## Abstract

The magnetic field response of a ^{87}Rb-^{129}Xe atomic spin comagnetometer operated in the spin-exchange relaxation-free (SERF) regime was investigated. The response model of the comagnetometer to the transverse magnetic fields along the *y*-axis and the *x*-axis considering the couple of electron spin and nuclear spin is presented. In the experiment, a high hybrid resonance peak near low-frequency was observed. By fitting the hybrid resonance with the presented response model, a pair of poles at 0.62 Hz and 1.8 Hz were obtained which correspond to the nuclear spin resonance and the electron spin resonance, respectively. The magnetic field response characteristic of the ^{87}Rb-^{129}Xe comagnetometer with different nuclear magnetic fields and electronic magnetic fields was simulated and analyzed. The simulation results indicate that the hybrid resonance frequency can be right-shifted by the larger nuclear magnetic field while the magnetic field suppression factor can be decreased by the larger electron magnetic field. This study is helpful to improve the performance of the atomic spin gyroscope based on the ^{87}Rb-^{129}Xe comagnetometer.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The alkali-metal noble-gas comagnetometer based on nuclear spin and electron spin can be used for inertial measurement as a gyroscope to measure the rotation rate [1–4]. Working in the spin-exchange relaxation-free (SERF) regime, the comagnetometer has opened additional possibilities for ultrasensitive measurement. In 2005, the nuclear spin gyroscope based on the K-^{3}He comagnetometer was first demonstrated by the Romalis group, and a rotation sensitivity of 5×10^{−7} rad/s/Hz^{1/2} has been reached [2]. Then the atomic spin gyroscopes based on the comagnetometer developed rapidly. Besides K-^{3}He [5,6], there are several choices of the spin species which have been demonstrated to be used as a gyroscope, such as Cs-^{129}Xe [7], K-Rb-^{21}Ne [8–10] and ^{87}Rb-^{129}Xe [11], etc.. Though the atomic spin gyroscope based on K-Rb-^{21}Ne comagnetometer has higher sensitivity than other spin specie combinations due to the small gyromagnetic ratio of ^{21}Ne, the polarization of ^{21}Ne atoms is quite difficult and time-consuming. However, the atomic spin gyroscope based on ^{87}Rb-^{129}Xe or Cs-^{129}Xe has characteristics of short start-up since ^{129}Xe atoms are easily optically pumped compared with ^{3}He atoms and ^{21}Ne atoms.

The response of the comagnetometer to magnetic field is a major source of noise in the atomic spin gyroscope, and it affects the performance of the gyroscope such as rotation sensitivity and long-term stability [12,13]. The magnetic field response of the K-Rb-^{21}Ne comagnetometer has been studied in deep. Due to the large electron magnetic field in the K-Rb-^{21}Ne comagnetometer, an obvious electron spin resonance about 188 Hz was found that it is far away from the nuclear spin resonance [14,15]. By locking the electron resonance of the K-Rb-^{21}Ne comagnetometer, the closed-loop control of the magnetic field compensation point was realized [15]. To study the ability of the magnetic-noise suppression, the magnetic field suppression factor of the K-Rb-^{21}Ne comagnetometer was defined as the ratio of the magnetic field response amplitude to the scale factor of rotation response [16]. However, the magnetic field response of the ^{87}Rb-^{129}Xe comagnetometer is quite different from that of the K-Rb-^{21}Ne comagnetometer, which is also ought to be studied thoroughly.

In this study, we systematically investigated the magnetic field response of the ^{87}Rb-^{129}Xe comagnetometer to the transverse oscillating magnetic field along the y-axis and y-axis. In Section 2, we present the magnetic field response model of the comagnetometer based on the coupled Bloch equations for the electron and nuclear spin polarization. In Section 3, the experiment setup of the ^{87}Rb-^{129}Xe comagnetometer was built and the experimental test was performed. In Section 4, the tested data were fitted and analyzed, and the simulation of magnetic field response with different nuclear magnetic fields and electron magnetic fields was performed. This study contributes to the improvement of SERF gyroscope performance based on the ^{87}Rb-^{129}Xe comagnetometer.

