## Abstract

We demonstrate a novel method of measuring magnetic field based on the transient signal of the K-Rb-^{21}Ne co-magnetometer operating in nuclear spin magnetization self-compensation magnetic field regime. The operation condition for self-compensation magnetic field by nuclear spin magnetization of ^{21}Ne in steady state is presented. We characterize the dynamics of the coupled spin ensembles by a set of Bloch equations, and formulate the expression of transient output signal. After verifying the stability of this method, the measurement range and error are studied. This method is also verified to be valid in various temperature and pumping light power density.

© 2017 Optical Society of America

## 1. Introduction

Precision measurements of magnetic field have been finding a wide range of applications, from searching for permanent electric dipole moments [1], geomagnetic mapping [2], to biomagnetism measurement [3,4]. The low-temperature super-conducting quantum interference device (SQUID) with sensitivity of about 1 fT/ Hz^{1/2} [5] has held the magnetic field sensitivity record for many years, hence it is dominant in the ultrasensitive magnetic field measurement. However, with the development of technology, such as the improvement in the performance of diode laser, the refinement of the production of demanding atomic vapor, and the progress on quantum manipulation, the atomic magnetometer has achieved ultrahigh sensitivity record [6], which surpasses the sensitivity of SQUID. Currently, the atomic optical magnetometer operating in spin-exchange relaxation-free (SERF) regime is the most sensitive magnetometer with the sensitivity of 0.16 fT/ Hz^{1/2} in a gradiometer arrangement [7] and the sensitivity of 0.54 fT/ Hz^{1/2} as a scalar magnetometer [8].

Co-magnetometers utilizing the precession of nuclear spin rather than electron are used in inertial rotation sensing and the test of fundamental symmetries [9], for example the K-Rb-^{21}Ne co-magnetometer is applied to search for tensor interactions violating the local Lorentz invariance [10] and the K-^{3}He co-magnetometer is a promising nuclear spin gyroscope with rotation sensitivity of 5 × 10^{−7} rad s^{−1} Hz^{−1/2} [11]. The measurement of magnetic field based on co-magnetometers has not been demonstrated, because co-magnetometers operating in nuclear spin magnetization self-compensation magnetic field regime are insensitive to magnetic field in steady state compared to the similar alkali metal magnetometers [12]. However, co-magnetometers operating in self-compensation magnetic field regime are still sensitive to magnetic field in transient process, thus the transient response could be used to measure magnetic field. Moreover, the study of transient signal would have benefit for revealing the dynamic performance of co-magnetometer and measuring optical parameters. In addition, the change of transverse residual magnetic field inside the alkali vapor cell of co-magnetometer would disturb the measurement of inertial rotation and the test of fundamental symmetries [12, 13]. Besides, the longitudinal compensation magnetic field of co-magnetometers would deviate from the self-compensation point due to the drift of co-magnetometer system, resulting in that the co-magnetometers cannot operate normally anymore because the nuclear spin magnetization of noble gas atoms cannot self-compensate small magnetic field [14]. Therefore, in-situ measurement of the change of the transverse residual magnetic fields inside the alkali vapor cell and the deviation from the self-compensation point based on the transient signal of co-magnetometer could be potentially used to rapidly and accurately compensate the change of the magnetic fields and the deviation, and would have significant benefit for the operation of co-magnetometers.

In this work, we study the self-compensation of magnetic field in steady state by nuclear spin magnetization of ^{21}Ne in the K-Rb-^{21}Ne co-magnetometer, formulate the expression of the transient output signal under external excitation, and measure the transient response to step change of the transverse magnetic field. We demonstrate a novel method of measuring magnetic field by fitting the transient signal of the co-magnetometer. Under various operation conditions the performance of the co-magnetometer is considered. The measurement range of this method is from 0.016 nT to 1.760 nT and the measurement error is lower than 0.01 nT, which can be potentially improved by suppressing the magnetic noise and increasing the stability of the pumping light. On one hand, this method could be potentially used to rapidly and accurately compensate the change of residual magnetic field and the derivation in co-magnetometers. On the other hand, it could be potentially used in high precision simultaneous measurement of magnetic field and inertial rotation in miniaturization applications.

