger time of TTL signal. The optimal vibration phase is indicated by green
arrow on the figure. The frequency of the fringe is obtained by fitting with
sine function.

Fig. S3(a) defines three delay times. $t_{d1}$ is the delay time between a
Transistor-Transistor Logic (TTL) signal and the beginning of piezoelectric
bender vibration. $t_{d2}$ is the delay time between the beginning of
piezoelectric bender vibration and the equilibrium point of the modulated
magnetic field. The equilibrium point is defined as the intensity of the
modulated magnetic field equivalent to $B_{\rm init}$. $t_{d3}$ is the delay
time between the equilibrium point of the modulated magnetic field and the
ending of the first $\pi/2$-pulse in the spin-echo sequence. The vibration
phase of the piezoelectric bender is depended on the three delay times,
compared to the first $\pi/2$-pulse in the spin-echo sequence.

$t_{d3}$ is essential for the spin-echo magnetometry. Since the three delay
times are fixed, we optimize $t_{d3}$ by changing the trigger time of the TTL
signal. To obtained the optimal sensitivity, we measured the fringe of the
signal $S$ as the function of an applied static magnetic field under the
different TTL signal delay, and calculated the frequency of the fringe. In
experiment, a TTL signal was sent to waveform generator to trigger
piezoelectric bender’s vibration. Fig. S3(b) shows the vibration phase
optimization of piezoelectric bender. The abscissa was calculated from the
trigger time divided the vibration period. The optimized trigger time for an
optimal vibration phase was obtained with the optimization.

###  I.5 Experimental setup

The 532-nm laser was provided by a high-power optically pumped semiconductor
laser (Coherent, Verdi G5). The microwave was generated by a microwave
frequency synthesizer (National Instrument, FSW-0010). The IQ mixer (Marki,
MLIQ0218L), the PIN (Mini-Circuits, ZASWA-2-50DRA+), the customized 20 W power
amplifier and the double split-ring microwave resonator were used for
microwave pulse sequence generation. The piezoelectric bender was driven by a
waveform generator (RIGOL, DG812). The home-built CRS based on Field-
Programmable Gate Array (FPGA) was used to generate TTL signals to AOM
(ISOMET, M1133-aQ80L-1.5), PIN and the waveform generator. The RF modulation
signal, generated by CRS, was applied to IQ mixer. The fluorescence was
detected by a PD (Thorlabs, SM05PD1A) after collected by a compound parabolic
concentrator (Edmund Optics, #65-441). The photocurrent was amplified by a
current amplifier (FEMTO, DHPCA-100) and finally sampled by the CRS.

###  I.6 Experimental method

The external magnetic field was applied by a coil placed near the probe. The
coil’s conversion coefficient of 4.1 $\mu$T/V was obtained using a Tunnel
Magnetoresistance (TMR) magnetometer (MultiDimension Technology, USB27053).
The 0.1 Hz test signal in sensitivity measurements was applied by another coil
via the 0.1 Hz alternating voltage fed into it. The intensity of the test
signal was 12 nT. The probe and coils for applying magnetic field were placed
in a magnetic shield. All the experiments were performed under the magnetic
shielding condition.

Fig. S4: Spin-echo sequence and the readout procedure used in this work.
$S_{1}$, $S_{2}$, $S_{3}$ and $S_{4}$ represents the integrated signal in the
specified readout duration, respectively. The final differential readout
signal $S$ can be obtained as $S=S_{1}/S_{2}-S_{3}/S_{4}$.

The readout procedure used in this work are shown by Fig. S4. The laser pulse
in the spin-echo sequence contained the initialization duration and the
readout duration. The intensity of the optical signal was recorded in readout
duration. A laser pulse contained one initialization duration and two readout
durations. Data acquired in the first readout duration was divided by data
acquired in second readout duration for normalization.

To suppress common mode noise, another differential method was used. In
experiment, the first microwave pulse sequence
$(\pi/2)_{x}-(\pi)_{x}-(\pi/2)_{x}$ and the second microwave pulse sequence
$(\pi/2)_{x}-(\pi)_{x}-(\pi/2)_{-x}$ were played alternately. The second
microwave pulse sequence was realized by reversing the phase of the second
$\pi/2$ pulse. Signal acquired from the first microwave pulse sequence was
subtracted by the signal of the second microwave pulse sequence to suppress
the common mode noise.

