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Analysis on the effect of electron spin polarization on a hybrid optically pumped K-Rb-21Ne co-magnetometer

Yang Fu, Wenfeng Fan, Jiasen Ruan, Ye Liu, Zhuo Wang, Xinxiu Zhou, and Wei Quan

Open Access Open Access

Abstract

In this paper, the effect of longitudinal electron spin polarization under the combined action of alkali metal density ratio and pump laser power density on the hybrid optically pumped co-magnetometer operated in the spin-exchange relaxation-free (SERF) regime is studied. The AC response model of rotation velocity and magnetic noise of the SERF co-magnetometer system is proposed, and the factors of frequency and system bandwidth are considered. Based on the proposed response model, the error equation of the system is obtained, and the relationship between alkali metal density ratio and pump laser power density and the system noise response is theoretically analyzed and experimentally tested. The results show that when the product of pumping rate and alkali metal density ratio is greater than the electron spin relaxation rate, there is a longitudinal electron spin polarization point that minimizes the system error. In addition, the range of minimum error calculated results obtained by changing the pumping rate for the cells with different alkali metal density ratios is within 5% of the average value, that is, their minimum error potential is roughly the same within a certain range. Under the experimental conditions in this paper, due to the limitation of the electron spin relaxation rate and the operating capacity of the pump laser, the optimal alkali metal density ratio range is about 1/100-1/300.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

SERF co-magnetometer has been widely applied in tests of Lorentz and CPT violation and searches for anomalous spin forces [1,2]. Furthermore, the coupling of alkali metal and noble gas in the SERF regime makes the co-magnetometer extremely sensitive to rotation and be a promising rotation sensor for the next generation of inertial navigation applications [3,4].

Romalis Group first realized the SERF co-magnetometer based on K-$^3$He in 2005 [3]. Subsequently, the research of SERF co-magnetometer based on Cs-$^{129}$Xe and Rb-$^{129}$Xe was carried out successively [5,6]. Through further investigation, it can be seen that the theoretical sensitivity of the SERF co-magnetometer is inversely proportional to the gyromagnetic ratios of nuclear spins, and the gyromagnetic ratio of $^{21}$Ne is nearly an order of magnitude smaller than that of $^{3}$He, so the SERF co-magnetometer based on $^{21}$Ne has higher sensitivity potential [7]. In order to realize more uniform polarization of alkali metal and more efficient pumping of noble gas, the hybrid optical pumping technique of alkali metal is studied [8–10]. That is, the K atmoic spins with thin optical depth directly polarized by pumped laser is used to polarize the Rb atomic spins with thick optical depth. Based on the above research results, the research of hybrid optically pumped K-Rb-$^{21}$Ne co-magnetometer is carried out [11–13]. However, the present performance of the SERF co-magnetometer is far from the theoretical accuracy [14,15], so further accurate research on the error model is needed.

On the one hand, the current output and error model of the SERF co-magnetometer is mainly based on the steady-state solution of the Bloch equation, without considering the factors of frequency responses [16,17]. The AC response of the SERF co-magnetometer to the magnetic field is related to the frequency in the low-frequency band [3,18,19]. The traditional model takes the steady-state response of the co-magnetometer to the magnetic field as the magnetic noise response coefficient, which is inaccurate. On the other hand, the longitudinal electron spin polarization of the hybrid optical pumped co-magnetometer is affected by the alkali metal density ratio and the pumping rate [20–22], and the action law needs to be further studied. According to the current research results, when the pumping rate is equal to the electron spin relaxation rate, the scale factor of the SERF co-magnetometer is the largest, so this point is regarded as the optimal point for pumping rate [23,24]. However, this conclusion does not consider the influence of magnetic noise on atomic ensemble.

In this paper, the effect of longitudinal electron spin polarization on the hybrid optically pumped K-Rb-$^{21}$Ne co-magnetometer under the combined action of alkali metal density ratio and pump laser power density is studied. This paper is organized as follows. Section 2 introduces the K-Rb-$^{21}$Ne co-magnetometer experimental setup built in this paper. In Sec. 3, the AC response model of the SERF co-magnetometer is established, and the output equation of the system is obtained. In Sec. 4, further theoretical analysis and experimental verification for the noise response of the system are made. Firstly, the accuracy of the established response model is verified by experiments. Through expansion and simplification, the error equation and expression of the response coefficient to various noises are obtained. Then, through the analysis and test of the noise response of the cells with different alkali metal density ratios under different pump laser power densities, the action law of longitudinal electron spin polarization on the SERF co-magnetometer and the applicable range of alkali metal density ratio is obtained.

2. Experimental setups

The experiment is performed in a K-Rb-$^{21}$Ne co-magnetometer, whose schematic is depicted in Fig. 1. The sensitive element of the co-magnetometer is a 10-mm-diameter spherical cell made from GE180 aluminosilicate glass with a mixture of K and Rb alkali metals in natural abundance, 2 atm of $^{21}$Ne (70$\%$ isotope enriched), and 33 Torr N$_2$ in it. To start the co-magnetometer, the cell is heated to a stable temperature of 185$^{\circ }$C by AC current-driven heating coils attached to the boron nitride oven. After several hours’ of polarizing by the pumping beam, which is a circularly polarized laser emitted by a distributed Bragg reflector (DBR) diode laser and expanded to a diameter of 8.5 mm, the polarization of the atomic ensembles reaches the equilibrium. The triaxial field coils can be used to compensate for the residual magnetic field in the shields and apply the excitation magnetic field to measure the magnetic field response of the system.

 figure: Fig. 1.

Fig. 1. Schematic of the SERF co-magnetometer. BSL: beam shaping lenses, M: reflection mirror, LCVR: liquid crystal variable retarder, PD: photodiode, ECU: electronic control unit, GT: Glan-Taylor polarizer, PD: photodiode, PBS: polarizing beam splitter, P: linear polarizer.

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The co-magnetometer is placed on a rotating platform with an accuracy of 0.001 $^\circ$/s, which can provide sinusoidal rotation in the frequency range of 0-5 Hz. The y-axis of the co-magnetometer is vertical and aligned with the rotating axis of the platform. When the rotation velocity vector ${\Omega _y}$ is sensed, the polarization vectors of electron spins and nuclear spins deviate slightly from the z-axis, generating a polarization component in x-direction. The probe laser in the x-direction, which is a linearly polarized laser with a diameter of 1.5 mm emitted by a distributed feedback (DFB) diode laser, is used to measure $P_{\rm {x}}^e$.

As a rotation sensor, the SERF co-magnetometer has an ultra-high theoretical bias stability of 10$^{-7}$ $^\circ$/h [5,25]. However, there is still a big gap between the actual accuracy and the theoretical level. Through the error analysis of the optical path system, temperature control system and the magnetic shielding system, it is concluded that the errors introduced by the magnetic noise and the probe background noise limit the further improvement of the accuracy of the co-magnetometer. In order to improve the precision of the co-magnetometer, the modeling analysis and experimental research of the system are carried out in this paper.