## 2. Basic principle

The K-Rb-^{21}Ne co-magnetometer principally consists of a spherical glass vapor filled with ^{21}Ne gas and a small mixed droplet of K-Rb as well as N_{2} gas (quenching and buffer gas). K atoms are spin polarized along the z-axis of xyz Cartesian coordinate system by circularly polarized pump light whose wavelength is locked on the D1 line of K atoms. Hybrid optical pumping [15, 16], concretely the spin-exchange collisions with K atoms, polarize Rb atoms along z-axis. The ^{21}Ne nuclear spin is polarized along z-axis by spin-exchange collisions with alkali atoms [17]. The linearly polarized probe light is utilized to measure the precession of Rb spin ensembles in the presence of magnetic field by optical rotation [18].

The K-Rb spin-exchange rate exceeds 10^{6} s^{−1} at typical densities of about 10^{14} cm^{−3} because of sufficiently large spin-exchange cross section, resulting in that the Rb and K atoms are in spin-temperature equilibrium with same spin polarization **P** [15]. The K-Rb spin-exchange interaction for Rb atoms could be represented by an equivalent polarization effect with an equivalent rate *R _{p}*, and the K-Rb spin-destruction interaction for Rb atoms is considered in the spin destruction rate of Rb atoms. The interactions between alkali atoms and

^{21}Ne atoms are determined by the imaginary part of spin-exchange cross section [19], and could be described by an effective magnetic field experienced by one spin species from the average magnetization of the other in a spherical cell,

**B**=

*λ*

**MP**, where

*λ*= 8

*π*

*k*

_{0}/3 [20]. The Fermi contact shift enhancement factor of the alkali-noble atoms pair

*k*

_{0}for Rb-

^{21}Ne pair and K-

^{21}Ne pair are 35.7 ± 3.7 and 30.8 ± 2.7 respectively [21].

**M**=

*μ**n*is the magnetization corresponding to fully polarization. Since the density ratio of K and Rb atoms is typically on the order of 10

^{−2}, the effective magnetic field of K-

^{21}Ne is two orders of magnitude smaller than that of Rb-

^{21}Ne. The K-

^{21}Ne spin destruction interaction for

^{21}Ne atoms is included in the spin destruction rate of

^{21}Ne atoms, which is two orders of magnitude smaller than that of Rb-

^{21}Ne. Therefore, the interaction of

^{21}Ne atoms and alkali atoms are dominated by coupled spin ensembles Rb-

^{21}Ne. The magnetic field generated by Rb electron spins and

^{21}Ne nuclear spins are denoted by

**B**

*and*

^{e}**B**

*respectively, and the magnetizations are designated by*

^{n}**M**

*and*

^{e}**M**

*. Moreover, the spin ensembles experience no effective magnetic field produced by themselves because of the symmetry of spherical vapor [19]. The co-magnetometer can be properly approximated by Bloch equations coupling the Rb ensemble polarization*

^{n}**P**

*with the*

^{e}^{21}Ne ensemble polarization

**P**

*[11,13,18]:*

^{n}Here *γ _{e}* and

*γ*are the gyromagnetic ratios of electron and

_{n}^{21}Ne nucleon.

*Q*is the slowing down factor due to hyperfine interaction between electron and nuclear spins of Rb atoms, and it depends on the nuclear spin

*I*and the polarization rate of the ensemble

*P*[22]. Since the natural-isotopic abundance of Rb (72.2%

^{85}Rb with nuclear spin

*I*= 5/2 and 27.8%

^{87}Rb with nuclear spin

*I*= 3/2) is used in our experiment and the Rb spin ensemble is in the regime of strong spin-exchange, the slowing down factor for the Rb spin ensemble is the hybrid of

*Q*(

*I*= 3/2) and

*Q*(

*I*= 5/2).

