1. Introduction

Sensitive atomic magnetometers and comagnetometers are used in various tests of fundamental physics experiments [1,2], including testing for CPT and Lorentz symmetries violation [3,4] and searching for exotic spin-dependent forces [5–7]. They are also used in precision measurements, such as inertial rotation measurement [8,9] and biomagnetism [10]. The spin-exchange (SE) relaxation often plays a dominant role in the broadening of their magnetic linewidths and thus limits the sensitivities [11]. However, when operating with high alkali-metal density and low magnetic field, the SE relaxation can be dramatically eliminated [12,13]. Operating in this spin-exchange relaxation-free (SERF) regime, the comagnetometer has achieved a sensitivity of $5\times 10^{-7}$ rad/s/Hz$^{1/2}$ based on a K-$^3\rm {He}$ comagnetometer, which initiated new possibilities for precision measurements [3,8].

To suppress the low-frequency magnetic noise, a compensation magnetic field is applied to operate the SERF comagnetometer in the self-compensation (SC) regime, where the interactions between alkali-metal and noble-gas atoms suppress the slow magnetic field changes, leaving the comagnetometer only sensitive to anomalous interactions and inertial rotation [8,14]. This compensation field is directed along the pumping direction with its value set to cancel the effective magnetic fields created by the magnetizations of noble-gas and alkali-metal spins, denoted by ${{\textbf B}^{\textbf e}}$ and ${{\textbf B}^{\textbf n}}$ respectively [8]. For the K-$^3$He comagnetometer, ${{\textbf B}^{\textbf e}}$ is very small (usually smaller than 1 nT), therefore K atoms experience a near-zero net magnetic field, $-{{\textbf B}^{\textbf e}}$, and work in the SERF regime [8]. In addition, to obtain the optimal sensitivity for typical SERF magnetometers and comagnetometers, the alkali-metal polarization should be equal to 0.5 [15,16].

However, we find these conditions do not hold in the K-Rb-$^{21}$Ne comagnetometer, where ${{\textbf B}^{\textbf e}}$ is approximately two orders of magnitude larger than in the K-$^3$He comagnetometer. When operating in the SC regime, the alkali spins in the K-Rb-$^{21}$Ne comagnetometer experience a net magnetic field in the order of 100 nT. Therefore, the SE relaxation is only partially suppressed, but not eliminated. It significantly broadens the magnetic linewidth and degrades the sensitivity of the SERF comagnetometer. In addition, the SE relaxation shifts the traditional optimal polarization value for sensitivity maximization. In this work, we investigate the broadening of the magnetic linewidth by the SE relaxation in the SC regime. An intuitive model is presented to describe the mechanism of the broadening by ${{\textbf B}^{\textbf e}}$. A new method is proposed for measuring the SE relaxation in the SC regime, where alkali-metal and noble-gas spin ensembles are coupled [8,14], which makes traditional methods unusable [11,17]. With different pump-light intensities and temperatures, the relationship between the SE relaxation and polarization is studied, and compared with the theoretical model at low polarization. We also study the shift of the optimal polarization under different conditions and compare it with the case without SE relaxation. Meanwhile, we experimentally find the SE relaxation is reduced by using $^{87}$Rb atoms and thus to improve the scale factor.

2. Theory

The K-Rb-$^{21}$Ne comagnetometer consists of a spherical cell that contains a small droplet of K-Rb alkali metal, $^{21}$Ne and N$_{2}$ gases. The molar fraction of K atoms in the K-Rb mixture is kept small, such that the density ratio of K to Rb $D_r=n_K/n_{Rb}$ is usually about 0.01. To reduce the polarization gradient of alkali spins and improve the polarization of noble-gas nuclear spins, the hybrid optical pumping is applied [18–20], i.e., the low-density K atoms are directly polarized by pump light and transfer polarization to high-density Rb atoms through SE interactions. The $^{21}$Ne nuclear spins are then polarized by SE interaction with alkali-metal atoms. The K-Rb-$^{21}$Ne comagnetometer can be characterized by the coupled Bloch equations for K, Rb and $^{21}$Ne polarizations [21,22]. Because the interactions between alkali-metal atoms and $^{21}$Ne are dominated by the Rb-$^{21}$Ne pair due to the small $D_r$, as well as K and Rb have the same polarization due to the rapid SE interactions, the Bloch equations can be simplified by including the equation of K into that of Rb, leading to the simplified equations coupling the Rb electron polarization $\textbf{P}^{\textbf{e}}$ and $^{21}$Ne nuclear polarization $\textbf{P}^{\textbf{n}}$ [6,9],