3. Theoretical analysis

3.1 Principle of the K-Rb-$^{21}$Ne co-magnetometer

The hybrid spin-exchange optical pumping technology is based on the K-Rb two kinds of alkali metal atoms. The time evolution of the coupled spin ensembles can be solved analytically with a complete set of Bloch equations [12,26]. The electron spin polarization ${{\bf {P}}^K}$ and ${{\bf {P}}^{Rb}}$, as well as nuclear spin polarization ${{\bf {P}}^{\bf {n}}}$ can be expressed as:

$$\scalebox{0.88}{$\begin{aligned} \frac{{\partial {{\bf{P}}^K}}}{{\partial t}} &= \frac{{{\gamma _e}}}{{Q\left( {{P^K}} \right)}}({\bf{B}} + {\lambda _{Kn}}{{\bf{M}}^{\bf{n}}}{{\bf{P}}^{\bf{n}}} + {{\bf{L}}^K}) \times {{\bf{P}}^K} - {\bf{\Omega }} \times {{\bf{P}}^K} + \frac{1}{{Q\left( {{P^K}} \right)}}(R_p^K{{\bf{s}}_{\bf{p}}} + R_{se}^{KRb}{{\bf{P}}^{Rb}} + R_{se}^{Kn}{{\bf{P}}^{\bf{n}}} - R_{tot}^K{{\bf{P}}^K}),\\ \frac{{\partial {{\bf{P}}^{Rb}}}}{{\partial t}} &= \frac{{{\gamma _e}}}{{Q\left( {{P^{Rb}}} \right)}}({\bf{B}} + {\lambda _{Rbn}}{{\bf{M}}^{\bf{n}}}{{\bf{P}}^{\bf{n}}} + {{\bf{L}}^{Rb}}) \times {{\bf{P}}^{Rb}} - {\bf{\Omega }} \times {{\bf{P}}^{Rb}} + \frac{1}{{Q\left( {{P^{Rb}}} \right)}}(R_{se}^{RbK}{{\bf{P}}^K} + R_m^{Rb}{{\bf{s}}_{\bf{m}}} + R_{se}^{Rbn}{{\bf{P}}^{\bf{n}}} - R_{tot}^{Rb}{{\bf{P}}^{Rb}}),\\ \frac{{\partial {{\bf{P}}^n}}}{{\partial t}}& = {\gamma _n}({\bf{B}} + {\lambda _{Kn}}{{\bf{M}}^K}{{\bf{P}}^K} + {\lambda _{Rbn}}{{\bf{M}}^{Rb}}{{\bf{P}}^{Rb}}) \times {{\bf{P}}^{\bf{n}}} - \Omega \times {{\bf{P}}^{\bf{n}}} + R_{se}^{nK}{{\bf{P}}^K} + R_{se}^{nRb}{{\bf{P}}^{Rb}} - R_{tot}^n{{\bf{P}}^{\bf{n}}}. \end{aligned}$}$$

Here, ${\gamma _{\rm {e}}}$ and ${\gamma _{\rm {n}}}$ are the gyromagnetic ratios of the electron and nuclear spins, respectively. $Q\left ( {{P^K}} \right )$ and $Q\left ( {{P^{Rb}}} \right )$ are the electron slowing down factors. ${\bf {B}}$ is the ambient magnetic field vector, while ${{\bf {L}}^K}$ and ${{\bf {L}}^{Rb}}$ are the fictitious magnetic fields of AC-Stark shifts. ${\bf {\Omega }}$ is the input rotation velocity vector. ${\lambda _{Kn}}{{\bf {M}}^K}$ and ${\lambda _{Rbn}}{{\bf {M}}^{Rb}}$ are the electron spin magnetic moment of K and Rb alkali metal atoms. ${\lambda _{Kn}}{{\bf {M}}^{\bf {n}}}$ and ${\lambda _{Rbn}}{{\bf {M}}^{\bf {n}}}$ are the nuclear spin magnetic moment of noble gas atoms. Here ${\lambda _{Kn}} = {{8\pi {\kappa _{Kn}}} \mathord {\left /{\vphantom {{8\pi {\kappa _{Kn}}} 3}} \right. } 3}$, ${\lambda _{Rbn}} = {{8\pi {\kappa _{Rbn}}} \mathord {\left /{\vphantom {{8\pi {\kappa _{Rbn}}} 3}} \right. } 3}$, ${\kappa _{Kn}}$ and ${\kappa _{Rbn}}$ is the Fermi constant. ${{\bf {M}}^K}$, ${{\bf {M}}^{Rb}}$ and ${{\bf {M}}^{\bf {n}}}$ are the magnetizations of K and Rb electron spins and nuclear spins corresponding to full spin polarization, respectively. $R_{se}^{Kn}$ and $R_{se}^{Rbn}$ are the alkali-metal–noble-gas spin-exchange rates for the K atomic electron spins and Rb atomic electron spins. $R_{se}^{nK}$ and $R_{se}^{nRb}$are the same rates for the $^{21}$Ne nuclear spins. ${{\bf {s}}_{\bf {p}}}$ and ${{\bf {s}}_{\bf {m}}}$ are the photon spin vectors of pump laser and probe laser respectively. Since the pump beam and the probe beam are circularly polarized in z direction and linearly polarized in x direction, ${{\bf {s}}_{\bf {p}}}$ and ${{\bf {s}}_{\bf {m}}}$ can be approximately $[0,0,1]^{\rm {T}}$ and $[0,0,0]^{\rm {T}}$ respectively. $R_{tot}^K$, $R_{tot}^{Rb}$ and $R_{tot}^n$ are the total relaxation rate of K and Rb electron spins and nuclear spins respectively. $R_p^K$ and $R_m^{Rb}$ are the pumping rate of the pump laser and the probe laser, respectively.

Due to the rapid spin-exchange collisions, The two kinds of alkali-metal atoms are hybridized together can be regarded as one alkali metal species [4,27]. Therefore, Eq. (1) can be approximated as follows:

$$\begin{aligned} \frac{{\partial {{\bf{P}}^{\bf{e}}}}}{{\partial t}} =& \frac{{{\gamma _e}}}{Q\left( {{P^{\rm{e}}}} \right)}({\bf{B}} + \lambda {{\bf{M}}^{\bf{n}}}{{\bf{P}}^{\bf{n}}} + {\bf{L}}) \times {{\bf{P}}^{\bf{e}}} - {\bf{\Omega }} \times {{\bf{P}}^{\bf{e}}} + \frac{1}{Q\left( {{P^{\rm{e}}}} \right)}({R_p}{{\bf{s}}_{\bf{p}}} + {R_m}{{\bf{s}}_{\bf{m}}} + R_{se}^{en}{{\bf{P}}^{\bf{n}}} - R_{tot}^e{{\bf{P}}^{\bf{e}}}),\\ \frac{{\partial {{\bf{P}}^n}}}{{\partial t}} =& {\gamma _n}({\bf{B}} + \lambda {{\bf{M}}^{\bf{e}}}{{\bf{P}}^{\bf{e}}}) \times {{\bf{P}}^{\bf{n}}} - \Omega \times {{\bf{P}}^{\bf{n}}} + R_{se}^{ne}{{\bf{P}}^{\bf{e}}} - R_{tot}^n{{\bf{P}}^{\bf{n}}}. \end{aligned}$$

Under ideal conditions, ${\bf {L}} ={{\bf {L}}^K}\frac {{{D_r}}}{{{D_r} + 1}} +{{\bf {L}}^{Rb}}\frac {1}{{{D_r} + 1}}$ can be offset to zero [28]. Where ${D_r} = {{{n_K}} \mathord {\left /{\vphantom {{{n_K}} {{n_{Rb}}}}} \right. } {{n_{Rb}}}}$ is the alkali metal density ratio of K and Rb. $Q\left ( {{P^e}} \right ) \approx Q\left ( {{P^{Rb}}} \right )$, $\lambda {{\bf {M}}^{\bf {e}}}{{\bf {P}}^{\bf {e}}} \approx {\lambda _{Rb - n}}{{\bf {M}}^{Rb}}{{\bf {P}}^{Rb}}$, and $\lambda {{\bf {M}}^{\bf {e}}}{{\bf {P}}^{\bf {e}}} \approx {\lambda _{Rb - n}}{{\bf {M}}^{Rb}}{{\bf {P}}^{Rb}}$. ${R_P}$ can be expressed as

$${R_P} = \frac{{{D_r}}}{{{D_r} + 1}}R_p^K = \frac{{{D_r}}}{{{D_r} + 1}}\frac{{{W_p}{r_e}cf}}{{h{\nu _K}}}\frac{{{\Gamma \mathord{\left/{\vphantom {\Gamma 2}} \right.} 2}}}{{{{\left( {{\nu _K} - {\nu _{K0}}} \right)}^2} + {{\left( {{\Gamma \mathord{\left/{\vphantom {\Gamma 2}} \right.} 2}} \right)}^2}}}.$$
Where ${W_p}$ is the pump laser power density. $c$ is the speed of light, and ${r_e}$ is the classical electron radius. $f$ is the oscillator strength, for alkali atoms, the oscillator strengths of D1 line and D2 line are approximately given by ${f_{D1}} \approx 1/3$ and ${f_{D2}} \approx 2/3$. $h$ is the Planck constant. ${\nu _{K0}}$ is the resonant transition frequency of K, $\Gamma$ is the pressure broadening of the pump laser at the resonance transition frequency of K, and ${\nu _{K}}$ is the pump laser working frequency.

