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Abstract
The transverse light-shift can induce non-negligible polarization error in the output signal of spin-exchange relaxation-free (SERF) co-magnetometer. In this paper, a novel method for rapid measurement of transverse light-shift based on the error of steady-state response of co-magnetometer is proposed firstly, then the sources of transverse light-shift in a compact SERF co-magnetometer is modeled and analyzed from three aspects: the non-ideal linear polarization of probe laser, the circular dichroism of the atomic spin ensembles, and the stress-induced birefringence effect of the cell wall. Furthermore, the decoupling and suppression methods of transverse light-shift based on a degree of circular polarization (DOCP) regulation scheme is presented, to realize the decoupling measurement of the transverse light-shift introduced by the whole co-magnetometer cell, and cancel it out with the non-ideal linear polarization of the probe laser. Eventually, the DOCP regulation scheme suggested in this paper achieves more than a 67% suppression ratio in transverse light-shift, and the short- and long-term performance of SERF co-magnetometer are improved due to the reduction of the coupling effect between the probe laser power and transverse field. Moreover, the measurement, decoupling and suppression methods provided in this paper also have the potential to be applied to other atomic sensors, such as the SERF magnetometers and nuclear spin co-magnetometers.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The SERF co-magnetometer contains two overlapping spin ensembles of alkali metal and noble gas inner a glass vapor cell [1]. With the effects of a resonant pump laser and high temperature, the electrons of alkali metal can be polarized and reach the SERF regime in an extremely weak magnetic field [2], in which the relaxation times and coherence are enhanced significantly [3,4]. Compared with nuclear magnetic resonance gyroscope, SERF co-magnetometer can achieve a higher sensitivity since the magnetic resonance linewidth cannot be broadened by the spin-exchange collisions. In addition, the magnetization produced by nuclear spin can follow and suppress the low frequency fluctuation on the transverse magnetic field such that SERF co-magnetometer has a self-compensation characteristic. Thus SERF co-magnetometer can be applied to inertial navigation as a high-precision rotation rate sensor [5–7]. Moreover, the SERF co-magnetometer has the potential to explore the frontiers of physics, like CPT and Lorentz violation [8–10], and search for spin-dependent forces [11–14].
The light-shift is produced when an off-resonant and circularly-polarized laser interacts with alkali metal atoms [15–17]. Since the light-shift cannot be consistently compensated by the coils as the residual field inner the shield barrel, the coupled spin ensembles with high sensitivity of magnetic field deviate from the ideal compensation point such that the response and performance are degraded [18–21]. In engineering, enabling an open-loop compensation of the magnetic field for light-shift can suppress the cross-talk effect introduced by light-shift in the dual-axis SERF co-magnetometers [22,23], and the optimization for density ratio of different alkali metal atoms can mix and zero the total light-shift of them in the hybrid pumping SERF co-magnetometers [24,25]. The other methods for suppression or elimination of the light-shift, for example the magic wavelength in atomic clock [26,27], are always not applicable for the atoms in SERF regime.
Along with the further improvement of accuracy and stability in SERF co-magnetometer, the negative effects of the previously neglected transverse light-shift along the probe beam begin to be revealed and start to be studied in recent years [20,28,29], and the best reported bias instability of 0.00846 deg/h has been achieved by the probe laser intensity working point optimization [30]. The accurate measurement of the transverse light-shift is the basis for its study. However, the transverse light-shift is not as easy to measure as the longitudinal one along the pump beam due to its lower magnitude [3]. When the pump beam is not activated, the zero filed response of the co-magnetometer to the probe light can be applied to estimate its pumping effects offline [2], but that response cannot reflect the real transverse light-shift in practice engineering as the introduction of strong coupling between the electrons and nuclear spins. Recently, researchers proposed a novel measurement method based on the correlation between self-compensation point and laser power [29], while it is time-consuming for stabilization of the apparatus and the accuracy cannot be guaranteed due to the excessive amount of fitting involved in the calculation. Therefore, there is an urgent need to develop and improve the rapidity and accuracy of the method for acquiring transverse light-shift.