## 2. Magnetic field response model

The alkali-metal noble-gas comagnetometer utilizes the electron spin and nuclear spin. The coupled Bloch equations for the electron and nuclear spin polarization, **P ^{e}** and

**P**, can be written as [2],

^{n}*γ*and

_{e}*γ*are the gyromagnetic ration of electron spin and nuclear spin, respectively. ${R}_{tot}^{e} = R_{p}+R_{m}+R_{sd}+R_{se}^{en}$ is the total relaxation rate of electron spin, where

_{n}*R*and

_{p}*R*are the pumping rate from the pump beam and probe beam,

_{m}*R*is the electron spin-destruction rate, and ${R}_{se}^{en}$ is the spin-exchange rate experienced by the electron spin. ${R}_{tot}^{n} = R_{se}^{ne}+R_{sd}^{n}$ is the total relaxation rate of nuclear spin, where ${R}_{se}^{ne}$ is the spin-exchange rate experienced by the nuclear spin, ${R}_{sd}^{n}$ is the nuclear spin-destruction rate.

_{sd}**B**

*=*

^{e}*λM*

^{e}**P**and

^{e}**B**

^{n}=λM^{n}**P**are the effective magnetic fields experienced by electron spin and nuclear spin due to the polarization of the other species, where

^{n}*M*and

^{e}*M*are the magnetizations of electron spin and nuclear spin.

^{n}*q*is the slowing down factor of the electron spin which is related to the electron spin polarization.

**and**

*s*_{p}**are the direction vector of pump light and probe light, respectively.**

*s*_{m}**B**is the ambient magnetic field,

**L**is the light shift experienced by electron spin, and

**Ω**is the rotation vector applied to the system.

To small transverse excitations, the longitudinal components of electron and nuclear spin polarization ${P}_{z}^{n}$ and ${P}_{z}^{e}$ can be regarded as two constants, which can be approximatively written as ${P}_{z}^{n}=R_{se}^{ne}/R_{tot}^{n},{P}_{z}^{e}=(R_{p}+{R}_{se}^{en} P_{z}^{n})/{R}_{tot}^{e}\approx{R}_{p}/{R}_{tot}^{e}$ [17]. To simplify Eq. (1), the rest four transverse polarization components can be noted by a matrix,

where ${P}_{x}^{e}$ and ${P}_{y}^{e}$ are the transverse components of electron spin polarization along the*x*-axis and the

*y*-axis, respectively. ${P}_{x}^{n}$ and ${P}_{y}^{n}$ are the transverse components of nuclear spin polarization along the

*x*-axis and the

*y*-axis, respectively.

Then the coupling equations described by Eq. (1) can be rewritten as the matrix form as follows,

The matrix**M**can be written as,

**N**can be written as,

**L**and pump rate of probe light

*R*are neglected since they can be set to zero by adjusting pump beam and probe beam [2], $\tilde{R}_{tot}^{e} = R_{tot}^{e}/q$, $\tilde{R}_{se}^{en} = R_{se}^{en}/q$, $\omega_{en}={\gamma}_{e} \lambda M^{n}P_{z}^{e}/q$, $\omega_{ne}={\gamma}_{n} \lambda M^{e}P_{z}^{n}$, $\omega_{e}={\gamma}_{e} (B_{z}+\lambda M^{n}P_{z}^{n})/q$, $\omega_{n}={\gamma}_{n} (B_{z}+\lambda M^{e}P_{z}^{e})$, $\tilde{b}_y^e = P_z^e({{\gamma_e}{B_y}/q - {\Omega _y}} )$, $\tilde{b}_x^e = P_z^e({{\gamma_e}{B_x}/q - {\Omega _x}} )$, ${b}_{y}^{n} = P_{z}^{n}({{\gamma_e}{B_y} - {\Omega _y}} )$, ${b}_x^n = P_z^n({{\gamma_n}{B_x} - {\Omega _x}} )$, ${\tilde{\gamma }_e} = {\gamma _e}/q$. And $B_c= -\lambda M^e P_z^e - \lambda M^n P_z^n$ is the compensation magnetic field which is applied to compensate the magnetic field produced by electron spin and nuclear spin along the z-axis [12]. When a transverse oscillating magnetic field ${\bf B} = ({{B_{0x}}\hat{x} + {B_{0y}}\hat{y}} )\cos ({\omega t} )$is applied where

_{m}*B*

_{0x}and

*B*

_{0y}are the amplitudes of the magnetic field along

*x*-axis and

*y*-axis, and

*ω*is the angular frequency of the magnetic field, the transverse electron polarization

*Pe x*can be obtained by solving Eq. (3).

*φ*

_{1}and

*φ*

_{2}are the phases which can be written as

*x*-axis and the

*y*-axis are also two oscillating functions with different amplitudes and phases.