Typically, the magnetic field gradients inside the magnetic shields are smaller than 1 nT/cm after degaussing, which makes the relaxation due to magnetic-field gradient negligible [15,23]. The relaxation attributed to wall collisions also can be ignored because of the high-pressure buffer gas [15]. Besides, the frequency of probe light is detuned away from the Rb D1 line by about 180 GHz, such that the relaxation due to the probe light is negligible. In addition, the SE rate between Rb and $^{21}$Ne atoms is very small compared with $D_r R_p$. Therefore, $R^e_1$ and $R^e_2$ are

Inputting relatively small transverse magnetic fields $B_{x/y}$ and inertial rotations $\Omega _{x/y}$, Rb transverse polarization $P^e_x$, which is proportional to the measured signal $S^e_x=\eta P^e_x$ by a factor $\eta$, is given as follows [25,26]:

For quasistatic input excitations, $P^e_x$ can be approximated by the steady part $P_x^{\textrm{steady}}$,

When the external magnetic fields are zeroed, the total field experienced by Rb electrons is the sum of the applied bias field $B_z$ and the effective field of the $^{21}$Ne atoms $B^n_z$. If $B_z$ is set to the zero-field point $\mathop B\nolimits _z^{\textrm{zero}} = - \mathop B\nolimits _z^n$, the effective field of $^{21}$Ne is canceled so that the net field experienced by Rb is zero. Therefore, $R^{\textrm{rel}}_{\textrm{se}}$ is eliminated and the comagnetometer operates in the SERF regime. Without $R^{\textrm{rel}}_{\textrm{se}}$, $R^e_2$ is equal to $R^e_1$, and the scale factor is $K=\gamma _eD_rR_p/(\gamma _n {R_1^e}^2)$. By solving $\delta K/\delta (\mathop D\nolimits _r \mathop R\nolimits _p ) = 0$, the optimal sensitivity occurs at $D_rR_p=R^e_{\textrm{sd}}$, leading to the corresponding optimal polarization $P^e_z=0.5$.

Generally, the comagnetometer is operated in the SC regime, where $B_z$ is set to the SC point $\mathop B\nolimits _z^c = - \mathop B\nolimits _z^e - \mathop B\nolimits _z^n$. Hence, $\delta \mathop B\nolimits _z$ equals zero and the response to the transverse magnetic fields is eliminated as expected in Eq. (5). However, in the SC regime, the net field for Rb, $-B^e_z$, is usually in the order of 100 nT in the K-Rb-$^{21}$Ne comagnetometer. This large field results in that $R^{\textrm{rel}}_{\textrm{se}}$ cannot be eliminated and the comagnetometer no longer works in the SERF regime. This is different from the SERF K-$^{3}$He comagnetometer, where $B^e_z$ is about two orders of magnitude smaller according to Eq. (3), because the enhancement factor $\kappa _{\textrm{0}}=5.9$ of K-$^{3}$He pair [14] is about 6 times smaller than $\kappa _{\textrm{0}}=35.7$ of Rb-$^{21}$Ne pair [24] and the density of K is smaller than Rb at typical temperature. Figure 1 shows the intuitive description of the SERF regime and the dephasing of atoms precession by SE collisions with a large $B^e_z$. In Fig. 1(a), the alkali atoms in the upper and lower hyperfine states of the ground state are denoted by vectors ${\textbf F}_{\textbf a}$ and ${\textbf F}_{\textbf b}$. The atoms are pumped towards the $\left | {F_a , m_F=F_a} \right \rangle$ end state, so a greater statistical weight of atoms is in the $\left | {F_a } \right \rangle$ state and the vector ${{\textbf F}_{\textbf a}}$ is larger. Initially they are aligned, and then are rotated towards opposite direction at rates $\omega _0$ and $-\omega _0$ respectively by the magnetic field $B$ during the time in between SE collisions. SE collisions between two atoms conserve the total angular momentum but redistribute the atoms in the hyperfine states. The atoms have a probability of switching hyperfine states and changing the precession direction with each SE collisions, leading to the dephasing of the precession of the spin ensemble. In Fig. 1(b), when $B$ is sufficiently small (for exemple $B^e_z$ in the K-$^3$He comagnetometer) and the SE rate $R_{\textrm{se}}$ is very high such that $R_{\textrm{se}}\gg \omega _0$, the atoms switch hyperfine states rapidly and precess only an infinitesimal angle in between each SE collisions. Thus the atoms are locked together and ${{\textbf F}_{\textbf a}}$ and ${\textbf{F}}_{\textbf b}$ are bounded to the sum vector ${{\textbf F}_{\textbf{a+b}}}$ to precess coherently at a slower rate $\omega =\omega _0/Q$ in the direction of ${{\textbf F}_{\textbf a}}$ due to the high statistical weight in the $\left | {F_a } \right \rangle$ state. Therefore, the SE collisions no longer cause dephasing of atoms precession. In Fig. 1(c), when $B^e_z$ becomes larger in the K-Rb-$^{21}$Ne comagnetometer, the precession angle of atoms in between each SE collisions becomes significant. The SE collisions bring ${{\textbf F}_{\textbf a}^{'}}$ and ${{\textbf F}_{\textbf b}^{'}}$ together along a new direction ${{\textbf F}_{\textbf{a+b}}^{'}}$ with the conservation of the total angular momentum ${{\textbf F}_{\textbf {a+b}}^{'}} = {{\textbf F}_{\textbf a}^{'}} + {{\textbf F}_{\textbf b}^{'}}$. But the length of the sum vector ${{\textbf F}_{\textbf {a+b}}^{'}}$ is smaller than the original sum vector ${{\textbf F}_{\textbf {a+b}}}$. Therefore, the SE collisions cause the dephasing of atoms precession and broaden the magnetic linewidth. Besides, $B^e_z$ rotates only the transverse component of the spins and thus causes only the broadening of $R^e_2$. $R^{\textrm{rel}}_{\textrm{se}}$ can be numerically analyzed based on the density matrix equation [17]. For low polarization and small magnetic field, it can also be approximated by [11]