##  Appendix II Further sensitivity improvement

###  II.1 Magnetic field sensitivity

The shot-noise-limited magnetic field sensitivity of the spin-echo protocol
can be defined as Barry2020 ; Degen2017

$\eta_{\rm spin-echo}^{\rm shot}\approx\frac{\pi}{2}\frac{\hbar}{\Delta
m_{s}g_{e}\mu_{B}}\frac{1}{e^{-\left(\tau/T_{2}\right)^{p}}C\sqrt{N}}\frac{\sqrt{t_{I}+\tau+t_{r}}}{\tau}$
(3)

where $\pi/2$ is corresponding to the spin-echo protocol Barry2020 . $\hbar$
is reduced Planck constant. $\Delta m_{s}$ is the difference in spin quantum
number between the two interferometry states. $g_{e}$ is the Landég-factor of
electron spin. $\mu_{B}$ is Bohr magneton. $\tau$ is the evolution time of the
NV center. $T_{2}$ is coherence time of the NV center. $p$ is stretched
exponential parameter depended on the origins of the dephasing Barry2020 . $C$
is the measurement contrast Barry2020 . $N$ is the average number of photons
detected per measurement. $t_{I}$ and $t_{r}$ are the duration of
initialization and readout respectively.

We set $\gamma_{e}={\hbar}/{g_{e}\mu_{B}}$ and $t_{m}=t_{r}+t_{I}+t_{d}$ for
simplifying the equation (3). $t_{d}$ is the multiple time delays in pulse
sequence with minor contribution to $t_{m}$. To match different types of NV-
magnetometry, the coefficient ${\pi}/{2}$ is replaced by $A$, and the
coherence time $T_{2}$ is replaced by $T_{\rm coh}$. $A=1$ ($A=\pi/2$) is
corresponding to the Ramsey protocol (spin-echo protocol) Barry2020 .
According to the previous work of the flux concentration Xie2021 , we add the
the magnification $G$ of FC and the angle factor $\alpha$. The modulation of
the magnetic field introduces a new parameter $E_{{}_{F}}$ pan12 , defined in
the equation (2). For the Ramsey-type magnetometry, we have $E_{{}_{F}}=1$
since the external magnetic field is not modulated. Because the system noise
must be larger than the shot noise, we define $n_{f}$ as the ratio between
overall system noise and the shot noise. With the above preconditions, the
equation (3) can be rewritten as

$\eta\approx A\frac{n_{f}}{G\alpha E_{{}_{F}}\gamma_{e}e^{-\left(\tau/T_{\rm
coh}\right)^{p}}C\sqrt{N}}\frac{\sqrt{t_{m}+\tau}}{\tau}\rm,$ (4)

where $\alpha$ is the angle factor used for describing the misalignment
between magnetic field and the NV symmetry axis Xie2021 . For the shot-noise
limited sensitivity, we have $n_{f}=1$. $\gamma_{e}$ is the gyromagnetic ratio
of electron. $T_{\rm coh}$ is coherence time of the NV centers.

According to the experimental data, the parameters of three methods in
equation (4) are shown in Table 1. The $p=1$ is used for the Ramsey-type
magnetometry Barry2020 . From the equation (4), the duty cycle
${{\tau}}/({t_{m}+\tau})$ is also important for the sensitivity enhancement.
From Table 1, we know the duty cycle increase from $<1\%$ to $\sim 40\%$. The
NV-magnetometry with FCM gives a sensitivity enhancement of 140 folds compared
to the Ramsey-type NV-magnetometry.

