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Effects of pump laser intensity on the cell temperature working point in a K-Rb-21Ne spin-exchange relaxation-free co-magnetometer

Jiasen Ruan, Lihong Duan, Yang Fu, Wenfeng Fan, and Wei Quan

Open Access Open Access

Abstract

The cell temperature working point optimization of the spin-exchange relaxation-free (SERF) co-magnetometer is studied theoretically and experimentally in this article. Based on the steady-state solution of the Bloch equations, the steady-state response model of the K-Rb-21Ne SERF co-magnetometer output signal with cell temperature is established in this paper. And combined with the model, a method to find the optimal working point of the cell temperature that incorporates the pump laser intensity is proposed. The scale factor of the co-magnetometer under different pump laser intensities and cell temperatures is obtained experimentally, and the long-term stability of the co-magnetometer at the different cell temperatures with corresponding pump laser intensities is measured. The results show that the bias instability of the co-magnetometer is reduced from 0.0311 deg/h to 0.0169 deg/h by obtaining the optimal working point of the cell temperature, which verifies the validity and accuracy of the theoretical derivation and the proposed method.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

With the development of quantum precision measurement technology, high-precision quantum sensors are widely used in basic physical testing [1–3], magnetic field measurement [4–6] and geophysics [7,8]. And the co-magnetometer based on the SERF regime has become an important development direction of the new generation of inertial measurement instruments due to its ultra-high theoretical precision and miniaturization potential [9–11]. The SERF co-magnetometer was first realized by the Romalis group of Princeton University in 2005 [12]. They used K-$^{3}$He as a sensitive atomic source and successfully achieved a bias drift of 0.04 deg/h. In the subsequent research, the SERF co-magnetometer with Rb-$^{129}$Xe and Cs-$^{129}$Xe as sensitive atomic sources were successively realized [13,14]. Compared to the optical pumping technology of a single alkali metal, the hybrid optical pumping technology of two alkali metals can obtain an order of magnitude higher polarization of nuclear spins [15]. Therefore, the SERF co-magnetometer using hybrid optical pumping technology has a higher theoretical precision. At present, the bias drift of a SERF co-magnetometer based on the K-Rb-$^{21}$Ne atomic source has reached the order of 10$^{-2}$ deg/h [16], but there is still a considerable gap between it and the theoretical precision. The recent research on the bias magnetic sensitivity of a SERF co-magnetometer has shown that the cell temperature is a limiting factor on its bias stability [17].

In the environment of high temperature and low magnetic field, the atoms in the SERF regime have a macro direction in the inertial space due to the polarization of the pump laser. When the angular rate is input, the polarized atoms follow with precession, changing the plane of polarisation of the linearly polarized probe laser. By detecting the angle of rotation, gyroscopic rotation measurement can be realised. As a sensitive medium, the polarization of atoms affects the response amplitude of the SERF co-magnetometer. The polarization of the atoms is related to the atomic density and the pumping rate of the pump laser, which are determined by the temperature of the cell and the intensity of the pump laser respectively. Therefore, the comprehensive study of the influence of cell temperature and pump laser intensity on the SERF co-magnetometer is of great significance to improve its sensitivity and bias stability. At present, many studies have been carried out on the temperature of the cell [18,19]. Fang et al. studied the influence of cell temperature on the sensitivity of the co-magnetometer in hybrid spin exchange optical pumping, but their simulation results are inconsistent with the measured results at high optical pumping power [20]. Sheng et al. found that the response of the frequency ratio to first order gradients is sensitive to temperature gradients across the cell [21]. Fan et al. found that increasing the cell temperature increases the bias magnetic sensitivity, which improves the bandwidth of the SERF co-magnetometer and also increases the magnetic noise error [17]. These studies did not give a method for selecting the cell temperature. Liu et al. studied the influence of cell temperature on the signal bias and established a steady-state response model related to cell temperature. The error caused by temperature fluctuations is suppressed by adjusting the temperature of the cell [22]. However, the influence of the pump laser intensity on the steady-state response was not analyzed.

In this paper, we study the influence of cell temperature on the SERF co-magnetometer signal, taking into account the pump laser intensity, and propose a method to select the optimal working point of the cell temperature. The remainder of this article is organized as follows. In Section 2, the relationship between the SERF co-magnetometer signal and the cell temperature is analyzed. In Section 3, the experimental setup of the SERF co-magnetometer is described. In Section 4, the proposed theory is verified by experiments, and the method for finding the optimal temperature working point considering the pump laser intensity is proposed. Finally, our conclusions are presented in Section 5.