Theoretically, the light-shift is mainly affected by the following three natures of laser: degree of circular polarization (DOCP), optical power, and frequency [31]. Thus, it is intuitive to consider the optimization of these three factors above to reduce the transverse light-shift. Actually, the relation between the transverse light-shift and laser frequency satisfies a dispersion curve [24,32], and the sole zero crossing point is at the resonant frequency of transition, which cannot be employed to the optical rotation angle detection, such that the transverse light-shift cannot be eliminated by a laser frequency optimization scheme in principle. Since the laser power is proportional to the light-shift, the small laser spot or low laser power scheme can reduce the transverse light-shift [30,33], but also limit the signal-to-noise ratio and sensitivity of SERF co-magnetometer. Eventually, in this paper, a DOCP regulation scheme is proposed for suppression the transverse light-shift. Furthermore, except to the non-ideal linearly polarization of probe laser [2,3,16,29], there is absence of the studies on other sources of DOCP, especially on the polarization error induced by the optical characteristics of co-magnetometer cell.
The structure of this paper is as follows. Section 2 proposes a novel measurement method for the transverse light-shift firstly, then models and analyzes the sources of transverse light-shift from three aspects: the probe laser, the circular dichroism of the atomic spin ensembles, and the stress-induced birefringence effect of the cell wall, according to which the decoupling and suppression methods for the total transverse light-shift based on the DOCP regulation scheme are further developed. Section 3 describes the experimental setup of SERF co-magnetometer and its key components. Section 4 and 5 demonstrate the experimental verification and related conclusions for the measurement principles as well as decoupling and suppression methods in Section 2, respectively.
2. Theoretical analysis of the transverse light-shift
2.1 Measurement method
The dynamic behaviors of the polarization of atomic spin ensembles in SERF co-magnetometer cell containing alkali metal and noble gas can be well described by the coupled Bloch equations [34,35].
The off-resonant and circularly-polarized laser interacts with atoms can induce light-shift, which can be regarded as a fictitious dc magnetic field experienced by atoms. Theoretically, due to the presence of a polarization beam splitter (PBS) in front of the cell, the linearly polarized laser can not produce the transverse light-shift. However, as the extinction ratio of the actual PBS can not be infinite, the incident probe laser always contains the circularly polarized laser power component that can not be completely eliminated, which is also the focus of the recent literature [28,29]. Considering the energy level splitting effect of the probe laser near the wavelength to the resonance of the D1 transition in Rb, the transverse light-shift can be expressed as follows [3].
where $\Phi$ is the photo flux proportional to the probe laser power $I_0$, i.e., $\Phi = I_0 / \hbar \nu$, and $\hbar \nu$ represents the energy of a single photon. $f$ is the oscillator strength of the D1 transition, $A$ is the cross sectional area of the incident laser, $\nu$ is the probe laser frequency, $\Gamma$ and $\nu _0$ are the full-width at half maximum and the central frequency of the D1 transition , respectively. $s_l$ is the DOCP of the incident probe laser and defined as the normalized difference in power for left-/right-handed circularly polarized laser [16].According to a series of theories around the self-compensation point regime [2,3,25], the coil around the co-magnetometer cell produces a longitudinal compensation field $B^{c}=-B^{e}_z-B^{n}_z$, and also applies the transverse fields to zero the ambient fields composed of the residual fields generated by the magnetic shield $\delta B_{x/y}$ and the transverse light-shift $L_x$ along the probe beam direction.
Furthermore, When considering to zero the magnetic field $B_x$ along the $x$-axis, the $B_xB_z$ dependence relation is derived from the steady-state solution of Eq. (1).
Considering to employ an asymmetric modulation with peak-peak amplitude of $\Delta B_z$ when the deviation from the compensation point $B_z$ is already close to zero, the corresponding transverse polarization satisfies that
The co-magnetometer output signal $S$ is determined by product of the transmission power $I$ and optical rotation angle $\theta$ as below, whereas the magnitude of coefficient $\kappa (\nu,N)$ is only affected by probe laser frequency $\nu$ and alkali metal density $N$.
