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Abstract
We investigate the effects of the density ratio of K-Rb hybrid cells on the alkali metal-noble gas comagnetometers. Bloch equations simplified with the density ratio and average-pumping-rate model are presented for numerical simulation, which simplifies equations of complete hybrid spin ensemble and problem of polarization gradient. The spin polarizations of electron and nucleon, total electronic relaxation rates, and the spin-exchange efficiencies are measured with cells of different density ratios. The results are in good agreement with our equivalent model. Based on our theoretical analysis, the K-Rb-21Ne comagnetometer achieves maximum output signal by optimizing the combination of density ratio and optical power density. The density ratio is critical to the homogeneity of spin polarization and efficiency of hyperpolarization. The method in this work finds a way to optimize the sensitivity of comagnetometers, which is significant for angular-rotation sensors and new physics research.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Spin-exchange relaxation free (SERF) magnetometers and comagnetometers are potent tools for detecting ultra-low field and non-magnetic interactions, respectively [1,2]. SERF magnetometers have been applied in bio-magnetic measurement, such as magnetoencephalography and magnetocardiography [3,4]. Meanwhile, SERF comagnetometers, which contain noble gases compared to magnetometers, are used in sensing angular-rotation and searching for preferred frames, fifth force, and axionic dark matter [5–8]. Higher sensitivity is required for those experimental studies. However, the polarization gradient and the low efficiency of hyperpolarized noble gases are limiting factors in magnetometers and comagnetometers of single alkali metal species [9,10]. To solve these problems, hybrid spin-exchange optical pumping (SEOP) technology has been applied to SERF magnetometers and comagnetometers [6,11], where pumping laser is used to polarize low-density alkali metal, and the other high-density alkali metal species are polarized through spin-exchange interactions with the low-density species.
The density ratio of two alkali metal species is the critical factor that affects the homogeneity of spin polarization and efficiency of hyperpolarization. Hybrid SEOP was used to improve the polarizing rate of $^{3}$He, which was important for magnetic resonance imaging, neutron spin filters and other precision measurements [12,13]. The works showed that spin-exchange efficiency was strongly dependent on density ratios of alkali metal mixtures. However, efficiency of hyperpolarization $^{21}$Ne was not stuided, and tens of watts of laser was not suitable for SERF magnetometers. Ito’s group has done many meaningful works on hybrid SEOP magnetometers and applied in magnetoencephalograms measurements [14–16]. They investigated the optimal densities in a K-Rb magnetometer considering the spatial distribution of spin polarization [14], and analyzed temperature characteristics depending on the density ratio [16]. However, the optimal conditions they calculated do not apply to comagnetometers affected by the interaction of electrons and nucleons. Romalis’s group used hybrid optical-pumping technology in the SERF comagnetometers for the first time [6]. Then many studies developed based on K-Rb-$^{21}$Ne comagnetometers [17–19]. The work [17] analyzed the effects of pump laser power density on comagnetometers and concluded the optimal working conditions. However, the conclusion is limited when the spin-exchange relaxation cannot be ignored [18]. The density ratio and the spin-exchange relaxation due to a large electron magnetic field are directly measured when comagnetometers work in the self-compensation regime [19]. Further studies are needed to investigate the effects of density ratio on comagnetometers.
In this study, we establish an effective electron model with the density ratio to substitute Bloch equations of two alkali metal species. Besides, average-pumping-rate model is used to simplify nonuniform polarization problem. The dynamic responses of the system are simulated with independent model of K-Rb Bloch equations and equivalent Bloch equations, respectively. Meanwhile, the signal intensity of comagnetometers is simulated as a function of density ratio and pump light power density. We use four cells of different density ratios to validate our model, experimentally. The spin polarizations of electron and nucleon, total electronic relaxation rates, and the spin-exchange efficiencies are measured. Finally, the equivalent rotation sensitivities are compared, which intuitively show the influence of density ratios on comagnetometers.
2. Principle
The dynamic characteristics of the comagnetometer are described using a set of Bloch equations [2,15]. In the K-Rb-$^{21}$Ne comagnetometer, the spin evolutions of two kinds of electron polarization and $^{21}$Ne nuclear polarization $\mathbf {P^n}$ are investigated. Because the comagnetometer is operated in the SERF regime, the spin-exchange rate between K and Rb atoms far outweighs electron spin-destruction rate. Moreover, the K and Rb atoms have almost the same spin polarization when the ensemble is in the spin-temperature distribution. So, the spin evolutions of alkali metals are approximated with electron polarization $\mathbf {P^e}$ , and the equivalent Bloch equations can be expressed as follows:
The Bloch equation integrates spin evolutions of hybrid alkali metal atoms through density ratios $Dr$ of K to Rb atoms. We define dimensionless factors $\varsigma {\rm {\ =\ }}{{{D_r}} \mathord {\left / {\vphantom {{{D_r}} {\left ( {1 + {D_r}} \right )}}} \right. } {\left ( {1 + {D_r}} \right )}}$ and ${\rm {\ }}\xi {\rm {\ =\ }}{1 \mathord {\left / {\vphantom {1 {\left ( {1 + {D_r}} \right )}}} \right.} {\left ( {1 + {D_r}} \right )}}$. Then, equivalent parameters can be described as Eq. (2). The superscripts K and Rb are designated as potassium atoms and rubidium atoms.