2. Principle

In the SERF co-magnetometer based on the K-Rb-$^{21}$Ne hybrid optical pumping, the evolution equations of the atomic ensemble can be represented by a set of Bloch equations as follows [23]:

$$\scalebox{0.9}{$\begin{aligned} \frac{{\partial {{\bf{P}}^{\bf{K}}}}}{{\partial t}} &= \frac{{{\gamma _e}}}{{{Q_K}}}({\bf{B}} + {{\bf{L}}^{\bf{K}}} + \lambda M_0^n{{\bf{P}}^{\bf{n}}}) \times {{\bf{P}}^{\bf{K}}} - {\bf{\Omega }} \times {{\bf{P}}^{\bf{K}}} + \frac{{R_p^K{{\bf{s}}_{\bf{p}}} + R_{se}^{en}{{\bf{P}}^{\bf{n}}} + R_{se}^{ee}({{\bf{P}}^{{\bf{Rb}}}} - {{\bf{P}}^{\bf{K}}}) - R_{tot}^e{{\bf{P}}^{\bf{K}}}}}{{{Q_K}}}\\ \frac{{\partial {{\bf{P}}^{{\bf{Rb}}}}}}{{\partial t}} &= \frac{{{\gamma _e}}}{{{Q_{Rb}}}}({\bf{B}} + {{\bf{L}}^{{\bf{Rb}}}} + \lambda M_0^n{{\bf{P}}^n}) \times {{\bf{P}}^{{\bf{Rb}}}} - {\bf{\Omega }} \times {{\bf{P}}^{{\bf{Rb}}}} + \frac{{R_m^{Rb}{{\bf{s}}_{\bf{m}}} + R_{se}^{en}{{\bf{P}}^{\bf{n}}} + R_{se}^{ee}({{\bf{P}}^{\bf{K}}} - {{\bf{P}}^{{\bf{Rb}}}}) - R_{tot}^e{{\bf{P}}^{{\bf{Rb}}}}}}{{{Q_{Rb}}}}\\ \frac{{\partial {{\bf{P}}^{\bf{n}}}}}{{\partial t}} &= {\gamma _n}({\bf{B}} + \lambda M_0^e{{\bf{P}}^{\bf{K}}} + \lambda M_0^e{{\bf{P}}^{{\bf{Rb}}}}) \times {{\bf{P}}^{\bf{n}}} - {\bf{\Omega }} \times {{\bf{P}}^{\bf{n}}} + R_{se}^{ee}({{\bf{P}}^{\bf{K}}} + {{\bf{P}}^{{\bf{Rb}}}}) - R_{tot}^n{{\bf{P}}^{\bf{n}}} \end{aligned}$}$$
where ${\gamma _e}$ and ${\gamma _n}$ are the gyromagnetic ratios of the electron and nuclear spins, respectively; ${Q_K}$ and ${Q_{Rb}}$ are the slowing-down factors for the K and Rb atoms, respectively; ${\bf {B}}$ is the environmental magnetic field; ${{\bf {L}}^{\bf {K}}}$ and ${{\bf {L}}^{\bf {Rb}}}$ are the light shifts felt by the electron spins of the K and Rb atoms, respectively; ${\bf {\Omega }}$ is the inertial rotation rate; $\lambda M_0^e$ and $\lambda M_0^n$ are the magnetizations of the electron and nuclear spins corresponding to full spin polarizations; ${{\bf {s}}_{\bf {p}}}$ and ${{\bf {s}}_{\bf {m}}}$ are the photon spin vectors of the pump laser and probe lasers, respectively; $R_p^K$ and $R_m^{Rb}$ are the pumping rates of the pump and probe laser, respectively; $R_{se}^{ee}$ is the alkali-metal–noble-gas spin-exchange rate for an alkali atom and $R_{se}^{en}$ is the alkali-metal–noble-gas spin-exchange rate for a noble-gas atom; $R_{tot}^e$ and $R_{tot}^n$ are the total spin relaxation rate for the electron and nuclear spins, respectively.