Table 1: Parameters used in the sensitivity evaluations Parameter | Ramsey | Ramsey with FC | FCM
---|---|---|---
$A$ | 1 | 1 |  $\pi$/2
$G$ | 1 | 85.1 | 85.1
$E_{{}_{F}}$ | 1 | 1 | 0.096
$N$ | 7.6 $\times$ 109 | 7.0 $\times$ 109 | 7.0 $\times$ 109
$C$ | 1.2 $\times$ 10-2 | 9.2 $\times$ 10-3 | 4.5 $\times$ 10-3
$T_{\rm coh}$ | 1.13 $\mu$s | 1.13 $\mu$s | 102 $\mu$s
$\tau$ | 0.7 $\mu$s | 0.7 $\mu$s | 92.7 $\mu$s
$t_{m}$ | 115 $\mu$s | 115 $\mu$s | 140 $\mu$s
$n_{f}$ | 12.2 | 15.6 | 19.2
$p$ | 1 | 1 | 1.24
$\gamma_{e}$ | 2$\pi$ $\times$ 28 GHz/T | 2$\pi$ $\times$ 28 GHz/T | 2$\pi$ $\times$ 28 GHz/T
$\alpha$ | 0.5774 | 0.5774 | 0.5774
$\eta$ (Experiment) | 4.6 nT/Hz1/2 | 76 pT/Hz1/2 | 32 pT/Hz1/2
$\eta$ (Evaluation) | 3.3 nT/Hz1/2 | 67 pT/Hz1/2 | 39 pT/Hz1/2

###  II.2 Improvement of the magnification ($G$)

Fig. S5:  The simulation magnification as the function of the gap width $d$
with the modulation chip nearby. Noticed that a modulation chip exists in the
simulation. $h$ is set to 40.4 $\mu$m. The impact of the imperfect modulation
chip will be discussed in the section II.3.

The magnification $G$ is decided by the geometry of FCs and the gap width
between FCs Xie2021 . Considering the future applications in biomagnetism, we
only optimize the gap between FCs to increase $G$. In experiment, the gap
width is 0.4 mm, which is the thickness of the diamond sample. Diameter of the
laser spot on diamond is about 40 $\mu$m. So the minimum gap width between FCs
can be reduced to 40 $\mu$m by thinning diamond sample.

The simulation magnification as the function of the gap width $d$ are shown in
Fig. S5. The magnification of the FCs increases rapidly with the decrease of
$d$. According to the simulation result, the magnification increases from 74
to 527 with the gap width reducing from 0.4 mm to 40 $\mu$m.

###  II.3 Improvement of the modulation efficiency ($E_{{}_{F}}$)

Fig. S6:  The surface morphology and the modulation efficiency. (a)-(b) The
surface morphology of the modulation chip and the FCs, respectively. The
relative height is measured by a position sensor and subtracted by minimum
relative height for a clear view. (c) The magnetic field in diamond versus
$h$. The simulation is conducted by assuming the ideal configuration of the
modulation chip and the FCs, which are perfectly parallel and smooth. (d) The
modulation efficiency as the function of the vibration amplitude. The
equilibrium position of the modulation chip is set as $h=$ 40.4 $\mu$m in the
simulation.

$E_{{}_{F}}$ is affected by the gap width between the modulation chip and the
FCs. In experiment, the surfaces of modulation chip and FCs are rough. The
surface roughness of the modulation chip and the FCs are shown in Fig. S6(a)
and Fig. S6(b). According to the figures, the surface roughness of modulation
chip and FC are larger than 30 $\mu$mRz and 7 $\mu$mRz, respectively. In
experiment, the width of gap $h$ is the minimum distance between modulation
chip’s lower surface and FCs’ upper surface. Fig. S6(c) shows the results of
the magnetic field in diamond between FCs as the function of $h$. The
simulation result in Fig. S6(c) is an ideal configuration of the modulation
chip and the FCs. It is noted that $h=0$ (simulation) corresponds to the
perfect contact of the modulation chip and the FCs, which leads to the
magnetic field intensity in diamond of nearly zero.

The other key parameter, which has the impact on the modulation efficiency, is
the vibration amplitude of piezoelectric bender. $B_{\rm max}$ and $B_{\rm
min}$ in (2) are decided by vibration amplitude as shown in Fig. S2(d). Fig.
S6(d) shows the modulation efficiency as the function of the vibration
amplitude. In Fig. S6(d), the equilibrium position of modulation chip is set
as $h=$ 40.4 $\mu$m. With the vibration amplitude reaching 40.4 $\mu$m, the
modulation efficiency of 56.8$\%$ can be achieved.