After the atomic ensemble reaches spin-temperature distribution, which means that ${{\partial \mathbf {P}^{\mathbf {e}}} \mathord {\left / {\vphantom {{\partial \mathbf {P}^{\mathbf {e}}} {\partial t{\rm {\ =\ }}0}}} \right. } {\partial t{\rm {\ =\ }}0}}$ and ${{\partial \mathbf {P}^{\mathbf {n}}} \mathord {\left / {\vphantom {{\partial \mathbf {P}^{\mathbf {n}}} {\partial t{\rm {\ =\ }}0}}} \right. } {\partial t{\rm {\ =\ }}0}}$, the polarization in the Z-axis can be solved as:
The transverse excitations in the SERF regime are small, so $P_z^e$ and $P_z^n$ can be approximated as constant. Hence, the Bloch equations can be linearized and transverse polarization component $P_x^e$ is solved, which contains input velocity signal ${\Omega _{y}}$. The spin precession due to the velocity is measured by optical rotation $\theta$ of probe beam. Photo-elastic modulator (PEM) is used because of high sensitivity to optical rotation angle. The demodulated signal can be derived using:
The efficiency of hyperpolarized $^{21}$Ne is a crucial parameter for SEOP. So spin-exchange efficiency, which is the ratio of nuclear-spin-polarized rate to electronic-spin-destroyed rate, is shown as [12,13]:
3. Experimental setup and numerical simulation
The experimental setup of the comagnetometer is shown in Fig. 1. A four-layer cylindrical $\mu$-metal shield and a ferrite barrel are used to suppress ambient magnetic fields. A three-axis coil placed inside the barrel compensates the residual fields and provides excitation signal. A 12-mm-diameter spherical vapor cell, made of aluminosilicate glass (GE-180), is heated to 200 $^{\rm {o}}$C by an AC electric heating oven. The cell is filled with a small droplet of K and Rb, approximately 3.3 amagat $^{21}$Ne (70% isotope enriched) and 50 torr N$_{2}$. Four cells with different density ratios are measured using laser-absorption-spectroscopy [24] and shown in Table 1.
Fig. 1. The experimental setup of the comagnetometer. The inset shows the schematic of the simplified interaction model with K and Rb hybrid cell. PBS (polarization beam splitter), BE (beam expander), M (reflective mirrors), GT (Glan-Taylor polarizer), PEM (photo elastic modulator), PD (photodiode).
Table 1. Densities of K and Rb at 200 $^{\rm {o}}$C used in the experiment
Hybrid optical-pumping technology is used in the K-Rb-$^{21}$Ne comagnetometer. The pump light, which is 770.108 nm and linear polarized, is expanded to cover the cell using beam expander. Then the light passes through $\lambda /4$ wave-plate and becomes circular polarization. The circularly polarized light pumps K atoms and the spin polarization is transferred to the Rb atoms via spin-exchange collisions. The linear polarized probe light, which is 795.497 nm and 2-mm diameter, propagates along the X-axis. The power of probe light is 12 mW and stabilized by a noise eater. The PEM is used to modulate probing light. The optical rotation angle of the Rb spin polarization is demodulated via a lock-in amplifier. The inset of Fig. 1 shows the simplified interaction model with K and Rb hybrid cells.
The four cells with different density ratios are assembled in the center of oven, respectively. The comagnetometer works at 200 $^{\rm {o}}$C, and the ensemble reaches spin-temperature distribution at different pump light power densities. The residual fields are compensated with the three-axis coils. Then, the comagnetometer is in the SERF regime. The characteristic parameters of comagnetometer are measured during the pumping process. The scale factor of the comagnetometer is measured through $B_y$-square-wave modulation. At last, the response signal is recorded, and the power spectral density is calculated to evaluate the equivalent rotation sensitivity of comagnetometers.
