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Abstract
This study presents a novel method for measuring the number density of K in K-Rb hybrid vapor cells using circularly polarized pump light on polarized alkali metal atoms. This proposed method eliminates the need for additional devices such as absorption spectroscopy, Faraday rotation, or resistance temperature detector technology. The modeling process involved considering wall loss, scattering loss, atomic absorption loss, and atomic saturation absorption, with experiments designed to identify the relevant parameters. The proposed method is real-time, highly stable, and a quantum nondemolition measurement that does not disrupt the spin-exchange relaxation-free (SERF) regime. Experimental results demonstrate the effectiveness of the proposed method, as the longitudinal electron spin polarization long-term stability increased by 204% and the transversal electron spin polarization long-term stability increased by 44.8%, as evaluated by the Allan variance.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Alkali vapor cell serves as a fundamental component of atomic sensors [1,2], with widespread use in ultra-high precision measurements [3] and frontier physics explorations [4,5], including atomic clocks [6], magnetometers [7,8], and gyroscopes [9,10]. Under high temperatures and extremely weak magnetic field conditions, the electrons of the alkali vapor, polarized by resonance pump light, exist in a spin-exchange relaxation-free (SERF) state [11,12]. Based on this principle, the SERF co-magnetometer has been developed as an inertial measurement device with ultra-high precision potential [13]. The assumption of the isothermal-isobaric ensemble is crucial, as it ensures that the temperature and pressure of each body in the cell remain constant [14]. This guarantees the total particle number of the alkali vapor remains constant, i.e., the alkali vapor density remains constant, which is vital for the ultra-high precision measurement of the atomic sensors [15–19].
A necessary condition for real-time stabilization of alkali vapor density is the real-time measurement of alkali vapor density. The traditional method of measuring alkali vapor saturation density is based on laser absorption spectroscopy, as Ito et al. measured the number density of $\mathrm {K}$ and $\mathrm {Rb}$ in a hybrid alkali vapor cell [20]. In addition, Vliegen et al. implemented the measurement of alkali vapor density using Faraday rotation, which is more suitable for conditions of high alkali vapor density and the presence of buffer gas compared to absorption spectroscopy [21]. The above methods are based on linearly polarized light, and applying these methods would limit the SERF co-magnetometer to perform dual-axis inertial measurements [22]. Furthermore, adding a new probe optical path would significantly increase the system complexity for single-axis inertial measurements of the SERF co-magnetometer. Wei et al. used the light shift of $\mathrm {K}$ and $\mathrm {Rb}$ in a hybrid alkali vapor cell to measure the number density ratio of $\mathrm {K}$-$\mathrm {Rb}$ hybrid vapor, which is limited by the mechanism that cannot measure alkali vapor density at low-density ratios [23]. In order to overcome these difficulties, this paper aims to develop a method for online measurement of alkali vapor density and apply it in a SERF co-magnetometer to stable alkali vapor density in real time.
The significant contributions of this paper are highlighted below. Firstly, this paper presents an innovative method for real-time optical measurement of alkali vapor density. This method models wall losses, scattering losses, atomic absorption losses, and atomic absorption saturation and achieves the measurement of alkali vapor density under circularly polarized light conditions. Additionally, the proposed method is successfully applied to measure the alkali vapor density in a SERF co-magnetometer system, achieving online stabilization of the alkali vapor density. Finally, the proposed approach significantly improves the longitudinal and transverse electron spin polarization stability of the SERF co-magnetometer.
The article is structured as follows. Section 2 discusses the basic principles of the measurement method. Section 3 provides details about the experimental setup. Section 4 verifies the effectiveness of the measurement method and applies it to the SERF co-magnetometer. Finally, Section 5 presents the conclusion.