In this study, the pump laser is circularly polarized light directed to the z-axis and is used to polarize K atoms. The probe laser, which is perpendicular to the pump laser, is linearly polarized light directed to the x-axis and detects the electron polarization component of the Rb atom along the x-axis. Due to the pumping effect of the pump laser, the angle of the atomic spin polarization deviating from the z-axis can be ignored. Therefore, the atomic spin polarization in the z-axis can be approximated to its steady-state solutions. Set the left-hand side of Eq. (1) to zero, and the steady-state solution of the electron polarization component in the x-axis $P_x^{Rb}$ can be obtained. It can be expressed as:

$$P_x^{Rb} = \frac{{{\gamma _e}P_z^KR_{tot}^e}}{{{{\left[ {{\gamma _e}\left( {{L_z} + \delta {B_z}} \right)} \right]}^2} + R{{_{tot}^e}^2}}}\left[ {{L_y} + \frac{{{\Omega _y}}}{{{\gamma _n}}} + \frac{{{\gamma _e}}}{{R_{tot}^e}}({L_z} + \delta {B_z}) \times \left( {{L_x} - \frac{{{\Omega _x}}}{{{\gamma _n}}}} \right)} \right]$$
Where $\delta {B_z} = {B_z} - B_z^c$, and $B_z^c$ is the magnetic field applied along the z-axis. Atoms are in the SERF regime when $B_z^c$ is set to the z-axis magnetic field felt by the electrons ${B_z}$. In addition, since the x-axis probe laser can only sense the y-axis rotation, so ${\Omega _x} = {\Omega _z} = 0$. ${L_x}$ and ${L_y}$ are considered negligible because the probe laser is linearly polarize. The electron spin polarization of the K atom in the z-axis can be expressed as $P_z^{K} = R_p^K/R_{tot}^e$ [24]. The total electron spin relaxation rate is $R_{tot}^e = {R_{rel}} + R_p^K = R_{rel}^{se} + R_{sd}^e + R_D^e + R_p^K + R_m^{Rb}$, where $R_{rel}^{se}$ is the spin-exchange relaxation rate, $R_{sd}^e$ is the spin-destruction collision relaxation rate, $R_D^e$ is the cell wall collision relaxation rate. So, Eq. (2) can be converted to:
$$P_x^{Rb} \approx \frac{{{\gamma _e}R_p^K}}{{{\gamma _e}^2{L_z}^2 + {{\left( {{R_{rel}} + R_p^K} \right)}^2}}}\left( {\frac{{{\Omega _y}}}{{{\gamma _n}}}} \right)$$

When linearly polarized light passes through the polarized atomic spin ensemble, the direction of its plane of polarisation will rotate due to circular birefringence. The angle of rotation $\theta$ can be expressed as [25]:

$$\theta = \frac{\pi }{2}l{r_e}{n_{Rb}}cP_x^{Rb}\left( { - f_{D1}^{Rb}{\mathop{\rm Im}\nolimits} \left[ {L\left( {{v_m} - v_{D1}^{Rb}} \right)} \right] + \frac{1}{2}f_{D2}^{Rb}{\mathop{\rm Im}\nolimits} \left[ {L\left( {{v_m} - v_{D2}^{Rb}} \right)} \right]} \right)$$
Where $l$ is the optical path of the probe laser passing through the cell, ${r_e}$ is the classical electron radius, ${n_{Rb}}$ is the density of Rb atoms, $c$ is the propagation speed of light, $f_{D1}^{Rb}$ is the oscillator strength of the D1 line for Rb atoms. $L\left ( {{v_m} - v_{D1}^{Rb}} \right )$ is the Lorentz dispersion function, where ${v_m}$ is the probe laser frequency, and $v_{D1}^{Rb}$ is the resonance transition frequency of the D1 line for Rb atoms.

Finally, the output signal of the co-magnetometer can be expressed as:

$$\begin{array}{l} S = \eta {I_{in}}\theta {e^{\left( { - OD} \right)}}\\ {\rm{ }} = \eta {I_{in}}\frac{\pi }{2}l{r_e}{n_{Rb}}c\frac{{{\gamma _e}R_p^K{e^{\left( { - OD} \right)}}{\Omega _y}}}{{{\gamma _n}\left[ {{\gamma _e}^2{L_z}^2 + {{\left( {{R_{rel}} + R_p^K} \right)}^2}} \right]}}\left( { - f_{D1}^{Rb}{\mathop{\rm Im}\nolimits} \left[ {L\left( {{v_m} - v_{D1}^{Rb}} \right)} \right] + \frac{1}{2}f_{D2}^{Rb}{\mathop{\rm Im}\nolimits} \left[ {L\left( {{v_m} - v_{D2}^{Rb}} \right)} \right]} \right) \end{array}$$
Where $\eta$ is the photoelectric conversion efficiency of the detector, ${I_{in}}$ is the light intensity before the probe laser enters the cell, and $OD = {n_{Rb}}l{r_e}c\left ( {f_{D1}^{Rb}{\mathop {\rm Re}\nolimits } [L\left ( {{v_m} - v_{D1}^{Rb}} \right )] + f_{D2}^{Rb}{\mathop {\rm Re}\nolimits } [L\left ( {{v_m} - v_{D2}^{Rb}} \right )]} \right )$ is the optical depth.

