Closed-loop atomic spin polarization detection based on EOAM modulation in K-Rb-21Ne co-magnetometers

Shimiao Fan; Wenfeng Fan; Qi Yuan; Xiaoping Li; Hongyu Pei; Wei Quan; Lihong Duan

doi:10.3788/COL202422.121201

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Abstract

We propose a high-sensitivity spin polarization detection scheme based on optical amplitude modulation in atomic co-magnetometers, which is different from the traditional configuration. A linearly polarized laser with intensity modulation interacts with electron spins, generating optical rotation angles. Through differential amplification and demodulation, atomic spin polarization information is extracted with high precision. The effectiveness of the proposed method is verified by applying external perturbations during testing. Compared to traditional detection methods, the proposed approach effectively enhances inertial measurement sensitivity. Specifically, at 1 Hz, the measurement sensitivity has improved from 9.7 × 10^-6 to 3.25 × 10^-6 deg/(s Hz^1/2). Furthermore, the proposed scheme is easily integrable and conducive to future research on miniaturizing co-magnetometers and can also be applied to many other related fields.

Keywords

amplitude modulation atomic spin polarization detection co-magnetometers inertial measurement sensitivity

1. Introduction

In recent years, with the development of quantum physics, the study of alkali metal and noble gas co-magnetometers operating in spin-exchange relaxation-free (SERF) states has undergone extensive exploration^[1,2]. These co-magnetometers find widespread applications in testing CPT violations^[3], searching for anomalous spin forces^[4], and other related fields^[5]. Notably, in the $K - Rb - {}^{21}{Ne}$ co-magnetometer, Rb atoms operate in the SERF state, endowing the co-magnetometer with exceptionally high sensitivity to inertial rotations^[6,7]. This has sparked considerable attention and research in the field of inertial navigation^[8–10] and is promising as a compact rotation sensor that can surpass existing technologies. However, due to the presence of systematic detection noise, the sensitivity achieved currently is lower than the theoretical sensitivity^[11]. In-depth research into the detection scheme is needed to enhance system performance.

Elevating low-frequency sensitivity can effectively enhance the performance and practicality of the co-magnetometers^[12,13]. In the previous research, the optical rotation angle of the linearly polarized beam is measured to detect atomic spin polarization using balanced differential detection, magnetic field modulation detection, or optical polarization modulation detection^[14,15]. The most applied approach is the balanced differential detection method, known for its structural simplicity and capability to suppress common-mode noise. However, it is unable to suppress $1 / f$ noise, leading to the proposal and research of other detection methods. The method of modulating the magnetic field along the $z$ -axis allows manipulation of alkali metal atoms without the need for additional equipment, but it elevates the rate of electron spin exchange relaxation, posing challenges to the sustained maintenance of the SERF state in the system during operation^[16]. Regarding optical polarization modulation detection methods, prior investigations have explored the applications of Faraday modulators^[17], photoelastic modulators^[18], and acousto-optic modulators^[19] for enhancing sensitivity. However, these methods involve modulating the polarization state of light, introducing certain polarization errors. Additionally, they are limited in terms of system size, complexity of optical pathways, or the introduction of supplementary magnetic fields.

It is worth noting that scholars have conducted some studies on atomic spin polarization detection schemes. However, due to the limitations of new approaches, the balanced differential detection method still remains the most widely used detection scheme for co-magnetometers to date^[11,20]. The inability of the balanced differential detection method to suppress $1 / f$ noise leads to a disparity between the measured low-frequency sensitivity and the theoretically optimal sensitivity. Therefore, there is still an urgent demand to devise new schemes to enhance low-frequency detection sensitivity.

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Herein, we present a closed-loop scheme for detecting atomic spin polarization based on a fiberized electro-optic amplitude modulator (EOAM), which is designed and implemented in a $K - Rb - {}^{21}{Ne}$ co-magnetometer. Compared to traditional approaches, our designed scheme directly modulates the intensity of the probe light rather than directly modulating the optical polarization, and a higher inertial measurement sensitivity is achieved in a portable prototype. The system can effectively reduce the impact of $1 / f$ noise and improve system integrability. Additionally, a closed-loop system is introduced into the modulation pathway to minimize the influence of physical factors, such as temperature on the drift of the EOAM operating point. This approach ensures the stability of the modulated light and ultimately achieves high-sensitivity detection of co-magnetometer signals.