2. Basic principle
This section studies the online measurement of the number density of $\mathrm {K}$ within the cell of a $\mathrm {K}$-$\mathrm {Rb}$-$\mathrm {^{21}Ne}$ co-magnetometer. The cell contains the $\mathrm {K}$-$\mathrm {Rb}$ hybrid alkali metal vapor, the noble gas $\mathrm {^{21}Ne}$, and the quenching gas $\mathrm {N_2}$. At the working temperature of the co-magnetometer, the number density $n_{\mathrm {K}}$ of the alkali metal $\mathrm {K}$ is relatively small compared to the number density $n_{\mathrm {Rb}}$ of the alkali metal $\mathrm {Rb}$, and the ratio between them is defined as the number density ratio $D_r$, with a typical value of the order of 1×10−2. Under this condition, the optical power $I$ of the pump light required to reach high longitudinal electron spin polarization is significantly minimized, compared to that of single alkali metal atoms, known as the Hybrid spin-exchange optical-pumping technique. Consider a circularly polarized pump light with D1 resonance frequency of $\mathrm {K}$ incident to a spherical glass cell of $\mathrm {K}$-$\mathrm {Rb}$-$\mathrm {^{21}Ne}$ coated with an anti-reflection film, up to the light exiting this cell. A differential equation [24] based on the Lambert-Beer law can describe this propagation process as
Here $c$ is the light speed in a vacuum, $r_e$ is the classical electron radius, $f$ is the oscillator strength, and $\Gamma$ is the broadening. The longitudinal electron spin polarization steady state of the $\mathrm {K}$ atoms is defined as $P^{\mathrm {K}}_{z0}$, which is obtained by solving a complex set of Bloch equations [23,25], specified as
where the effective longitudinal electronic spin relaxation $\bar {R}_{1}^{\mathrm {K}}$ of $\mathrm {K}$ isHere, $R_{\mathrm {se}}^{\mathrm {Rb}-\mathrm {K}}$ represents the rate at which $\mathrm {Rb}$ spins are transferred to $\mathrm {K}$ spins. The longitudinal electron spin relaxation rates of $\mathrm {K}$ and $\mathrm {Rb}$ are expressed as $R_{1}^{\mathrm {K}}$ and $R_{1}^{\mathrm {Rb}}$, respectively. By substituting Eq. (3) to Eq. (1), the following formulas can be obtained as
Reformulating Eq. (5) and taking the number density of $\mathrm {K}$ to the left side of the equation yields
This equation is the measurement equation proposed in this paper for $n_{\mathrm {K}}$. By substituting the values of $R_p\left (0\right )$, $R_p\left (z\right )$ and $\bar {R}_{1}^{\mathrm {K}}$, $n_{\mathrm {K}}$ can be obtained in real time. Next, we address the measurement of the values of $R_p\left (0\right )$, $R_p\left (z\right )$ and $\bar {R}_{1}^{\mathrm {K}}$.
The pump rate is determined by the optical power of the pump light, and the relationship is shown as follows
where $A$ is the effective light area, approximated as 28 mm2. $h$ is Planck’s constant, and $v$ is the frequency of the pump laser. $\alpha$ is defined as the proportionality coefficient between the optical power of the pump light and the pump rate.Assuming that the spherical cell is coated with an anti-reflective film, the effect of spherical reflection can be neglected. The optical power attenuation caused by the thick walls of the cell as the pump laser passes through the cell needs to be modeled, and this portion of the optical loss is defined as wall loss. This portion of optical power attenuation needs to be accounted for twice, as the pump light passes through the cell wall during both incident and exit from the cell. Furthermore, scattering effects should be considered, as noble and quenching gas molecules interact with photons, causing partial absorption or re-radiation of photons, resulting in a loss of optical power. This is defined as scattering loss. In the operational condition of the SERF co-magnetometer, the alkali metal atom $\mathrm {K}$ absorbs photons at the resonance frequency, causing a transition to a higher energy level. This portion of optical power loss is defined as atomic absorption loss. The propagation process of optical power and the pump rate inside the cell can be obtained with the modeling of each type of optical loss inside the cell, as shown in Fig. 1(a).
Fig. 1. The propagation of pump laser. (a) The propagation of pump light incident into a spherical cell. (b) The propagation of pump light incident into a cylindrical cell is simplified by the spherical cell.
The pumped light is composed of countless photons, and the path distance $L$ travels from each photon entering to exiting the cell are variable, which makes the analysis complicated. In the macroscopic world, we measure statistical information about photons, such as optical power, rather than information about individual photons. Therefore, in modeling and analysis, the path distance $L$ of each photon can be simplified to the average path distance $\bar {L}$ of the photon, simplifying the analysis’s complexity. Consider a column coordinate system where the spherical cell’s sphere center coincides with the coordinate system’s origin. A pump light with beam radius $r$ propagates along the z-axis in the positive direction, and its axis coincides with the z-axis. The volume of the intersection between the beam and the spherical cell is the sensitive volume, where the photons and the medium inside the cell interact. With the beam radius more minor than the cell radius, the effective cross section between the photon and the atom is equivalent to the transverse cross section $A$, as shown in Fig. 1(b). Hence, we can calculate the average path distance of the photon as
Thus, it can be concluded that the propagation of a pump beam with a radius of $r$ incident on the spherical cell with a radius of $R$ can be simplified as the propagation of the beam in a cylindrical cell with a radius $r$ and length $\bar {L}$.
