The atomic magnetometer, as a new generation of high-precision and sensitive magnetometry, is attractive for vast applications, including fundamental physics^[1], biological sensing^[2,3], materials inspection^[4], geophysics^[5], and navigation^[6]. An increasing number of studies have focused on all kinds of specific and challenging application scenarios^[7]. Parameters such as accuracy, sensitivity, bandwidth, dynamical range, size, and power consumption need to be balanced and optimized in a targeted manner. Here, we consider applications such as unknown magnetic detection and large-strength variations where fast magnetic field search and locking scheme is required.

In an optically pumped atomic magnetometer, the magnetic field can be measured by probing the free induction decay (FID) signal of the atomic spin polarization^[8]. However, this is determined by the frequency counter’s bandwidth and accuracy or the FID signal fitting, which is not quite suitable for weak fields and high sensitivity requirements. Another widely used method is to apply modulation techniques and balanced polarimetry methods for polarization rotation measurement^[9,10]. In order to achieve high sensitivity, the resonance linewidth of the magnetometer should be narrow, which limits the bandwidth and dynamical range^[11,12]. Some researchers turn to closed-loop configurations for tracking the resonant Larmor frequency, including self-oscillating techniques^[13–15], real-time magnetic compensation^[16,17], and active modulation feedback^[18,19]. Among these methods, there exist separate pumping and probe laser beams or additional oscillating or compensation coils, which usually increase the sensors’ complexity and lead to systematic error.

Here, we present a feedback method to search and measure the scalar magnetic field using a nonlinear magneto-optical rotation (NMOR) magnetometer^[11] with single-beam intensity modulated light^[10,20]. Based on the in-phase response of the atomic polarization rotation, we apply the successive approximation algorithm to find an unknown magnetic field within a time duration of 0.8 s and with an accuracy of (0.2%±4)nT. The measurement range is 10 to 104nT. In contrast, current atomic magnetometer studies mainly deal with either near-zero magnetic fields or strong magnetic fields. After reaching the linear phase response region of the field, the 3-dB locking bandwidth is about 87 Hz, and the sensitivity is about 0.4pT/Hz1/2 @10 Hz, both of which are comparable with current commercial magnetometers. Our all-optical single-beam configuration also has the advantages of compactness and no influence on the measured magnetic field. The scheme should be valuable in future practical applications that require precise and fast unknown magnetic field measurements that cover both the weak and strong field regimes.

As illustrated in Fig. 1, an all-optical single-beam atomic magnetometer is applied. One 795 nm diode laser (Newport-TLB-6814) is used as the light source. The light is controlled by an acousto-optic modulator (AOM, AA-MT80-B30A1-IR) to generate a square wave with a 50% duty cycle. Its frequency is 80 MHz red detuning to the Rb87 D1-line transition |F=2⟩→|F′=1⟩. The intensity modulation frequency ωm of the pulse shape is controlled by a direct digital synthesizer (DDS). The DDS is generated through a multifunction data acquisition card (NI-6363). The light pulse is then linearly polarized and propagated through a cylindrical paraffin-coated vapor cell. The averaged light power is 40 µW, and the collimated beam diameter is 3 mm. The vapor cell has a diameter of 25 mm and a length of 40 mm. It is filled with isotopically pure Rb87 atoms without buffer gas. In our experiment, the vapor cell is working at room temperature and placed inside a five-layer μ-metal magnetic shield to attenuate the ambient magnetic field. The residual magnetic field is compensated down to the nT level by additional three-dimensional coils. A pair of Helmholtz coils are used to generate the constant magnetic field B along the direction of the light beam for measurement.

After atom–light interaction in the vapor cell, the polarization of the transmitted light is rotated by a half-wave plate (HWP) and detected by a balanced photodetector (BPD, Thorlabs-PDB210A/M). The BPD output signal reflects the precession of the atomic polarization in the presence of the magnetic field. The signal is sent to a lock-in amplifier (LIA, SRS-SR830) together with the DDS signal as a reference to demodulate the field strength B. The in-phase and quadrature signals are sent to our data acquisition and processing system (DAPS). The sampling rate is 335 Hz. Based on the detected signals, we use feedback to adjust the DDS frequency to track the resonant Larmor frequency and achieve the magnetic field search and locking tasks. The DAPS system also sends instructions to a direct current (DC) current supply to adjust the field strength B.