In Eq. (5), ${R_{rel}}$, ${n_{Rb}}$ and $R_p^K$ are the cell temperature related terms in the rotation angle $\theta$, and the same term in the optical depth $OD$ is ${n_{Rb}}$. Among the ${R_{rel}}$, the spin destruction collision relaxation rate $R_{sd}^e$ accounts for the largest proportion, and its expression is:

$${R_{rel}} \approx R_{sd}^e = \frac{{{n_K}}}{{{n_{Rb}}}}R_{sd}^K + R_{sd}^{Rb} = \frac{{{n_K}}}{{{n_{Rb}}}}\sigma _{sd}^{K - K}\bar v{n_K} + \sigma _{sd}^{Rb - Rb}\bar v{n_{Rb}}$$
Where $\sigma _{sd}^{K - K}$, $\sigma _{sd}^{Rb - Rb}$ are the spin-destruction cross sections for K atoms self and Rb atoms self respectively; ${n_K}$ is the density of K atoms; $\bar v$ is the relative thermal velocity of the atoms, and its rate of change with temperature can be ignored. The change in atomic density affects the optical depth of the pump light and thus the pumping rate. The other temperature independent terms are defined as $U$, which can be expressed as
$$U = \eta {I_{in}}\frac{\pi }{2}l{r_e}c\left( { - f_{D1}^{Rb}{\rm{Im}}\left[ {L\left( {{v_m} - v_{D1}^{Rb}} \right)} \right] + \frac{1}{2}f_{D2}^{Rb}{\rm{Im}}\left[ {L\left( {{v_m} - v_{D2}^{Rb}} \right)} \right]} \right)\frac{{{\gamma _e}}}{{{\gamma _n}}}$$

The relationship between saturation atomic density and temperature can be expressed by the empirical formula [26]:

$${n^S} = \frac{{{{10}^{21.866 + A - B/T}}}}{T}$$
Where $T$ is the thermodynamic temperature, $A$ and $B$ are constants related to the type of atom. $A$ and $B$ of the K atom are 4.402 and 4453 respectively; $A$ and $B$ of the Rb atom are 4.312 and 4040 respectively. According to Raoult’s law, the density of Rb atoms in the cell is ${n_{Rb}} = \alpha n_{Rb}^S$, and the density of K atoms in the cell is ${n_K} = \left ( {1 - \alpha } \right )n_K^S$. The value of $\alpha$ is calculated by measuring the density of alkali metals and $0 \le \alpha \le 1$ [27].

Therefore, the relationship between the output signal of the co-magnetometer and the temperature can be obtained by combining Eq. (5) – (8):

$$S = UR_p^K\frac{{{n_{Rb}}{e^{ - l\sigma \left( {{v_m}} \right){n_{Rb}}}}}}{{{{\left( {{\gamma _e}{L_z}} \right)}^2} + {{\left( {\sigma _{sd}^{K - K}\bar v\frac{{{n_K}^2}}{{{n_{Rb}}}} + \sigma _{sd}^{Rb - Rb}\bar v{n_{Rb}} + R_p^K} \right)}^2}}}{\Omega _y}$$

For ease of expression, Eq. (9) can be simplified to $S = {K_\Omega } \cdot {\Omega _y}$, where ${K_\Omega }$ is the scale factor of the co-magnetometer.

The model between the output signal and the cell temperature is established in the above analysis. It can be seen from Eq. (9) that the light shift ${L_z}$ and the pumping rates of the pump laser $R_p^K$ cannot be ignored when analyzing the influence of cell temperature on the SERF co-magnetometer, both of which are proportional to the pump laser intensity ${W_{pump}}$. The scale factor ${K_\Omega }$ represents the sensitivity of the SERF co-magnetometer to rotation, so it is used to evaluate the working point of the cell temperature. In the following, the relationship between the scale factor and the cell temperature, which consider the intensity of the pump laser, will be analyzed through experiments, so as to find the optimum working point of the cell temperature for the co-magnetometer to achieve the best performance.