First, we model the optical power loss caused by the cell wall, and this loss can be described as
Here $\beta$ is the absorption factor, determined by the cell wall material. $d_{cell}$ is the thickness of the cell wall. The optical loss caused by the cell wall is proportional to the optical power of the incident light. Thus a loss efficiency $\eta$ can be defined. The first equation in Eq. (9) is the optical loss model for light incident into the cell from the outside, and the second is the optical loss model for light exit from the cell to the outside.
Next, we model the optical losses of light propagation inside the cell. These losses consist of two parts: scattering losses caused by scattering effects and absorption losses caused by atomic absorption.
To model the scattering loss [26], we consider the effect of elastic scattering of photons by atoms within the gas cell and define the scattering cross section as $\sigma _s$, which is a physical quantity used to describe the probability of particles being scattered. A simplified scattering model developed to describe the scattering of light as it passes through the gas medium is defined as
We consider the probability of atoms absorbing photons to model the absorption loss. It can be quantified by the Lambert-Beer law, considering the effect of atomic polarization on absorption. An absorption model can be given as
Here, the function $W$ is the Lambert W function. The optical loss due to atomic absorption during the whole propagation process can be described as
Assuming that the absorption loss and scattering loss are independent, the optical power propagation model under SERF co-magnetometer operating conditions can be given as
With this, we can present the optical power at point $a$ and point $b$ as
Finally, the measurement equation for $n_{\mathrm {K}}$ can be described as
3. Experimental setup
Figure 2 shows the experimental setup utilized in this study. A boron nitride ceramic oven containing a spherical cell, which is heated to the SERF co-magnetometer operating conditions using an alternating current electrical heater, is the centerpiece of the setup. This spherical cell has a diameter of 8 millimeters and a wall thickness of 1 millimeter, and it contains an alkali metal mixture of $\mathrm {K}$ and $\mathrm {Rb}$ with a number density ratio of approximately 1:91.4, noble gas $\mathrm {^{21}Ne}$ at a pressure of about 1.76 amagat, and quenching gas $\mathrm {N_2}$ at a pressure of about 50 Torr. The spherical cell is placed inside a triple magnetic shielding cylinder to eliminate the interference of the ambient magnetic field. A triaxial magnetic field coil compensates for the residual magnetic field and puts the SERF co-magnetometer system in a nuclear spin self-compensating state [27].
Real-time monitoring of the optical power data of incident and exit light is carried out by photodetectors (PD). The incident optical power data is calculated by splitting the branched optical power obtained from the pumped light by the polarization beam splitter (PBS). The data acquisition (DAQ) card collects and transfers the data to a personal computer (PC) for processing. The arbitrary wavelength generator (AWG) is commanded by the PC and outputs signals to the liquid crystal variable retarder (LCVR) and the alternating current electric heater, which adjust the incident light power and the cell temperature. The pump light is generated by a distributed Bragg reflector (DBR) laser device, whose frequency is locked to the D1 resonance of $\mathrm {K}$ using the saturated absorption technique. The beam radius of the pump light is expanded by a beam expander (BE). The probe light is generated by a distributed feedback (DFB) laser device with a frequency near the D1 resonance of $\mathrm {Rb}$. It is applied to measure the transverse spin polarization of the SERF co-magnetometer. The $\lambda /2$ and $\lambda /4$ devices are half-wave and quarter-wave plates, respectively.
4. Measurement method and experimental verification
The online measurement of the number density of $\mathrm {K}$ requires modeling each type of optical loss within the cell and measuring the optical power of the incident and exit light. Based on Eq. (11), we can fit the relevant parameters of the wall loss and scattering loss of the cell. Figure 3(a) demonstrates the exit optical power with different incident optical power under room temperature conditions to construct a model for optical loss apart from atomic absorption. This is attributed to the liquid state of the alkali metal atom $\mathrm {K}$ under room temperature conditions. In this case, Fitting Curve A considers both scattering and wall loss, while Fitting Curve B only takes into account wall loss. Fitting Curve B has been widely used in SERF magnetometers [28]. Scattering loss is a crucial component in modeling the optical losses of a cell containing noble gas at several standard atmospheric pressures, but to our knowledge, it has not been discussed in previous studies. Figure 3(b) shows the optical power propagation for different incident optical powers. The high incident optical powers lead to high scattering losses, meaning that scattering losses are a significant optical loss that must be considered when applying a high-power pump laser. The results for Fitting Curve A are that the parameter $\eta$ of wall loss is 0.5051 and the parameter $\varphi$ of scattering loss is 0.0133. For Fitting Curve B, the parameter $\eta$ of wall loss is 0.456. With this, we have obtained the models of optical loss, excluding the atomic absorption loss.