The dynamic responses of the system are analyzed with independent model of K-Rb Bloch equations [19] and equivalent Bloch equation, respectively. The parameters used for numerical calculations are listed in Table 2. For the convenience of analysis, $R_P^K$ is set to enable $P_z^e=0.5$ and $D_r$ is 1/100. The atomic number density of K and Rb can be calculated with saturated vapor pressure empirical formula [25], and the slow-down factor is related to electronic polarization [20]. The amplitude of the transverse magnetic field is 0.5 nT, which is a small disturbance for the longitudinal polarization. As shown in Fig. 2(a), the polarization process of coupled model is similar to those of independent model. The size of $P_z^e$ at steady state is the weighted average of $P_z^K$ and $P_z^{Rb}$. AC responses to transverse sinusoidal magnetic fields are plotted in Fig. 2(b). After the transients decay, the steady states of $P_x^K$ and $P_x^{Rb}$ are approximate sine curves with a small phase differences. Meanwhile, the response form of $P_x^e$ is a sine curve. In short, the equivalent Bloch equation can accurately describe the coupling relationship of K-Rb mixed alkali metals, which significantly reduces the complexity of independent K-Rb Bloch equations.
Fig. 2. (a) Comparison of electron longitudinal polarization process using K-Rb Bloch equations and equivalent Bloch equation, respectively. (b) AC response to transverse magnetic fields ${B_x} = {B_0}\sin (\omega t)$.
Table 2. Parameters for numerical calculations
Figure 3 shows the simulation results of the homogeneity ratio, which is the ratio of $\bar R_p$ to $R_p(0)$. The simulation parameters are consistent with typical experimental conditions. The absorption of pumping light depends on both atomic number density of alkali metals and photon polarization. At low pump light power density and high density ratio, the homogeneity ratio is close to zero, which means the polarization gradient is large. However, at low density ratio and high pump light power density, the average pumping rate is more uniform.
Fig. 3. The simulation results of the homogeneity ratio, which is the ratio of $\bar R_p$ to $R_p(0)$. The average pumping rate is calculated from Eqs. (5) at different density ratios and pump light power densities.
Figure 4 shows the calculated results for the signal amplitude $S$ as a function of density ratio and pump light power density $I_{pump}$. The simulation is based on Eqs. (5)–(7) and (9). As shown in Fig. 4, at a certain density ratio, the comagnetometer can achieve a relatively large amplitude with optimizing pump light power density. However, to gain the maximum signal amplitude, the effect of density ratio should be considered. The optimal signal in Fig. 4 is achieved when $D_r$ is 1/196 and $I_{pump}$ is 0.254 W/cm$^2$. Therefore, it is meaningful to optimize the density ratio of the cell to improve the sensitivity of the comagnetometer.
Fig. 4. Calculated results for the signal amplitude $S$ as a function of density ratio $D_r$ and pump light power density $I_{pump}$.
4. Results and discussion
At different pump light power densities, the system reaches polarization equilibrium and the magnetic field $\mathbf {B}$ is zeroed with three-axis coils. Then, the ensemble is in a self-compensating condition and the magnetic field applied by Z-coil is ${B^c} = - B_z^e - B_z^n - \delta {B_z}$. Nuclear magnetic field $B_z^n$ is measured by finding the minimum amplitude in the steady-state system response to a sinusoidal magnetic field $B_x$ [26]. At compensation point, the residual field $\delta B_z$ is typically small to be ignored, so $B_z^e$ is obtained by $B_z^e=-B^c-B_z^n$. Then, $P_z^e$ and $P_z^n$ can be calculated with Eq. (7) [27]. With the change of light intensity, the measured effective magnetic fields and spin polarization are plotted in Fig. 5. Both $B_z^e$ and $B_z^n$ have a positive association with $I_{pump}$. At the same light intensity, the cell with larger $D_r$ has larger $B_z^e$ and $B_z^n$. A high density ratio causes larger $\bar R_p$ and smaller $R_{sd}^e$, leading to higher $P_z^e$. According to Eq. (6), $P_z^n$ is proportional to $P_z^e$ and the measured data are fitted with $y=k x /(x+a)$. In cell 4, $P_z^e$ is $0.19 \pm 0.01$ at 748 mW/cm$^2$ and to reach higher value, larger light intensity is needed. The effective magnetic field $B_z^e$ is on the order of tens to hundreds of nanotesla because of large ${\kappa _{Rb - Ne}}$. So $R_{rel}^{se}$ cannot be ignored in the K-Rb-$^{21}$Ne system.
