# Experimental Constraints on Exotic Spin-Dependent Interactions by a
Magnetometer with Ensembles of Nitrogen-Vacancy Centers in Diamond

Hang Liang  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
Man Jiao  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
Yue Huang  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
Pei Yu  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
Xiangyu Ye  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
Ya Wang  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
Yijin Xie  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
Yi-Fu Cai  CAS Key Laboratory for Research in Galaxies and Cosmology,
Department of Astronomy, University of Science and Technology of China, Hefei
230026, China  School of Astronomy and Space Science, University of Science
and Technology of China, Hefei 230026, China  Xing Rong<EMAIL_ADDRESS>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  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  CAS Center for Excellence in Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei 230026,
China

###### Abstract

Improved constraints on exotic spin-dependent interactions are established at
the micrometer scale by a magnetometer with ensembles of nitrogen-vacancy (NV)
centers in diamond. A thin layer of NV electronic spin ensembles is utilized
as the sensor, and a lead sphere is taken as the source of the nucleons. The
exotic spin-dependent interactions are explored by detecting the possible
effective magnetic fields by the sensor. Stringent bounds on an exotic parity-
odd spin- and velocity-dependent interaction are set within the force range
from 5 to 500 $\mu$m. The upper limit of the corresponding coupling constant,
$g_{A}^{e}g_{V}^{N}$, is improved by more than three orders of magnitude at
330 $\mu$m. Improved constraints of $P,T$-violating scalar-pseudoscalar
nucleon-electron interactions, are established within the force range from 6
to 45 $\mu$m. The limit of the corresponding coupling constant,
$g_{S}^{N}g_{P}^{e}$, is improved by more than one order of magnitude at 30
$\mu$m. Our result shows that a magnetometer with a NV ensemble can be a
powerful platform for probing exotic spin-dependent interactions.

New light bosons as dark matter candidates have attracted attention in
astronomy and particle physics recently Dailey2021 ; Vermeulen2021 ; Afach2021
. Among the most well-motivated new light bosons, axion addresses the strong
CP mysteries peccei1977cp ; weinberg1978new , hierarchy problem Graham2015 ,
and is a generic prediction of string theories Svrcek2006 ; Gorghetto2021 .
The exotic spin-dependent interactions mediated by new bosons were first
proposed by Moody and Wilczek Moody1984  and have been analyzed according to
polarization and relative velocity between fermions dobrescu2006spin ;
Fadeev2019 . The exotic spin-dependent interactions characterized by scalar,
pseudoscalar, vector and axial-vector couplings lead to effective magnetic
fields applied to polarized spins, which enables us to search for exotic spin-
dependent interactions via precision measurement experiments in laboratory
safronova2018search .

Experimental searches for exotic spin-dependent interactions have been
conducted using various kinds of sensors, such as the electron-spin polarized
torsion pendulum Heckel2008 , superconducting quantum interference device
(SQUID) Ni1999 , and spin-exchange-relaxation-free (SERF) atomic magnetometer
Kim2019 . Recently, single nitrogen-vacancy (NV) centers in diamond have been
developed as single-spin magnetometers Maze2008  to search for exotic spin-
dependent interactions at micrometer scales Rong2018 ; Jiao2021PRL ;
Rong2018PRL . An ensemble-NV-diamond magnetometer has been suggested to be
utilized to conduct the search for exotic spin-dependent interactions
Jiao2021PRL  due to its superior magnetic field sensitivity Taylor2008 ;
Xie2021 ; Zhang2021 ; Barry2017 ; Barry2020 . Here we experimentally conducted
searches for exotic spin-dependent interactions by an ensemble-NV-diamond
magnetometer.

In this work, we focus on two types of exotic interactions between polarized
electron spin and unpolarized nucleon, which are described as dobrescu2006spin
,

$V_{AV}(\textbf{r})=g_{A}^{e}g_{V}^{N}\frac{\hbar}{4\pi}(\frac{e^{{-\frac{r}{\lambda}}}}{r})\bm{\sigma}\cdot\bm{v},$
(1)

$V_{SP}(\textbf{r})=g_{S}^{N}g_{P}^{e}\frac{\hbar^{2}}{8\pi
m_{e}}(\frac{1}{\lambda
r}+\frac{1}{r^{2}})e^{{-\frac{r}{\lambda}}}\bm{\sigma}\cdot\bm{e_{r}},$ (2)

where $g_{A}^{e}$ ($g_{V}^{N}$) is the axial-vector (vector) coupling constant
of new bosons to electrons (nucleons), and $g_{P}^{e}$ ($g_{S}^{N}$) is the
pseudoscalar (scalar) coupling constant. $\lambda=\hbar/mc$ is the force range
with $m$ being the mass of the hypothetical particle and $c$ the speed of
light, $\hbar$ is the reduced Planck’s constant. $m_{e}$ is mass of the
electron, $\bm{\sigma}$ is Pauli vector of the electron spin, r is the
displacement vector between the electron and nucleon, $r=|\textbf{r}|$ and
$\bm{e_{r}}=\textbf{r}/r$, $\bm{v}$ is the relative velocity. These
interactions induce effective magnetic fields sensed by the electron spins of
the NV centers,