2. Methods

In the inertial measurements based on the SERF co-magnetometer, the polarization evolution of electron spins, as well as the nuclear spin, can be modeled with the classical Bloch equation^[20]. The first term of the Bloch equations describes the polarization of the polarization vectors of the electron spins $P^{e} = {[P_{x}^{e}, P_{y}^{e}, P_{z}^{e}]}^{T}$ and the inert gas nuclear spin $P^{n} = {[P_{x}^{n}, P_{y}^{n}, P_{z}^{n}]}^{T}$ in external fields, such as magnetic field $B = [B_{x}, B_{y}, B_{z}]^{T}$ , light shift $L = {[L_{x}, L_{y}, L_{z}]}^{T}$ , and the effective magnetic fields $λ M^{e / n} P^{e / n}$ due to the polarization of electron spins and nuclear spins, respectively. The precise frequency is determined by the respective gyromagnetic ratios $γ_{e / n}$ and the slowing-down factor of the electron spin polarization $Q (P_{z}^{e})$ . The second term describes the relative inertial changes of the electron and nuclear spin polarization as they rotate in the inertial space with angular velocity $Ω = {[Ω_{x}, Ω_{y}, Ω_{z}]}^{T}$ . The third term accounts for the impact of various pumping processes on polarization. $R_{p}$ and $R_{m}$ are the pumping rates for the pump and probe light, respectively. $s_{p}$ and $s_{m}$ denote the polarization and propagation directions of the pump and probe light. $R_{s e}^{e n}$ and $R_{s e}^{n e}$ are the spin-exchange collision rates between electron spins and nuclear spins. The last term describes the relaxation mechanisms for the electron and nuclear spins. The transverse relaxation rate for the electron spins and nuclear spins is $R_{2}^{e / n}$ , and the longitudinal relaxation rate is $R_{1}^{e / n}$ , ${\begin{matrix} \frac{\partial P^{e}}{\partial t} = \frac{γ_{e}}{Q (P_{z}^{e})} (B + λ M_{n} P^{n} + L) \times P^{e} - Ω \times P^{e} \\ + \frac{(R_{p} s_{p} + R_{m} s_{m} + R_{s e}^{e n} P^{n})}{Q (P_{z}^{e})} - \frac{P^{e}}{Q (P_{z}^{e})} ⊙ {[R_{2}^{e}, R_{2}^{e}, R_{1}^{e}]}^{T} \\ \frac{\partial P^{n}}{\partial t} = γ_{n} (B + λ M_{e} P^{e}) \times P^{n} - Ω \times P^{n} + R_{s e}^{n e} P^{e} \\ - P^{n} ⊙ {[R_{2}^{n}, R_{2}^{n}, R_{1}^{n}]}^{T} \end{matrix} .$ (1)

Based on Eq. (1), the steady-state values of the electron spin polarization vectors under a transverse angular velocity input are determined as $P_{x}^{e} = \frac{P_{z}^{e} [Q (P_{z}^{e}) γ_{n} - γ_{e}]}{R_{2}^{e} γ_{n}} Ω_{y} .$ (2)

The linearly polarized probe light, after passing through the polarized atomic vapor cell, experiences a rotation of the polarization plane. This is reflected in the change in the optical phase of the beam, resulting in an optical rotation angle $θ \propto P_{x}^{e}$ induced by the polarized alkali metal. Additionally, in the process of converting the optical rotation angle into an output signal, various processes, such as atomic absorption of laser light and photoelectric conversion, are involved. Therefore, a power-stable optical rotation angle probe system with modulation and demodulation functions is necessary to isolate the $1 / f$ noise introduced during the optical rotation angle probe process.