 

Fig. 1. Vector diagram illustrating the dephasing of atoms precession by SE collisions with a large $B^e_z$. [a] Due to SE collisions, the atoms switch between hyperfine states and change the precession direction, leading to the dephasing of atoms precession. [b] When the magnetic field $B$ is sufficiently small that $R_{\textrm{se}} \gg \omega _0$, the atoms are locked together by the rapid SE collisions and precess coherently at a slower rate $\omega = \omega _0 / Q$. [c] When the magnetic field $B^e_z$ is large, the precession of atoms in between each SE collisions is significant, such that SE collisions would cause the dephasing of atoms precession again.

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When $R^{\textrm{rel}}_{\textrm{se}}$ is non-negligible, the magnetic linewidth $\mathop R\nolimits _2^e /({\textrm{2}}\pi Q)$ is broadened to be larger than $R_1^e /({\textrm{2}}\pi Q)$. The scale factor becomes $K=\gamma _eD_rR_p/[\gamma _n R_1^e(R_1^e+QR^{\textrm{rel}}_{\textrm{se}})]$, which is smaller than $K$ in the SERF regime, leading the degradation of sensitivity and the shift of the optimal $P^e_z$ from 0.5. The relationship between $QR^{\textrm{rel}}_{\textrm{se}}$ and $D_rR_p$ is very complicated, where $QR^{\textrm{rel}}_{\textrm{se}}$ increases with the square of $B^e_z$ that increases with $D_rR_p$, but $QR^{\textrm{rel}}_{\textrm{se}}$ decreases nonlinearly with $P^e_z$ that increases with $D_rR_p$. Thus, the complete analytic solution for the optimal condition by solving $\delta K/\delta ( D_r R_p ) = 0$ would be very complicated and not illuminating. We experimentally find $QR^{\textrm{rel}}_{\textrm{se}}$ is of the same magnitude as $D_rR_p$ at typical conditions, thus the new optimal $P^e_z$ is still around 0.5. It is convenient to ignore the dependence of $R^{\textrm{rel}}_{\textrm{se}}$ on $D_rR_p$ around $P^e_z=0.5$ to get an approximate condition for the optimal $P^e_z$, $D_r R_p \approx \sqrt {{{ R_{\textrm{sd}}^e }^2} + R_{\textrm{sd}}^e Q R_{\textrm{se}}^{\textrm{rel}} }$. Therefore, $P^e_z$ for maximizing the sensitivity is larger than 0.5 according to Eq. (6).

The $R^{\textrm{rel}}_{\textrm{se}}$ in the magnetometer can be obtained by the fit of $R^e_2$ as a function of different magnetic field, which is measured by the synchronous pumping technique [11] or the fit of the response to different frequency transverse fields [17] with different bias magnetic field. However, in the K-Rb-$^{21}$Ne comagnetometer, the alkali-metal and noble-gas spins are coupled around the SC point. Their transverse relaxation rates $R^e_2$ and $R^n_2$ are mixed together and depend on $B_z$, such that the traditional methods in the magnetometer cannot be applied to measure $R^e_2$ separately. The typical method to measure $R^e_2$ in the comagnetometer is based on the fit of the response to $\Delta B_y$ modulation as $B_z$ changing around $B_z^c$ [15],