From Eq. (8), ESSR can directly reflect the magnitude and sign of $B_x$. Ideally, if there is the absence of the transverse light-shift term in $B_x$, a constant field $B^{c}_x$ produced by the coil can achieve the compensation for the residual magnetic field $\delta B_x$ along the $x$-axis, and the relation of $I_0$-$\Delta S$ can be converted into a linear equation from a quadratic function. In that case, the probe laser power $I_0$ will have no effect on ESSR $\Delta S$ when $B_x$ is completely compensated to zero.
Therefore, even though the transverse field has been compensated perfectly already, ESSR will still vary once the fluctuation of laser power occur due to the existence of the transverse light-shift. For illustrating this phenomenon, a reasonable response profile of SERF co-magnetometer is plotted in Fig. (1). In additional, the polynomial coefficients of quadratic function (8) do not explicitly contain the transverse light-shift $L_x = \mathcal {G}I_0$, which means that one cannot obtain the exact information of $L_x$ by a simple polynomial fitting on the $I_0$-$\Delta S$ curve.
Fig. 1. The response profile under an applied asymmetric modulation field with peak-peak amplitude of $\Delta B_z$: (a) $I=I_0$, and the ambient field $B_x$ has been compensated perfectly by coil already. (b) The magnitude of transverse light-shift $L_x$ decreases since the probe laser switches to a lower power level $I<I_0$ and (c) increases when it switches to a higher one $I>I_0$. Though both two cases (b) and (c) can result in $B_x\ne 0$, they still can be identified by ESSR $\Delta S$.
Thereby we come up with an ingenious measurement method: first calibrate ESSR versus the ambient magnetic field $B_x$ near the transverse compensation point of SERF co-magnetometer under $M$ different probe laser powers $I_k$ ($k=1,2,\ldots,M$).
Based on the calibration results $\mathcal {C}_k$, one can measure the ESSR $\Delta S_1$ at the compensation point $B^{c}_{x1}$ of SERF co-magnetometer corresponding to the first laser power point $I_1$, then fix the compensation point at $B^{c}_{x1}$ and rapidly switch to the $M$-th laser power point in sequence to acquire their corresponding $\Delta S_k$, and the ratio $\mathcal {R}_k=\Delta S_k/\mathcal {C}_k$ denotes the deviation of the ambient magnetic field $B_{xk}$ of SERF co-magnetometer from the zero field under the $k$-th laser power point.
Ultimately, the transverse light-shift of SERF co-magnetometer under $k$-th probe laser power $I_k$ can be obtained as follows, the experimental procedure and measurement results based on the above measurement method are stated in detail in Section 4.
2.2 Decoupling and suppression methods
The co-magnetometer based on SERF regime is sensitive to weak fluctuations of the ambient magnetic field [18,36], yet the presence of transverse light-shift causes the coupling between the probe laser power and the transverse magnetic field, which means that fluctuations in the laser power unexpectedly introduce instability in the transverse magnetic field, thus making the long-term stability of SERF co-magnetometer further degraded. More than that, $L_x$ also results in variation in the transverse polarization $P^{e}_x$, which reduces the sensitivity of SERF co-magnetometer to the magnetic field.
Therefore, besides proposing the measurement method of transverse light-shift, we proceed to develop the decoupling and suppression methods against the transverse light-shift based on a DOCP regulation scheme.
According to the numerator of Eq. (2), the achievement of the elimination of transverse light-shift must satisfies $\Phi (\nu -\nu _0)s_l=0$. Considering that neither $\Phi =0$ nor $\nu =\nu _0$ can realize the detection of optical rotation, $s_l=0$ seems to be the only approach to completely eliminate $L_x$. However, due to the circular dichroism of the polarized atomic spin ensembles [2,3], whose different absorption for the left-/right-handed circularly polarized components will also result in elliptical polarization of the probe beam as it propagates within the cell. By the selection rule [37], the populations $\rho _{\pm }$ of the ground states $m_s=\pm 1/2$ only interact with $\sigma _{\mp }$ laser components, and the imaginary part of the refractive index $n_{\mp }$ of the atomic spin ensembles from the macroscopic point of view is shown below.
In this case, the circular dichroism expressed in Eq. (16) leads to a phase retardation $\delta$ in the vertical component of the beam with respect to the horizontal direction, which indicates the inevitable introduction of elliptical polarization of the probe beam by the circular dichroism of atomic spin ensembles.