The magnetic field is usually obtained by sweeping the modulation frequency ωm and fitting the line shape of the polarization process signal. A typical response curve of the LIA in-phase (X) and quadrature (Y) components for our atomic magnetometry is shown in Fig. 2(a). Here, we set the magnetic field Bset at 500 nT and change the intensity modulation frequency ωm at around twice the resonant Larmor frequency ΩL · ΩL = γB and γ is the gyromagnetic ratio. The in-phase value X0 at ωm=2ΩL is set exactly at zero by adjusting the offset of the LIA. At the resonant point, the quadrature signal (Y) reaches its maximum. The field strength B can be obtained by searching the peak position of Y. For the in-phase component, there is a narrow linear region (δω∼0.2kHz) near the resonance frequency 2ΩL, showing a narrow dispersive relation of the polarization rotation with the magnetic field. The slope kslop is about 1.9 V/kHz. Within this linear region, we can calculate the field B=(ωm−X/kslop)/2γ. Since such a narrow region corresponds only to about 10 nT, it is time-consuming to find an unknown magnetic field by means of step scanning ωm^[21]. Considering a large search range, such as 104nT for example, hundreds of steps are generally required. This is not available for a magnetic field that is not constant but changes with time.

Focusing on how to quickly find the linear region, we check the in-phase value X0 at the resonant position ωm=2ΩL for the different Bset in Fig. 2(b). The inverse of the slope rate kslop is also calculated. We fit them linearly between partitions for simplicity and use the fitting results as the calibration in our search and locking scheme discussed in Sec. 4. The value X0 is not always zero but changes with the magnetic field. Remembering that our magnetometer is a single-beam configuration, the light pulse acts as both pumping and probe of the atomic polarization. The pumping efficiency depends both on the magnetic field and the laser beam intensity. The larger the magnetic field and the stronger the laser power, the better of the atomic polarization. However, strong laser power also contributes more noise and thus affects the sensitivity. The light power is a tradeoff between pumping efficiency and sensitivity. In the experiment, we fix the light power averaged at 40 µW. Thus, the polarization rotation is maximized at a certain magnetic field. For a different magnetic field, change of the atomic polarization leads to the observed drift of X0.

To see clearly, we show variation examples of the BPD output signal Sout for each cyclic light pulse in Fig. 2(c). It reflects the time evolution of the atomic spin polarization. For each cyclic light pulse, the aligned polarization gradually increases when the light pulse is on and damps when the light pulse is off. Thus, Sout varies accordingly. We further select the minimum and maximum values Smin and Smax for different Bset and plot them in Fig. 2(d). When the magnetic field approaches zero, the pumping process is slow and does not reach the maximum polarization. As the magnetic field increases, the atomic spin polarization first increases to the maximum at around 500 nT and then decreases slowly, resulting in a different time variation shape of Sout(t). Thus, the product output of the signal and reference through the LIA changes for different magnetic fields. Because of the X0 variation, the usual PID (proportion-integration-differentiation) locking method cannot be directly applied over a large magnetic range. As mentioned above, to find the peak of the quadrature Y by sweeping ωm is time-consuming^[21]. Instead, based on the measured in-phase values X(ωm), we present one novel scheme for quick searching and locking the magnetic field B in the following section.

We plot the flowchart of our scheme in Fig. 3, including the magnetic search, locking, and output parts. Large-scale scanning curves are shown in the upper-right of the figure when setting the magnetic field B at 500 and 4000 nT. The in-phase signal X(ωm) is almost odd symmetric, corresponding to the resonant position ωm=2ΩL. The task is how to find the symmetry point X0 quickly. In our scheme, we first select two modulation frequencies ωL and ωR and obtain the values XL and XR. If the signs of XL and XR are opposite corresponding to the precalibrated resonant value X0(ωc) [ωc=(ωL+ωR)/2] as shown in Fig. 2(b), then the unknown magnetic field is in between ωL/2γ and ωR/2γ. As X0(ωc) may be far from the real resonant value, resulting in misjudgment, we add another criterion to compare the difference between XR and XL. If the difference is larger than δω×kslop, then B should be within the search range. δω is the width of the linear region. Then, we measure Xc at the modulation frequency ωc and compare it with (ωL, XL) in the same way. Based on the product sign of XL×Xc, we cut the search region in half. If XL×Xc>0, then we replace ωL with ωc. Otherwise, we replace ωR with ωc. Repeating the above operation several times till ωR−ωL<δω, we will reach the linear region at around the resonant frequency 2ΩL. This iterative approaching algorithm converges quickly, requiring only several steps [see Fig. 4(a)].

After reaching the linear region, we turn to calculate the resonant modulation frequency ωm and the output of the magnetic strength B=(ωc−δX/kslop)/2γ. δX is the difference between the measured signal Xc and the precalibrated reference X0(ωc). Continuously, we get the LIA signal (Xc, Yc) and check the value δX. If |δX|<δω×kslop, then we output the magnetic field B accordingly. Otherwise, the magnetic field is out of the linear region. We need to find it again. Because of the possibility that the changing magnetic field is not in the linear region but is still on the slope of X(ωm), we update the modulation frequency to ωc−δX/kslop and then measure and recheck the new value of Xc. If |δX|<δω×kslop, then the magnetic field can be relocked. Otherwise, we determine the direction of the magnetic field change by the sign of δX. If δX>0 (δX<0), then the actual magnetic field becomes larger (smaller). Accordingly, we set a new magnetic field search region [ωL,ωR] and jump to the search part.