## 3. Experiment setup and results

The experimental setup of the ^{87}Rb-^{129}Xe comagnetometer is shown in Fig. 1. In the center of the setup is an 8 mm ×8 mm× 8 mm cubic cell. There are a few drops of ^{87}Rb, 20 torr of ^{129}Xe and 700 torr of N_{2} as the quenching gas filled in the vapor cell. The cell is heated to 160 °C by a heating plate with a 100 kHz AC current, where the Rb vapor atoms fulfill the whole cell. There is a set of three-dimensional coils displaced out of the vapor cell, which is used to compensate the ambient magnetic field and the magnetic field produced by nuclear spin and electron spin. The three-dimensional coils are driven by two high precision function generators (Keysight 33500B) with two channels. The cell and the coil are placed into a four layers magnetic shield, which is used to screen the influence of the geomagnetic field.

A linearly polarized light emitting from a distributed Bragg reflector (DBR) laser diode is used to pump the alkali-metal atoms along the *z*-axis. The pump laser is protected by a Faraday isolator and the power of the laser is stabilized by a half-wave plate (λ/2) and a noise eater (Thorlabs, NEM03L). Before linearly polarized light propagates through the vapor cell, it is transformed into a circularly polarized light by a quarter-wave plate (λ/4), which is used to polarize the Rb atoms. The pump power is chosen as 40 mW, and the wavelength is set on the center of the ^{87}Rb D1 resonance line. The transverse polarization component ${P}_{x}^{e}$ is detected a probe laser with 3 mW and tuned away 0.2 nm of the ^{87}Rb D1 resonance line. Similar to the pump laser, the probe laser is also protected by a Faraday isolator and stabilized by a half-wave plate and a noise eater. The output signal is detected by a differential detection module which contains a polarization balance splitter (PBS) and two photodiodes (PD1 and PD2). Another half-wave plate is placed before the differential detection module to adjust the polarization plane of the linearly polarized probe light.

It takes hours to heat the cell to 160 °C and realize the polarization of atoms. In order to promote the polarization of ^{129}Xe, a DC magnetic field is applied along the *z*-axis. After the polarization reaches the equilibrium state, the compensation magnetic field is carried out. The total Rb electron relaxation rate $R_{tot}^{e}$ is measured by changing the magnetic field along *z-*axis *B _{z}* according to the response of the comagnetometer to square wave modulation magnetic field

*B*. The tested result is shown in Fig. 2. The total electron relaxation rate can be obtained by fitting the tested data to the function described in Ref. [18],

_{y}*B*and

_{y}*δB*=

_{z}*B*is the total residual field along

_{z}-B_{c}*z*-axis where

*B*is the compensation magnetic field.

_{c}*L*is the light shift along the z-axis. The total electron relaxation rate $R_{tot}^{e}$ is 2813 s

_{z}^{-1}which is highly suppressed from the spin-exchange rate $T_{SE}^{-1}$ of 3.9×10

^{5}s

^{-1}, so SERF regime is realized. The spin-exchange rate $T_{SE}^{-1}$ can be obtained by $T_{SE}^{-1}=nv \sigma_{SE}$ [19] where

*n*is alkali-metal atom density,

*v*is the thermal velocity,

*σ*is the spin-exchange cross-section which is 1.9×10

_{SE}^{−14}cm

^{2}for

^{87}Rb atoms [20]. The thermal velocity

*v*can be obtained by

*v=*(8

*RT*/π

*M*)

_{mol}^{1/2}where

*R*=8.31 (J mol

^{-1}K

^{-1}) is the gas constant,

*T*is the temperature in Kelvin,

*M*is molar mass of

_{mol}^{87}Rb atom. And the typical values of the three parameters mentioned above are listed in Table 1.

By adjusting the comagnetometer into proper condition, it can be used to measure the rotation rate of the system. Here, we focus on the magnetic field response of the comagnetometer when the comagnetometer works at the magnetic field compensation point, especially the transverse oscillating magnetic field. A series of magnetic fields with different frequencies and the same amplitude of 1.0 nT are applied to the *y*-axis and x-axis, respectively. The amplitude-frequency response of the comagnetometer is depicted in Fig. 3. It can be seen there is a high resonance peak both in y and x axis in the tested frequency ranging from 0.05 Hz to 100 Hz. Then, the input frequency is extended to 1000 Hz, and there is no other resonance peak found in the frequency spectrum. Because the resonance is near low-frequency, it is obviously thought to be a nuclear resonance. However, the actual situation is more complex, which is discussed in Section 4.

