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Abstract
Broad-dynamic-range magnetometers are demanded in practical applications and fundamental research. We experimentally demonstrate a parametrically modulated atomic magnetometer with a large dynamic range by taking advantage of the high-order resonance effects. With the increase of the strength of the modulation field, both low-order and high-order resonances are well resolved and used to measure the DC or AC magnetic fields. The experimentally demonstrated sensitivity of the magnetometer based on the zeroth-order resonance is $1.5~\textrm{pT}/\sqrt{\textrm{Hz}}$, and those based on the high-order resonances are below $3~\textrm{pT}/\sqrt{\textrm{Hz}}$, making the measurement of high magnetic fields feasible under an open-loop operation. Moreover, we also demonstrated the measurement of high-frequency large AC magnetic field with the high-order resonances, and the sensitivity for the AC magnetic field based on the first-order resonance is $7~\textrm{pT}/\sqrt{\textrm{Hz}}$. Our scheme provides a new path for the development of broad-dynamic-range and miniaturized atomic magnetometers.
© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The precision measurement of magnetic fields plays an important role in the areas of fundamental physics [1–4], material science [5,6], and biomedical sensing [7–10]. Optically pumped atomic magnetometers have advantages over other magnetometers in cost, volume and sensitivity [11–15]. Especially in recent years, optically pumped magnetometers based on parametric resonance, where parameters such as resonant frequency or decay rate are modulated, have reached excellent levels of sensitivity and have widely been used for the measurement of magnetic field gradients [16,17], medical imaging of biological currents [13,18], and fundamental measurements [19,20].
Generally, two schemes of parametric modulation atomic magnetometer are defined according to the direction of the modulation field. One is called the $M_z$ scheme, where the modulation field is along the pumping beam. The other one is called the $M_x$ scheme, where the modulation field is perpendicular to the pumping beam [21]. For $M_z$ scheme magnetometer, it generally requires two beams and can measure two or three components of magnetic fields simultaneously [22–25]. While for $M_x$ scheme magnetometer, based on the same principle as the Hanle effect, a single pumping beam is sufficient to measure the weak magnetic fields perpendicular to the direction of spin polarization, which is helpful for the miniaturization [26–30].
For a particular parametrically modulated atomic magnetometer at low field, either the $M_z$ scheme or $M_x$ scheme, the zeroth-order or first-order resonance is used, while the other high-order resonances are neglected [21,25,31–34]. However, for strong DC or AC magnetic field measurements, this method cannot work well unless some closed-loop controls are employed to solve the dynamic range problem. The closed-loop controls will inevitably increase the volume of the system and cost more physical resources, which are disadvantageous to the magnetometer miniaturization.
Here, we experimentally demonstrate a single-beam $M_x$ scheme atomic magnetometer with a broad dynamic range under an open-loop operation. In the weak-field regime, the magnetic fields can be detected by using the low-order resonances, and the demonstrated sensitivity of the atomic magnetometer based on the zeroth-order resonance is $1.5~\textrm{pT}/\sqrt{\textrm{Hz}}$. To measure the strong DC magnetic field, we increase the strength of the modulation field and take advantage of the high-order resonance effects. In this manner, the large DC magnetic field can be detected with $n$ times smaller modulation frequency than that detected with typical parametric modulation atomic magnetometers such as Bell-Bloom type atomic magnetometers [35–37], which greatly reduces the bandwidth requirement of the coils and detectors. By optimizing the parameters such as modulation depth, the sensitivities for high-order resonances are below $3~\textrm{pT}/\sqrt{\textrm{Hz}}$. The high-order resonance effects can also allow the detection of the AC magnetic field by observing the amplitude response of the resonances of atomic spins to the strength of the modulation field, and the sensitivity based on the first-order resonance is $7~\textrm{pT}/\sqrt{\textrm{Hz}}$. Moreover, high-order resonance effects also provide a new method to calibrate the coil constants for single-beam atomic magnetometers.