Fig. 3. Modeling of optical loss under room temperature conditions. (a) The exit light power corresponds to different incident light powers. (b) The propagation simulation of optical loss based on the Fitting curve A.
Next, we model the atomic absorption loss based on Eq. (11) and (14). The difference in exit optical power between the working environment of the SERF magnetometer and room temperature conditions is attributed to atomic absorption loss. As shown in Fig. 4(a), we measured the atomic absorption loss in the cell for temperatures ranging from 433 K to 463 K with a step size of 2 K and fitted the data using Eq. (13). Typically, we choose a high incident optical power as the working state of the SERF magnetometer to meet the high longitudinal electron spin polarization condition, which significantly suppresses the spin polarization gradient in the cell. As shown in Fig. 4(b), we obtained the values of parameter $\bar {R}^{\mathrm {K}}_1/\alpha$ under different cell temperatures and fitted them using a cubic polynomial. With these results, we have obtained all the necessary parameters for online measurement of the number density $n_\mathrm {K}$.
Fig. 4. Modeling of atomic absorption losses under SERF co-magnetometer operating conditions. (a) The atomic absorption loss corresponds to different incident optical powers under various cell temperatures. (b) $\bar {R}^{\mathrm {K}}_1/\alpha$ corresponding to different cell temperatures.
To verify the efficacy of the proposed measurement method, we compared the proposed model with the classical empirical equation for saturated vapor pressure [29], as illustrated in Fig. 5(a). The results demonstrated that the proposed method could be applied successfully for measuring the number density of alkali metals $\mathrm {K}$. In Fig. 5(b), we compare our proposed Model 1 with two other models against the classical empirical equation for saturated vapor pressure. The green curve in Fig. 5(a) and Fig. 5(b) was calculated based on the formula for saturated vapor pressure in [29], which can be expressed as
where $T$ is the cell temperature in Kelvin. $A = 4.312$ and $B = 4040$ are coefficients for liquid Rb. Number density ratio for alkali metal K and Rb $D_r \approx 1/91.4$ can be calibrated using the method proposed in [23]. Model 2 neglects scattering losses and is suitable for describing situations where the alkali vapor cell contains only a small amount of noble gas, such as in SERF magnetometers [28]. Experimental results showed that using Model 2 for measuring number density $n_\mathrm {K}$ is unreliable due to a significant standard deviation of the measurement results. This is because the alkali vapor cell in $\mathrm {K}$-$\mathrm {Rb}$-$\mathrm {^{21}Ne}$ co-magnetometer contains a large amount of noble gas, which makes scattering losses significant and cannot be neglected. Model 3 neglects the effect of longitudinal electron spin polarization on atomic absorption and is commonly used for measuring alkali metal number densities using linearly polarized light [22]. Experimental results indicate that Model 3 cannot be used to measure number density $n_\mathrm {K}$ using circularly polarized light because circularly polarized light polarizes alkali metal atoms and reduces atomic absorption loss. We have now completed the experimental validation of our proposed method for measuring the number density of $\mathrm {K}$. Next, we will discuss how this method can be applied for online measurement of the number density $n_\mathrm {K}$ and stable control in the SERF co-magnetometer.Fig. 5. The real time measurement results of the number density of $\mathrm {K}$. (a) The real time measurement results of Model 1 are proposed in this paper under different cell temperatures. (b) The real time measurement results of Models 2 and 3 were proposed in previous studies under different cell temperatures.