3. Experimental setup

The K-Rb-$^{21}$Ne co-magnetometer used in the experiment is shown in the Fig. 1. We use a 10 mm diameter spherical glass cell made of GE180 aluminosilicate as the sensitive core. The glass cell is filled with a certain proportion of K metal and Rb metal in natural abundance, 1520 Torr of $^{21}$Ne and 30 Torr of N$_{2}$. The oven is used to fix and heat the glass cell. The heating coils covered on the surface of the oven use alternating current at 100 kHz to heat the glass cell. The Pt1000 platinum resistor, placed near the glass cell, is used to monitor the temperature of the glass cell. When the cell temperature changes, the amplitude of the alternating current through the heating coils changes at the same time. During system operation, the temperature fluctuation of the glass cell fed back by the Pt1000 platinum resistor is within 0.01 $^{\circ }\textrm {C}$, which is controlled by the proportional integral derivative (PID) controller. The multi-layer magnetic shields are used to isolate the environmental magnetic field and the triaxial field coils are used to compensate for the residual magnetic field, making the glass cell in a low magnetic field environment.

 figure: Fig. 1.

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

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The K atoms in the glass cell are polarized by the laser emitted from a distributed Bragg reflector (DBR) laser. The laser beam is expanded to a diameter of 8 mm by the beam shaping lenses. The pump laser intensity is stabilized and adjusted by an LCVR actuator, and the split beam from the Glan-Taylor polarizer in front of the polarizer is used as a feedback signal. Finally, the pump laser is converted into circularly polarized light by a quarter-wave plate. The pump laser power density acting on the atoms is freely adjustable from 19.9 mW/cm$^{2}$ to 79.6 mW/cm$^{2}$, and the laser wavelength is stabilized at the D1 line of K atoms by the saturated absorption system.

The spin precession of Rb atoms is accurately measured by using linearly polarized light emitted by a distributed feedback (DFB) laser. The diameter of the probe laser is 1.5 mm, and the laser intensity is stabilized in the same way as the pump laser. In this paper, the probe laser power density acting on the atoms is set to 56.6 mW/cm$^{2}$. After the glass cell, the probe laser through the half-wave plate and the polarization beam splitter (PBS) is detected by PD1 and PD2. The light intensity detected by PD1 and PD2 has been differentiated in the signal acquisition system, which can reduce the common mode noise.

4. Results and discussion

In order to achieve the atomic density required for the SERF regime, the cell generally needs to be heated to 160 $^{\circ }\textrm {C}\;\sim$ 190 $^{\circ }\textrm {C}$ [28]. Within this temperature range, the law between cell temperature and scale factor can be obtained by measuring the scale factor at different cell temperatures. Due to the limitations of the experimental conditions, the scale factor is calibrated by the Bx modulation method [12]. First, the atoms are placed in a zero magnetic field by applying the three-axis magnetic field. Then a sinusoidal magnetic field with an amplitude of 0.6 nT is applied to the x-direction under the frequency of 0.01Hz $\sim$ 0.04Hz, and the signal amplitude is measured as shown in Fig. 2. Finally, the scale factor ${K_\Omega }$ can be obtained by linear fitting, as shown in Fig. 3.

 figure: Fig. 2.

Fig. 2. Signal response of the co-magnetometer when a sinusoidal magnetic field of different frequency is applied in the x-direction. The frequency of the sinusoidal wave is: 0.01 Hz at 0 s $\sim$ 280 s, 0.02 Hz at 280 s $\sim$ 550 s, 0.03 Hz at 550 s $\sim$ 820 s and 0.04 Hz at 820 s $\sim$ 1100 s.

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 figure: Fig. 3.

Fig. 3. Linear fit of signal response amplitude and sine wave frequency. The slope of the line is the scale factor ${K_\Omega }$.