**B**and

**Ω**are external magnetic field and inertial rotation.

**L**is the light shift from pump and probe lasers, and can be set to zero as long as the pump light is tuned to optical resonance and the probe light is linearly polarized [12].

*R*and

_{p}*R*are the pumping rate of pump and probe lights, while

_{m}*S*and

_{p}*S*are their photon spins. ${R}_{\mathit{se}}^{\mathit{ne}}$ is the spin-exchange rate from

_{m}^{21}Ne atoms to Rb atoms, and ${R}_{\mathit{se}}^{\mathit{en}}$ is the spin-exchange rate from Rb to

^{21}Ne. For Rb spin ensemble, the longitudinal and transverse relaxation rates are ${\left(Q{T}_{1}\right)}^{-1}=\left({R}_{m}+{R}_{p}+{R}_{\mathit{se}}^{\mathit{ne}}+{R}_{\mathit{sd}}^{e}\right)/Q$ and ${\left(Q{T}_{2}\right)}^{-1}={\left(Q{T}_{1}\right)}^{-1}+{R}_{\mathit{se}}^{ee}$ respectively, where ${R}_{\mathit{sd}}^{e}$ is the spin destruction rate of Rb atoms. The spin-exchange relaxation rate of Rb-Rb, ${R}_{\mathit{se}}^{ee}={\omega}_{0}^{2}{T}_{\mathit{se}}\left[{Q}^{2}-{\left(2I-1\right)}^{2}\right]/2$ [22, 23], cannot be ignored for large

*B*, which is negligible in SERF regime [11, 19], where

^{e}*ω*

_{0}is the reduced precession frequency,

*T*is the spin-exchange time of Rb-Rb,

_{se}*I*is the nuclear spin of Rb atoms, and ${T}_{2}^{-1}$ is denoted by ${R}_{\mathit{tot}}^{e}$. ${R}_{\mathit{tot}}^{n}={R}_{\mathit{se}}^{\mathit{en}}+{R}_{\mathit{quad}}^{n}+{R}_{\mathit{sd}}^{n}$ is the total spin relaxation rate of

^{21}Ne, where ${R}_{\mathit{quad}}^{n}$ is the quadrupole relaxation rate of

^{21}Ne, ${R}_{\mathit{sd}}^{n}$ is the spin destruction rate.

With small transverse excitation, the longitudinal polarization
${P}_{z}^{e}$ and
${P}_{z}^{n}$ nearly remain constant, therefore the Bloch equations could be linearized. By solving Eq. (1) we can obtain the complete output signal *S _{x}* proportional to the transverse polarization
${P}_{x}^{e}$ including transient and steady components in Eq. (2).

*K*is the factor that converts the transverse polarization of Rb spin ensemble to output voltage signal by the probe system;

_{d}*λ*

_{1r},

*λ*

_{1i},

*λ*

_{2r}and

*λ*

_{2i}are determined by the system itself;

*P*

_{1r},

*P*

_{1i},

*P*

_{2r},

*P*

_{2i}and ${S}_{x}^{\mathit{steady}}$ depend on the input signals.

As inputting transverse magnetic field *B _{y}*, the basic operation of this co-magnetometer is that a bias magnetic field

*B*produced by coils is applied along z-axis whose value is set to

_{z}*B*= −

_{c}*B*−

^{e}*B*, which is defined as the compensation point. At this point, the nuclear magnetization of

^{n}^{21}Ne could arbitrarily cancel slowly changing

*B*in steady state [11], leading to the steady signal of x-axis no longer depending on

_{y}*B*, while the transient signal of x-axis still related to

_{y}*B*. Hence, the transient signal can be used to obtain the input magnetic field. The intuitive compensation principle is illustrated in Fig. 1 and will be discussed in more detail below. At the compensation point, the parameters