According to the above discussion, the following methods can enhance the
modulation efficiency. The first one is polishing of the modulation chip and
the FCs, and second one is improving the vibration amplitude of the
piezoelectric bender. The polishing of the modulation chip and the FCs enables
the perfect contact of the two objects. The electron beam uno07  and micron-
sized diamonds kubota20  could be used to reduce the surface roughness of the
two objects to several nanometers. Under the same excitation voltage, the
product of the vibration amplitude and the resonance frequency of the
piezoelectric bender can be an approximate invariant, depended on
piezoelectric coefficients and quality factor wang98 . PZT8 is a low damping
piezoelectric material with high piezoelectric coefficients $d_{13}$ of 240
pC/N. As a reference, a piezoelectric bender is fabricated by cuboid cut PZT8
fastened to silicon substrate via Au-In metal bonding and realized resonance
frequency over 8 kHz with vibration amplitude about 15 $\mu$m du19 . Thus a
piezoelectric bender with the resonance frequency over 3 kHz and the vibration
amplitude over 40 $\mu$m can be achievable by using PZT8. With the above
improvements, the modulation efficiency about 56.8$\%$ can be achieved
according the simulation as shown in Fig. S6(d).

###  II.4 Improvement of the coherence time ($T_{\rm coh}$)

For the NV-magnetometry with FCM, the coherence time $T_{2}$ is regarded as
$T_{\rm coh}$. Due to the low applied magnetic field in this work, no 13C
revival was obtained as shown in Fig. 1(a) of main text. Under the applied
magnetic field of low strength, the natural abundance of 13C decrease $T_{2}$.
With 12C enrich technique, the effect of 13C can be suppressed Bauch2020 .

According to the detailed work on diamond samples Barry2020 ; Bauch2018 , the
expected $T_{2}$ of $\sim$700 $\mu$s is achievable with the nitrogen
concentration of about 0.05 ppm. The nitrogen-to-NV conversion efficiency of
the diamond used in this work is about 37.5% (Element Six, DNV B1). The NV
concentration of 0.019 ppm is reachable for the nitrogen concentration of
about 0.05 ppm in the future.

###  II.5 Improvement of the average number of photons detected per
measurement ($N$)

The number $N$ is depended on the photoluminescence rate $R$ of single NV
center and the the number of the NV centers being excited. $R$ can be
estimated as Chapman2011

$R=R_{\infty}(k)\frac{I}{I+I_{s}(\sigma,k)},$ (5)

where $R_{\infty}$ is maximum detected photoluminescence rate of single NV
center. $I$ is the power density of the laser. $I_{s}$ is saturation intensity
of the laser. Because of the NV centers’ absorption to laser,
$I=I_{0}e^{-\beta L}$ decreases with the path length $L$ of excitation laser,
where $I_{0}$ is the incident power density of the laser. $\beta$ is the
absorption coefficient for the laser. The absorption cross-section
$\sigma_{\rm NV}$ of NV center is about 1 $\times$ 10-16 cm2 Chapman2011 .
Thus, $N$ can be estimated by the integral of $R$ for all NV centers being
excited in the laser, and we have

$N\approx\int_{0}^{L_{\rm max}}Rt_{r}n_{{}_{\rm NV}}\pi r^{2}dL,$ (6)

where $L_{\rm max}$ is the maximum path length of the excitation laser in the
diamond. $t_{r}$ is the readout time per measurement. $n_{\rm NV}$ is the
concentration of the NV centers in diamond. $r$ is the radius of the laser
spot.

In this work, the laser power was 0.375 W. $t_{r}$ was 9 $\mu$s. $r$ was 20
$\mu$m. The area of the laser spot was about $1.3\times 10^{3}$ $\mu$m2.
$n_{\rm NV}$ was about 0.3 ppm. The attenuation coefficient $\beta=\sigma_{\rm
NV}n_{\rm NV}$ was 528 m-1. $L_{\rm max}$ was 1 mm for the single-pass
configuration. According to the experiment, the fluorescence intensity is
about 0.22 mW. The average wavelength of the fluorescence is 680 nm. To fit
the experimental result, $R_{\infty}=44.8$ kHz is obtained with the assumption
of $I_{s}=650$ MW/m2 plakhotnik10 . Thus $N$ is calculated as 7.0 $\times$
109.