The DOCP introduced by circular dichroism diminishes with the detuning of the probe laser, as shown in Fig. 2(a), but the magnitude of the pressure in the cell is proportional to the broadening of the linewidth, which makes the DOCP at the far detuning still non-negligible. Not only that, the length of the optical path through the cell also affects the DOCP, as shown in Fig. 2(b).
In addition to the above-mentioned circular dichroism, the spherical co-magnetometer cell wall can also introduce the elliptical polarization of the probe beam due to the stress-induced birefringence effect and the high temperature-induced thermal expansion effect [2]. Since the thermal expansion effect only slightly affects the optical path at the cell wall where the probe beam crosses , the birefringence effect is mainly considered in this section.
Considering the pressure difference between the inside and outside of the cell wall as the principal stress $\sigma _p$, a stress-optic analysis is performed on the micro wall volume at the incident point, as shown in Fig. 3. Define the normal vector of the surface of the micro wall volume as $\vec {n}$, and the probe beam with power $I$ enters the wall at an angle $\beta$ with $\vec {n}$ ($\beta$ = 0 when normal incidence), then the effective laser power component affected by stress-induced birefringence [38–40] is $I_{e} = I\sin ^{2}\beta$ . The refractive indices of wall and air are $n_w$ and $n_0$, respectively, which can derive the refraction angle $\beta '$. The residual stress contained in the cell wall material is $\sigma _r$.
Then according to the Wertheim law [41–43] and neglecting the Poisson’s effect, the stress-induced optical retardation can be deduced as follows.
where $\alpha$ is the stress optical coefficient and $d$ is the cell wall thickness. Hence the DOCP $s_w$ can be expressed further in the case that $\delta _\sigma$ and $\beta \approx 0$.In summary, considering the terms of DOCPs respectively introduced by the probe laser, circular dichroism of the atomic spin ensembles, and stress-induced birefringence effect of the cell wall simultaneously, all three DOCPs will contribute to the transverse light-shift with equal weight, thus the total one $L^{tot}_x$ experienced by the electrons can be modeled as below.
- 1. when at the decoupling point, i.e., $s_l=0$, the transverse light-shift introduced by probe laser is suppressed, but the total one $L_x^{tot}\ne 0$, which is consistent with the experimental results stated in Section 5. In this case, one can decouple and measure the value of $L^{cell}_x$ component introduced by the entire co-magnetometer cell.
- 2. when at the suppression point, i.e., $s_l=-s_w-\int _0^{\mathcal {L}}s_a(l)\mathrm {d}l$, the transverse light-shift introduced by the probe laser cancels the entire cell-induced one out and the zeroing of the total term $L_x^{tot}=0$ can be achieved.
Fig. 4. The theoretical results of variation curves of total transverse light-shift with DOCP at different probe laser powers. The colored solid and dashed lines represent the cases of probe laser frequency of $\nu =\nu _0\pm \Delta \nu$, respectively, where $\nu _0$ denotes the D1 transition of Rubidium 87 (about 377.1075 THz) and the applied detuning is $\Delta \nu =50$ GHz.
3. Experimental setup
In order to verify the validity and theoretical conclusions of the measurement, decoupling and suppression methods proposed in Section 2, we perform experiments in a compact SERF co-magnetometer as shown in Fig. 5. The structure of the apparatus is similar to those in our previous work [36,44], and the difference lies in the insertion of a rotatable quarter-wave plate (QWP) in front of the cell in the probe optical path in order to realize the DOCP regulation scheme. The 8 mm diameter spherical cell contains about 1.77 amagat of $^{21}$Ne, 50 Torr of nitrogen quenching gas and a mixture of potassium and rubidium. The density ratio between potassium and rubidium is about 1:180 obtained by the alkali metal absorption spectroscopy [45]. The co-magnetometer cell is heated to 458 K within the oven made of boron nitride, and the three-axis magnetic field coils can compensate the magnetic field inner the triple magnetic shields to an expected compensation point. Through the pump optical path, a distributed Bragg reflector (DBR) laser device generates the pump laser with a center frequency of the D1 transition of Potassium (about 389.2862 THz), which is locked by saturated absorption technology [46,47]. PBS7 and QWP are applied to produce a left-handed circularly polarized laser. A pair of planoconvex lenses PL1 and PL2 can expand the spot diameter to 8 mm to ensure that the pump beam can cover the entire cell. The remaining parts form the laser power stabilization system [34,48]. Then for the probe optical path, besides the same laser power stabilization system as above, the balanced polarimeter can realize the high-sensitivity measurement of the optical rotation angle. A QWP is mounted on the high-precision rotation mount with the part number PRM05, from Thorlabs to achieve the angle adjustment with the resolution of 10 arcmin.