The longitudinal electron spin polarization can be expressed as $P_z^e = {{{R_P}} \mathord {\left /{\vphantom {{{R_P}} {R_{tot}^e}}} \right. } {R_{tot}^e}}$. Here, $R_{tot}^e \approx {R_p} + R_{se}^{ee} + R_{se}^{en} + R_{sd}^e + {R_D}$. The last four items are the relaxation rates due to alkali-alkali spin-exchange collisions, alkali-noble gas spin-exchange collisions, spin-destruction collisions, diffusion to the walls in order [23]. For convenience of expression, the sum of other terms in $R_{tot}^e$ except ${R_p}$ is written as ${R_e}$. ${R_e} = \frac {{{D_r}}}{{{D_r} + 1}}{R_K} + \frac {1}{{{D_r} + 1}}{R_{Rb}}$ [29]. Here ${R_K}$ and ${R_{Rb}}$ are ${R_e}$ when the alkali metal e is potassium and rubidium respectively. When ${n_K} \ll {n_{Rb}}$,

$${R_e} \approx \frac{1}{{{D_r} + 1}}{R_{Rb}}.$$
Therefore, The longitudinal electron spin polarization can be expressed as
$$P_z^e = \frac{{{D_r}R_p^K}}{{{D_r}R_p^K + {R_{Rb}}}}.$$

3.2 the AC response model and the output equation

After the hybrid atomic spin ensembles are polarized to the equilibrium state, their longitudinal polarizations $P_z^{e}$ and $P_z^{n}$ are assumed to be constant and not affected by small transverse excitation. Under these conditions, the system can be approximated as a linear time-invariant system and the system state equation can be written as

$${\mathbf{\dot X}} = {\mathbf{AX}} + {\mathbf{BU}}.$$
Here, the state vector can be expressed as ${\bf {X}} = {\left [ {P_x^e,P_y^e,P_x^n,P_y^n} \right ]^{\mathop {\rm T}\nolimits } }$. This paper focuses on the response model of the system to radial magnetic noise and the measured quantity $\Omega _y$, so the input vector is set as ${\bf {U}} = {\left [ {{B_x},{B_y},{\Omega _y}} \right ]^{\mathop {\rm T}\nolimits } }$. The matrix ${\mathbf {A}}$ can be expressed as
$$\textrm{A} = \left[ {\begin{array}{cccc} { - \frac{{R_{{\rm{tot}}}^{\rm{e}}}}{Q}} & {\frac{{{\gamma ^{\rm{e}}}}}{Q}\lambda M_{}^eP_z^e} & {\frac{{R_{{\rm{se}}}^{{\rm{en}}}}}{Q}} & {\frac{{{\gamma ^{\rm{e}}}}}{Q}\lambda M_{}^{\rm{n}}P_z^{\rm{e}}}\\ { - \frac{{{\gamma ^{\rm{e}}}}}{Q}\lambda M_{}^eP_z^e} & { - \frac{{R_{{\rm{tot}}}^{\rm{e}}}}{Q}} & { - \frac{{{\gamma ^{\rm{e}}}}}{Q}\lambda M_{}^{\rm{n}}P_z^{\rm{e}}} & {\frac{{R_{{\rm{se}}}^{{\rm{en}}}}}{Q}}\\ {R_{{\rm{se}}}^{{\rm{ne}}}} & {{\gamma ^n}\lambda M_{}^{\rm{e}}P_z^{\rm{n}}} & { - R_{{\rm{tot}}}^{\rm{n}}} & { - {\gamma ^{\rm{n}}}\lambda M_{}^nP_z^n}\\ {{\rm{ - }}{\gamma ^{\rm{n}}}\lambda M_{}^{\rm{e}}P_z^{\rm{n}}} & {R_{{\rm{se}}}^{{\rm{ne}}}} & {{\gamma ^{\rm{n}}}\lambda M_{}^nP_z^n} & { - R_{{\rm{tot}}}^{\rm{n}}} \end{array}} \right].$$

The items whose absolute value is more than two orders of magnitude lower than the main effect items are ignored, and the eigenvalues of ${\mathbf {A}}$ can be simplified to

$${\lambda _{1,2}} ={-} \frac{{R_{{\rm{tot}}}^{\rm{e}}}}{Q} \pm i\frac{{{\gamma ^e}}}{Q}\lambda M_{}^nP_z^n, {\lambda _{3,4}} ={-} R_{{\rm{tot}}}^{\rm{n}} \pm i{\gamma ^n}\lambda M_{}^{\rm{e}}P_z^e.$$

In the Laplace domain, the characteristic equation of the system can be expressed as:

$$D(s) = \left( {{{\left( {s + \frac{{R_{{\rm{tot}}}^{\rm{e}}}}{Q}} \right)}^2} + {{\left( {\frac{{{\gamma ^e}}}{Q}\lambda M_{}^nP_z^n} \right)}^2}} \right) \left( {{{\left( {s + R_{{\rm{tot}}}^{\rm{n}}} \right)}^2} + {{\left( {{\gamma ^n}\lambda M_{}^eP_z^e} \right)}^2}} \right).$$

The matrix ${\mathbf {B}}$ can be expressed as

$${\bf{B}} = \left[ {\begin{array}{ccc} 0 & {{{{\gamma ^e}P_z^e} \mathord{\left/ {\vphantom {{{\gamma ^e}P_z^e} Q}} \right. } Q}} & { - P_z^e}\\ {{{ - {\gamma ^e}P_z^e} \mathord{\left/ {\vphantom {{ - {\gamma ^e}P_z^e} Q}} \right. } Q}} & 0 & 0\\ 0 & {{\gamma ^n}P_z^n} & { - P_z^n}\\ { - {\gamma ^n}P_z^n} & 0 & 0 \end{array}} \right].$$

The transfer function of the system can be expressed as ${\bf {G}}(s) = {(sI - {\bf {A}})^{ - 1}}{\bf {B}}$. After simplification, the transfer function between ${B_x}$ and $P_x^e$ can be written as