$\textbf{B}_{{eff,~{}AV}}(\textbf{r})=\frac{g_{A}^{e}g_{V}^{N}}{2\pi\gamma_{e}}(\frac{e^{{-\frac{r}{\lambda}}}}{r})\bm{v},$
(3) $\textbf{B}_{{eff,~{}SP}}(\textbf{r})=g_{S}^{N}g_{P}^{e}\frac{\hbar}{4\pi
m_{e}\gamma_{e}}(\frac{1}{\lambda
r}+\frac{1}{r^{2}})e^{{-\frac{r}{\lambda}}}\bm{e_{r}},$ (4)

where $\gamma_{e}$ is the gyromagnetic ratio of the electron spin.

Figure 1: Experimental detection scheme. (a) Schematic experimental
parameters. The lead sphere with the radius being R is denoted as M. The
amplitude of the vibration is A, and $d_{0}$ is the minimal distance between
the bottom of M and the surface of the diamond. h is the thickness of layer
with doped NV centers. v is the relative velocity vector between the sensor
and the lead sphere. A static magnetic field $B_{0}$ is applied along the
symmetry axis of NV centers. $\theta$ is the angle between $B_{0}$ and the
velocity vector. (b) The time evolution of the distance d(t), the velocity
v(t), the simulated magnetic fields $B_{{AV}}(t)$ and $B_{SP}(t)$,
respectively. (c) and (d) are Fourier transform spectrum of $B_{{AV}}(t)$ and
$B_{SP}(t)$, respectively. For $B_{{AV}}$, the none-zero component in the real
part corresponds to $a_{{AV}}^{(1)}$ . For $B_{SP}$, the none-zero component
in the imaginary part corresponds to $b_{SP}^{(1)}$.

The detection scheme is shown in Fig. 1. The schematic diagram of nucleon
source and NV ensemble are shown in Fig. 1(a). The time-dependent distance
between the bottom of the sphere and the surface of the diamond is
$d(t)=d_{0}+A[1+\cos(2\pi f_{M}t)]$, where $d_{0}$ is the minimal distance
between the bottom of the sphere and the surface of the diamond. A and $f_{M}$
are the amplitude and frequency of the vibration, respectively. The velocity
of the sphere can be written as $v(t)=-2\pi f_{M}A\sin(2\pi f_{M}t)$. The
possible effective magnetic field $B_{{AV}}$ and $B_{SP}$ sensed by the sensor
can be estimated by integrating Eq. (3) and (4) over all nucleons in the
sphere and electrons in the NV doped layer ( the detailed data and analysis
have been included in the Appendix). The time evolution of the effective
magnetic field $B_{{AV}}(t)$ and $B_{SP}(t)$ is shown in Fig. 1(b), together
with evolution of the distance d(t) and the velocity v(t). The effective
magnetic field $B_{{AV}}$ and $B_{SP}$ can be decomposed into

$B_{{AV}}=\sum_{n}[a_{{AV}}^{(n)}~{}\sin(2\pi
nf_{M}t)+b_{{AV}}^{(n)}~{}\cos(2\pi nf_{M}t)],$ (5)
$B_{SP}=\sum_{n}[a_{SP}^{(n)}~{}\sin(2\pi nf_{M}t)+b_{SP}^{(n)}~{}\cos(2\pi
nf_{M}t)],$ (6)

where $a_{{AV}({SP})}^{(n)}$ and $b_{{AV}({SP})}^{(n)}$ are Fourier
coefficients of the $n$-th harmonic. We take $g_{A}^{e}g_{V}^{N}=1\times
10^{-20}$, $g_{S}^{N}g_{P}^{e}=10^{-20}$, $\lambda=10^{-4}$ m and
$f_{M}=1.953~{}$kHz as an example to numerically simulate the effective
magnetic fields. The none-zero amplitudes of fourier series of $B_{{AV}}$ and
$B_{SP}$ mainly lie in $a_{{AV}}^{(1)}$ and $b_{SP}^{(1)}$ components,
according to our experimental parameters SM. The component with frequency
being $1.953~{}$kHz in the real (imaginary) part of Fourier transform spectrum
corresponds to $a_{{AV}}^{(1)}$ ($b_{SP}^{(1)}$) as shown in Fig. 1(c) and
(d). Thus our method can be utilized to explore the two types of exotic spin-
dependent interactions simultaneously.