## 4. Discussions

By fitting the tested data with Eq. (6) to Fig. 3(a), it is found that the resonance is consists of the two poles, the nuclear resonance and the electron resonance, respectively. Therefore, the resonance is called the hybrid resonance. For the typical electron spin polarization 50%, the slowing factor should be *q*≈5.2. The two resonances frequencies of electron spin and nuclear spin *ω*_{0}*=γ _{n}B^{n}* and

*ω*

_{e}_{0}

*=γ*/

_{e}B^{e}*q*are 0.62 Hz and 1.8 Hz, which corresponds to

*B*53 nT and

^{n}=*B*0.33 nT, respectively. $\tilde{R}_{tot}^e = R_{tot}^e/q$ is about 14 Hz, which means the relaxation rate $R_{tot}^{e}$ is 2461 s

^{e}=^{-1}. It is a little lower than the value obtained from the data shown in Fig. 2 by fitting with Eq. (9). The discrepancy between the two values is due to the different treatments of light shift

*L*. Then the same kind of oscillating magnetic field is implemented on the

_{z}*x*-axis, and the tested data and the fitting curve is shown in Fig. 3(b). The fitting results are almost equal to the mentioned above, which confirms the reliability of the results. As shown in Figs. 3(a) and (b), the fitting results on the two sides of the hybrid resonance peak is some bad. It can be contributed to the approximation conditions of the presented response model, such as the constant approximations of $P_{z}^{n}$ and $P_{z}^{e}$, and the ignorance of the light shift

**L**and pump rate

*R*. In the comagnetometer, the pump laser and the probe light are along the z-axis and x-axis, respectively. Based on Eq. (6), even though the response to

_{m}*B*and

_{x}*B*have the same pole structure in the dominator

_{y}*K*, the nominators are different. Therefore, the spin polarization projections on the x-axis induced by Bx and By are different, which lead to different line shape for the response to

_{1}K_{2}*B*and

_{x}*B*. As shown in Fig. 3, the hybrid resonance peak for

_{y}*B*response is not obvious and that for

_{y}*B*response cannot be found.

_{x}Compared with the resonance peak of the K-^{3}He comagnetometer reported in Ref. [1], the resonance peak of the ^{87}Rb-^{129}Xe comagnetometer reported in this paper is much closer to low-frequency. This can be contributed to the low ^{129}Xe spin precession frequency *ω*_{0}*=γ _{n}B^{n}*, because the gyromagnetic ratio

*γ*of

_{n}^{129}Xe is about three times smaller than that of

^{3}He, and

*B*53 nT is a little lower than that of

^{n}=^{3}He. Different to the nuclear magnetic resonance gyroscope (NMRG) where the electron spin and nuclear spin are decoupled and resonances are separated far away from each other, the spin species in the comagnetometer strongly couple together and the shape of the coupling is different from normal resonance. As shown in Fig. 3, the right edge of the resonance does not drop immediately but extends to about 10 Hz since the two resonances have different widths. As a gyroscope, all the magnetic noises should be reduced. The magnetic noise near the hybrid resonant frequency (0.8 Hz) should be especially taken care of, which would lead to much larger magnetic field noise than that of the other frequencies.

According to Eq. (6) in Section 2, *B ^{n}* and

*B*are two main factors affecting the hybrid resonance peak. To investigate the influences of

^{e}*B*and

^{n}*B*to amplitude-frequency response, we changed

^{e}*B*or

^{n}*B*in the normal range and simulate the amplitude-frequency response when the other was kept unchanged. The unchanged parameters in the simulation were obtained by fitting the tested data with Eq. (6) and the frequency ranges from 0.02 Hz to 1000 Hz. The simulation results of magnetic field response along the y-axis with the different nuclear magnetic fields are shown in Fig. 4(a). The blue curve is corresponding to the fitting curve in Fig. 3(a). It was observed that the shape and the amplitude of the hybrid resonance peaks for

^{e}*B*responses did not change when nuclear magnetic field changes from 13 nT to 93 nT, but the resonant frequency is shifted by the nuclear magnetic field. As shown in Fig. 4(b), the nuclear resonant frequency is right-shifted with the increase the nuclear magnetic field, and the hybrid resonance peaks for