3. Experimental results and discussion

The experimental setup is present in our previous work [22,25]. A $\Phi = 12$ mm spherical aluminosilicate glass cell filled with a small droplet of K-Rb alkali metal atoms (density ratio $\mathop D\nolimits _r \approx 1/100$), 2430 Torr of $^{21}$Ne (70$\%$ isotope enriched), and 54 Torr of N$_2$ (quenching gas) is heated using a PID-controlled 110 kHz AC electrical heater in a boron nitride ceramic oven. The oven is installed in a vacuum vessel made of PEEK to reduce air convection. The vacuum vessel is also the framework of the three-axis magnetic field coils and water-cooling tubes. It is enclosed in a magnetic shield consisting of a ferrite barrel and a 5-layer cylindrical $\mu$-metal shield. The magnetic field gradients inside the shield are all smaller than 1 nT/cm after degaussing, with the advantage of no need gradient coils. The low-density K atoms are pumped along $\hat {z}$ axis by a circularly polarized K D1 light, which is generated by a distributed feedback diode laser and amplified by a tapered amplifier (Toptica Photonics). The transverse polarization of Rb $\mathop P\nolimits _x^e$ is probed by a linearly polarized Rb D1 light along $\hat {x}$ axis. The probe light is measured by the photodetector after passing through a polarizer. To reduce the low-frequency noise, the probe light is modulated by a photo-elastic modulator at 50 kHz (Hinds Instruments), and demodulated by a lock-in amplifier (Zurich Instruments).

Figure 2 shows the measurement of $R^{\textrm{rel}}_{\textrm{se}}$ based on this new method. $B^c_z$ is measured as follows: scanning $\delta B_z$, the difference $\Delta S^e_x$ between the steady signals before and after applying a small step field $B_y$ is measured and fitted by Eq. (8) to obtain $B^c_z$. As shown in Fig. 2(a) by left-$\hat {y}$ axis, $\mathop B\nolimits _z^c {\textrm{ = 425}}{\textrm{.8}} \pm 0.1$ nT is obtained by fitting $\Delta S^e_x$ with a temperature 190 $^{\textrm{o}}$C and pump power density of 340 mW/cm$^2$. $\mathop B\nolimits _z^{\textrm{zero}} = 324 \pm 12$ nT is acquired by the frequency shift of the noble-gas spins precession (FSNP) method [26]: when the spin polarization of the Rb is flipped by reversing the polarization of the pump light, the shift in the precession frequency of $^{21}$Ne atoms $\Delta \omega = 2\mathop \gamma \nolimits _n \mathop B\nolimits _z^e$ can be used to obtain $B^e_z$; and then $\mathop B\nolimits _z^{\textrm{zero}} = \mathop B\nolimits _z^c + \mathop B\nolimits _z^e$. The transient precession of $P^e_x$ induced by a small step $B_y$ is fitted by Eq. (4) to acquire the decay rates, which are used to calculate $R^e_2/Q$ based on Eq. (9). Figure 2(b) depicts the measured $R^e_2/Q$ with different $B_z$. As expected, $R^e_2/Q$ reaches the minimum value, $276 \pm 23$ 1/s, at $B_z^{\textrm{zero}}$ point, because Rb atoms experience zero net field so that $R^{\textrm{rel}}_{\textrm{se}}$ is eliminated. While at $B_z^c$ point, ${{\mathop R\nolimits _2^e } \mathord {\left / {\vphantom {{\mathop R\nolimits _2^e } Q}} \right.} Q} = 455 \pm 59$ 1/s is larger than that at $B_z^{\textrm{zero}}$ point. This is consistent with the above analysis that $R^{\textrm{rel}}_{\textrm{se}}$ cannot be ignored in the SC regime and significantly broadens $R^e_2/Q$. The more $B_z$ deviates from $B_z^{\textrm{zero}}$, the larger the contribution of $R^{\textrm{rel}}_{\textrm{se}}$ to $R^e_2/Q$ is. The difference between the two $R^e_2/Q$ at $B_z^c$ and $B_z^{\textrm{zero}}$ points is $R^{\textrm{rel}}_{\textrm{se}}=179\pm 8$ 1/s in the SC regime. When the Larmor precession frequency is much less than the SE rate, $\omega \ll \mathop R_{\textrm{se}}$, $R^{\textrm{rel}}_{\textrm{se}}$ contributes only to $R^e_2/Q$ in the second order of the net magnetic field [11,17]. In this study, $\omega / R_{\textrm{se}}$ is approximately 0.01 $-$ 0.001, thus we fit the measured $R^e_2/Q$ with ${{\mathop R\nolimits _2^e } \mathord {\left / {\vphantom {{\mathop R\nolimits _2^e } Q}} \right.} Q} = a{(\mathop B\nolimits _z - \mathop B\nolimits _z^{\textrm{0}} )^2} + \mathop R\nolimits _1^e /Q$. The R-square of the fit is better than 0.98 and the fitted $B_z^{\textrm{0}}=322 \pm 5$ nT is consistent with the measured $B_z^{\textrm{zero}}$.