Our experiment was conducted with an ensemble-NV-diamond magnetometer as shown
in Fig. 2. A high-purity lead sphere with a radius of R = 978(3) $\mu$m was
taken as the nucleon source with the density of nucleons being $6.8\times
10^{30}~{}m^{-3}$ Lee2018 . The lead sphere was attached to a piezoelectric
bender, which can vibrate with a frequency $f_{M}=1.953$ kHz. The magnetometer
is an ensemble of NV centers in a 23-$\mu$m-thick layer at the surface of the
diamond. The substrate was a high purity electronic-grade $<$100$>$ oriented
single crystal diamond with ppb nitrogen density. A 23-$\mu$m-thick nitrogen-
rich layer was grown on the surface by chemical vapour deposition method.
After electron irradiation and thermal annealing, a layer of NV ensemble with
concentration of 14 ppm was obtained. A 532-nm laser with a diameter of 0.8 mm
illuminated the NV doped layer via the flank of the diamond. The red
fluorescence from the NV centers was collected by a compound parabolic
concentrator Wolf2015 , filtered by a long-pass filter, and detected by a
photodetector (PD). An external magnetic field of 20 Gauss along one of four
symmetry axes of NV centers was applied. Resonant microwave was applied to NV
ensemble via a double-split-ring-resonator Bayat2014 . The fluorescence of NV
ensemble varies as the external magnetic field changes, thus the NV ensemble
can serve as a magnetometer. Recently, NV ensembles have been employed for
high-sensitivity measurements of magnetic fields Zhang2021 ; Barry2017 . In
our experiment, to overcome low frequency noise, the frequency of microwave
from the synthesizer was modulated with FM being 87.975 kHz. The signal of PD,
which detected the fluorescence from NV centers, was demodulated by the first
lock-in amplifier (LIA1) with demodulated frequency being $87.975$ kHz and the
time constant being 8 $\mu$s. Another PD was used to monitor the power
fluctuation of the laser for further noise cancellation. Then the output of
the LIA1 was demodulated by the second lock-in amplifier (LIA2) with a
reference signal $V_{ref}=V_{0}\cos(2\pi f_{M}t+\phi)]$, where $\phi=-32\pm
9~{}^{\circ}$ was the experimental calibrated phase shift between output
signal of magnetometer and d(t). The specific information of the devices
utilized in our setup and the calibration of the phase shift are included in
Appendix. The time constant of LIA2 was set to be 10 ms. With such detection
method, the in-phase and quadrature components of demodulated signal from LIA2
are corresponding to $B_{{AV}}$ and $B_{SP}$, respectively.

Figure 2:  The schematic of our experimental setup. The 532-nm laser
illuminates the layer of ensemble NV centers at the diamond surface. The red
fluorescence is collected by a compound parabolic concentrator at the
bottom.$\lambda$/2 denotes half-wave plate. PBS is the polarizing beam
splitter. PD is the photodiode. CPC is the compound parabolic concentrator.
LPF is the long-pass filter. FM is the modulation frequency of the microwave.
$f_{M}$ is the modulation frequency of the nucleon source.

In our experiment, the sensitivity of the ensemble-NV-diamond magnetometer is
1.4 nT/Hz1/2 within the frequency range from 0.4 to 2 kHz. The vibration
amplitude was $A=718(7)~{}$nm, and the minimal distance between the bottom of
M and the surface of diamond was $d_{0}=9.3(5)~{}\mu$m. In Fig. 3(a), the in-
phase and quadrature parts of the output from LIA2 have been presented with
the time duration being 120 seconds. The total experimental measurement was
performed for 291.9 hours to suppress statistical fluctuations. The histograms
of experimentally measured effective magnetic fields were shown in Fig. 3(b)
and (c). The fits with Gaussian distribution to the histograms for 291.9-hour
data provide the mean values and the standard errors of the effective magnetic
fields. The experimental $B_{{AV}}$ is (0.1 $\pm$ 1.4) pT, and the measured
$B_{SP}$ is (-1.3 $\pm$ 1.4) pT. Our results show no evidence of the existence
of exotic spin-dependent interactions, and experimental limits on both
interactions can be obtained.