$${G_{{B_x}}}(s) = {K_{{B_x}}}\left( {s - {z_{{B_x}1}}} \right)\left( {s - {z_{{B_x}2}}} \right)/D(s).$$
Where ${K_{{B_x}}} = {\left ( {{{{\gamma ^{\rm {e}}}} \mathord {\left / {\vphantom {{{\gamma ^{\rm {e}}}} Q}} \right.} Q}} \right )^2}\lambda M_{}^{\rm {n}}P_{\rm {z}}^{\rm {e}}P_{\rm {z}}^n, {z_{{B_x}1}} = - \frac {{\lambda M_{}^{\rm {n}}}}{{\lambda M_{}^{\rm {e}}}}R_{{\rm {se}}}^{{\rm {ne}}} - R_{{\rm {tot}}}^{\rm {n}} \approx - R_{{\rm {tot}}}^{\rm {n}}, {z_{{B_x}2}} =$$- \frac {QR{{_{{\rm {se}}}^{{\rm {ne}}}}^2}{\gamma ^n}\left ( {{\gamma ^n}R_{{\rm {tot}}}^e + {\gamma ^e}R_{{\rm {tot}}}^n} \right ) + {\gamma ^e}R_{{\rm {tot}}}^n\left ( {{\gamma ^n}R_{{\rm {se}}}^{{\rm {ne}}}R_{{\rm {se}}}^{en} + {\gamma ^e}R{{_{{\rm {tot}}}^n}^2}} \right )}{{{Q^2}R{{_{{\rm {se}}}^{{\rm {ne}}}}^2}{\gamma ^n}^2 + R{{_{{\rm {tot}}}^{\rm {n}}}^2}{\gamma ^e}^2}} \approx- R_{{\rm {tot}}}^{\rm {n}}.$ The transfer function between ${B_y}$ and $P_x^e$ can be written as
$${G_{{B_y}}}(s) = {K_{{B_y}}}({s^2} + {b_{{B_y}1}}s + {b_{{B_y}2}})(s - {z_{{B_y}1}})/D(s).$$
Where ${K_{{B_y}}} = \frac {{{\gamma ^{\rm {e}}}}}{Q}P_{\rm {z}}^{\rm {e}},{b_{{B_y}1}} = R_{{\rm {tot}}}^{\rm {n}} + \frac {{R_{{\rm {tot}}}^{\rm {e}}}}{Q}, {b_{{B_y}2}} = \frac {{R_{{\rm {tot}}}^{\rm {n}}R_{{\rm {tot}}}^{\rm {e}} - \left ( {{\gamma ^n}\lambda M_{}^{\rm {e}}P_z^{\rm {e}}} \right )\left ( {{\gamma ^e}\lambda M_{}^nP_z^n} \right )}}{Q}, {z_{{B_y}1}} = - R_{{\rm {tot}}}^{\rm {n}}.\\$ Through the simplification of high-order terms and non-major terms, the transfer function between ${\Omega _y}$ and $P_x^e$ can be written as
$${G_{{\Omega _y}}}\left( s \right) = \frac{{{\gamma _e}P_z^eB_z^n}}{{R_{tot}^e{\gamma _n}B_z^n + 2\left( {Q{\gamma _n}B_z^n + {\gamma _e}B_z^e} \right)s}}.$$
According to Eq. (13), the working frequency band of the co-magnetometer can be expressed as $\left [ {0,\frac {{R_{tot}^e{\gamma _n}B_z^n}}{{2\left ( {Q{\gamma _n}B_z^n + {\gamma _e}B_z^e} \right )}}} \right ]$.

The system noise of SERF co-magnetometer can be divided into two types: the noise that causes the response of polarization and the noise that does not cause the response of polarization. Where, the noise causing the response of polarization is mainly the magnetic noise $\delta B\left ( \omega \right )$ according to Eq. (3). The noise that does not cause the response of the polarization is the probe background noise ${N_{probe}}\left ( \omega \right )$, which mainly includes the power and polarization fluctuation of the probe laser, the circuit noise of the acquisition system, the device noise caused by temperature fluctuation and so on. Therefore, the output equation of the SERF co-magnetometer can be expressed as

$$S\left( \omega \right) = K \cdot P_x^{\rm{e}}\left( \omega \right) = K \cdot \left[ {\left| {{G_{{\Omega _{\rm{y}}}}}\left( \omega \right)} \right|{\Omega _{\rm{y}}}\left( \omega \right) + \left| {{G_{{B_x}}}\left( \omega \right)} \right|\delta {B_x}\left( \omega \right) + \left| {{G_{{B_y}}}\left( \omega \right)} \right|\delta {B_y}\left( \omega \right) + {N_{probe}}\left( \omega \right)} \right]$$
Where $K$ refers to the magnification of the probe and acquisition system of the SERF co-magnetometer. With the probe laser path of Fig. 1, the coefficient $K$ can be expressed as [30]
$$K = 2\eta {W_m} \cdot {e^{ - OD\left( {{\nu _m}} \right)}} \cdot \frac{\pi }{2}l{r_e}{n_{Rb}}c \left( { - {f_{D1}} {L\left( {{\nu _m} - {\nu _{D1}}} \right)} + \frac{1}{2}{f_{D2}} {L\left( {{\nu _m} - {\nu _{D2}}} \right)}} \right).$$
Where $\eta$ is the photoelectric conversion efficiency of the detector, and $OD\left ( {{\nu _{\rm {m}}}} \right )$ is the optical depth. $L\left ( {{\nu _m} - {\nu _D}} \right ) =\frac {{{{{\Gamma _D}} \mathord {\left /{\vphantom {{{\Gamma _D}} {2\pi }}} \right. } {2\pi }}}}{{{{\left ( {{\nu _m} - {\nu _D}} \right )}^2} + {{\left ( {{{{\Gamma _D}} \mathord {\left /{\vphantom {{{\Gamma _D}} 2}} \right. } 2}} \right )}^2}}}$ is the Lorentz dispersion function, ${\nu _{D1}}$ and ${\nu _{D2}}$ are the resonant transition frequency of the alkali atom, and ${\nu _m}$ is the probe laser working frequency. The alkali metal density ratio is adjusted by changing ${n_{K}}$, and ${n_{Rb}}$ is the density of saturated vapor pressure, so the coefficient $K$ of can be regarded as a constant under the experimental conditions of this paper.

Next, the proposed model in this section will be verified by experiments, and the influence of electron spin polarization on the output of the co-magnetometer will be analyzed on this basis.

4. Results and discussion

In this paper, four cells with different alkali metal density ratios are fabricated for a comparative test. The density ratios ${D_r}$ of the four cells is shown in Table 1. In order to control the variables, the other condition parameters of the cells selected in this paper are consistent except ${D_r}$.

Tables Icon

Table 1. Alkali metal density ratio of different cells.

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In Eq. (14), ${\left | {{G_{{\Omega _{\rm {y}}}}}\left ( \omega \right )} \right |}$ and ${\left | {{G_B}\left ( \omega \right )} \right |}$ determine the response of the co-magnetometer to the measured quantity and system noise, thus affecting the measurement accuracy of the co-magnetometer. In order to further explore the relevant change mechanism, several experiments are carried out in this section.

4.1 Error equation and the noise response coefficient

Firstly, in order to verify the accuracy of Eq. (11)- (13), the amplitude-frequency response of the SERF co-magnetometer to the magnetic field and rotation velocity is measured. The sinusoidal magnetic field with an amplitude of 0.15 nT is provided to the x and y directions respectively, and the amplitude frequency response of the signal is recorded. The sinusoidal rotation with an amplitude of 0.1 $^\circ$/s is provided to the y-direction, and the amplitude-frequency response of the signal is also recorded.

The amplitude-frequency response of the SERF co-magnetometer to magnetic field and rotation velocity is shown in Fig. 2. Here, Take the test results of cell A with pump laser power density of 120 mW/cm$^2$ as an example. The measurement result is shown by asterisks, and the fitting curve according to Eq. (11)- (13) agrees well with the experimental data. In the fitting results, $R_{tot}^e$ and $R_{tot}^n$ are 582.6 s$^{-1}$ and 0.0292 s$^{-1}$ respectively, $P_z^e$ and $P_z^n$ are 0.852 and 0.00369 respectively. In the working frequency band of the SERF co-magnetometer, the response coefficient of the system to the measured quantity ${\Omega _y}$ can be expressed as a constant, that is, the scale factor of the system which can be expressed as

$${k_{{\Omega _{\rm{y}}}}} = K \cdot \left| {{G_{{\Omega _{\rm{y}}}}}\left( 0 \right)} \right| = K \cdot \frac{{{\gamma _e}P_z^e}}{{{\gamma _n}R_{tot}^e}}.$$

 figure: Fig. 2.

Fig. 2. (a) Amplitude frequency response to ${\Omega _y}$. (b) Amplitude frequency response to ${B_x}$ and ${B_y}$.

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Through all the test results of the four cells selected in this paper under different conditions, it can be seen that in the working frequency band of the SERF co-magnetometer, the response of the system to the x-axis magnetic field is more than one order of magnitude larger than that of the y-axis magnetic field, which is consistent with the existing research conclusions [3,18]. Therefore, $\left | {{G_B}\left ( \omega \right )} \right |$ in this paper only considers the magnetic field response in the x-direction $\left | {{G_{{B_x}}}\left ( \omega \right )} \right |$. The transfer function of the system to the x-direction magnetic field includes a proportional component, a second-order differentiation element and two second-order integration elements. In the working frequency band of the system, the proportional component and the second-order differentiation element play a leading role. Therefore, the response of the system to the magnetic field can be simplified to

$${k_B}\left( \omega \right) = \frac{{\left| {{G_{{B_x}}}\left( \omega \right)} \right|}}{{\left| {{G_{{\Omega _y}}}\left( 0 \right)} \right|}} \approx \frac{1}{{\lambda {M_n}P_z^n}}\left( {R_{tot}^{\rm{n}} + \frac{{{\omega ^2}}}{{R_{tot}^{\rm{n}}}}} \right) = \frac{{R_1^{\rm{n}}}}{{\lambda {M_n}P_z^eR_{se}^{{\rm{ne}}}}}\left( {R_{tot}^{\rm{n}} + \frac{{{\omega ^2}}}{{R_{tot}^{\rm{n}}}}} \right).$$
Where $R_1^{\rm {n}}$ is the longitudinal nuclear spin relaxation rate.