Fig. 5. Schematic diagram of the SERF co-magnetometer. PBS, polarization beam splitter; LCVR, liquid crystal variable retarder; PD, photodiode; PL, planoconvex lenses.
4. Measurement method and experimental verification
According to the theoretical analysis of the measurement method in Section 2, the first step is to carry out the calibration of a series of coefficients $\mathcal {C}_k$. One can adjust the feedback voltage setting of photodiode PD1 to reach the switching between different incident probe laser power, for which 10 different laser power points to be tested are determined and listed in Table 1.
Table 1. Measurement data of the total transverse light-shift $L_x^{tot}$ at different probe laser power points.
The calibration results $\mathcal {C}_k$ can be obtained by determining the slope of the curve that ESSR versus the ambient field $B_x$, and Fig. 6(a) depicts the calibration process under the probe laser power of 3.54 mW. Theoretically, Eq. (10) indicates that the magnitude of calibration results $|\mathcal {C}_k|$ should be proportional to the probe laser power $I_k$, whereas in practice, the probe-induced pumping effect is amplified as the power increases, which shapes the profile closer to be quadratic as Fig. 6(b) shown. This quasi-quadratic correlation can also be manifested in the profile of scale factor versus probe laser power of the co-magnetometer.
Fig. 6. The calibration process and results of coefficients $\mathcal {C}_k$ ($k=1,2,\ldots,10$): (a) ESSR $\Delta S$ varies with the ambient magnetic field $B_x$ at the probe laser power point $I=3.54$ mW, and $\mathcal {C}_k$ can be obtained by fitting the slope of $B_x$-$\Delta S$ curve. (b) The magnitudes of experiment results of calibration coefficients $|\mathcal {C}_k|$ and scale factors both perform a nonlinear correlation with respect to the probe laser power.
Then, we choose the maximum laser power to be tested of $I_1=10.44$ mW as the starting point of $L_x$ measurement and implement the compensation field on the transverse and longitudinal directions based on it. After that, an asymmetric modulation with peak-peak amplitude of $\Delta B_z=0.8$ nT is employed to generate an ESSR signal. In that case, one can switch the probe laser power from $I_1$ to $I_N$ in sequence and record the corresponding ESSR data. According to the measurement method proposed in Section 2, whose nature is to achieve the measurement of the relative variation of $L_x$ between different laser power points, the relevant measurement results are shown in Fig. 7(a): The hollow circle and dashed line respectively represent the measurement results of the relative variation of transverse light-shift $\delta L_x$ and corresponding fitting line, where $\delta L_{x1}=0$ since $I_1$ is the start point. Ultimately, the $I$-$\delta L_x$ line can be translated to cross the origin to obtain the estimated results of total transverse light-shift, which is listed in Table 1, and the measurement error of $L_{x}$ is related to the fitting precision of the intercept of the $I$-$\delta L_x$ line, generally speaking, the error bars increase with the slope of the $I$-$\delta L_x$ line, i.e., the DOCP of probe laser $\partial L_x/\partial \Phi =s_l$. Afterwards, we have repeated the experiment 3 times in order to verify the results, and the mean values of repeated measurement results are as shown in Fig. 7(b).
Fig. 7. The measurement results of the total transverse light-shift. (a) The $I$-$\delta L_x$ line can be translated to cross the origin to obtain the estimated results of total transverse light-shift. (b) The mean values of 3 times repeated measurement results.