Figure 3:  (a) Measurement of the two effective magnetic fields. The red (blue) line corresponds to $B_{{AV}}$ ($B_{SP}$) which is from the in-phase (quadrature) component of the output of the second lock-in amplifier. (b) and (c) The histograms of experimental results for 291.9-hour data. The red and blue solid lines are fits to the Gaussian distributions. The averages and the standard errors of the $B_{{AV}}$ and $B_{SP}$ are (0.1 $\pm$ 1.4) pT and (-1.3 $\pm$ 1.4) pT, respectively.  Table 1: Summary of the systematic errors. The corrections to the constraint on $g_{A}^{e}g_{V}^{N}$ with $\lambda=330~{}\mu$m and $g_{S}^{N}g_{P}^{e}$ with $\lambda=30~{}\mu$m are listed. |  | $\Delta g_{A}^{e}g_{V}^{N}$ | $\Delta g_{S}^{N}g_{P}^{e}$
---|---|---|---
Parameter | Value | $(\times 10^{-25})$ | $(\times 10^{-21})$
Diamagnetism | $-1.6\times 10^{-5}$ | $\pm 0.3$ | $\pm 2.9$
$\theta$ | $54.7\pm 1.3^{\circ}$ | ${}^{+2.9}_{-2.8}$ | $\pm 0.4$
Distance |  $9.3\pm 0.5~{}\mu$m | $\pm 0.2$ | $\pm 0.4$
Radius |  $978\pm 3~{}\mu$m | $\pm 0.2$ | $\pm 0.3$
Thickness |  $23\pm 1~{}\mu$m | $\pm 0.2$ | ${}^{+0.3}_{-0.4}$
Amplitude | $718\pm 7~{}nm$ | ${}^{+0.8}_{-1.0}$ | ${}^{+0.3}_{-0.4}$
Deviation | $0\pm 10~{}\mu m$ | $\pm 0.2$ | ${}^{+0.3}_{-0.4}$
Phase delay $\phi$ | $-32\pm 9^{\circ}$ | ${}^{+2.6}_{-0.6}$ | $\pm 0.3$
Calib. Const. | $2.29\pm 0.03\times 10^{5}V/T$ | $\pm 1.2$ | $\pm 0.3$
Final |  | $\pm 4.3$ | $\pm 3.1$

The systematic errors of our experiment are summarized in Table 1, where we
take $\lambda=330~{}\mu$m for $g_{A}^{e}g_{V}^{N}$ and $\lambda=30~{}\mu$m for
$g_{S}^{N}g_{P}^{e}$. The systematic errors in our experiment mainly come from
uncertainties of parameters of the setup, such as the angle between the
effective magnetic field and the NV axis, the distance between the bottom of M
and the surface of the diamond, the radius of M and the vibration amplitude.
We also consider the uncertainties of the thickness of NV layer and the
misalignment between the center of diamond and the lead sphere in the x-y
plane. Since our experiment has been performed in an external magnetic field
being $20~{}$Gauss, one possible systematic error is due to the diamagnetism
of M. The observable magnetic field variation due to diamagnetism of M is
estimated to be less than 0.5 pT, which is smaller than standard error so of
the measured magnetic under the current statistics in our experiment. A
detailed analysis of systematic errors is included in the Appendix. Combing
the systematic errors in quadrature, the total systematic error for
$g_{A}^{e}g_{V}^{N}$ ($g_{S}^{N}g_{P}^{e}$) is derived to be $\pm 4.3\times
10^{-25}$ ($\pm 3.1\times 10^{-21}$). The bound for the coupling constant
$g_{A}^{e}g_{V}^{N}$ with $\lambda=330~{}\mu$m is $|g_{A}^{e}g_{V}^{N}|\leq
2.5\times 10^{-22}$ with a 95$\%$ confidence level when both statistical and
systematic errors are taken into account. The limit for the coupling constant
$g_{S}^{N}g_{P}^{e}$ with $\lambda=30~{}\mu$m is $|g_{S}^{N}g_{P}^{e}|\leq
2.5\times 10^{-20}$ with a 95$\%$ confidence level. The other values of upper
bound with different values of force range can de derived with the same
procedure.

Figure 4: (a) Upper limit on $g_{A}^{e}g_{V}^{N}$, as a function of the force
range $\lambda$ and mass of the bosons $m$. Black lines are upper limits
established by experiments in Refs.Jiao2021PRL ; Kim2019 . The red line is the
upper bound obtained from our experiment, which establishes an improved
laboratory bound in the force range from 5 to 500 $\mu$m. (b) Upper limit on
$g_{S}^{N}g_{P}^{e}$. Black lines are upper limits established by experiments
in Refs.Stadnik2018 ; Hoedl2011 . Our experiment set the most stringent
constraints in the force range from 5 to 45 $\mu$m.

Fig. 4(a) shows the constraints on $g_{A}^{e}g_{V}^{N}$ set by this work
together with limits established by previous experiments. The excluded values
of coupling constant are presented as grey filled areas. The constraints of
$g_{A}^{e}g_{V}^{N}$ for the force range $\lambda<5~{}\mu$m were established
with single electron spin sensor by Jiao _et al._ Jiao2021PRL . For force
range $\lambda>500~{}\mu$m, constraints were set by Kim _et al_. Kim2019 ,
when an optically polarized vapor magnetometer was utilized to detect possible
effective magnetic field from a BGO crystal. For the force range from 5 to
500$~{}\mu$m, the improved experimental limit is established by this work as
the red line shown in Fig. 4(a). The upper bound for the force range
$\lambda=330~{}\mu$m, is $|g_{A}^{e}g_{V}^{N}|\leq 2.5\times 10^{-22}$, which
is more than three orders of magnitude more stringent than the bound
established by the previous result Jiao2021PRL .