_{y}*P*

_{1r},

*P*

_{1i},

*P*

_{2r},

*P*

_{2i}and ${S}_{x}^{\mathit{steady}}$ of Eq. (2) can be specifically rewritten with different types of change in transverse magnetic field, for example step-like change and sinusoidal change. When the change is step-like with amplitude

*B*

_{y0}, these parameters can be expressed by Eq. (3), and with other types of change they can be similarly rewritten:

Here Ω* _{y}*, projection of earth rotation along y-axis, is invariable equaling to 5.586×10

^{−5}rad/s because the y-axis is directed along the direction of the earth rotation horizontal component, hence ${K}_{1r}^{0}{\mathrm{\Omega}}_{y}$, ${K}_{1i}^{0}{\mathrm{\Omega}}_{y}$, ${K}_{2r}^{0}{\mathrm{\Omega}}_{y}$ and ${K}_{2i}^{0}{\mathrm{\Omega}}_{y}$ are constants. Since ${K}_{d}{\gamma}_{e}{P}_{z}^{e}/Q$ could be obtained by the calibration procedure [13],

*K*and

_{r}*K*can be determined by the system parameters

_{i}*λ*

_{1r},

*λ*

_{1i},

*λ*

_{2r}and

*λ*

_{2i}.

*K*is the scale factor of steady state value, and ${S}_{x}^{\mathit{steady}}$ is constant only depending on Ω

_{steady}*because*

_{y}*δ*

*B*=

_{z}*B*−

_{z}*B*equals to zero at the compensation point. Therefore, after determining the system parameters, the magnetic field

_{c}*B*

_{y0}can be acquired by fitting the transient signal based on Eq. (2) and Eq. (3).

## 3. Experimental setup

The schematic of the co-magnetometer is shown in Fig. 2, which is similar to our previous apparatus [14,24]. There is a 14 mm diameter spherical aluminosilicate glass vapor in the centre of the device, which contains a mixture droplet of K-Rb (natural abundance) alkali metals, 2020 Torr ^{21}Ne gas (70% isotope enriched) and 31 Torr N_{2}. The cell is installed in a boron nitride ceramic oven and heated by a 110 kHz AC electrical heater. A PID temperature control system is applied to control the current of the 110 kHz AC electrical heater to reduce temperature fluctuation of the cell. The oven is enclosed by 5-layers cylindrical mu-metal magnetic shields and a ferrite barrel for shielding the ambient magnetic field, and the residual magnetic field is further compensated by a set of three-axis Helmholtz coils. A water-cooling system is used to reduce the cross-ventilation due to the temperature difference between the oven and the room.

The electron spin of K atoms is polarized along z-axis by the circularly polarized pumping light, which is produced by a external cavity diode laser (ECDL) and amplified by a tapered amplifier (TA). The wavelength of the pumping light is tuned to D1 resonance of K and the maximum light power is about 1W. The pumping light beam is expanded by beam expander (BE) and selected by aperture, consequently the cell is covered by light with relatively uniform power density. The precession of the electron-nuclear spin ensembles in the presence of magnetic field can be measured by the linearly polarized probe light based on optical rotation. The probe light along x-axis is formed by a distributed feedback (DFB) laser, whose wavelength is 0.3 nm red detuning from D1 line of ^{85}Rb, for detecting the x-component of electron polarization. The power density of the probe light and the pumping light are stabilized by electrically controlled half waveplate respectively. A photo-elastic modulator (Hinds Instrument) modulates the probe light with resonance frequency of about 50 kHz and modulation amplitude about 0.08 rad. And the signal measured by a photodetector is demodulated by a lock-in amplifier (Stanford Research SR830) and the first harmonic of the signal is recorded [25].