5. Decoupling and suppression methods and experimental verification
The theory presented in Section 2 indicates that there are three sources of the transverse light-shift of SERF co-magnetometer: the probe laser, the circular dichroism of the atomic spin ensembles, and the stress-induced birefringence effect of the cell wall. Therefore, a single minimization of the magnitude of DOCP of the probe beam to approximate $s_l=0$ may not achieve the complete elimination of $L_x^{tot}$. To validate this point of view, a QWP is inserted into the optical path between PBS3 and cell, and a high-precision rotation mount is implemented to achieve the angle adjustment with the resolution of 10 arcmin such that one can finely regulate the DOCP $s_l$ of the incident probe laser.
Before we integrate these elements above on the compact SERF co-magnetometer, the fast axis of the QWP is roughly aligned with the $0^{\circ }$ scale of the mount, and the $0^{\circ }$ direction of the fast axis is defined as the $z$-axis direction, thus the scale value of the mount $\theta _M$ can coarsely represent the fast axis angle $\theta _{FA}$ of the QWP. Then the DOCP regulation function was examined offline by the rotating quarter-wave plate method [49], the relation between $\theta _M$ and DOCP $s_l$ of incident laser in a narrow range around $\theta _M=90^{\circ }$ is calibrated, and the results are shown in Fig. 8. Theoretically, the DOCP $s_l$ of the probe laser is perfectly sinusoidal with respect to the fast axis angle $\theta _{FA}$ of the QWP, and the derivative of the theoretical curve (black line) at $\theta _{FA}=90^{\circ }$ is $\pi /90$, which is agree with the slope of the fitting curve (red dash line) of the experimental data. According to that, DOCP $s_l$ can be considered to vary linearly in the range of $\theta _M=90^{\circ }\pm 4^{\circ }$. Due to device performance and mounting errors, etc., the actual test results indicate a deviation of about $0.5^{\circ }$ between $\theta _M$ and $\theta _{FA}$, thus the zero crossing point (green pentagram) implies that $\theta _M=90^{\circ } 33'$ is the actual decoupling point.
Fig. 8. The offline measurement for the DOCP of the probe laser by the rotating quarter-wave plate method, where the DOCP can be regulated by a QWP with rotation mount.
The rotation angle of the mount of QWP, i.e., $\theta _M$, is regulated to adjust the DOCP $s_l$ of the probe laser, based on which the total transverse light-shift $L^{tot}_x$ is measured under different laser power. In order to make the results more intuitive, the logarithmic axes are employed to plot them as Fig. 9(a) depicted, which demonstrates that the magnitude of transverse light-shift grows with the probe laser power, and the direction of the variation is related to the positive or negative sign of the introduced DOCP $s_l$. Then the relation between $\theta _M$ and $L^{tot}_x$ is as shown in Fig. 9(b), which agrees with the theoretical profiles plotted in Fig. 4, Section 2. It is noteworthy that a certain experimental point $\theta _M=90^{\circ } 50'$ (purple hexagram) is very close to the theoretical location of the suppression point, and the location of decoupling point is obtained from the zero crossing point in Fig. 8, i.e., $\theta _M=90^{\circ } 33'$ (green pentagram). In addition, the error bars of the measurement results are improved with the decreasing of the magnitude of DOCP due to the fitting precision.
Fig. 9. The measurement results of the total transverse light-shift under different conditions: (a) The relation between $L^{tot}_x$ and probe laser power under different DOCP $s_l$, the DOCP $s_l$ is regulated by $\theta _M$. The measurement result of the green dash line is obtained from Section 4 and there is absence of QWP. (b) The relation between $L^{tot}_x$ and $\theta _M$ under different probe laser power, where the decoupling point is obtained from the fitted zero crossing point in Fig. 8 and $\theta _M=90^{\circ } 50'$ is very close to the theoretical location of the suppression point.
According to Fig. 9(a), by regulating the DOCP of the probe laser into the suppression point, the transverse light-shift is clearly reduced and related data is recorded in Table 2. Then from Fig. 9(b), one can directly read out the decoupling measurement results of the transverse light-shift introduced by the whole cell of the compact SERF co-magnetometer at each laser power point under the fitted decoupling point $\theta _M=90^{\circ } 33'$, which satisfies the expression $L^{cell}_x = I\cdot -43.86$ pT/mW.
Table 2. Measurement data of the total transverse light-shift $L_x^{tot}$ at decoupling point and suppression point.