As shown in Fig. 4(b), our work established improved constraints of
$g_{S}^{N}g_{P}^{e}$ in the force range from 6 to 45$~{}\mu$m. Recent
experiments set limits with EDM experiments Stadnik2018  ($\lambda<6~{}\mu$m)
and with torsion pendulum Hoedl2011  ($\lambda>45~{}\mu$m). The upper bound by
our experiment at the force range $\lambda=30~{}\mu$m is
$|g_{S}^{N}g_{P}^{e}|\leq 2.5\times 10^{-20}$, which is more than one order of
magnitude better than previous bound established by Ref.Stadnik2018 .

In summary, experimental searches for two types of exotic spin-dependent
interactions have been performed by an ensemble-NV-diamond magnetometer.
Improved constraints of two types of coupling constant have been established.
The current searching sensitivity is mainly limited by the sensitivity of the
magnetometer. In future, the sensitivity of our magnetometer can be improved.
Firstly, the concentration of NV centers in diamond can be optimized to
enhance the coherence time Barry2020 . Secondly, the double resonance
Fescenko2020  and the hyperfine structure Barry2017  could be used to further
improve the sensitivity. Thirdly, other miscellaneous upgrades are helpful for
the advancement, such as increasing the optical pumping rate Jensen2013 ,
using silicon carbide heatsink to cool the diamond to increase the signal
contrast Schloss2018 . Our setup can also be utilized to search for other
exotic spin-dependent interactions, such as an exotic parity-even spin- and
velocity-dependent interaction between polarized electrons and nucleons, and
interactions between polarized electrons. This work shows that an ensemble-NV-
diamond magnetometer has great potential of exploring physics beyond the
standard model.

This work was supported by the Chinese Academy of Sciences (Grants No.
XDC07000000, No. GJJSTD20200001, No. QYZDY-SSW-SLH004, No. QYZDB-SSW-SLH005),
the National Key R$\&$D Program of China (Grant No. 2018YFA0306600, N0.
2021YFC2203100), Anhui Initiative in Quantum Information Technologies (Grant
No. AHY050000) ,and NSFC (11961131007, 11653002). X. R. thanks the Youth
Innovation Promotion Association of Chinese Academy of Sciences for the
support. Y. F. C. and Y. W. thank the Fundamental Research Funds for Central
Universities. Y. F. C. is supported in part by the CSC Innovation Talent
Funds, by the USTC Fellowship for International Cooperation, and by the USTC
Research Funds of the Double First-Class Initiative.

Hang Liang, Man Jiao and Yue Huang contributed equally to this work.

##  Appendix A Numerical Simulations of the Effective magnetic fields

In this section, we perform numerical simulations of the possible effective
magnetic fields due to exotic spin-dependent interactions. The effective
magnetic fields between polarized electron spin and nucleon are shown in
Equ.(3) and (4) in the main text. By Integrating over the volume of both the
lead sphere and NV layer, we derive the possible effective magnetic fields
$\textbf{B}_{{AV}}$ and $\textbf{B}_{{SP}}$ sensed by the NV ensemble as
follows

$\displaystyle\textbf{B}_{{AV}}=\frac{1}{{V_{S}}}\int_{{V_{S}}}\int_{{V_{M}}}\textbf{B}_{{eff,AV}}(\textbf{r})\rho_{M}d{V_{M}}d{V_{S}},$
(7)
$\displaystyle\textbf{B}_{{SP}}=\frac{1}{{V_{S}}}\int_{{V_{S}}}\int_{{V_{M}}}\textbf{B}_{{eff,SP}}(\textbf{r})\rho_{M}d{V_{M}}d{V_{S}},$
(8)

where $\rho_{M}=6.8\times 10^{30}~{}m^{-3}$ is the nucleon density of the lead
sphere. $V_{S}$ and $V_{M}$ are integral volume of NV layer and lead sphere,
respectively. The radius of the lead sphere is $R=978(3)~{}\mu m$ . The size
of NV layer is $660\times 661\times 23~{}\mu m^{3}$. The minimal distance
$d_{0}$ is 9.3(5) $\mu$m.

The Monte Carlo method can be utilized to numerically calculate the effective
magnetic fields to avoid complex calculations due to high integral dimensions
Kim2019 . The algorithm of Monte Carlo integral is performed as follows:

  * (1)

$N_{{MC}}=2^{20}$ random pairs of points inside both the volumes of the lead
sphere and the NV ensemble are generated.

  * (2)

The effective magnetic field $B_{{eff,AV}}^{i}$ ($B_{{eff,SP}}^{i}$) between a
randomly generated pair of points is calculated with a given force range.

$\displaystyle
B_{{eff,AV}}^{i}=\frac{g^{e}_{A}g^{N}_{V}}{2\pi\gamma_{e}}\frac{e^{-\frac{r}{\lambda}}}{r}vcos\theta,$
(9) $\displaystyle B_{{eff,SP}}^{i}=g^{N}_{S}g^{e}_{P}\frac{\hbar}{4\pi
m_{e}\gamma_{e}}(\frac{1}{\lambda
r}+\frac{1}{r^{2}})e^{-\frac{r}{\lambda}}\frac{z}{r}cos\theta,$ (10)

where $\theta=arccos(1/\sqrt{3})$ is the angle between the direction of the
velocity v and the NV axis. z is the component of distance vector r along the
direction of the velocity v .