2. Experimental setup and method
The experimental setup is shown in Fig. 1. A cubic cell (4 mm $\times$ 4 mm $\times$ 4 mm) containing $^{87}\textrm{Rb}$ atoms, 680 Torr $\textrm{N}_2$ as quenching gas, and 50 Torr $^{4}\textrm{He}$ as buffer gas is placed in four-layer cylindrical magnetic shields which used for shielding magnetic fields of the environment. In addition, two pairs of saddle coils and a pair of Helmholtz coil are applied along the $x,~y$, and $~z$ directions to compensate the residual magnetic fields and generate the static magnetic field $B_0$ and the modulation field $B_1\cos (\omega t)$ along the $x$ direction. To increase the vapor density, the cell is also heated to $90~^{\circ }\textrm{C}$ in our experiment.
Fig. 1. Experimental setup: PBS: Polarization beam splitter; QW: Quarter waveplate; PD: Photoelectric detector; LIA: Lock-in amplifier.
The optical configuration is simple. A circularly polarized pumping beam with wavelength 795 nm (on resonance with D1 line of $^{87}$Rb atoms) and intensity about 360 $\mu \textrm{W}/\textrm{cm}^2$, is applied along the $z$ direction to polarize the $^{87}$Rb atoms. The spin polarization of $^{87}$Rb atoms is measured by the changes in the intensity of the transmitted pumping beam, which is detected by a photoelectric detector. Finally, the detected signal is analyzed with a lock-in amplifier.
The dynamical evolution of the spin polarization $\mathbf {S}$ of $^{87}$Rb atoms under a magnetic field $\mathbf {B}$ can be phenomenally described by the Bloch equation [38,39]
Under the normal experimental conditions, the first-order harmonic of $S_z$ gives the most dominant signal component. For this reason, the transmitted pumping beam is demodulated at frequency $\omega$ with a lock-in amplifier, and the demodulated in-phase $X_n$ component and quadrature $Y_n$ component for the $n$th order resonance are
The detected in-phase component $X$ is $\sum _{n}X_n$, and quadrature component $Y$ is $\sum _{n}Y_n$. It is noticed that the $Y_n$ component gives a dispersion curve with respect to $\omega _0$ near $\omega _0=n\omega$. Therefore, the DC magnetic field can be detected within the linewidth determined by the relaxation rate $\Gamma$.
3. Results and analysis
In this section, we present the practical applications of both the low-order and high-order resonances of atomic spins in magnetic field sensing. In the weak-field regime, the DC fields and AC magnetic fields with low frequencies can be measured with the zeroth-order resonance of atomic spins. To avoid closed-loop control in high field sensing, we increase modulation depth $u$ and use the observed high-order resonance to measure a high DC field. In addition, the amplitude of the $n$-th order resonance varies with the modulation depth, and this behavior can be used to measure the strength of an unknown AC magnetic field with high frequency. Finally, we discuss the problem of coil constant calibration with the high-order resonances, which also plays an important role in magnetic field sensing.
3.1 DC magnetic field and low-frequency AC magnetic field sensing
According to Eq. (5), the demodulated $Y_n$ component shows a dispersion curve with respect to $\omega _0$ at the center of $\omega _0=n\omega$. It is clear to find that from Fig. 2(a), for each order resonance, the dependence of its $Y$ component on $\omega _0$ within the linewidth $\Gamma$ can be used for magnetic field sensing.
Fig. 2. Magnetic resonance signals of the $M_x$ scheme atomic magnetometer. (a) Demodulated $X,~Y$, and $~R=\sqrt{X^2+Y^2}$ components with the first harmonic of the modulation frequency ($\omega =~2\pi \times 20~$kHz) as a function of $\omega _0$, and $n=-1$, $n=0$ and $n=1$ resonances are observed. (b) The sensitivity of the magnetometer based on the $n=0$ resonance, where a 2.2 nT magnetic field is added at 60 Hz.
Weak magnetic fields can be measured with the zeroth-order resonance of atomic spin under open-loop operation. By optimizing the parameters such as pumping intensity and modulation depth $u$, the magnetometer based on the zeroth-order resonance achieves a sensitivity of $1.5~\textrm{pT}/\sqrt{\textrm{Hz}}$ in our experiment, which can be used to measure weak DC fields and AC field with low frequencies [see Fig. 2(b)]. Since the linewidth of the zeroth-order resonance in our experiment is 346.6 Hz, then the dynamic range of the magnetometer based on zeroth-order resonance is [−24.8 nT, 24.8 nT].