Therefore, the output equation of the SERF co-magnetometer can be expressed as

$$S\left( \omega \right) = \frac{{K \cdot {\gamma _e}P_z^e}}{{{\gamma _n}R_{tot}^e}}\left[ {{\Omega _{\rm{y}}}\left( \omega \right) + \frac{{R_1^{\rm{n}}}}{{\lambda {M_n}P_z^eR_{se}^{{\rm{ne}}}}}\left( {R_{tot}^{\rm{n}} + \frac{{{\omega ^2}}}{{R_{tot}^{\rm{n}}}}} \right)\delta B\left( \omega \right)} \right] + K \cdot {N_{probe}}\left( \omega \right).$$
The error equation of the SERF co-magnetometer can be expressed as
$$N\left( \omega \right) = \frac{{R_1^{\rm{n}}}}{{\lambda {M_n}P_z^eR_{se}^{{\rm{ne}}}}}\left( {R_{tot}^{\rm{n}} + \frac{{{\omega ^2}}}{{R_{tot}^{\rm{n}}}}} \right)\delta B\left( \omega \right) + \frac{{{\gamma _n}R_{tot}^e}}{{{\gamma _e}P_z^e}}{N_{probe}}\left( \omega \right).$$
By simultaneously suppressing the magnetic field response coefficient ${k_B}\left ( \omega \right ) = \frac {{R_1^{\rm {n}}}}{{\lambda {M_n}P_z^eR_{se}^{{\rm {ne}}}}}\left ( {R_{tot}^{\rm {n}} + \frac {{{\omega ^2}}}{{R_{tot}^{\rm {n}}}}} \right )$ and the probe background noise response coefficient ${k_{pro}} = \frac {{{\gamma _n}R_{tot}^e}}{{{\gamma _e}P_z^e}}$ , the suppression of system error and the improvement of test accuracy can be realized.

4.2 Response coefficient of probe background noise ${k_{pro}}$

Substituting Eq. (5) into the expression of ${k_{pro}}$ can obtain ${k_{pro}} = \frac {{{\gamma _n}{{\left ( {{D_r}R_p^K + {R_{Rb}}} \right )}^2}}}{{{\gamma _e}{D_r}R_p^K}}$. It can be seen from Eq. (3) that the pumping rate $R_p^K$ is directly proportional to the pump laser power density ${W_p}$. Here, set $R_p^K = {K_p} \cdot {W_p}$. Therefore, ${k_{pro}}$ can be expressed as

$${k_{pro}} = \frac{{{\gamma _n}{{\left( {{D_r} \cdot {K_p} \cdot {W_p} + {R_{Rb}}} \right)}^2}}}{{{\gamma _e}{D_r} \cdot {K_p} \cdot {W_p}}}$$

To study the change law of ${k_{pro}}$, the values of the parameters ${K_p}$ and $R_{Rb}$ should be measured firstly. Here, the values of these two parameters are calculated by the measured results of ${R_{tot}^e}$. The value of ${R_{tot}^{e}}$, which can be pressed as $R_{tot}^e = {D_r} \cdot {K_p} \cdot {W_p} + {R_{Rb}}$, is measured by the square wave modulation method [4]. Fig. 3 shows the relationship between ${R_{tot}^e}$ and pump laser power density in different cells. Here, the marks are the measurement results and the curve is the fitting result. In the fitting results, the value of $R_{Rb}$ is 73$\pm$4 s$^{-1}$, and the value of ${K_p}$ is 311$\pm$5. According to Eq. (16), the relationship between ${k_{pro}}$ and scale factor ${k_{{\Omega _{\rm {y}}}}}$ can be expressed as ${k_{pro}} = \frac {K}{{{k_{{\Omega _{\rm {y}}}}}}}$. The measurement results of three groups scale factors for each cell are substituted into the above equation, and the value of $K$ can be calculated to be 1.4V.

 figure: Fig. 3.

Fig. 3. Relationship between pump laser power density ${W_p}$ and ${R_{tot}^e}$ in cells with different alkali metal density ratio ${D_r}$.

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Figure 4 shows the relationship between ${k_{pro}}$ and pump laser power density ${W_p}$ in different cells. The measurement results are shown by different marks, which are obtained by dividing the coefficient $K$ by the measurement result scale factor ${k_{{\Omega _{\rm {y}}}}}$. The curve, which is calculated by Eq. (20), agrees well with the experimental data.

 figure: Fig. 4.

Fig. 4. Relationship between pump laser power density ${W_p}$ and ${k_{pro}}$ in cells with different alkali metal density ratio ${D_r}$.

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The derivation of Eq. (20) shows that when ${D_r} \cdot {K_p} \cdot {W_p} = {R_{Rb}}$, ${k_{pro}}$ gets the minimum value and the SERF co-magnetometer has the largest scale factor ${k_{{\Omega _{\rm {y}}}}}$. The symbol of star in Fig. 4 indicates the value of ${k_{pro}}$ when ${D_r} \cdot {K_p} \cdot {W_p} = {R_{Rb}}$. It can be seen that there is a minimum point for ${k_{pro}}$ in each cell, and the minimum value of ${k_{pro}}$ in different cells is basically the same. The pump optical power density ${W_p}$ at the minimum ${k_{pro}}$ increases with the decrease of ${D_r}$. The theoretical results are consistent with the experimental results.

4.3 Response coefficient of magnetic noise ${k_B}\left ( \omega \right )$

It can be seen from Eq. (17) that the increase of longitudinal electron spin polarization $P_z^e$ can suppress the response to magnetic noise. Both ${D_r}$ and ${W_p}$ are positively correlated with longitudinal electron spin polarization $P_z^e$. The relationship between ${W_p}$ of the cells with different alkali metal density ratios and $P_z^e$ is shown in Fig. 5. Here, the measurement results, which are obtained by fitting the relationship curve between z-direction magnetic field and resonance frequency [29,31], are shown by corresponding marks, and the curve is the calculation result obtained by substituting the value of parameters ${K_p}$ and $R_{Rb}$ into Eq. (5).

 figure: Fig. 5.

Fig. 5. Relationship between pump laser power density ${W_p}$ and ${P_z^e}$ in cells with different alkali metal density ratio ${D_r}$.

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${k_B}\left ( \omega \right )$ at 0.01Hz is selected as an example to discuss the relationship between the ${k_B}\left ( \omega \right )$ and $P_z^e$. The magnetic field response coefficient ${k_B}\left ( \omega \right )$ of each cell under different pump laser power densities ${W_p}$ is shown in Fig. 6. The measurement result is shown by corresponding marks, which is obtained by applying a magnetic field with an amplitude of 0.15 nT and frequency of 0.01Hz to the x-direction and dividing the response amplitude of the signal by the applied magnetic field amplitude. The curve is the fitting result according to Eq. (17). Here, the expression of $P_z^e$ is the curve obtained in Fig. 5, and other parts in Eq. (17) $\frac {{R_1^{\rm {n}}}}{{\lambda {M_n}R_{se}^{{\rm {ne}}}}}\left ( {R_{tot}^{\rm {n}} + \frac {{{\omega ^2}}}{{R_{tot}^{\rm {n}}}}} \right )$ are fitted as a constant factor. It can be seen that the value of ${k_B}\left ( \omega \right )$ is inversely proportional to $P_z^e$. The symbol of star in Fig. 6 indicates the value of ${k_B}\left ( \omega \right )$ when ${D_r} \cdot {K_p} \cdot {W_p} = {R_{Rb}}$, and the values of ${k_B}\left ( \omega \right )$ corresponding to each cell are basically the same.

 figure: Fig. 6.