  * (3)

All the contributions to the effective magnetic field are summed and
normalized to give the average magnetic field generated by the lead sphere and
sensed by the NV ensemble:

$\displaystyle
B_{{AV}}=N_{{nucleon}}\frac{1}{N_{{MC}}}\sum_{i}^{N_{{MC}}}B_{{eff,AV}}^{i},$
(11) $\displaystyle
B_{{SP}}=N_{{nucleon}}\frac{1}{N_{{MC}}}\sum_{i}^{N_{{MC}}}B_{{eff,SP}}^{i},$
(12)

where $N_{{nucleon}}$ is the total number of nucleons in lead sphere.

The magnetic field $B_{{AV}}$ and $B_{{SP}}$ can be decomposed into orthogonal
components of Fourier series as shown in Equ.(5) and (6) in the main text, the
coefficients can be derived as

$\displaystyle a_{{AV(SP)}}^{(n)}=\frac{2}{T}\int_{0}^{T}\sin(2\pi
nf_{M}t)B_{{AV(SP)}}(t)dt,$ (14) $\displaystyle
b_{{AV(SP)}}^{(n)}=\frac{2}{T}\int_{0}^{T}\cos(2\pi
nf_{M}t)B_{{AV(SP)}}(t)dt,$ (15)

We take ${g^{e}_{A}g^{N}_{V}}=1\times{10}^{-20}$, and $\lambda={10}^{-4}~{}m$
as an example, the coefficients of $B_{{AV}}$ at first three harmonic
frequencies are listed as follows

n | 1 | 2 | 3
---|---|---|---
$a_{{AV}}^{(n)}$ (pT) | 9.62 | 0.02 | 0.00
$b_{{AV}}^{(n)}$ (pT) | 0 | 0 | 0

The amplitude of the first harmonic coefficient $a_{{AV}}^{(1)}$ is much
larger than higher order harmonic coefficients $a_{{AV}}^{(2)}$ and
$a_{{AV}}^{(3)}$. Values of $b_{{AV}}^{(n)}$ are zero. Similarly, the
calculated coefficients of $B_{{SP}}$ at the first three harmonic frequencies
are

n | 1 | 2 | 3
---|---|---|---
$a_{{SP}}^{(n)}$(pT) | 0 | 0 | 0
$b_{{SP}}^{(n)}$(pT) | 5.24 | -0.06 | -0.06

when ${g^{N}_{S}g^{e}_{P}}=1\times{10}^{-20}$, and $\lambda={10}^{-4}~{}m$.
The amplitude of the first harmonic coefficient $b_{{SP}}^{(1)}$ is much
larger than higher order harmonic coefficients $b_{{SP}}^{(2)}$ and
$b_{{SP}}^{(3)}$, and values of $a_{{SP}}^{(n)}$ are zero. The possible
effective magnetic fields mainly lie in the first order harmonic coefficients
with the experimental parameters in our setup.

##  Appendix B Calibration of the phase delay

The phase delay $\phi$ between the output signal of the ensemble-NV-diamond
magnetometer and $d(t)$ can be calibrated by a given signal as discussed in
Ref. Su2021 . A thin copper wire carrying a DC current was stuck to the front
section of the piezoelectric bender. The magnetic field generated by the
copper wire was modulated by the vibration of the piezoelectric bender. In our
experiment, a calibration magnetic field oscillating with an amplitude of (18
$\pm$ 2) nT was applied, and the phase delay $\phi$ was obtained as
$\phi=(-32\pm 9)~{}^{\circ}$.

##  Appendix C The performance of the ensemble-NV-diamond magnetometer

The ensemble-NV-diamond magnetometer used in our experiment was based on the
continuous-wave (CW) method, wherein laser and microwave fields are
continuously applied to NV centers. The NV centers of $|m_{s}=0\rangle$ state
can be transmitted to $|m_{s}=+1\rangle$ by a resonance microwave with an
angular frequency $\omega_{e}=D+\gamma_{e}B_{0}$, where $D=2\pi\times 2.87$
GHz is the ground-state zero splitting, $\gamma_{e}=2\pi\times 28$ GHz/T is
the gyromagnetic ratio of the electron spin, and $B_{0}$ is the external
magnetic field along the symmetry axis of NV centers generated by a solenoid
coil. When the external magnetic field varies, the population on
$|m_{s}=+1\rangle$ states decreases, resulting in changes in fluorescence
which could be detected.

In order to avoid flicker noise, the frequency modulation technique was used
in our experiment. The frequency of microwave from the synthesizer was
modulated by a lock-in amplifier(LIA1 in Fig. 2 of the main text) with FM
being 87.975 kHz, and the signal of PD, which detected the fluorescence from
NV centers was demodulated with the same frequency. For further noise
cancellation, the signal of a reference PD used to monitor the power
fluctuation of the laser was also demodulated by LIA1 with a frequency of
87.975 kHz. The laser intensity noise was canceled by scaling and subtracting
the demodulated reference signal from the demodulated fluorescence signal,
with a cancellation coefficient of about 2 Schloss2018 .