Fig. 5. (a) The solid symbols represent measured effective magnetic fields of alkali metal electrons in four cells, which are the functions of pump light power density. The hollow symbols represent calculated results of electron spin polarization according to Eq. (7), which are plotted against the right y-axis. (b) The solid symbols represent nuclear effective magnetic fields in four cells, and hollow symbols represent calculated nuclear spin polarization according to Eq. (7). Besides, the solid curves in (a) and (b) are fitting curves based on $y= kx/(x+a)$.
The pumping rate experienced by spin ensemble $R_{pe}$ is calculated by substituting the measured $P_z^e$ and calculated $R_{sd}^e$ into Eq. (6). Figure 6 shows $R_{pe}$ of cell 2 and 3 under different pump light power densities. The lines in Fig. 6 are simulation of $\bar R_p$ based on Eq. (5). The measured $R_{pe}$ are consistent with $\bar R_p$ in the two cells, which indicates that $\bar R_p$ is effective approximation to solving longitudinal polarization gradient. The deviation comes from the error of measured $I_{pump}$ and $P_z^e$.
Fig. 6. Measured pumping rates under different pump light power densities, which are experienced by spin ensemble of cell 2 and 3. The solid lines are calculated average pumping rates by Eq. (5), which match well with experimental data.
After measuring $P_z^e$ and $P_z^n$ of equilibrium state, the system response to a small $B_y$ square wave modulation is studied under different pump light power densities. According to Eq. (8), $\Delta S$ as a function of $\delta B_z$ has the form of dispersion curve. The measured $\Delta S$ and fitting curves are depicted in Fig. 7(a), using data of Cell 2 as an example. The slope of dispersion curve at its zero-crossing characterizes the size of scale factor $K_B$, which is the comagnetometer response to a magnetic field change experienced by the atoms in the Z- axis. $K_B$ reaches its maximum at 502.5 mW/cm$^2$ from Fig. 7(a). $R_{tot}^e$ can be derived from fitting curves and is plotted in Fig. 7(b). $R_{tot}^e$ increases with $I_{pump}$ and raises slightly at large intensity. At the same $I_{pump}$, $R_{tot}^e$ grows rapidly with $D_r$ due to rapid increment of $\bar R_p$. $R_{tot}^e$ of Cell 1 is more than double that of Cell 2 at 607 mW/cm$^2$. $R_{sd}^e$ can be estimated from $R_{tot}^e$ when $I_{pump}$ is zero, so linear fitting is used with the first three experimental data. The intercepts are $479 \pm 167$ s$^{-1}$, $583 \pm 142$ s$^{-1}$, $669 \pm 135$ s$^{-1}$, and $619 \pm 108$ s$^{-1}$ for Cell 1-4, respectively,which are consistent with theoretically calculated $R_{sd}^e$ 572 s$^{-1}$, 663 s$^{-1}$, 676 s$^{-1}$, and 681 s$^{-1}$, respectively. $R_{tot}^e$ rises slowly at large light intensity due to nonlinearity of $R_{rel}^{se}$, which might cause the abnormal growth trend of last data in Cell 1. Besides, the heating effect of the strong pump light at the high density ratio should be considered [27]. The model of spin-exchange relaxation should be further studied at high electron polarization and large electron magnetic field.
Fig. 7. (a) Difference of system response to the $B_y$ square wave modulation when cell 2 works at different light intensities. The fitting curves are based on Eq. (8). (b) The total electronic relaxation rate $R_{tot}^e$ derived from fitting curves of (a). The first three data are fitted with linear function. The intercepts of Y axis are the electron spin-destruction relaxation rate $R_{sd}^e$.
According to Eq. (8), the scale factor $K_s$, which is the coefficient transforming input angular velocity ${\Omega _{y}}$ to the comagnetometer response $S$, can be obtained with ${K_s} = {{{K_B}B_z^n} \mathord {\left /{\vphantom {{{K_B}B_z^n} {\Delta {B_y}}}} \right.} {\Delta {B_y}}}{\gamma ^n}$. $\Delta B_y$ is the amplitudes of the $B_y$ square wave. As shown in Fig. 8(a), the size of $K_s$ is affected by $I_{pump}$ and $D_r$. The highest $K_s$ is achieved with Cell 2 when $I_{pump}$=408 mW/cm$^2$. To reach the same level of electron polarization, larger light intensity is required when $D_r$ is smaller. The trend of $K_s$ in Cell 4 is upward but the experiment is limited by the maximum output $I_{pump}$. With the increase of $D_r$, peak value of $K_s$ is shown at lower light intensity. Consistent with the theoretical simulation shown in Fig. 4, the maximal signal amplitude is achieved when $D_r$ reaches near 200 or so. To further validate the effect of density ratio on comagnetometer sensitivity, the ability to suppress magnetic noise is studied when comagnetometers work at the maximum scale factor. A sinusoidal magnetic field with frequency $\omega$ is applied to the comagnetometer along X-axis. The suppression factor to the magnetic response is defined as [28]:
Fig. 8. (a) The relationships between scale factor $K_s$ and pump light power density in four cells. (b) Suppression factor to sinusoidal oscillating magnetic field in X-axis, when the four cells work at the maximum scale factor, respectively. The fitting curves are based on Eq. (10).