To improve $N$, we are going to use the total internal reflection
Clevenson2015  to increase the number of the NV centers being excited. With
$n_{\rm NV}$ of 0.019 ppm in the future, we have the attenuation coefficient
$\beta=\sigma_{\rm NV}n_{\rm NV}$ of 33.44 m-1. $L_{\rm max}$ will be $\sim$32
mm, while the whole diamond with the size of 0.04 mm $\times$ 1 mm $\times$ 1
mm is excited. According to the simulation result, the collection efficiency
can be enhanced by 2.2 folds with the coating technique yu20  and a better
fabrication technique of diamond and CPC. Thus $R_{\infty}$ of 98.6 kHz is
available with the improvements. It should be noticed that a long-pass
dielectric film is required to prevent the intensity loss of the laser. The
dielectric film should be able to reflect the laser, and coated on the diamond
surface contacted with CPC. To avoid effects on other parameters, we do not
change the laser power and the diameter of the laser spot in the calculation.
With $I_{s}=650$ MW/m2 plakhotnik10  and $R_{\infty}=98.6$ kHz, the $N$ can be
improved from 7.0 $\times$ 109 to 2.6 $\times$ 1010.

###  II.6 Improvement of the measurement contrast ($C$)

For the Ramsey-type NV-magnetometry, $C$ was about 1.2$\times$10-2. For the
NV-magnetometry with FCM, $C$ was about 4.5$\times$10-3. The decrease mainly
comes from the magnetic field inhomogeneity in diamond. We describe a model
for the impact of the magnetic field inhomogeneity. The signal of the
magnetometry based on the ensemble of NV centers can be represented as

$S=\int dG^{\prime}\int dB_{r}P_{0}(G^{\prime})P_{1}(B_{r})S^{\prime}(B),$ (7)

where $B=G^{\prime}B_{a}+B_{r}$ is the magnetic field acting on the NV
centers, and $S^{\prime}(B)$ is the readout signal of the NV centers under
$B$. $G^{\prime}$ is the magnification of the FCs for $B_{a}$. $B_{a}$ is the
applied magnetic field to the probe. $B_{r}$ is a static magnetic field, which
could come from the remanence of the FCs. $P_{0}(G^{\prime})$ and
$P_{1}(B_{r})$ are the distributions of the parameters.

According to previous discussions in the solid-state spins Bauch2020 ;
Rong2015 ; Boss2017 , we suppose that $P_{0}(G^{\prime})$ and $P_{1}(B_{r})$
are satisfied with the widely used normal distribution. Thus the two
distributions are characterized as

$\displaystyle\begin{cases}P_{0}(G^{\prime})&=\frac{1}{\sqrt{2\pi}kG}exp\left[-\frac{(G^{\prime}-G)^{2}}{2(kG)^{2}}\right]\\\
P_{1}(B_{r})&=\frac{1}{\sqrt{2\pi}mB_{r0}}exp\left[-\frac{(B_{r}-B_{r0})^{2}}{2(mB_{r0})^{2}}\right]\\\
\end{cases}$ (8)

where $k$ and $m$ are relative scale parameters of the corresponding
distributions. $G$ and $B_{r0}$ are mean values of the corresponding
distributions.

Fig. S7:  The simulation of $C$ under different conditions. (a) The signal as
the function of $B_{a}$ under different $k$. Factor $m$ was set to be 0.01 in
the simulation. (b) The signal as the function of $B_{a}$ under different $m$.
Factor $k$ was set to be 0.01 in the simulation. (c) The comparison between
the experiment and the simulation. In the simulation with the practical
condition, $k$ and $m$ were set to be 0.015 and 0.109, respectively. In the
simulation with the ideal condition, $k$ and $m$ were both set to be 0.01. (d)
The simulation result of $C$ as the function of the averaged remanence of the
FCs $B_{r0}$. $k$ and $m$ were set to be 0.015 and 0.109 as in the simulation
of the practical condition.