Fig. 6. Relationship between pump laser power density ${W_p}$ and ${k_B}\left ( \omega \right )$ in cells with different alkali metal density ratio ${D_r}$.

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4.4 Relationship between $P_z^e$ and the system error

Substituting Eq. (5) into Eq. (19), the expression of the error equation can be obtained as follows:

$$N\left( \omega \right) = \frac{{R_1^{\rm{n}}}}{{\lambda {M_n}R_{se}^{{\rm{ne}}}}}\frac{{{D_r}{K_p}{W_p} + {R_{Rb}}}}{{{D_r}{K_p}{W_p}}}\left( {R_{tot}^{\rm{n}} + \frac{{{\omega ^2}}}{{R_{tot}^{\rm{n}}}}} \right)\delta B\left( \omega \right) + \frac{{{\gamma _n}}}{{{\gamma _e}}}\frac{{{{\left( {{D_r}{K_p}{W_p} + {R_{Rb}}} \right)}^2}}}{{{D_r}{K_p}{W_p}}}{N_{probe}}\left( \omega \right).$$

Taking the derivative of Eq. (21) can obtain that when ${D_r}{K_p}{W_p} = \left ( {R_{Rb}}^2 +\right.$$\left.\frac {{{\gamma _e}R_1^{\rm {n}}\left ( {R{{_{tot}^{\rm {n}}}^2} + {\omega ^2}} \right )\delta B\left ( \omega \right )}}{{{\gamma _n}\lambda {M_n}R_{se}^{{\rm {ne}}}R_{tot}^{\rm {n}}{N_{probe}}\left ( \omega \right )}}{R_{Rb}} \right )^{\frac {1}{2}} > {R_{Rb}}$, the system error reaches the minimum value.

In order to further analyze the influence of $P_z^e$ on the SERF co-magnetometer, the magnetic noise $\delta B\left ( \omega \right )$ and the probe background noise ${N_{probe}}\left ( \omega \right )$ of the system are calibrated. The probe background noise ${N_{probe}}\left ( \omega \right )$ can be measured by applying a large magnetic field of 1000 nT to the z direction. The magnetic noise $\delta B\left ( \omega \right )$ can be measured by using the cell of K-Rb-$^4$He with the system shown in Fig. 1. The data sampling rate is 200 Hz and data collection length is 3 hours. The Allan deviation results of $\delta B\left ( \omega \right )$ and ${N_{probe}}\left ( \omega \right )$ are shown in Fig. 7(a) and Fig. 7(b) respectively.

 figure: Fig. 7.

Fig. 7. (a)The Allan variances of $\delta B\left ( \omega \right )$.(b)The Allan variances of ${N_{probe}}\left ( \omega \right )$.

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The Allan variances of ${N_{probe}}\left ( \omega \right )$ and $\delta B\left ( \omega \right )$ at 100 s are 0.0026 $^\circ$/h and $2.27 \times {10^{ - 5}}$ nT, respectively. Substituting the above test results into Eq. (21), the system error curve of each cell at 100 s can be obtained, as shown in Fig. 8.

 figure: Fig. 8.

Fig. 8. Relationship between pump laser power density and the system error in cells with different alkali metal density ratio ${D_r}$.

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In Fig. 8, the symbol of star indicates the calculation result of the system error $N\left ( \omega \right )$ at the maximum point of the scale factor of each cell, and the symbol of dot indicates the calculation result of the minimum value of the system error $N\left ( \omega \right )$ of each cell. Due to the existence of magnetic noise, ${W_p}$ corresponding to the lowest system error point of each cell is greater than that corresponding to the maximum scale factor point. The alkali metal density ratio ${D_r}$ of the co-magnetometer cell is inversely proportional to the value of ${W_p}$ corresponding to the lowest system error, and the relationship between the two is shown in the black curve in Fig. 10. The lowest system error level of cells with different ${D_r}$, which can be achieved by changing the value of ${W_p}$, is roughly the same. Choosing the appropriate point of ${W_p}$ can suppress the system noise by more than half.

To verify the above conclusions, the Allan deviation of the co-magnetometer under different pump laser power densities with different cells is compared, as shown in Fig. 9. The data sampling rate is 200 Hz and data collection length is 3 hours. It can be seen that the pump laser power density point corresponding to the minimum value of $\sigma \left ( \tau \right )$ at 100 s is greater than that corresponding to the maximum value of the scale factor. The optimal Allan deviation analysis results of the two cells are roughly the same, and the corresponding pump laser power density points ${W_p}$ are related to the alkali metal density ratio ${D_r}$. The experimental results are consistent with the theoretical analysis. According to the different noise characteristics such as Markov noise and rate slope in different Allan variance analysis curve, it can be seen that the accuracy of the SERF co-magnetometer is greatly affected by the conditions of test environment, which needs further research and suppression. In addition, it is worth noting that since the magnetic noise of the system at different frequency points are different, the specific values of the pump laser power density points corresponding to the optimal long-term stability of each cell and that corresponding to the optimal sensitivity will be slightly different. Therefore, we need to further suppress the magnetic noise of the system, so that the SERF co-magnetometer can obtain optimal long-term stability and sensitivity at the same pump laser power density point.

 figure: Fig. 9.

Fig. 9. (a) Alllan variance of the co-magnetometer signals under different pump laser power densities of cell A. (b) The changing law of scale factor and $\sigma \left ( \tau \right )@100{\rm {s}}$ under different pump laser power densities of cell A. (c) Alllan variance of the co-magnetometer signals under different pump laser power densities of cell B. (d) The changing law of scale factor and $\sigma \left ( \tau \right )@100{\rm {s}}$ under different pump laser power densities of cell B.

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However, in practical use, there is an upper limit on the value of ${D_r}$ that can make Eq. (4) valid. When ${D_r}$ increases to a certain extent, ${n_K}$ cannot be ignored, and ${R_K}$ in ${R_e}$ cannot be ignored. In addition, the increase of ${n_K}$ will increase the optical depth and the gradient of the pump laser on the propagation path. The gradient of the pumping rate causes alkali metal atoms at different positions in the cell to feel different magnetic fields and have different precession frequencies, resulting in electron spin relaxation [32,33]. When the electron spin relaxation rate ${R_e}$ increases, both ${k_{pro}}$ and ${k_B}\left ( \omega \right )$ will increase, and the scale factor ${k_{{\Omega _{\rm {y}}}}}$ will decrease, resulting in the increase of system error and the decrease of sensitivity. In order to estimate the range of ${D_r}$ applicable to the conclusions proposed in this paper, ${R_e}$ under two conditions ${D_r} = 1/21$ and ${D_r} = 1/77$ are supplementarily tested. The test results of the relationship curve between ${D_r}$ and ${R_e}$ are shown in the blue curve in Fig. 10. It can be seen from the test results that when ${D_r}$ is less than about 1/100, the change of electron spin relaxation rate caused by the change of ${n_K}$ can be ignored. In addition, due to the performance limitation of the pumped optical laser, there is an upper limit on the working range of the pumped laser power density. Taking the system in this paper as an example, the DBR(770 nm) laser used in this paper can work in the range of 0-130 mW/cm$^2$. For the system in this paper, the appropriate range of alkali metal density ratio is about 1/100-1/300. The specific numerical conclusions obtained in this paper will be different for different systems, but the proposed evaluation method is universal for the SERF co-magnetometer system. In a word, the upper limit of the suitable application range of the alkali metal density ratio is mainly determined by the electron spin relaxation rate, and the lower limit is mainly determined by the working range of the pumping laser.

 figure: Fig. 10.

Fig. 10. The black curve is the relationship between the alkali metal density ratio ${D_r}$ and the pump optical power density ${W_p}$ that minimizes the system error of the SERF co-magnetometer. The blue curve is the relationship between alkali metal density ratio ${D_r}$ and electron spin relaxation rate ${R_e}$.