According to Eq. (9), spin-exchange efficiency of hyperpolarized $^{21}$Ne ${\eta _{se}}$ is calculated and shown in Fig. 9. ${\eta _{se}}$ is inversely proportional to $I_{pump}$, because the increment of $I_{pump}$ leads to a rise of $\bar R_p$ on the denominator. As shown in Fig. 5, $P_z^e$ of Cell 1 is almost as high as 0.9 at 755 mW/cm$^2$, but $P_z^n$ of Cell 1 only reaches 0.13. Nuclear polarization is difficult to further raise because of the low spin-exchange efficiency. The ${\eta _{se}}$ of Cell 4 is the highest of the four cells at the same $I_{pump}$. However, $P_z^n$ of Cell 4 is small, because small $D_r$ leads to low electron polarization. Considering the spin-exchange efficiency, the $I_{pump}$ should not be arbitrarily increased, and the size of $D_r$ should be moderate to achieve the appropriate system parameters. The spin-exchange efficiency of K-Rb-$^{21}$Ne is an order of magnitude lower than that of Rb- K-$^{3}$He used for neutron spin filters, which indicates big promotion space. To raise ${\eta _{se}}$, the density of noble gases and polarization gradient should be further optimized in future work.
Fig. 9. The spin-exchange efficiency of K-Rb-$^{21}$Ne, as a function of pump light power density, in four cells.
The comagnetometer signal is recorded for 900 s, and the power spectral density of the output signal is calculated and averaged in each 0.1 Hz bin. The equivalent rotation sensitivity is obtained by converting the voltage signal with the scale factor $K_s$. Sensitivities of the Cell 2 operated at different pump laser power densities are shown in Fig. 10(a). The highest rotation sensitivity is achieved at 408 mW/cm$^2$, and sensitivity decreases with a decline of $K_s$, which indicates that the optimal sensitivity is related to the largest scale factor. So, the sensitivities of four cells are compared and plotted in Fig. 10(b), when working at the maximum scale factor, respectively. Limited by the performance of vibration isolation platform of experimental device, there are some resonance peaks below 5 Hz. Cell 2 achieves a sensitivity of ${\rm {4}}{\rm {.2}} \times {\rm {1}}{{\rm {0}}^{ - 7}}$ ${\rm {rad/s/H}}{{\rm {z}}^{ - 1/2}}{\rm {@}}1$ ${\rm {Hz}}$, which is the optimal rotation sensitivity among four cells. Combined with the scale factor and the suppression of magnetic noise, the results indicate the condition that $D_r$ is 1/196 is optimal for the K-Rb-$^{21}$Ne comagnetometer.
Fig. 10. (a) The equivalent rotation sensitivity and the probe background noise of the Cell 2 operated at different pump laser power densities. (b) Comparison of the equivalent rotation sensitivity, when the four cells work at the maximum scale factor, respectively.
5. Conclusion
In conclusion, we have systematically investigated spin polarization characteristics of K-Rb-$^{21}$Ne comagnetometers with different density ratios. A set of equivalent Bloch equations considering the density ratios of two alkali metal species have been established. Meanwhile, an average-pumping-rate model is proposed to solve nonuniform polarization problem in hybrid SEOP process. Simulation calculations and experiments based on cells of different density ratios are implemented to validate our model. Multiple parameters of polarization process affected by density ratios are measured and compared. Cell 2 of $D_r$=1/137 has maximum scale factor and magnetic noise suppression ability, experimentally. Therefore, an optimal sensitivity of ${\rm {4}}{\rm {.2}} \times {\rm {1}}{{\rm {0}}^{ - 7}}$ ${\rm {rad/s/H}}{{\rm {z}}^{ - 1/2}}{\rm {@}}1$ ${\rm {Hz}}$ is achieved with Cell 2. Based on our theoretical analysis, the optimal combination of density ratio and optical power density can realize the maximum output signal of comagnetometers, which provides theoretical guidance for the manufacture of K and Rb hybrid cells. Optimization of density ratios is of great significance for a deep understanding of hybrid SEOP technology. The theory and method presented here can be applied to promote the sensitivity of SERF comagnetometers.
Funding
National Natural Science Foundation of China (62003020).
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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