The simulation is performed by changing applied magnetic field $B_{a}$. Fig.
S7(a) and Fig. S7(b) show the signal as the function of $B_{a}$ with different
$k$ and $m$, respectively. The contrast of the signal decreases with factor
$k$ increasing at the large $B_{a}$. The holistic decrease of the contrast can
be observed with the increase of $m$. It can be summarized as that the
stronger magnetic field inhomogeneity leads to the larger decrease of $C$.

To match with the experimental data, we adjust $k$ and $m$. Fig. S7(c) shows
the comparison between the experiment and the simulation, which is perfectly
matching. It should be noticed that the spectra is not symmetry about
$B_{a}=0$ due to $B_{r0}$. All the parameters used in the simulation are
listed in Table 2.

According to the above simulations, the reduction of FC’s remanence is helpful
in the improvement of $C$. Fig. S7(d) shows the simulation result of $C$ as
the function $B_{r0}$. To increase $C$ to about 1.2$\times$10-2, we have to
suppress $B_{r0}$ to less than 2 $\mu$T from Fig. S7(d). The elimination of
FC’s remanence can be realized by applying an ac magnetic field to the FC
gerdroodbari18 .

Table 2: Parameters used in simulation for Fig. S7(c) Parameter | Value | Parameter | Value
---|---|---|---
$k$ | 0.015 (practical) / 0.01 (ideal) | $G$ | 85.1
$m$ | 0.109 (practical) / 0.01 (ideal) | $B_{r0}$ | 25.0 $\mu$T
$C$ | 1.2 $\times$ 10-2 | $T_{2}$ | 102 $\mu$s
$p$ | 1.24 | $\tau$ | 92.7 $\mu$s

###  II.7 The potentially achievable sensitivity

According to the above discussions, the shot-noise-limited sensitivity of the
NV-magnetometry with FCM can reach $\sim$3 fT/Hz1/2, evaluated by the equation
(4). For an achievable $n_{f}$ of 3, the sensitivity of femtotesla level is
promising. All the parameters used in the evaluation are listed in Table 3.

From Table 3, we can see the essential enhancements coming from the
magnification $G$ of the FCs, the modulation efficiency $E_{{}_{F}}$, and the
noise ratio $n_{f}$. The improvements of $G$ and $E_{{}_{F}}$ require the
micromachining technologies. The improvement of $n_{f}$ requires further
efforts on the readout system optimization. Although a sensitivity of
femtotesla level is proposed, more efforts on the excitation and the quantum
sensing protocols could lead to a more promising result. This $T_{2}$-limited
dc quantum magnetometry is worth further exploration.

Table 3: Parameters used in sensitivity calculation. Parameter | Present value | Improved value | Enhancement for sensitivity | Reference
---|---|---|---|---
$G$ | 85.1 | 527 | 6.2 | Simulation
$E_{{}_{F}}$ | 0.096 | 0.568 | 5.9 |  uno07 , kubota20 , du19 , Simulation
$T_{2}$ | 102 $\mu$s |  $\sim$ 694 $\mu$s |  $\sim$ 1.6 | Bauch2020
$N$ | 7.0 $\times$ 109 | 2.6 $\times$ 1010 | 1.9 |  Clevenson2015 , yu20
$C$ | 4.5 $\times$ 10-3 |  $\sim$ 1.2 $\times$ 10-2 |  $\sim$ 2.7 |  gerdroodbari18 , Simulation
$\tau$ | 92.7 $\mu$s | 333.6 $\mu$s | 2.5 | Experiment
$t_{m}$ | 140 $\mu$s | 140 $\mu$s | $/$ | Experiment
$n_{f}$ | 19.2 | 19.2 | $/$ | Experiment
$p$ | 1.24 | 1.24 | $/$ | Experiment
$\gamma_{e}$ | 2$\pi$ $\times$ 28 GHz/T | 2$\pi$ $\times$ 28 GHz/T | $/$ | $/$
$\alpha$ | 0.5774 | 0.5774 | $/$ | $/$
$\eta$ (Evaluation) | 39 pT/Hz1/2 |  $\sim$ 50 fT/Hz1/2 |  $\sim$ 750.6 | Calculation
$\eta$ (Shot-noise-limited) | 2 pT/Hz1/2 |  $\sim$ 3 fT/Hz1/2 |  $\sim$ 750.6 | Calculation