The closed-loop atomic spin polarization detection system proposed in this Letter comprises an optical intensity modulation module and an optical rotation angle $θ$ demodulation module, as indicated by the green and pink boxes in Fig. 1. In the optical intensity modulation module, the probe light is eliminated from the fiber distributed Bragg reflector (DBR) laser and then enters the fiber EOAM. In the EOAM, the amplitude modulation of the probe light is performed. Assuming the light intensity output from the probe laser is $I_{0}$ , the optical power after passing through the EOAM is as shown in the following equation: $I_{EOAM} = I_{0} {1 + \cos {\frac{π}{V_{π}} [V_{\mod} \sin (ω t) + V_{DC}]}},$ (3)where $V_{π}$ is the half-wave voltage of the EOAM, $V_{\mod} \sin (ω t)$ is the control voltage applied to the radio frequency (RF) port of the EOAM for the optical intensity modulation of the probe laser, and $V_{DC}$ is the voltage value applied to the direct current (DC) port of the EOAM for operating point setting.

Figure 1.Schematic of the closed-loop atomic spin polarization detection system. EOAM, electro-optic amplitude modulator; CO, coupler; PL, polarizer; λ/2, half-wave plate; PBS, polarizing beam splitter; PD, photodiode.

Then, the probe light passes through a fiber collimator, a polarizer, a half-wave plate, and a PBS. The beam splitter separates the light into two linearly polarized beams. One beam enters the cell and interacts with the alkali atoms, producing the rotation angle $θ$ shift of the linearly polarized light. The other beam enters PD1, converting into an electrical signal for closed-loop control of the modulation amplitude of the probe light. The light intensity signal detected by PD1 is described as $I_{PD 1} = \frac{1}{4} I_{0} {1 + \cos {\frac{π}{V_{π}} [V_{\mod} \sin (ω t) + V_{DC}]}}$ (4)

Furthermore, the EOAM modulation closed-loop control is implemented through the lock-in amplifier (LIA) channel 2 and a PID controller. The LIA channel 2 extracts the first harmonic signal from the PD1 intensity signal and feeds this signal as an input to the PID controller. The output from the PID controller is employed as the modulation amplitude $V_{\mod}$ , which is applied to the RF port of the EOAM, ensuring the stability of the modulation amplitude for the probe laser injected into the cell. The first harmonic signal extracted by LIA channel 2 is given by $I_{{LIA}_{2}} = - \frac{1}{2} I_{0} \sin (\frac{π}{V_{π}} V_{DC}) J_{1} (\frac{π}{V_{π}} V_{\mod}),$ (5)where $J_{1}$ is the first-order Bessel coefficient. The output power of the semiconductor lasers, the polarization devices, and the optical modulation devices undergo significant drift due to environmental changes, directly affecting the stability of measurements. The optical intensity modulation module proposed in this Letter dynamically adjusts the $V_{\mod}$ to stabilize the optical intensity amplitude $I_{LIA 2}$ received by PD1 at a set value, thereby generating a stable optical intensity amplitude for precise optical rotation angle measurements.

The optical rotation angle demodulation module contains balanced differential extraction and demodulation functions of optical rotation angle $θ$ . The linearly polarized light, after interacting with the electron spin in the cell, passes through a half-wave plate and a polarizing beam splitter. It is then split into two beams of probe light. These beams enter PD2 and PD3, converting into two electrical signals, which, after differential processing, are connected to the input of LIA channel 1. The output of the EOAM working modulation closed-loop control system serves as the reference signal for the LIA. The differential signal undergoes the first harmonic demodulation to output the rotation signal of the SERF co-magnetometer. The intensity-modulated probe laser, after passing through the cell, generates a linearly polarized optical rotation angle $θ$ . The light is detected by the balanced differential PD amplifier and then is converted into an electrical signal as $V_{in} = \frac{1}{4} I_{0} {1 + \cos {\frac{π}{V_{π}} [V_{\mod} \sin (ω t) + V_{DC}]}} \sin (2 θ) .$ (6)