The calibrated constant of magnetic field to output of magnetometer was
determined by the max slope of the continuous-wave spectrum, which was 0.816
$\pm$ 0.009 V/MHz as shown in Fig. 5(a), and corresponds to a calibrated
constant of $2.29\pm 0.03\times 10^{5}$ V/T with the gyromagnetic ratio
$\gamma_{e}=2\pi\times 28$ GHz/T. The sensitivity of 1.4 nT/Hz1/2 from 0.4 to
2 kHz was achieved, as shown in Fig. 5(b).

Figure 5: The performance of the magnetometer.(a) The CW spectrum. The red
line is the linear fitting to give the max slope. (b) Magnetic sensitivity of
ensemble-NV-diamond magnetometer. The peaks of red line at 50 Hz and harmonics
are the magnetic noise due to power supply of the equipment. The vibration
frequency of lead sphere $f_{M}$ = 1.953 kHz is also displayed.

##  Appendix D List of experiment Instruments

Table 2: Detailed information about the devices in our experimental setup.  Instrument | Company | Model
---|---|---
Lock-in Amplifier1 and 2 | Zurich Instruments | HF2LI
Laser | Cobolt | 0532-05-01-1500-700
Synthesizer | National Instruments | FSW-0010
MW Amplifier | CIQTEK | GYPA2530-42
PD | Thorlabs | SM05PD1A
Piezoelectric Bender | Harbin Core Tomorrow | NAC2223

The schematic of experimental setup has been shown in Fig. 2 in the main text.
Table 2 shows the companies and models of devices used in our experiment.

##  Appendix E Systematic Error Analysis

Diamagnetism of the lead sphere

In the external magnetic field of $B_{0}~{}=~{}20$ Gauss, the diamagnetism of
the lead sphere causes a magnetic field $B_{diam}$ on NV ensemble.

$\mathbf{B}_{{diam}}=\frac{1}{V_{S}}\int_{V_{S}}~{}\int_{V_{M}}\frac{\chi}{4\pi}\left[\frac{3\mathbf{r}\left(\mathbf{B}_{0}\cdot\mathbf{r}\right)}{r^{5}}-\frac{\mathbf{B}_{0}}{r^{3}}\right]dV_{M}dV_{S},$
(16)

where $\chi=-16\times 10^{-6}$, is magnetic susceptibility of the lead sphere
CRC . $V_{M}$ is the integral volume of lead sphere, $V_{S}$ is the volume of
NV ensemble.

Since the large zero filed splitting of NV centers, the magnetic field
perpendicular to NV axis can be ignored. The magnetic field parallel to NV
axis due to diamagnetism is denoted as $B_{diam,\parallel}$. The distribution
of $B_{diam,\parallel}$ in NV ensemble is shown in Fig. 6. The x-y plane is
perpendicular to the vibration of piezoelectric bender, x-axis stands for the
direction perpendicular to the NV axis.

Figure 6: Contour plot of $B_{diam,\parallel}$ on NV ensemble in x-y plane.

As shown in Fig. 6, $B_{diam,\parallel}$ has a sign reversal along y-axis.
Thus the averaged magnetic field sensed by the NV ensemble due to diamagnetism
is in the range from 0.738 pT to 0.740 pT during the vibration of lead sphere,
corresponding to a magnetic field of 0.002 pT, which is much less than the
standard error of the measured field under current statistics. The possible
misalignment of the lead sphere and diamond is also taken into consideration.
The misalignment is estimated to be $\pm 10~{}\mu m$, and the maximum
variation of magnetic field due to diamagnetism is 0.5 pT, which is taken as a
systematic error. The correction to the constraint on $g^{e}_{A}g^{N}_{V}$ is
(0.0$\pm 0.3)\times 10^{-25}$ at $\lambda=330~{}\mu$m. The correction to the
constraint on $g^{N}_{S}g^{e}_{P}$ is (0.0$\pm 2.9)\times 10^{-21}$ at
$\lambda=30~{}\mu$m.

Uncertainty in $d_{0}$

The minimal distance between the bottom of the lead sphere and the diamond
$d_{0}$ is measured to be 9.3(5) $\mu$m. To estimate the corrections to the
constraints on $g^{e}_{A}g^{N}_{V}$ and $g^{N}_{S}g^{e}_{P}$ due to the
uncertainty in $d_{0}$ , ${10}^{5}$ samples for $d_{0}$ was randomly taken,
which satisfied a Gaussian distribution
$P_{d_{0}}(d_{0,i})=\frac{1}{\sqrt{2\pi}\sigma_{d_{0}}}exp[-\frac{(d_{0,i}-\mu_{d_{0}})^{2}}{2{\sigma_{d_{0}}}^{2}}]$.
$\mu_{d_{0}}=9.3~{}\mu$m and $\sigma_{d_{0}}=0.5~{}\mu$m are the mean value
and uncertainty of measured $d_{0}$. The correction to $g^{e}_{A}g^{N}_{V}$ is
$(0.0\pm 0.2)\times 10^{-25}$ at $\lambda=330~{}\mu$m. The correction to
$g^{N}_{S}g^{e}_{P}$ is $(0\pm 0.4)\times 10^{-21}$ at $\lambda=30~{}\mu$m.