A comparative experiment was conducted at the cell temperature of 448.15 K using the platinum resistance thermometer (PRT) method based on the positive temperature coefficient effect described in [30], which is a commonly used stable solution for measuring the number density of $\mathrm {K}$. In this method, a platinum resistance thermometer is placed near the cell wall to estimate the temperature and stabilize the $\mathrm {K}$ vapor density inside the cell. The stability of transversal electron spin polarization, a crucial physical quantity utilized in SERF co-magnetometers for inertial measurements, is a critical performance. The stability of the longitudinal electron spin polarization is also critical, as it is positively proportional to the scalar factor of the SERF co-magnetometer. Therefore, we assess the stability of both transversal and longitudinal electron spin polarizations using the proposed method and conventional methods in the context of the SERF co-magnetometer. We apply the Allan standard deviation, a widely used technique to evaluate the stability of long-time measurements, for this purpose. Figure 6(a) shows a comparison of the number density stabilities of $\mathrm {K}$ in the SERF co-magnetometer using the proposed method and the PRT method. The study reveals that the proposed method exhibits superior short-term stability for $n_\mathrm {K}$ in the $\tau \le 10 \,\mathrm {s}$ compared to the PRT method. Additionally, it significantly enhances long-term stability for $n_\mathrm {K}$ in the range of $\tau \ge 60 \,\mathrm {s}$ compared to the PRT method. On the other hand, the PRT method outperforms the proposed method in the range of $10 \,\mathrm {s}< \tau < 60 \,\mathrm {s}$. The dashed lines indicate the $99\%$ confidence level of the obtained Allan standard deviation based on the student’s t distribution [31]. Figure 6(b) demonstrates the experimental results by comparing the proposed and PRT methods. We define the stability ratio as the Allan standard deviation $\sigma _o$ ratio of the PRT method to the Allan standard deviation $\sigma _n$ of the proposed method. This ratio is convenient to demonstrate the effectiveness of the methods. The experimental results show that the proposed method improves the stability of both transversal and longitudinal electron spin polarizations compared to the PRT method. The short-term stability of the longitudinal electron spin polarization is improved by $158\%$ at $\tau = 0.13 \,\mathrm {s}$, and the long-term stability is improved by $204\%$ at $\tau = 1967 \,\mathrm {s}$. More importantly, the long-term stability of the transversal electron spin polarization is improved by $44.8\%$ at $\tau = 2994 \,\mathrm {s}$, indicating that the proposed method can significantly contribute to the critical performance of the SERF co-magnetometer.
Fig. 6. The comparative experimental results of number density stabilization control. (a) Allan standard deviation of the proposed method and the comparison method. (b) Allan standard deviation ratios of transversal and longitudinal electron spin polarization.
In addition, the conventional method relies on a DC bridge to drive the platinum resistance thermometer for measuring the temperature inside the cell. However, the galvanomagnetic effect inevitably generates a magnetic field interference inside the cell, which destroys the SERF regime, leading to system errors and affecting the reliability of the measurement results. Therefore, we determined and listed five different cell temperature points to be tested in Table 1. The comparative experimental results based on the three-dimensional magnetic field compensation technique demonstrate that the conventional method generates a magnetic field interference of 0.331(68) nT, which is a norm of the magnetic field vector $\mathrm {d}\mathbf {B}$. Based on the step response, comparative experiments were carried out, and the results show that the conventional method generates an equivalent angular velocity of 0.008 38(21) ° s−1 of rotation around the y-axis.
Table 1. Evaluation results of the proposed method compared with the conventional method at different cell temperatures
5. Conclusion
In conclusion, the proposed real-time measurement method using circularly polarized pump light offers a promising approach for accurately and efficiently measuring alkali vapor density in the SERF co-magnetometer. Compared to traditional absorption spectroscopy, Faraday rotation, and PRT techniques, this method eliminates the need for additional devices and provides real-time measurement and stabilization capabilities. Our experimental findings demonstrate the effectiveness of the proposed method in accurately measuring the number density of $n_\mathrm {K}$ and achieving real-time stabilization of this quantity. Specifically, using the proposed method, our results significantly improve the stability of the longitudinal and transversal electron spin polarization. One of the reasons why the proposed method has better long-term stability than the PRT method is that the sensor used by the PRT method is intrinsically separated from the alkali vapor cell, and it can only ensure temperature stability at the measurement point outside the cell. By eliminating the need for platinum resistance thermometers, this method provides a better achievement of the SERF regime, which is essential for various applications in atomic sensors, quantum computation, and related fields. Overall, this study contributes to developing efficient and reliable methods for measuring alkali vapor density in real time. This method is not limited to the K-Rb mixed alkali metal ensemble, but can be extended to other types of mixed alkali metal ensembles as well.
Funding
National Natural Science Foundation of China (61673041, 62103026); Special Project for Research and Development in Key areas of Guangdong Province (2021B0101410005); China Postdoctoral Science Foundation (2021M703049).
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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