Utilizing a lock-in amplification method to extract the first harmonic of the electrical signal with LIA channel 1, the co-magnetometer output signal is obtained, $V_{sig} = - \frac{1}{2} I_{0} \sin (\frac{π}{V_{π}} V_{DC}) J_{1} (\frac{π}{V_{π}} V_{\mod}) \sin (2 θ) \approx - I_{0} \sin (\frac{π}{V_{π}} V_{DC}) J_{1} (\frac{π}{V_{π}} V_{\mod}) θ .$ (7)

3. Experimental Setup

The experimental schematic diagram in the $K - Rb - {}^{21}{Ne}$ co-magnetometer is shown in Fig. 2. The experimental setup mainly consists of the atomic vapor cell and its heating module, the magnetic shielding and compensation module, as well as the pump laser path module and probe laser path module. The atomic vapor cell, housed within a cylindrical aluminum nitride oven, is a spherical glass container with a diameter of 10 mm. It contains a mixture of K-Rb hybrid alkali metal atoms (with a density ratio of approximately 1:160), inert gas ${}^{21}{Ne}$ , and quenching gas $N_{2}$ . To enhance temperature stability, Pt1000 resistors used for measuring real-time temperature around the vapor chamber and heating coils used to generate alternating current for temperature control of the cell are, respectively, attached to the inner and outer sides of the oven. The pump laser is a beam of polarized light whose frequency is stabilized at the K atom’s D1 line through the saturated absorption. It is extracted via a DBR laser fiber, and its power is stabilized at 90 mW through closed-loop control of the tapered amplifier (TA) current. Subsequently, it passes through a quarter-wave plate to become circularly polarized light. Then, the pump laser is used to polarize the alkali metal atoms inside the vapor cell. The probe laser is linearly polarized light detuned from the Rb atom’s D1 line after passing through the intensity modulation module designed in this study, and it enters the cell to measure the optical rotation angle. Additionally, to shield against external magnetic field interference, the atomic vapor cell and its heating module are placed within a magnetic shielding system composed of a three-layer permalloy and a single-layer MnZn ferrite. Three-axis magnetic field coils are installed within this magnetic shielding system to compensate for any residual magnetic fields.

Figure 2.Experimental setup of SERF co-magnetometer. TA, tapered amplifier; λ/4, quarter-wave plate; GL, Glan–Taylor prism.

4. Results and Discussion

To validate the usability of the closed-loop detection system proposed in this Letter, we conduct tests to evaluate the basic performance of the co-magnetometer. After the orthogonal adjustment of the pump light and probe light, the pump light is activated to polarize the atomic ensemble. Following polarization completion, residual magnetization compensation is achieved through cross-modulation. Subsequently, the co-magnetometer is positioned on a single-axis rate turntable with an accuracy of $Ω_{y} = 0.001 \deg / s$ , and the $y$ -axis of the co-magnetometer is aligned parallel to the rotation axis of the rate turntable. Figure 3 shows the results of the angular velocity test, where the horizontal axis represents the rotation rate of the rate table, and the vertical axis represents the corresponding output signal of the co-magnetometer. From the test results, it can be observed that the system’s response to the angular velocity is linear, indicating the capability for angular velocity measurement. Additionally, the system’s scale factor is determined through fitting to be 5.09 V/(deg s⁻¹).

Figure 3.Testing of co-magnetometer angular velocity signal and scale factor based on closed-loop EOAM modulation detection system.

A perturbation test is conducted based on the established detection system for evaluating the closed-loop performance of the detection system proposed in the presence of external disturbances. We actively apply a voltage bias signal to the DC port of the EOAM to simulate the drift of the operating point caused by temperature changes. The output signal of the co-magnetometer is obtained in real-time through LIA channel 1, and simultaneously, the control voltage output by the PID controller is recorded. Temperature is a key factor affecting the drift of the operating point. According to previous research and our experiment, the drift can be approximately 0.007 V/°C^[21]. In a laboratory environment, the temperature fluctuation is typically around $\pm 2 ° C$ . To more effectively assess our system’s closed-loop control capability in real-world environments, we apply a DC signal from the signal generator and artificially apply a bias voltage ranging from $- 0.1$ to $+ 0.1 V$ to the DC port, resulting in experimental data as shown in Fig. 4.