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5. Conclusion

In conclusion, the influence of the longitudinal electron spin polarization on the hybrid optical pumped K-Rb-$^{21}$Ne co-magnetometer is studied in this paper. The Bloch equations are solved by the transfer function method, and the AC response model of the SERF co-magnetometer considering magnetic noise and system bandwidth is proposed for the first time. The error equation of the system is obtained according to the established response model, and the effects of longitudinal electron spin polarization on the response to the system noise are analyzed through experimental research. The results indicate that due to the influence of magnetic noise, the product of the alkali metal density ratio and the pumping rate of potassium corresponding to the lowest system error point is greater than the electron spin relaxation rate, which is inconsistent with the maximum point of the scale factor. In addition, due to the limitation of the electron spin relaxation rate and the working range of pump laser, the optimal selection range of alkali metal density ratio is about 1/100 - 1/300. The theory and method proposed in this paper will be significant for optimizing the performance of the SERF co-magnetometer and other high-precision measurement applications using hybrid optical pumping technology.

Funding

National Natural Science Foundation of China (61873020, 62103026, 62122009); National Science Fund for Distinguished Young Scholars (61925301).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (10)
Fig. 1.
Fig. 1. Schematic of the SERF co-magnetometer. BSL: beam shaping lenses, M: reflection mirror, LCVR: liquid crystal variable retarder, PD: photodiode, ECU: electronic control unit, GT: Glan-Taylor polarizer, PD: photodiode, PBS: polarizing beam splitter, P: linear polarizer.

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Fig. 2.
Fig. 2. (a) Amplitude frequency response to ${\Omega _y}$. (b) Amplitude frequency response to ${B_x}$ and ${B_y}$.

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Fig. 3.
Fig. 3. Relationship between pump laser power density ${W_p}$ and ${R_{tot}^e}$ in cells with different alkali metal density ratio ${D_r}$.

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Fig. 4.
Fig. 4. Relationship between pump laser power density ${W_p}$ and ${k_{pro}}$ in cells with different alkali metal density ratio ${D_r}$.

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Fig. 5.
Fig. 5. Relationship between pump laser power density ${W_p}$ and ${P_z^e}$ in cells with different alkali metal density ratio ${D_r}$.

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Fig. 6.
Fig. 6. Relationship between pump laser power density ${W_p}$ and ${k_B}\left ( \omega \right )$ in cells with different alkali metal density ratio ${D_r}$.

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Fig. 7.
Fig. 7. (a)The Allan variances of $\delta B\left ( \omega \right )$.(b)The Allan variances of ${N_{probe}}\left ( \omega \right )$.

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Fig. 8.
Fig. 8. Relationship between pump laser power density and the system error in cells with different alkali metal density ratio ${D_r}$.

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Fig. 9.
Fig. 9. (a) Alllan variance of the co-magnetometer signals under different pump laser power densities of cell A. (b) The changing law of scale factor and $\sigma \left ( \tau \right )@100{\rm {s}}$ under different pump laser power densities of cell A. (c) Alllan variance of the co-magnetometer signals under different pump laser power densities of cell B. (d) The changing law of scale factor and $\sigma \left ( \tau \right )@100{\rm {s}}$ under different pump laser power densities of cell B.

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Fig. 10.
Fig. 10. The black curve is the relationship between the alkali metal density ratio ${D_r}$ and the pump optical power density ${W_p}$ that minimizes the system error of the SERF co-magnetometer. The blue curve is the relationship between alkali metal density ratio ${D_r}$ and electron spin relaxation rate ${R_e}$.

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Tables (1)

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Table 1. Alkali metal density ratio of different cells.

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Equations (21)