This co-magnetometer operates with the cell temperature set to 456K, at which the densities of K and Rb atoms are about 3.8 × 10^{12} cm^{−3} and 3 × 10^{14} cm^{−3} respectively. And the pumping light power density is 81.3 mW cm^{−2}. Applying a series of procedures for zeroing residual magnetic field [12], the residual magnetic fields are found to be approximate to 1.586 ± 0.001 nT and 3.626 ± 0.002 nT in x and y axis respectively, while the compensation point is found to be *B _{c}* = −312.7 nT. Utilizing the calibration procedure [11,13], the light shift

*L*is measured to be 0.127 nT, ${R}_{\mathit{tot}}^{e}/{\gamma}_{e}$ is 31.84 nT, and the steady state scale factor

_{z}*K*is 22.1 V/nT. A sensitivity of 2.1 × 10

_{steady}^{−8}rad s

^{−1}Hz

^{−1/2}for the K-Rb-

^{21}Ne co-magnetometer has been demonstrated in our previous work [14], corresponding to a magnetic field sensitivity of 1.0 fT Hz

^{−1/2}.

## 4. Results and discussions

The responses to transverse input magnetic field with various bias magnetic field are shown in Fig. 3. The output signals *S _{x}*, following a small step magnetic field

*B*

_{y0}= 0.32 nT formed by the three-axis Helmholtz coils, are measured when the bias magnetic field

*B*is set to compensation point or non-compensation point, and are fitted with Eq. (2). As shown in Fig. 3(a), when

_{z}*B*is set to

_{z}*B*, the response signal would damp to the initial offset value ${S}_{x}^{0}=0.1286\hspace{0.17em}\text{V}$ after a short-time oscillation, that means

_{c}*B*completely compensated by the nuclear magnetization of

_{y}^{21}Ne in steady state. In concrete term, the offset value ${S}_{x}^{0}=0.1286\hspace{0.17em}\text{V}$ is steady response to the projection of earth rotation along y-axis, and the characteristic time determining the time of transient process is approximately equals to 0.45 second. In Fig. 3(b), when

*B*tuned to non-compensation point, the transient oscillation signal attenuates to the steady state value unequal to the initial offset value, and the time of transient process is longer than the one at the compensation point. The ratio of the difference between the initial value and the steady value $\mathrm{\Delta}{S}_{x}^{0}$ to the initial value is −0.493, indicating

_{z}*B*incompletely compensated in steady state. Specifically, the difference values $\mathrm{\Delta}{S}_{x}^{0}$ with same input magnetic field

_{y}*B*

_{y0}= 0.32 nT and various bias magnetic field are illustrated in Fig. 3(c). At the compensation point

*δ*

*B*=0, the steady response to magnetic field is entirely suppressed, and $\mathrm{\Delta}{S}_{x}^{0}$ increases along with the growth of deviation of compensation point

_{z}*δ*

*B*with the increase rate declining as expected with Eq. (3).

_{z}At the compensation point, the determined system parameters cannot be utilized to fit the transient signal to acquire the input magnetic field, unless the system parameters are stable over time and unaffected by inputting signals. The stability of the system parameters is studied and shown in Fig. 4. A series of response signals to the same input step magnetic field *B*_{y0} = 0.32 nT are recorded every half hour, and these 2-second transient signals are fitted by Eq. (2) as shown in Fig. 4(a). The coefficients of determination R-square (the goodness of fit) are superior to 0.99.

The average values of the system parameters (*λ*_{1i}, *λ*_{1r} , *λ*_{2i}, *λ*_{2r}) are −1.78, −2.24, −8.3, −45.5 respectively, and the standard deviations of the system parameters are less than 0.03, 0.02, 0.1 and 0.8 respectively. The corresponding relative uncertainty (percent uncertainty) of the system parameters is less than 1.5%, 0.8%, 1.5%, 1.8% respectively, proving that the system parameters are constant immune to time. To verify that different input signals have no effect on the system parameters,
${\lambda}_{1i}^{0}=-1.78$,
${\lambda}_{1r}^{0}=-2.24$,
${\lambda}_{2i}^{0}=-8.3$,
${\lambda}_{2r}^{0}=-45.5$, obtained from the fitting of the response signal to step magnetic field *B*_{y0} = 0.32 nT, are applied to fit the responses of magnetic fields with various amplitudes. In Fig. 4(b), the corresponding R-square are all better than 0.99, indicating that the fitting curves are in good agreement with the measured signals. The goodness of fit of the signals with smaller input amplitudes are slightly poorer than those with larger input amplitudes, probably because their response amplitudes are relatively small that would be influenced by the fluctuation of the system more easily.