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In the range of 160 $^{\circ }\textrm {C}\;\sim$ 190 $^{\circ }\textrm {C}$, the scale factor was measured at different cell temperatures. Figure 4 shows the relationship between the scale factor and the cell temperature when the pump laser power density is 39.8mW/cm$^{2}$. The markers are the measurement results and the solid line is the fitted curve according to Eq. (9). It can be seen that the measurement data agrees well with the fitted curve. The scale factor first increases and then decreases with increasing cell temperature, and a turning point (maximum value) occurs when the cell temperature is near 172$^{\circ }\textrm {C}$. The SERF co-magnetometer has maximum signal value and minimum sensitivity to temperature variations when the cell temperature is 172$^{\circ }\textrm {C}$. So, this temperature is the optimum cell temperature under pump laser power density is 39.8mW/cm$^{2}$.

 figure: Fig. 4.

Fig. 4. Relationship between the scale factor and the cell temperature under the pump laser power density is 39.8mW/cm$^{2}$.

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According to Eq. (9), the optimum cell temperature ${T_{op}}$ satisfies the formula $\partial S\left ( {{T_{op}}} \right )/\partial {T} = 0$, which can be expressed as $R_{sd}^e({T_{op}}) \approx R_p^K$ by simplifying the term of a relatively small order of magnitude. $R_p^K$ decreases with increasing temperature and $R_{sd}^e$ changes in the opposite direction. At lower temperature, the atomic density within the cell is lower and as the temperature increases more atoms are polarized, thus increasing the scale factor of the device. At higher temperature, however, the atomic density increases and the smaller pump light is not sufficient to polarize all the atoms, resulting in a subsequent decrease in the scale factor. The scale factor therefore changes with temperature. However, we have only discussed the selection of the optimal cell temperature when the pump laser intensity is unchanged. Changing the pump laser intensity ${W_{pump}}$ will lead to a change in $R_p^K$ and thus affect the value of ${T_{op}}$. Therefore, we need to discuss the effects of the pump laser intensity on the relationship between the scale factor and the cell temperature.

Figure 5 shows the relationship between the scale factor and the cell temperature at different pump laser power densities. The markers are the measurement results, and the solid lines are the fitted curves according to Eq. (9). The results show that the scale factor has a maximum value with cell temperature change at each pump laser power density determined. As the pump laser power density increases, the maximum value of the scale factor first increases and then decreases, reaching an extremum when the pump laser power density is 59.7 mW/cm$^{2}$. At the same time, the cell temperature corresponding to the maximum scale factor increases as the pump laser power density increases. The scale factor reaches a maximum value corresponding to the cell temperature is 174$^{\circ }\textrm {C}$. It indicates that 174$^{\circ }\textrm {C}$ is the optimal working temperature for this cell to achieve the best performance in the SERF co-magnetometer.

 figure: Fig. 5.

Fig. 5. Relationship between the scale factor and the cell temperature under different pump laser power densities.

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In order to obtain the optimal working temperature of the cell quickly and accurately, bring ${W_{pump}}\left ( T \right )$ satisfying $R_p^K = R_{sd}^e$ into Eq. (9) and fit the relationship curve between the scale factor and the cell temperature, the results are shown with the curve and asterisks in Fig. 6. It can be seen that the measured results agree well with the fitted curve. Therefore, we can quickly and accurately find the optimal cell temperature using the established model compared to the traditional ergodic method.

 figure: Fig. 6.

Fig. 6. Relationship between scale factor and cell temperature for pump laser power density satisfying $R_p^K = R_{sd}^e$.

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To further analyze the influence of the cell temperature on the long-term stability of the SERF co-magnetometer, the Allan variance of the co-magnetometer signal was used for comparison. Five groups of signals are collected under different cell temperatures and corresponding pump laser power densities. The signal sampling frequency is 1200Hz and the acquisition time is 2h. The Allan variance results are shown in the Fig. 7. In the Allan variance, the lowest point represents the bias instability, which is usually used to evaluate the long-term stability of the co-magnetometer. The results show that the law of variation of the bias instability with increasing cell temperature first decreases and then increases. The bias instability is 0.0169 deg/h when the cell temperature is 174$^{\circ }\textrm {C}$, which is smaller than those under other cell temperatures, indicating that the co-magnetometer has the best long-term stability at this temperature.

 figure: Fig. 7.

Fig. 7. Allan deviation of the co-magnetometer signal under different cell temperatures.

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The above experimental results were all obtained under the conditions given in the paper. Changes in hardware, such as changing the volume of the cell, the design of the oven, and the location of the temperature monitoring point, may affect the selection of the optimum temperature point. However, the method presented in this paper is generic.