 

Fig. 2. [a] With different $B_z$, the difference $\Delta S^e_x$ between the steady signals before and after applying a small step field $B_y$ is measured and fitted by Eq. (8) to obtain $B^c_z$. The measured precession frequency of Rb atoms with different $B_z$ is fitted by $\omega ^e_0 = \gamma _e B^e_{\textrm{total}} /Q$ against right $\hat {y}$ axis. [b] The decay rates obtained by fitting the transient precession of $P^e_x$ based on Eq. (4) are used to calculate $R^e_2/Q$ based on Eq. (9). The measured $R^e_2/Q$ are fitted by $R^e_2/Q = 0.02353 (B_z-322)^2 + 258$. $R^e_2/Q$ reaches the minimum value near $B^{\textrm{zero}}_z$ point as expected.

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To varify the validity of the new method, $R^e_2/Q$ at $B_z^c$ point is measured by the fit of the response to $\Delta B_y$ modulation as $B_z$ changing around $B_z^c$, and compared with the result at the $B_z^c$ point measured above. Meanwhile, $R^e_1/Q$ is calculated based on parameters in Ref. [27] and measured values, and then compared with the result at the $B_z^{\textrm{zero}}$ point measured above. At $B_z^c$ point, $R^e_2= 2868 \pm 75$ 1/s is obtained by the traditional method as shown in Fig. 2(a) by left-$\hat {y}$ axis. Substitution of the Rb density, which is measured in situ to include the heating effect of the strong pump light [25], into the expression of $B_z^e$ yields $\mathop P\nolimits _z^e = 0.65 \pm 0.07$ [25,26]. $R^e_{\textrm{sd}}$ is calculated using parameters in Ref. [27] and then substituted in $\mathop R\nolimits _1^e {\textrm{ = }}\mathop R\nolimits _{\textrm{sd}}^e /(1 - \mathop P\nolimits _z^e )$ to get $R^e_1= 1602 \pm 172$ 1/s. The slowing-down factor $Q$ is measured by the resonance frequency of Rb atoms $\omega ^e_0=\gamma _e B^e_{\textrm{total}}/Q$. Changing $B_z$ around $B_z^c$, the resonance frequencies are measured and shown in Fig. 2(a) by right-$\hat {y}$ axis. The slope of the fitting line is $\gamma _e /Q$, giving $Q=6.99 \pm 0.37$. Therefore, at $B^c_z$ point $R^e_2/Q=410 \pm 11$ 1/s and $R^e_1/Q=229 \pm 12$ are in good agreement with the $R^e_2/Q$ measured at $B_z^c$ and $B_z^{\textrm{zero}}$ points by the new method respectively. That further verifies that at $B_z^{\textrm{zero}}$ point, $R^{\textrm{rel}}_{\textrm{se}}$ is eliminated and $R^e_2$ equals $R^e_1$, while at $B_z^c$ point $R^e_2$ is significantly broadened by the SE relaxation.

With different pump power intensities and temperatures, $R^e_2$ at $B^c_z$ and $B_z^{\textrm{zero}}$ points are measured. For clarity, we denote the $R^e_2$ at $B_z^{\textrm{zero}}$ point as $R^{e*}_2$. In Fig. 3, $R^e_2$ and $R^{e*}_2$ are plotted by the solid and hollow dots respectively. Because $D_rR_p$ is linear with the pump power density $I$, $R^e_2$ should increase linearly with $I$ according to Eq. (2), when $R^{\textrm{rel}}_{\textrm{se}}$ is eliminated in the SERF regime. However, $R^e_2$ at $B^c_z$ point increases with $I$ by a decreasing rate. This discrepancy is because $R^e_2$ in the SC regime is broadened by $QR^{\textrm{rel}}_{\textrm{se}}$, which increases with $I$ nonlinearly. When $I$ is small, $QR^{\textrm{rel}}_{\textrm{se}}$ due to $-B^e_z$ is small, such that the relationship between $QR^{\textrm{rel}}_{\textrm{se}}$ and $I$ can be described by the first-order approximation. Therefore $R^e_2$ at $B^c_z$ point with small $I$ can be fitted by $\mathop R\nolimits _2^e = {a_2} I + {b_2}$ as shown by solid lines in Fig. 3. Based on Eq. (2), the $\hat {y}$-intercept of the fitting line is $R^e_{\textrm{sd}}$. The fitted ${b_2} = 428 \pm 67$ 1/s, ${b_2} = 501 \pm 172$ 1/s and ${b_2} = 679 \pm 158$ 1/s for 180 $^{\textrm{o}}$C, 190 $^{\textrm{o}}$C, and 200 $^{\textrm{o}}$C respectively are consistent with the theoretically calculated counterparts $R^e_{\textrm{sd}} = 420 \pm 15$ 1/s, $R^e_{\textrm{sd}} = 566 \pm 27$ 1/s and $R^e_{\textrm{sd}} = 768 \pm 24$ 1/s. As expected, $R^{e*}_2$ at $B_z^{\textrm{zero}}$ point increases linearly with $I$ and are fitted well by $R^{e*}_2 = {a_1} I + {b_1}$ (dashed line) with R-square all better than 0.98. $R^e_2$ at $B^c_z$ point are significantly larger than $R^{e*}_2$ at $B_z^{\textrm{zero}}$ point and their difference increases with temperature. This is because the net field experienced by Rb increases with temperature, leading to the related $QR^{\textrm{rel}}_{\textrm{se}}$ increases with temperature.