Figure 4(a) shows two examples of our magnetic field search when Bset=500 and 5000 nT. The initial searching region is between 20 and 2000 nT. Labeled with ① to ❿ in Fig. 4(a), the linear region is found within ten trial measurements at different ωm. Each measurement takes about 90 ms in our system, so it is about 0.8 s in total to reach the linear region at around the actual magnetic field. The time is mainly spent on the DDS signal update and the LIA frequency discrimination, and we take an average of 10 ms of data sample to obtain an accurate XL(R) for the region selection. The sampling rate and searching speed can be further improved using a field programmable gate array (FPGA) and fast digital LIA technique. As shown by the response time of atomic pumping polarization in Fig. 5(b), the time consumption for each trial has the potential to be improved within 10 ms. In Fig. 4(b), we measure the searching accuracy by comparing the set and measured values Bset and Bout. The error bar is the statistic of five searching measurements. It shows good linearity, and the deviation between them is no more than (0.2%±4)nT. The accuracy is determined by the magnetometer’s stability and calibration parameters, as measured in Fig. 2(b). System noises and fluctuations, such as vapor cell temperature and laser frequency and power, will cause unwanted drift of X0 and kslop. Thus, active stabilization of these factors is required for long-term accuracy. We can also improve the accuracy via the search of the maximum quadrature value Y after reaching the linear region δω. As mentioned in Sec. 3, the single-beam magnetometer needs to deal with the pumping and probe tradeoff of the atomic polarization, which results in the large variation of X0 for different magnetic fields. We may apply temporal or spatial decoupling configurations, i.e., strong-pump weak-probe or forward-pump backward-probe methods^[22,23], to decrease the variation of X0. The accuracy can be optimized better than the current results. In this work, we measure field B in the range of thousands of nT due to the limited sampling rate and current supply of our system. The dynamic range can be increased up to the geomagnetic field level without any principal barriers.

Further, the response bandwidth of our magnetometer is measured in Fig. 5. We first check the magnetic following response in Fig. 5(a) when Bset is pulled up linearly on a large scale. The sweep rate is 100 nT/s, and Bout can basically follow Bset with a time delay. The delay is mainly due to the slow response of the LIA. When the actual field is far away from the linear region before the LIA frequency discrimination is done, the field locking fails. In order to check the searching limit, we observe the BPD output signal Sout when abruptly switching the magnetic field from 500 to 1000 nT [Fig. 5(b)]. The modulation frequency ωm equals 7 kHz. It takes about 6 ms to reach a stable value. This reflects the response time of the atomic polarization. We estimate that the magnetic field following rate can be further improved by an order of magnitude up to 1000 nT/s. In addition, we measure the bandwidth by applying a sinusoidal time-varying magnetic field, where B=B0+ΔBsin(2πft). Here, we set B0=500nT, ΔB=3nT, and ω_m = 7 kHz. Figure 5(c) is an example of when the frequency f=60Hz. The magnetic oscillation range is on the slope of the in-phase cure X(ωm). There is a delay and amplitude attenuation between Bout and Bset. We calculate the amplitude ratio between Bout and Bset and plot it in Fig. 5(d). The data is fitted by a function of f3dB/f2+f3dB2. The 3-dB bandwidth f3dB is about 87 Hz.

We measure the sensitivity as the yellow line shown in Fig. 6. The magnetic field is set at 500 nT and the magnetometer is feedback locked on. The sensitivity is about 2pT/Hz1/2 @ 1 Hz and 0.4pT/Hz1/2 @ 10 Hz. Considering similar dynamical measurement ranges of commercial magnetometers available today, the sensitivity of our magnetometer is comparable or even better. In Fig. 6, we also measure the power spectral density of the in-phase signal of the two cases of laser off (black line) and laser on with constant 40 µW light power (red line). The black line represents the background noise of the balanced photodetector at around 10fT/Hz1/2. Since our laser is free running, the noises and fluctuations of the laser frequency and power are the main noise source both at low and high frequency zones. Current supply for the stray magnetic field compensation and the set magnetic field 500 nT is another main noise source, with low-frequency noise from resistance fluctuation and high-frequency noise from commercial power. Imperfect intensity modulation of the laser beam also provides some noises. With effective methods to suppress these noises, the sensitivity can be improved to below 100fT/Hz1/2.

In this Letter, we demonstrate a simple magnetic field search and measurement scheme for all-optical single-beam atomic magnetometry. We observe that the central point of the LIA signal X drifts with the magnetic field B. After calibrating this offset, we quickly find the magnetic field within one second by iteratively narrowing the search region over a range of 10 to 104nT. The searching accuracy is (0.2%±4)nT, and the following capability of the linear field change is close to 100 nT/s. Our closed-loop locking scheme maintains good bandwidth and sensitivity performance of the magnetometer. All the parameters can be further optimized as discussed in Sec. 5. Our scheme should benefit those practical applications that involve measurements of unknown and changing magnetic fields, covering both the weak and strong field regimes.