After determining the system parameters (
${\lambda}_{1r}^{0}$,
${\lambda}_{1i}^{0}$,
${\lambda}_{2r}^{0}$,
${\lambda}_{2i}^{0}$), the input magnetic field *B _{y}* can be acquired by fitting the response signal based on Eq. (2) and Eq. (3) with

*K*= 5.96 and ${K}_{1r}^{0}\hspace{0.17em}{\mathrm{\Omega}}_{y}=0.034$. The responses to input signals varying from 0.008 nT to 2.080 nT are measured. As shown in Fig. 5(a), the fitted

_{r}*B*with R-square superior to 0.99 is proportional to the input magnetic field, except for

_{y}*B*=1.920 nT and

_{y}*B*=2.080 nT whose fitted

_{y}*B*are smaller than the input ones respectively with their responses exceeding the measurement range of the lock-in amplifier. In Fig. 5(b), the corresponding residuals are smaller than the relevant input

_{y}*B*by at least two orders of magnitude, excluding the one of

_{y}*B*= 0.008 nT whose residual is only one order smaller than the relevant input

_{y}*B*. The exception of

_{y}*B*= 0.008 nT probably results from the response is relative small and affected by fluctuation of the signal significantly. Therefore, the measurement range of this method is from 0.016 nT to 1.760 nT, and the reverse input

_{y}*B*can also be fitted, and the measurement error is less than 0.01 nT. To further examine the transient response to small magnetic field, a high precision function generator (Agilent 33522A, minimum output 1 mV), which is attenuated by a 100 kΩ resistor and drives the Helmholtz coils directly (generating a minimum magnetic field about 1.6 pT), is utilized. The 0.5-second responses to small input magnetic fields (16 pT, 8 pT and 1.6 pT) are measured and fitted. As shown in Fig. 5(c), the measured signals are fitted with ${\lambda}_{1r}^{0}$, ${\lambda}_{1i}^{0}$, ${\lambda}_{2r}^{0}$, ${\lambda}_{2i}^{0}$,

_{y}*K*= 5.96 and ${K}_{1r}^{0}\hspace{0.17em}{\mathrm{\Omega}}_{y}=0.034$ based on Eq. (2) and Eq. (3), and the fitted curves are consistent with the measured signals with R-square superior to 0.9. The transient response model could still effectively describe the response to small magnetic field on the order of pT. The 0.5-second probe signal without input signal is denoted by black solid curve in Fig. 5(c), and the noise peak-to-peak value of this probe signal, which is primarily caused by magnetic field noise inside the magnetic shields, pumping light and probe light power density noises, operation temperature noise and electronic noise, is an order of magnitude smaller than the peak value of response to

_{r}*B*

_{y0}= 1.6 pT, hence the minimum measurement range could be potentially reduced to approach the noise level of probe signal without input signal. In addition, the transient signal model can also be applied to fit the short time transient signal to acquire the input signal as long as there is sufficient data to characterize the characteristic of the transient process. And the characteristic time of the transient process approximates to 0.45 second in this K-Rb-

^{21}Ne co-magnetometer, that means the required transient signal could be shorter than the 2-second signal in Fig. 5(a). The characteristic time of this K-Rb-

^{21}Ne co-magnetometer could be potentially reduced by optimizing the operation condition. And the time of transient process could be significant reduced by using K-

^{3}He co-magnetometer, whose characteristic time approximates to 0.05 second [19]. Therefore, the measurement time can be potentially diminished by reducing the characteristic time of co-magnetometer and fitting a short time transient signal.