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$$\scalebox{0.88}{$\begin{aligned} \frac{{\partial {{\bf{P}}^K}}}{{\partial t}} &= \frac{{{\gamma _e}}}{{Q\left( {{P^K}} \right)}}({\bf{B}} + {\lambda _{Kn}}{{\bf{M}}^{\bf{n}}}{{\bf{P}}^{\bf{n}}} + {{\bf{L}}^K}) \times {{\bf{P}}^K} - {\bf{\Omega }} \times {{\bf{P}}^K} + \frac{1}{{Q\left( {{P^K}} \right)}}(R_p^K{{\bf{s}}_{\bf{p}}} + R_{se}^{KRb}{{\bf{P}}^{Rb}} + R_{se}^{Kn}{{\bf{P}}^{\bf{n}}} - R_{tot}^K{{\bf{P}}^K}),\\ \frac{{\partial {{\bf{P}}^{Rb}}}}{{\partial t}} &= \frac{{{\gamma _e}}}{{Q\left( {{P^{Rb}}} \right)}}({\bf{B}} + {\lambda _{Rbn}}{{\bf{M}}^{\bf{n}}}{{\bf{P}}^{\bf{n}}} + {{\bf{L}}^{Rb}}) \times {{\bf{P}}^{Rb}} - {\bf{\Omega }} \times {{\bf{P}}^{Rb}} + \frac{1}{{Q\left( {{P^{Rb}}} \right)}}(R_{se}^{RbK}{{\bf{P}}^K} + R_m^{Rb}{{\bf{s}}_{\bf{m}}} + R_{se}^{Rbn}{{\bf{P}}^{\bf{n}}} - R_{tot}^{Rb}{{\bf{P}}^{Rb}}),\\ \frac{{\partial {{\bf{P}}^n}}}{{\partial t}}& = {\gamma _n}({\bf{B}} + {\lambda _{Kn}}{{\bf{M}}^K}{{\bf{P}}^K} + {\lambda _{Rbn}}{{\bf{M}}^{Rb}}{{\bf{P}}^{Rb}}) \times {{\bf{P}}^{\bf{n}}} - \Omega \times {{\bf{P}}^{\bf{n}}} + R_{se}^{nK}{{\bf{P}}^K} + R_{se}^{nRb}{{\bf{P}}^{Rb}} - R_{tot}^n{{\bf{P}}^{\bf{n}}}. \end{aligned}$}$$
$$\begin{aligned} \frac{{\partial {{\bf{P}}^{\bf{e}}}}}{{\partial t}} =& \frac{{{\gamma _e}}}{Q\left( {{P^{\rm{e}}}} \right)}({\bf{B}} + \lambda {{\bf{M}}^{\bf{n}}}{{\bf{P}}^{\bf{n}}} + {\bf{L}}) \times {{\bf{P}}^{\bf{e}}} - {\bf{\Omega }} \times {{\bf{P}}^{\bf{e}}} + \frac{1}{Q\left( {{P^{\rm{e}}}} \right)}({R_p}{{\bf{s}}_{\bf{p}}} + {R_m}{{\bf{s}}_{\bf{m}}} + R_{se}^{en}{{\bf{P}}^{\bf{n}}} - R_{tot}^e{{\bf{P}}^{\bf{e}}}),\\ \frac{{\partial {{\bf{P}}^n}}}{{\partial t}} =& {\gamma _n}({\bf{B}} + \lambda {{\bf{M}}^{\bf{e}}}{{\bf{P}}^{\bf{e}}}) \times {{\bf{P}}^{\bf{n}}} - \Omega \times {{\bf{P}}^{\bf{n}}} + R_{se}^{ne}{{\bf{P}}^{\bf{e}}} - R_{tot}^n{{\bf{P}}^{\bf{n}}}. \end{aligned}$$
$${R_P} = \frac{{{D_r}}}{{{D_r} + 1}}R_p^K = \frac{{{D_r}}}{{{D_r} + 1}}\frac{{{W_p}{r_e}cf}}{{h{\nu _K}}}\frac{{{\Gamma \mathord{\left/{\vphantom {\Gamma 2}} \right.} 2}}}{{{{\left( {{\nu _K} - {\nu _{K0}}} \right)}^2} + {{\left( {{\Gamma \mathord{\left/{\vphantom {\Gamma 2}} \right.} 2}} \right)}^2}}}.$$
$${R_e} \approx \frac{1}{{{D_r} + 1}}{R_{Rb}}.$$
$$P_z^e = \frac{{{D_r}R_p^K}}{{{D_r}R_p^K + {R_{Rb}}}}.$$
$${\mathbf{\dot X}} = {\mathbf{AX}} + {\mathbf{BU}}.$$
$$\textrm{A} = \left[ {\begin{array}{cccc} { - \frac{{R_{{\rm{tot}}}^{\rm{e}}}}{Q}} & {\frac{{{\gamma ^{\rm{e}}}}}{Q}\lambda M_{}^eP_z^e} & {\frac{{R_{{\rm{se}}}^{{\rm{en}}}}}{Q}} & {\frac{{{\gamma ^{\rm{e}}}}}{Q}\lambda M_{}^{\rm{n}}P_z^{\rm{e}}}\\ { - \frac{{{\gamma ^{\rm{e}}}}}{Q}\lambda M_{}^eP_z^e} & { - \frac{{R_{{\rm{tot}}}^{\rm{e}}}}{Q}} & { - \frac{{{\gamma ^{\rm{e}}}}}{Q}\lambda M_{}^{\rm{n}}P_z^{\rm{e}}} & {\frac{{R_{{\rm{se}}}^{{\rm{en}}}}}{Q}}\\ {R_{{\rm{se}}}^{{\rm{ne}}}} & {{\gamma ^n}\lambda M_{}^{\rm{e}}P_z^{\rm{n}}} & { - R_{{\rm{tot}}}^{\rm{n}}} & { - {\gamma ^{\rm{n}}}\lambda M_{}^nP_z^n}\\ {{\rm{ - }}{\gamma ^{\rm{n}}}\lambda M_{}^{\rm{e}}P_z^{\rm{n}}} & {R_{{\rm{se}}}^{{\rm{ne}}}} & {{\gamma ^{\rm{n}}}\lambda M_{}^nP_z^n} & { - R_{{\rm{tot}}}^{\rm{n}}} \end{array}} \right].$$
$${\lambda _{1,2}} ={-} \frac{{R_{{\rm{tot}}}^{\rm{e}}}}{Q} \pm i\frac{{{\gamma ^e}}}{Q}\lambda M_{}^nP_z^n, {\lambda _{3,4}} ={-} R_{{\rm{tot}}}^{\rm{n}} \pm i{\gamma ^n}\lambda M_{}^{\rm{e}}P_z^e.$$
$$D(s) = \left( {{{\left( {s + \frac{{R_{{\rm{tot}}}^{\rm{e}}}}{Q}} \right)}^2} + {{\left( {\frac{{{\gamma ^e}}}{Q}\lambda M_{}^nP_z^n} \right)}^2}} \right) \left( {{{\left( {s + R_{{\rm{tot}}}^{\rm{n}}} \right)}^2} + {{\left( {{\gamma ^n}\lambda M_{}^eP_z^e} \right)}^2}} \right).$$
$${\bf{B}} = \left[ {\begin{array}{ccc} 0 & {{{{\gamma ^e}P_z^e} \mathord{\left/ {\vphantom {{{\gamma ^e}P_z^e} Q}} \right. } Q}} & { - P_z^e}\\ {{{ - {\gamma ^e}P_z^e} \mathord{\left/ {\vphantom {{ - {\gamma ^e}P_z^e} Q}} \right. } Q}} & 0 & 0\\ 0 & {{\gamma ^n}P_z^n} & { - P_z^n}\\ { - {\gamma ^n}P_z^n} & 0 & 0 \end{array}} \right].$$
$${G_{{B_x}}}(s) = {K_{{B_x}}}\left( {s - {z_{{B_x}1}}} \right)\left( {s - {z_{{B_x}2}}} \right)/D(s).$$
$${G_{{B_y}}}(s) = {K_{{B_y}}}({s^2} + {b_{{B_y}1}}s + {b_{{B_y}2}})(s - {z_{{B_y}1}})/D(s).$$
$${G_{{\Omega _y}}}\left( s \right) = \frac{{{\gamma _e}P_z^eB_z^n}}{{R_{tot}^e{\gamma _n}B_z^n + 2\left( {Q{\gamma _n}B_z^n + {\gamma _e}B_z^e} \right)s}}.$$
$$S\left( \omega \right) = K \cdot P_x^{\rm{e}}\left( \omega \right) = K \cdot \left[ {\left| {{G_{{\Omega _{\rm{y}}}}}\left( \omega \right)} \right|{\Omega _{\rm{y}}}\left( \omega \right) + \left| {{G_{{B_x}}}\left( \omega \right)} \right|\delta {B_x}\left( \omega \right) + \left| {{G_{{B_y}}}\left( \omega \right)} \right|\delta {B_y}\left( \omega \right) + {N_{probe}}\left( \omega \right)} \right]$$
$$K = 2\eta {W_m} \cdot {e^{ - OD\left( {{\nu _m}} \right)}} \cdot \frac{\pi }{2}l{r_e}{n_{Rb}}c \left( { - {f_{D1}} {L\left( {{\nu _m} - {\nu _{D1}}} \right)} + \frac{1}{2}{f_{D2}} {L\left( {{\nu _m} - {\nu _{D2}}} \right)}} \right).$$
$${k_{{\Omega _{\rm{y}}}}} = K \cdot \left| {{G_{{\Omega _{\rm{y}}}}}\left( 0 \right)} \right| = K \cdot \frac{{{\gamma _e}P_z^e}}{{{\gamma _n}R_{tot}^e}}.$$
$${k_B}\left( \omega \right) = \frac{{\left| {{G_{{B_x}}}\left( \omega \right)} \right|}}{{\left| {{G_{{\Omega _y}}}\left( 0 \right)} \right|}} \approx \frac{1}{{\lambda {M_n}P_z^n}}\left( {R_{tot}^{\rm{n}} + \frac{{{\omega ^2}}}{{R_{tot}^{\rm{n}}}}} \right) = \frac{{R_1^{\rm{n}}}}{{\lambda {M_n}P_z^eR_{se}^{{\rm{ne}}}}}\left( {R_{tot}^{\rm{n}} + \frac{{{\omega ^2}}}{{R_{tot}^{\rm{n}}}}} \right).$$
$$S\left( \omega \right) = \frac{{K \cdot {\gamma _e}P_z^e}}{{{\gamma _n}R_{tot}^e}}\left[ {{\Omega _{\rm{y}}}\left( \omega \right) + \frac{{R_1^{\rm{n}}}}{{\lambda {M_n}P_z^eR_{se}^{{\rm{ne}}}}}\left( {R_{tot}^{\rm{n}} + \frac{{{\omega ^2}}}{{R_{tot}^{\rm{n}}}}} \right)\delta B\left( \omega \right)} \right] + K \cdot {N_{probe}}\left( \omega \right).$$
$$N\left( \omega \right) = \frac{{R_1^{\rm{n}}}}{{\lambda {M_n}P_z^eR_{se}^{{\rm{ne}}}}}\left( {R_{tot}^{\rm{n}} + \frac{{{\omega ^2}}}{{R_{tot}^{\rm{n}}}}} \right)\delta B\left( \omega \right) + \frac{{{\gamma _n}R_{tot}^e}}{{{\gamma _e}P_z^e}}{N_{probe}}\left( \omega \right).$$
$${k_{pro}} = \frac{{{\gamma _n}{{\left( {{D_r} \cdot {K_p} \cdot {W_p} + {R_{Rb}}} \right)}^2}}}{{{\gamma _e}{D_r} \cdot {K_p} \cdot {W_p}}}$$
$$N\left( \omega \right) = \frac{{R_1^{\rm{n}}}}{{\lambda {M_n}R_{se}^{{\rm{ne}}}}}\frac{{{D_r}{K_p}{W_p} + {R_{Rb}}}}{{{D_r}{K_p}{W_p}}}\left( {R_{tot}^{\rm{n}} + \frac{{{\omega ^2}}}{{R_{tot}^{\rm{n}}}}} \right)\delta B\left( \omega \right) + \frac{{{\gamma _n}}}{{{\gamma _e}}}\frac{{{{\left( {{D_r}{K_p}{W_p} + {R_{Rb}}} \right)}^2}}}{{{D_r}{K_p}{W_p}}}{N_{probe}}\left( \omega \right).$$
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