Figure 4.Response testing of the closed-loop detection system to external perturbation. Case A: Using EOAM modulation only, with the closed-loop control system output turned off. Case B: The closed-loop EOAM modulation detection system designed in this Letter.

From the experimental results shown in Fig. 4, it is clear that when the detection system is in an open-loop state (Case A), fluctuations in the external environment directly result in signal drift. In contrast, with the closed-loop detection system designed in this Letter (Case B), the closed-loop system generates a control signal to counteract external perturbation. This leads to a relatively stable output of the co-magnetometer signal obtained through demodulation, suppressing detection errors caused by environmental disturbances. It can be verified that the proposed closed-loop detection system significantly reduces the impact of external environmental changes on the detection system.

Furthermore, the inertial measurement sensitivity (a normalized power spectral density) is provided to validate the noise and drift suppression brought by the closed-loop detection method. A $K - Rb - {}^{21}{Ne}$ co-magnetometer based on both traditional balanced differential detection systems and EOAM modulation detection systems is constructed for comparison. Initially, the output voltage signal undergoes conversion into angular velocities using the scale factor. Subsequently, the power spectral density is computed utilizing the fast Fourier transform (FFT) algorithm. Each dataset used for calculation spans a duration of 1800 s. Within the test data segment, after employing the closed-loop detection proposed in this Letter, the relative fluctuation of optical intensity amplitude received by PD1 has improved from 1.6% to 0.05%.

Figure 5 presents the experimental results. It is evident that compared to conventional approaches and open-loop EOAM modulation detection, there is a notable improvement in sensitivity across the frequency range of 0.01 to 1 Hz when employing the closed-loop detection method. Noise sources, such as probe laser noise and temperature fluctuations in the system, couple into the co-magnetometer signal at low frequencies (0.01–1 Hz). With the adoption of EOAM closed-loop modulation detection, the impact of these noise sources is significantly reduced. Quantitatively, the sensitivity of the co-magnetometer at 1 Hz has improved from 9.7 × 10⁻⁶ to 3.25 × 10⁻⁶ deg/(s Hz^1/2), and the sensitivity at 0.1 Hz has improved from 9.46 × 10⁻⁵ to 1.05 × 10⁻⁵ deg/(s Hz^1/2), indicating effective suppression of low-frequency noise. In engineering applications, the angle random walk for high-precision gyroscopes is typically better than 0.0015 deg/h^1/2, equivalent to an inertial measurement sensitivity better than 2.5 × 10⁻⁵ deg/(s Hz^1/2)^[22,23]. We have achieved a high sensitivity comparable to other gyroscopes in engineering applications, and there is still potential for further development. The improvement also demonstrates the development potential of SERF co-magnetometers in the field of inertial navigation, paving the way towards further advancements in high-precision gyroscopes.

Figure 5.Comparison of inertial measurement sensitivity among different detection methods.

5. Conclusion

In summary, we have proposed an atomic spin polarization detection scheme based on closed-loop EOAM modulation. We analyze the feasibility of the proposed approach theoretically and demonstrate its ability to suppress disturbances such as operating point drift. The experimental results demonstrate a significant improvement in the sensitivity of the system compared to traditional detection methods within the frequency range of 0.01 to 1 Hz. At 1 Hz, the sensitivity has improved from $9.7 \times 10^{- 6}$ to 3.25 × 10⁻⁶ deg/(s Hz^1/2), representing a 67% enhancement. This improvement can be attributed to two main factors: modulating noise to high frequencies via modulation-demodulation techniques and filtering it out through demodulation by employing closed-loop control of optical intensity modulation amplitude to counteract optical system drift caused by external environmental factors. This system holds great potential for high sensitivity and miniaturized design in co-magnetometers, and it is also applicable to enhancing detection capabilities in areas such as testing CPT violations and searching for anomalous spin forces.

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