Finally, the validity of this method in different operation temperature and pumping light power density is also studied. In Fig. 6(a) and 6(b), the pumping light power density changes from 58.2 mW cm^{−2}, 81.3 mW cm^{−2} to 97.3 mW cm^{−2}, while the temperature holds at 459 K. In Fig. 6(c) and 6(d), the temperature ranges within typical experimental conditions from 437 K, 448 K to 459 K with the pumping light power density fixed on 81.3 mW cm^{−2}. Under each of these conditions, the magnetic field zeroing procedure is executed to eliminate the residual magnetic field and find the new compensation point after the system reaching equilibrium state. The system parameters, determined by the fitting of the response signal to step magnetic field *B*_{y0} = 0.32nT under each of these conditions, are used to fit the responses of various input magnetic fields. As shown in Fig. 6, the responses to various input magnetic field are recorded and fitted with Eq. (2) and Eq. (3). The fitted curves are consistent with the measured signals because the corresponding goodness of fit are superior to 0.99 in Fig. 6(a) and (c). The reason why the goodness of fit of signals with smaller input magnetic field are slightly poor is identical to the one described above. In Fig. 6(b) and 6(d), the function of the black solid lines is *y* = *x*, and the lines are plotted to indicate the deviation of the fitted magnetic field and the input magnetic field. The fitted magnetic field values are all close to the solid lines, which indicate that the fitted results are approximate to the input signals. And for clarity, these fitted magnetic fields are fitted by *y* = *ax* + *b*. The slopes of the fitting lines for the results in Fig. 6(b) with pumping light power density 58.2 mW cm^{−2}, 81.3 mW cm^{−2} and 97.3 mW cm^{−2} are 1.048, 1.013 and 0.9677 respectively, and the corresponding R-square are all better than 0.99. The slopes of the fitting lines for the results in Fig. 6(d) with temperature 437 K, 448 K and 459 K are 1.093, 1.081 and 1.044 respectively, and the corresponding R-square are also all better than 0.99. Therefore, this method is still feasible in different typical temperature and pumping light power density.

## 5. Conclusion

In conclusion, we have characterized the dynamic performances of coupled spin ensembles in K-Rb-^{21}Ne co-magnetometer using Bloch equations. We have also studied the operation of the system as a self-compensation co-magnetometer that can be used to measure magnetic field in transient process, as well as the theoretical expression of the transient signal. The stability of this model to different input signals over time and the linear relationship between input signals and fitting results have been verified. Meanwhile, this method has been proved to be valid in a set of typical temperature and pumping light power density. The measurement range of this method is from 0.016 nT to 1.760 nT, and the measurement error is lower than 0.01 nT, which can be further possibly improved by reducing the noise of the magnetic field and the fluctuation of the pumping light power. This method can be utilized in small magnetic field measurement situation demanding high precision and miniaturization. The parameter *λ*_{1r}, acquired by fitting the transient signal to step transverse magnetic field, is a function of *δ* *B _{z}* and could be potentially utilized to zero

*δ*

*B*to find a better compensation condition for co-magnetometers. And it is capable of improving the sensitivity of inertial rotation measurement based on this co-magnetometer by accurately eliminating the disturbance of magnetic field. Moreover, this method could be possibly developed to measure magnetic field and inertial rotation simultaneously.

_{z}## Appendix

Here detailed solutions to the Bloch equations are provided. With small transverse excitation, the longitudinal polarization ${P}_{z}^{e}$ and ${P}_{z}^{n}$ nearly remain constant, therefore the resulting 4 × 4 system of Bloch equations for the transverse components ${P}_{x}^{e}$, ${P}_{y}^{e}$, ${P}_{x}^{n}$ and ${P}_{y}^{n}$, becomes linear [11, 13]. Utilizing ${\tilde{P}}^{e}={P}_{x}^{e}+i{P}_{y}^{e}$ and ${\tilde{P}}^{n}={P}_{x}^{n}+i{P}_{y}^{n}$, the 4 × 4 system of equations can be written in the 2 × 2 system of equations:

The standard methods of solving differential equations can be used, and the formal solution is given by:

*P*

_{1}=

*P*

_{1r}+

*i P*

_{1i},

*P*

_{2}=

*P*

_{2r}+

*i P*

_{2i},

*λ*

_{1}=

*λ*

_{1r}+

*i*

*λ*

_{1i}and

*λ*

_{2}=

*λ*

_{2r}+

*i*

*λ*

_{2i}.