Uncertainty in $R$

The radius of the lead sphere is measured to be $R=978(3)~{}\mu$m. The
corrections to the constraints on $g^{e}_{A}g^{N}_{V}$ and
$g^{N}_{S}g^{e}_{P}$ are in the same procedure as for the uncertainty in
$d_{0}$. The correction to the constraint on $g^{e}_{A}g^{N}_{V}$ is (0.0$\pm
0.2)\times 10^{-25}$ at $\lambda=330~{}\mu$m. The correction to the constraint
on$g^{N}_{S}g^{e}_{P}$ is (0.0$\pm 0.3)\times 10^{-21}$ at
$\lambda=30~{}\mu$m.

Uncertainty in $\theta$

The uncertainty in $\theta$ is estimated to be 1.3∘. The correction to the
constraint on $g^{e}_{A}g^{N}_{V}$ is from $-2.8\times 10^{-25}$ to $2.9\times
10^{-25}$ at $\lambda=330~{}\mu$m. The correction to the constraint on
$g^{N}_{S}g^{e}_{P}$ is (0.0$\pm 0.4)\times 10^{-21}$ at $\lambda=30~{}\mu$m.

Uncertainty in $h$

The thickness of the NV layer is estimated by measuring the thickness of the
diamond before and after the growth of NV layer. The original thickness of the
diamond is 551(1) $\mu m$. After growth of the NV layer, the thickness of the
diamond is 574(1) $\mu m$. The thickness of the NV layer is measured to be
$h=23(1)~{}\mu$m. The correction to the constraint on $g^{e}_{A}g^{N}_{V}$ is
(0.0$\pm 0.2)\times 10^{-25}$ at $\lambda=330~{}\mu$m. The correction to the
constraint on $g^{N}_{S}g^{e}_{P}$ is from $-0.4\times 10^{-21}$ to $0.3\times
10^{-21}$ at $\lambda=30~{}\mu$m.

Uncertainty in $A$

The vibration amplitude is measured to be $A=718(7)$ nm. The correction to the
constraint on $g^{e}_{A}g^{N}_{V}$ is from $-1.0\times 10^{-25}$ to $0.8\times
10^{-25}$ at $\lambda=330~{}\mu$m. The correction to the constraint on
$g^{N}_{S}g^{e}_{P}$ is $-0.4\times 10^{-21}$ to $0.3\times 10^{-21}$ at
$\lambda=30~{}\mu$m.

Deviation in x-y plane.

The deviation in x-y plane is measured to be $(0\pm 10)~{}\mu$m. The
correction to the constraint on $g^{e}_{A}g^{N}_{V}$ is (0.0$\pm 0.2)\times
10^{-25}$ at $\lambda=330~{}\mu$m. The correction to the constraint on
$g^{N}_{S}g^{e}_{P}$ is $-0.4\times 10^{-21}$ to $0.3\times 10^{-21}$ at
$\lambda=30~{}\mu$m.

Uncertainty in $\phi$

The phase delay is measured to be $\phi=(-32\pm 9)$ degree. The correction to
the constraint on $g^{e}_{A}g^{N}_{V}$ is from $-0.6\times 10^{-25}$ to
$2.6\times 10^{-25}$ at $\lambda=330~{}\mu$m. The correction to the constraint
on $g^{N}_{S}g^{e}_{P}$ is (0.0$\pm 0.3)\times 10^{-21}$ at
$\lambda=30~{}\mu$m.

Uncertainty in Calib. Const.

The Calib. Const. is measured to be $(2.29\pm 0.03)\times{10}^{5}$ V/T. The
correction to the constraint on $g^{e}_{A}g^{N}_{V}$ is (0.0$\pm 1.2)\times
10^{-25}$ at $\lambda=330~{}\mu$m. The correction to the constraint on
$g^{N}_{S}g^{e}_{P}$ is (0.0$\pm 0.3)\times 10^{-21}$ at $\lambda=30~{}\mu$m.

With the fiducial probability of 95$\%$, the upper bound
$|g_{A}^{e}g_{V}^{N}|\leq 2.5\times 10^{-22}$ for the force range $\lambda$ =
330$~{}\mu$m and the upper bound $|g_{S}^{N}g_{P}^{e}\leq 2.5\times 10^{-20}$
for the force range $\lambda$ = 30 $\mu$m were obtained, taking both
statistical and systematic errors into account.

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