5. Conclusion

In this study, we have investigated the effect of the pump laser intensity on the optimal cell temperature working point in the SERF co-magnetometer. The model of the relationship between the signal output and the cell temperature is deduced. Through the model, a method for finding the optimal cell temperature of the co-magnetometer considering the pump laser intensity is proposed. Then, the bias instability of the SERF co-magnetometer under different cell temperatures and corresponding pump laser power densities was experimentally measured to verify the proposed method. The experimental results show that the bias instability of the co-magnetometer can be reduced from 0.0311 deg/h to 0.0169 deg/h by optimizing the appropriate cell temperature in the range of 160$\sim$190$^{\circ }\textrm {C}$. According to the method proposed in this paper, the optimal working point of the cell temperature can be selected quickly and accurately, which enables the co-magnetometer to ensure better performance and greatly shortens the working point optimization time after the instrument is started up.

Funding

National Science Fund for Distinguished Young Scholars (61925301); National Natural Science Foundation of China (62003024, 62103026).

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 (7)
Fig. 1.
Fig. 1. Schematic of the SERF co-magnetometer. ECU: electronic control unit, M: reflection mirror, GT: Glan-Taylor polarizer, PD: photodiode, P: Linear polarizer, LCVR: liquid crystal variable retarder, BSL: beam shaping lenses, PBS: polarization beam splitter, and RTD: resistance temperature detector.

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Fig. 2.
Fig. 2. Signal response of the co-magnetometer when a sinusoidal magnetic field of different frequency is applied in the x-direction. The frequency of the sinusoidal wave is: 0.01 Hz at 0 s $\sim$ 280 s, 0.02 Hz at 280 s $\sim$ 550 s, 0.03 Hz at 550 s $\sim$ 820 s and 0.04 Hz at 820 s $\sim$ 1100 s.

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Fig. 3.
Fig. 3. Linear fit of signal response amplitude and sine wave frequency. The slope of the line is the scale factor ${K_\Omega }$.

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Fig. 4.
Fig. 4. Relationship between the scale factor and the cell temperature under the pump laser power density is 39.8mW/cm$^{2}$.

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Fig. 5.
Fig. 5. Relationship between the scale factor and the cell temperature under different pump laser power densities.

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Fig. 6.
Fig. 6. Relationship between scale factor and cell temperature for pump laser power density satisfying $R_p^K = R_{sd}^e$.

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Fig. 7.
Fig. 7. Allan deviation of the co-magnetometer signal under different cell temperatures.

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Equations (9)