For the experimentally realized K-Rb-^{21}Ne co-magnetometer,
${R}_{\mathit{se}}^{\mathit{en}}\ll {\gamma}_{n}\hspace{0.17em}\lambda \hspace{0.17em}{M}^{e}\hspace{0.17em}{P}_{z}^{n}$,
${R}_{\mathit{se}}^{\mathit{ne}}\ll {\gamma}_{e}\hspace{0.17em}\lambda \hspace{0.17em}{M}^{n}\hspace{0.17em}{P}_{z}^{e}$ and
${\gamma}_{e}\left({B}_{z}+\lambda {M}^{n}+{P}_{z}^{n}+{L}_{z}\right)/Q\gg {\mathrm{\Omega}}_{z}$, hence
${R}_{\mathit{se}}^{\mathit{en}}$,
${R}_{\mathit{se}}^{\mathit{ne}}$ and Ω* _{z}* are set to zero. Since
${R}_{\mathit{tot}}^{e}/Q\gg {R}_{\mathit{tot}}^{n}$ and
${B}_{z}=-\lambda \hspace{0.17em}{M}^{n}\hspace{0.17em}{P}_{z}^{n}-\lambda {M}^{e}\hspace{0.17em}{P}_{z}^{n}$ at compensation point, the

*λ*

_{1}and

*λ*

_{2}are represented by:

The general form of output signal *S _{x}*, which is proportional to the transverse polarization
${P}_{x}^{e}$ including transient and steady components, is given by:

*K*is the factor that converts the transverse polarization of Rb spin ensemble to output voltage signal by the probe system ( ${S}_{y}={K}_{d}{P}_{y}^{e}$);

_{d}*λ*

_{1r},

*λ*

_{1i},

*λ*

_{2r}and

*λ*

_{2i}are determined by the system itself;

*P*

_{1r},

*P*

_{1i},

*P*

_{2r},

*P*

_{2i}and ${S}_{x}^{\mathit{steady}}$ depend on the input signals.

At the compensation point, the parameters *P*_{1r}, *P*_{1i}, *P*_{2r} , *P*_{2i} and
${S}_{x}^{\mathit{steady}}$ can be specifically rewritten with different types of change in transverse magnetic field, for example step-like change and sinusoidal change. When the change is step-like with amplitude *B*_{y0} (here we primarily consider the change of *B _{y}* and ignore the changes of

*B*, Ω

_{x}*and Ω*

_{x}*for the benefit of simplicity) and the horizontal component of earth rotation Ω*

_{y}*is along the y-axis, these parameters can be specifically rewritten.*

_{y}By setting *∂ P̃ ^{e}*

*/∂t*= 0 and

*∂ P̃*

^{n}*/∂t*= 0, the steady signal is:

*δ*

*B*=

_{z}*B*−

_{z}*B*.

_{c}And *P*_{1r} , *P*_{1i}, *P*_{2r} , *P*_{2i} can be found by substituting the initial values (
${S}_{x}(0)={S}_{x}^{0}$,
${S}_{y}(0)={S}_{y}^{0}$,
${\frac{\partial {S}_{x}}{\partial t}|}_{t=0}=\left(\frac{{\gamma}_{e}{B}_{y0}}{Q}-{\mathrm{\Omega}}_{y}\right){K}_{d}{P}_{z}^{e}$ and
${\frac{\partial {S}_{y}}{\partial t}|}_{t=0}=0$), and are given by:

*K*and

_{r}*K*.

_{i}## Funding

National Natural Science Foundation of China (NSFC) (61227902, 61374210); National Key R&D Program of China (2016YFB0501601).

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