 

Fig. 3. The solid (hollow) square, circular and triangular dots are measured $R^e_2$ ( $R^{e*}_2$) at $B^c_z$ ( $B_z^{\textrm{zero}}$) point with 180 $^{\textrm{o}}$C, 190 $^{\textrm{o}}$C, and 200 $^{\textrm{o}}$C respectively. The first-five $R^e_2$ data points at each temperature are fitted by the $R^e_2= a_2 I + b_2$, as shown by the solid lines. The $\hat {y}$-intercept of the fitting lines are ${b_2} = 428 \pm 67$ 1/s, ${b_2} = 501 \pm 172$ 1/s and ${b_2} = 679 \pm 158$ 1/s for 180 $^{\textrm{o}}$C, 190 $^{\textrm{o}}$C, and 200 $^{\textrm{o}}$C respectively, which are consistent with the theoretically calculated counterparts $R^e_{\textrm{sd}} = 420 \pm 15$ 1/s, $R^e_{\textrm{sd}} = 566 \pm 27$ 1/s and $R^e_{\textrm{sd}} = 768 \pm 24$ 1/s. As expected, $R^{e*}_2$ increases linearly with $I$. Hence, $R^{e*}_2$ data points are fitted well by $R^{e*}_2= a_1 I + b_1$ as shown by the dashed lines.

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To further study the relationship between $QR^{\textrm{rel}}_{\textrm{se}}$ and $P^e_z$, the measured results under different pump power intensities and temperatures are shown in Fig. 4. In Fig. 4(a), $P^e_z$ are fitted well by $\mathop P\nolimits _z^e {\textrm{ = }} a \mathop D\nolimits _r \mathop R\nolimits _p /\mathop R\nolimits _1^e$ with R-square all better than 0.98. In Fig. 4(b), $QR^{\textrm{rel}}_{\textrm{se}}$ scales with $P^e_z$ at the beginning, because the net field $-B^e_z$ is proportional to $P^e_z$ and the SE relaxation increases with the square of $-B^e_z$. Whereas $P^e_z$ continues to increase, $QR^{\textrm{rel}}_{\textrm{se}}$ begins to decrease inversely. This reduction results from that most of the Rb atoms are pumped into the $\left | {F = I + 1/2, m_F = I + 1/2} \right \rangle$ end state by the strong pump light; SE collisions between two atoms preserve the total angular momentum, such that the atoms in the end state are still in this end state after collisions and precess in the same direction with the same frequency; thus SE collisions cause no relaxation. This is similar to the light narrowing of magnetic resonance [28]. $QR^{\textrm{rel}}_{\textrm{se}}$ at low polarization and small magnetic field situation can be described by Eq. (7), thus we can substitute the measured electron polarization $P^e_z$ and the effective magnetic field $B^e_z$ into Eq. (7) to calculate $QR^{\textrm{rel}}_{\textrm{se}}$, and compare the calculated values with the directly measured results. In our experiment, natural Rb atoms ( 72.2 $\%$ $^{85}$Rb and 27.8 $\%$ $^{87}$Rb) are used and $^{85}$Rb ($I=5/2$) and $^{87}$Rb ($I=3/2$) atoms are bound together by rapid SE collisions. Therefore, $QR^{\textrm{rel}}_{\textrm{se}}$ of the naturally abundant Rb is the weighted average of $^{85}$Rb and $^{87}$Rb atoms according to their density ratio, $Q\mathop R\nolimits _{\textrm{se}}^{\textrm{rel}} {\textrm{ = 0}}{\textrm{.722}}\mathop R\nolimits _{{{\textrm se}\_5/2}}^{\textrm{rel}} {Q_{5/2}} + 0.278\mathop R\nolimits _{{{\textrm se}\_3/2}}^{\textrm{rel}} {Q_{3/2}}$. We substitute the measured $P^e_z$ and $B^e_z$ into Eq. (7) to calculate $QR^{\textrm{rel}}_{\textrm{se}}$. The calculated results are shown in Fig. 4(b) with dashed-dotted lines. Although the calculated $QR^{\textrm{rel}}_{\textrm{se}}$ are slightly smaller than the measured values, the relationships between the calculated $QR^{\textrm{rel}}_{\textrm{se}}$ and $P^e_z$ are consistent with those of the measured results.