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$$\scalebox{0.9}{$\begin{aligned} \frac{{\partial {{\bf{P}}^{\bf{K}}}}}{{\partial t}} &= \frac{{{\gamma _e}}}{{{Q_K}}}({\bf{B}} + {{\bf{L}}^{\bf{K}}} + \lambda M_0^n{{\bf{P}}^{\bf{n}}}) \times {{\bf{P}}^{\bf{K}}} - {\bf{\Omega }} \times {{\bf{P}}^{\bf{K}}} + \frac{{R_p^K{{\bf{s}}_{\bf{p}}} + R_{se}^{en}{{\bf{P}}^{\bf{n}}} + R_{se}^{ee}({{\bf{P}}^{{\bf{Rb}}}} - {{\bf{P}}^{\bf{K}}}) - R_{tot}^e{{\bf{P}}^{\bf{K}}}}}{{{Q_K}}}\\ \frac{{\partial {{\bf{P}}^{{\bf{Rb}}}}}}{{\partial t}} &= \frac{{{\gamma _e}}}{{{Q_{Rb}}}}({\bf{B}} + {{\bf{L}}^{{\bf{Rb}}}} + \lambda M_0^n{{\bf{P}}^n}) \times {{\bf{P}}^{{\bf{Rb}}}} - {\bf{\Omega }} \times {{\bf{P}}^{{\bf{Rb}}}} + \frac{{R_m^{Rb}{{\bf{s}}_{\bf{m}}} + R_{se}^{en}{{\bf{P}}^{\bf{n}}} + R_{se}^{ee}({{\bf{P}}^{\bf{K}}} - {{\bf{P}}^{{\bf{Rb}}}}) - R_{tot}^e{{\bf{P}}^{{\bf{Rb}}}}}}{{{Q_{Rb}}}}\\ \frac{{\partial {{\bf{P}}^{\bf{n}}}}}{{\partial t}} &= {\gamma _n}({\bf{B}} + \lambda M_0^e{{\bf{P}}^{\bf{K}}} + \lambda M_0^e{{\bf{P}}^{{\bf{Rb}}}}) \times {{\bf{P}}^{\bf{n}}} - {\bf{\Omega }} \times {{\bf{P}}^{\bf{n}}} + R_{se}^{ee}({{\bf{P}}^{\bf{K}}} + {{\bf{P}}^{{\bf{Rb}}}}) - R_{tot}^n{{\bf{P}}^{\bf{n}}} \end{aligned}$}$$
$$P_x^{Rb} = \frac{{{\gamma _e}P_z^KR_{tot}^e}}{{{{\left[ {{\gamma _e}\left( {{L_z} + \delta {B_z}} \right)} \right]}^2} + R{{_{tot}^e}^2}}}\left[ {{L_y} + \frac{{{\Omega _y}}}{{{\gamma _n}}} + \frac{{{\gamma _e}}}{{R_{tot}^e}}({L_z} + \delta {B_z}) \times \left( {{L_x} - \frac{{{\Omega _x}}}{{{\gamma _n}}}} \right)} \right]$$
$$P_x^{Rb} \approx \frac{{{\gamma _e}R_p^K}}{{{\gamma _e}^2{L_z}^2 + {{\left( {{R_{rel}} + R_p^K} \right)}^2}}}\left( {\frac{{{\Omega _y}}}{{{\gamma _n}}}} \right)$$
$$\theta = \frac{\pi }{2}l{r_e}{n_{Rb}}cP_x^{Rb}\left( { - f_{D1}^{Rb}{\mathop{\rm Im}\nolimits} \left[ {L\left( {{v_m} - v_{D1}^{Rb}} \right)} \right] + \frac{1}{2}f_{D2}^{Rb}{\mathop{\rm Im}\nolimits} \left[ {L\left( {{v_m} - v_{D2}^{Rb}} \right)} \right]} \right)$$
$$\begin{array}{l} S = \eta {I_{in}}\theta {e^{\left( { - OD} \right)}}\\ {\rm{ }} = \eta {I_{in}}\frac{\pi }{2}l{r_e}{n_{Rb}}c\frac{{{\gamma _e}R_p^K{e^{\left( { - OD} \right)}}{\Omega _y}}}{{{\gamma _n}\left[ {{\gamma _e}^2{L_z}^2 + {{\left( {{R_{rel}} + R_p^K} \right)}^2}} \right]}}\left( { - f_{D1}^{Rb}{\mathop{\rm Im}\nolimits} \left[ {L\left( {{v_m} - v_{D1}^{Rb}} \right)} \right] + \frac{1}{2}f_{D2}^{Rb}{\mathop{\rm Im}\nolimits} \left[ {L\left( {{v_m} - v_{D2}^{Rb}} \right)} \right]} \right) \end{array}$$
$${R_{rel}} \approx R_{sd}^e = \frac{{{n_K}}}{{{n_{Rb}}}}R_{sd}^K + R_{sd}^{Rb} = \frac{{{n_K}}}{{{n_{Rb}}}}\sigma _{sd}^{K - K}\bar v{n_K} + \sigma _{sd}^{Rb - Rb}\bar v{n_{Rb}}$$
$$U = \eta {I_{in}}\frac{\pi }{2}l{r_e}c\left( { - f_{D1}^{Rb}{\rm{Im}}\left[ {L\left( {{v_m} - v_{D1}^{Rb}} \right)} \right] + \frac{1}{2}f_{D2}^{Rb}{\rm{Im}}\left[ {L\left( {{v_m} - v_{D2}^{Rb}} \right)} \right]} \right)\frac{{{\gamma _e}}}{{{\gamma _n}}}$$
$${n^S} = \frac{{{{10}^{21.866 + A - B/T}}}}{T}$$
$$S = UR_p^K\frac{{{n_{Rb}}{e^{ - l\sigma \left( {{v_m}} \right){n_{Rb}}}}}}{{{{\left( {{\gamma _e}{L_z}} \right)}^2} + {{\left( {\sigma _{sd}^{K - K}\bar v\frac{{{n_K}^2}}{{{n_{Rb}}}} + \sigma _{sd}^{Rb - Rb}\bar v{n_{Rb}} + R_p^K} \right)}^2}}}{\Omega _y}$$
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