_{y}*B*responses is still not found. Therefore, the magnetic field noise in near-low frequency can be reduced by the right-shift of resonant frequency. This indicates that B

_{x}^{n}has a little effect on hybrid resonance peak because the nuclear spin precession frequencies

*ω*

_{0}=

*γ*changes littile dut to the small gyromagnetic ratio

_{n}B^{n}*γ*

_{n}_{.}

The simulation results of magnetic field response along the y-axis and the x-axis with various electron magnetic field are shown in Fig. 5. It was observed that the linewidth of the hybrid resonance peak is broadened with the increase of the electron magnetic field. The electron resonance and nuclear resonance separate with each other and the large electron resonance peak appears when the electron magnetic field reaches about 10 nT. Though the simulation result with large *B ^{e}* is different from the experimental result, it matches with the report of K-Rb-

^{21}Ne comagnetometer with the large electron magnetic field [14]. This also proves the correctness of the simulation. The simulation results indicate the

*B*has a large effect on hybrid resonance peak because the change of the electron spin precession frequencies

^{e}*ω*

_{e}_{0}

*=γ*/

_{e}B^{e}*q*is large with the electron gyromagnetic ratio

*γ*of 28 Hz/nT when

_{e}*B*range from to 0.2 nT to 10 nT. With the change of the Be, the hybrid resonance peaks are obvious both in the response to By and Bx. There are some differences between the response to By and Bx as shown in Figs. 5(a) and (b) due to the different nominators in Eq. (6) of the two components.

^{e}The electron magnetic field *B _{z}^{e}* is proportional to the polarization of electron

*P*[22], that is

_{z}^{e}*B*=

_{z}^{e}*k*

_{1}

*P*where

_{z}^{e}*k*

_{1}is the conversion coefficient between

*B*and

_{z}^{e}*P*, and the polarization of electron

_{z}^{e}*P*, is proportional to the scale of gyroscope rotation

_{z}^{e}*K*[18], that is

_{scal}*K*=

_{scal}*k*

_{2}

*P*where

_{z}^{e}*k*

_{2}is the related coefficient. Therefore, the change of electron magnetic field will cause the change of gyroscope scale factor. In order to better evaluate the effect of the electronic magnetic field on gyroscope performance, the magnetic field suppression factor is used which is defined as the ratio of magnetic field response to the gyroscope scale [16]. The simulations of the suppression factor along the

*y*-axis and

*x*-axis were conducted. As shown in Fig. 6, the suppression factor obviously decreases with the increase of the electron magnetic field, especially near the hybrid resonance frequency. This suggested that we can reduce magnetic field noise by increasing the electron magnetic field. There is no direct correlation between the nuclear magnetic field and the gyroscope scale, so the magnetic field suppression factor is consistent with the magnetic field response when the nuclear magnetic field is changed, as shown in Fig. 4.

## 5. Conclusion

In conclusion, the magnetic field response of the ^{87}Rb-^{129}Xe comagnetometer at the magnetic field compensation point was theoretically and experimentally investigated. By testing the amplitude-frequency response of the ^{87}Rb-^{129}Xe comagnetometer to the transverse oscillating magnetic field, one high hybrid resonance peak in near-low frequency was found in the frequency range extending to 1000 Hz. By fitting the resonance with the presented model, a pair of poles at 0.62 Hz and 1.8 Hz are obtained which correspond to the nuclear and electron spin magnetic field of 53 nT and 0.33 nT, respectively. In order to further investigate the response characteristics of the comagnetometer, the magnetic field response of the comagnetometer with different nuclear magnetic fields and electron magnetic fields was simulated with the presented model. The simulation results indicate that the hybrid resonance frequency can be right-shifted by increasing the nuclear magnetic field and the suppression factor can be extremely decreased by the larger electron magnetic field. Therefore, the increase of both the nuclear magnetic field and the electron magnetic field can suppress the response of the ^{87}Rb-^{129}Xe comagnetometer to magnetic field. This work provides a theoretical and experimental basis for the research of the ^{87}Rb-^{129}Xe comagnetometer to magnetic field response and has great significance in the performance enhancement of gyroscope based the ^{87}Rb-^{129}Xe comagnetometer.

## Funding

National Key Research and Development Program of China (2016YFB0501601); Youth Fund Project of National Natural Science Foundation of China (61601017).

## Acknowledgments

The authors would like to thank Professor Haifeng Dong and Dr. Tianshun Wang for useful discussions.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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