By this DOCP regulation scheme, the sensitivity of the transverse light-shift to the probe laser power fluctuations is also significantly suppressed, which can be examined by the measurement of ESSR and as shown in Fig. 10. The variation of the transverse magnetic field caused by the fluctuation of probe laser power can significantly result in the ESSR, while this magnetic-optic coupling effect is well suppressed by the canceling of $L_x^{tot}$ at the suppression point. Furthermore, the specific theoretic expression of the quadratic curves in Fig. 10 is mentioned as Eq. (8).
Fig. 10. The quadratic variation curves of ESSR with probe laser power under different DOCP $s_l$. The inset shows the quadratic profiles of ESSR on the linear axes.
Fig. 11. The long-term stability and sensitivity performance of SERF co-magnetometer under different DOCP $s_l$ and different probe laser power. (a) The bias stability of SERF co-magnetometer output signal under different conditions, which can be represented by the flat line’s corresponding Allan standard deviation value [51]. (b) The sensitivity values of SERF co-magnetometer output signal at a frequency of 1 Hz under different conditions. (c) The Allan standard deviation curves with a certain probe laser power $I=3.54$ mW under different DOCP $s_l$. (d) The sensitivity curves with a certain probe laser power $I=10.44$ mW under different DOCP $s_l$.
Furthermore, DOCP can also affect the long-term stability and sensitivity performance of SERF co-magnetometer, as depicted in Fig. 11. The total transverse light-shift $L_x^{tot}$ increases apparently as the improvement of DOCP $s_l$, which leads the transverse field to deviate from zero and reduces the sensitivity of SERF co-magnetometer, as confirmed by the experiment results in Fig. 11(b) and 11(d). In order to consider both the efficiency of the experiment and the accurate assessment of long-term stability, the output signal of SERF co-magnetometer is collected for 2 hours under different conditions, according to which the Allan deviation can be obtained to present the bias instability of SERF co-magnetometer. As shown in Fig. 11(a) and 11(c), when under the same probe laser power, the increasing of DOCP $s_l$ can explicitly result in a "hump" at the cluster time of about 10 s,which is consistent with some properties of exponentially correlated (Markov) noise [50], such that the bias instability gets worse. And at the cluster time of 500 s and beyond, Allan variance is dominated by the manifested rate ramp, which is mainly correlated to the slow change in temperature (including the cell temperature and ambient temperature) instead of the magnetic-optic coupling effect induced by the transverse light-shift. Therefore, by regulating the DOCP of probe laser to the suppression point under any probe laser power, the long-term stability and sensitivity performance of SERF co-magnetometer can both be effectively improved.
6. Conclusion
In conclusion, in this paper, we firstly proposed a novel measurement method for fast measurement of the transverse light-shift based on the ESSR of co-magnetometers, whose error depends only on DOCP of the probe laser. Then we model and analyze the sources of transverse light-shift in a compact SERF co-magnetometer from three aspects: the non-ideal linear polarization of probe laser, the circular dichroism of the atomic spin ensembles, and the stress-induced birefringence effect of the cell wall. According to the theory model, we further suggest the decoupling and suppression methods of transverse light-shift based on a DOCP regulation scheme. When at the decoupling point, the transverse light-shift is induced by the whole co-magnetometer cell only, such that we can obtain the decoupling measurement results $L_x^{cell}=I\cdot -43.86$ pT/mW, and at the suppression point, the transverse light-shift introduced by the probe laser and the whole cell cancel each other out, such that the total transverse light-shift can be zeroing theoretically. The experimental results demonstrate that our method achieves the total transverse light-shift suppression of more than $67\%$, and the short- and long-term performance of SERF co-magnetometer are improved due to the reduction of the coupling effect between the probe laser power and transverse field. In practical engineering, further reduction of transverse light-shift can be realized by employing a cubic cell wall instead of the spherical one, and the properly adjustment of the pressure in the cell and the probe laser frequency. Moreover, the measurement, decoupling and suppression methods provided in this paper also have the potential to be applied to other atomic sensors, such as the SERF magnetometers and nuclear spin co-magnetometers.
Funding
Special Project for Research and Development in Key areas of Guangdong Province (2021B0101410005); National Natural Science Foundation of China (62103026, 61673041).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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