 

Fig. 4. [a] The measured $P^e_z$ at different temperatures are denoted by dots and fitted well by the solid curves. [b] $QR^{\textrm{rel}}_{\textrm{se}}$ as functions of $P^e_z$ at different temperatures. The solid dots represent the measured $QR^{\textrm{rel}}_{\textrm{se}}$, which scales with $P^e_z$ at the beginning, because $B^e_z$ is proportional to $P^e_z$. Whereas $P^e_z$ continues to increase, $QR^{\textrm{rel}}_{\textrm{se}}$ begins to decrease inversely, which is because most of the Rb atoms are pumped into the $\left | {F = I + 1/2,{m_F} = I + 1/2} \right \rangle$ end state by the strong pump light. The dashed-dotted curves are the calculated $QR^{\textrm{rel}}_{\textrm{se}}$, which are calculated by substituting the measured $P^e_z$ and $B^e_z$ into Eq. (7).

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Figure 5(a) shows the relationships between the scale factors $K$ and $P^e_z$ at different temperatures and pump light intensities. The values of $P^e_z$ for maximizing $K$ are all larger than the traditional value, 0.5, for different temperatures, which is consistent with the above analysis that the optimal $P^e_z$ is shifted to be larger than 0.5 by the SE relaxation . Moreover, the shift of the optimal $P^e_z$ increases significantly with temperature. This is potentially caused by that $B^e_z$ increases with temperature; thus $QR^{\textrm{rel}}_{\textrm{se}}$ increases with $B^e_z$ and the approximated optimal pumping rate $D_r R_p \approx \sqrt {{{ R_{\textrm{sd}}^e }^2} + R_{\textrm{sd}}^e Q R_{\textrm{se}}^{\textrm{rel}} }$ becomes larger, leading to a higher optimal $P^e_z$ according to Eq. (6). Besides, this is probably related to that the polarization gradient is larger at higher temeprature due to a stronger light absorption, such that a higher $P^e_z$ is needed to reduce the polarization gradient and thus improve the scale factor [15]. When $P^e_z$ is higher than the optimal value, the scale factors for 180 $^{\textrm{o}}$C, 190 $^{\textrm{o}}$C begin to fall down as expected. Due to the large optimal $P^e_z$ (around 0.7) and the small maximum achievable $P^e_z$ (around 0.75), there is no significant falling of the scale factor for 200 $^{\textrm{o}}$C in the limited range of polarization. Because the suppression of low-frequency magnetic noise in the comagnetometer increases with the noble-gas polarization $P^n_z$ in the SC regime [8], this higher optimal $P^e_z$ can generate a higher $P^n_z$ according to Eq. (6) and improve the suppression ability. To illustrate the influence of SE relaxation on $K$, we calculate the scale factor $K^*$ in SERF regime and compare it with the measured $K$. According to Eq. (5), in the SC regime $\delta B_z=0$, the scale factor is $K^* = P_{z0}^e \gamma _e / (\gamma _n R_2^{e*})$. We replace the actual $R^e_2$ in the working condition with $R^{e*}_2$ measured at $B_z^{\textrm{zero}}$ point, which is free from the SE relaxation and euqal to $R^e_1$, to calculate $K^* = K R_2^e / R^{e*}_2$. As shown in Fig. 5(a) against right-$\hat {y}$ axis (dashed-dotted curve), $K^*$ at 190 $^{\textrm{o}}$C are nearly twice larger than the measured $K$ at 190 $^{\textrm{o}}$C, indicating the sensitivity is significantly degraded by the SE relaxation. Although the optimal $P^e_z$ for $K^*$ is smaller than that of $K$, but not equal to 0.5 as expected, which is potentially due to the polarization gradient as discussed above.

 

Fig. 5. [a] The relationships between $K$ and $P^e_z$ at different temperatures are plotted against left-$\hat {y}$ axis. The optimal $P^e_z$ for maximizing $K$ are all larger than 0.5 as expected. To compare $K$ with scale factor $K^*$ in SERF regime, the $K^*=K R^e_2 / R^{e*}_2$ calculated by replacing $R^e_2$ with $R^{e*}_2$ is shown against right-$\hat {y}$ axis by dashed-dotted curve. The $K^*$ is significantly larger than $K$ as expected, indicating that the scale factor is degraded by the SE relaxation. [b] The measured $QR^{\textrm{rel}}_{\textrm{se}}$ of the $^{87}$Rb cell and the natural Rb cell at 200 $^{\textrm{o}}$C are plotted against left-$\hat {y}$ axis by solid dots, while the corresponding $K$ are plotted against right-$\hat {y}$ axis. At typical operation pump power densities, $QR^{\textrm{rel}}_{\textrm{se}}$ is reduced by replacing natural Rb atoms with $^{87}$Rb atoms, leading to a higher scale factor.

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To improve the sensitivity, it is necessary to reduce the SE relaxation. The slowing-down factors of $^{85}$Rb and $^{87}$Rb atoms are $Q_{5/2} = (38 + 52{{P_z^e }^2} + 6{{ P_z^e }^4})/ (3 + 10{{ P_z^e }^2} + 3{{ P_z^e }^4})$ and $Q_{3/2} = (6 + 2{{\mathop P\nolimits _z^e }^2})/(1 + {{\mathop P\nolimits _z^e }^2})$, which range from 6 to 12.7 and from 4 to 6 for different $P^e_z$ respectively. According to Eq. (7), $R^{\textrm{rel}}_{\textrm{se}}$ is inversely proportional to the square of $Q$, thus replacement of natural Rb with $^{87}$Rb would reduce $R^{\textrm{rel}}_{\textrm{se}}$ and improve the sensitivity. Another $\Phi = 14$ mm $^{\textrm{87}}$Rb cell, containing a mixed droplet of K-$^{\textrm{87}}$Rb metal atoms ($D_r \approx 81$ at 200 $^{\textrm{o}}$C), 2460 Torr $^{\textrm{21}}$Ne (70$\%$ isotope enriched) and 49 Torr N$_2$, which has the similar composition as the natural Rb cell except the Rb atoms, is applied to compare with the natural Rb cell. As shown in Fig. 5(b), at the same temperature 200 $^{\textrm{o}}$C, $QR^{\textrm{rel}}_{\textrm{se}}$ of the $^{\textrm{87}}$Rb cell is significantly smaller than the natural Rb cell at higher pump power density. Similarly, the scale factor $K$ of the $^{\textrm{87}}$Rb cell is larger than the natural Rb cell at higher pump power density. Therefore, replacing natural Rb with $^{\textrm{87}}$Rb can reduce $QR^{\textrm{rel}}_{\textrm{se}}$ and improve the sensitivity at the typical operating pump power density. At low pump power density, $QR^{\textrm{rel}}_{\textrm{se}}$ is small for both the natural Rb cell and the $^{\textrm{87}}$Rb cell, such that the contribution of $QR^{\textrm{rel}}_{\textrm{se}}$ to the total relaxation is insignificant compared with other relaxations. The influence of $QR^{\textrm{rel}}_{\textrm{se}}$ on $K$ is insignificant. Hence, although $QR^{\textrm{rel}}_{\textrm{se}}$ of the two cells are slightly different, their scale factors $K$ are similar under low pump power density.

4. Conclusion

We demonstrate the broadening of the magnetic linewidth of the K-Rb-$^{21}$Ne comagnetometer by the SE relaxation in the SC regime. The SE relaxation causes significant degradation of sensitivity and shift of the optimal alkali-metal polarization. We present an intuitive model to illustrate the mechanism of the broadening. A new method is proposed to measure the SE relaxation in the SC regime, and is verified using the theoretically calculated values. Under various conditions, the relationships between the SE relaxation, the scale factor and the alkali-metal polarization are studied. The theoretical model of SE relaxation for low-polarization situation is applied to evaluate the measured SE relaxation at low polarization. We also study the reduction of the SE relaxation by replacing the natural Rb atoms with the $^{87}$Rb atoms to improve the sensitivity. The new method to directly measure the SE relaxation rate is important for evaluating the contribution of SE relaxation to the transverse relaxation. Besides, the study of the cause of the SE relaxation and it’s dependence on the polarization is beneficial to search SE relaxation reducing method. The discovery of the influences of SE relaxation in the SC regime is important for improving the sensitivity of the K-Rb-$^{21}$Ne comagnetometer.