For a strong DC magnetic field $B_0$ beyond the dynamic range of the zeroth-order resonance, there are generally two methods to realize magnetic field measurement. The first one is based on closed-loop control, where the magnetic field is compensated by a control loop such that the quadrature $Y$ is locked to the resonant point $\omega _0=0$, and the magnetic field $B_0$ is obtained by the feedback signal. The other one is based on the first-order resonance effect, where the modulation frequency is swept to determine the first-order resonant point $\omega =\omega _0$. For the first method, it will increase the volume of the system and cost more power, which is unsuitable for the atomic magnetometer miniaturization. For the second method, the measurement of the magnetic field $B_0$ is limited by the frequency of the modulation field, i.e., the stronger the magnetic field $B_0$, the higher the modulation frequency $\omega$ is required.
The high-order resonance effects of atomic spins can overcome the disadvantages of these two methods. To cancel the effect of demodulated phase on the measurement, we focus on the behavior of the $R$ component,
since it is phase-independent. Figure 3 shows the behavior of the $R$ component, where $R$ achieves its extreme points at $\omega _0=n\omega$. For a small modulation depth $u$ ($u<0.1$), only the zeroth-order ($n=0$) and first-order ($n=\pm 1$) resonances are observed, and the resonant points of the first-order resonances change with the modulation frequency [as shown in Fig. 3(a)]. While for a large modulation depth $u$, high-order resonances are observed at $\omega _0=n\omega$ even though the modulation frequency is $\omega$, as if there is a pseudomagnetic field with frequency $n\omega$ along the $x$ direction [as shown in Fig. 3(b)]. The high-order resonances can also be used for magnetic field sensing. Figure 4(a) shows the sensitivities of the magnetometer obtained with $n=1-6$ resonances. By optimizing experimental parameters such as $u$, all the sensitivities are below $3~\textrm{pT}/\sqrt{\textrm{Hz}}$. Meanwhile, we also measured the bandwidth of the magnetometer [as shown in Fig. 4(b)], and the bandwidths are about 100 Hz, meaning that the maximum frequency of the AC magnetic field measured based on a single resonance is below 100 Hz.Fig. 3. Amplitude behavior of the demodulated $R$ component. (a) Demodulated $R$ component as a function of $\omega _0$ under different modulation frequencies. (b) Demodulated $R$ component as a function of $\omega _0$ under different modulation depths, and the high order resonances are observed as the modulation depth $u$ increases. The frequency of the modulation field is $\omega =2\pi \times$5 kHz.
Fig. 4. Sensitivities and bandwidths of the magnetometer obtained with high-order resonances. (a) Sensitivities of the magnetometer obtained with $n=1-6$ resonances. A 3.4 nT peak-to-peak magnetic field at 55 Hz, 60 Hz, 65 Hz, 70 Hz, 75 Hz and 80 Hz are applied for $n=1-6$ resonances, respectively. (b) The bandwidths of the magnetometer for $n=1-6$ resonances. They are obtained by sweeping the frequency of an oscillation field and observing the amplitude response of the magnetometer.
The existence of high-order resonances enables us to measure a strong magnetic field $B_0$ with a low modulation frequency $\omega =\gamma B_0/n$ provided that $\omega =\gamma B_0/n \gg \Gamma$, i.e., the adjacent high-order resonances should be well resolved. To experimentally demonstrate our scheme, we apply a magnetic field $B_0 \approx 3571.4~\textrm{nT}$, corresponding to $\omega _0\approx 2\pi \times 25~\textrm{kHz}$, in the $x$ direction, and then slowly scan the modulation frequency $\omega$ under different strength $\omega _1$. The experiment shows high-order resonances become observable at $\omega =\omega _0/n$ as the modulation depth increases (see Fig. 5). This indicates that the large DC magnetic field can be measured with a much smaller modulation frequency $\omega =\omega _0/n$. The comparison of experimental data and fitting results are shown in Fig. 6(a), where the resonant points $\omega _n$ of high-order resonances are extracted, and the magnetic field is obtained with the relation $B_0=n\omega _n/\gamma$ in principle. Figure 6(b) gives the measured value of $B'_0$ with different high-order resonances. Although the measurement error increases with the resonance order $n$, which attributes to the measurement error of $\omega _{\textrm{n}}$, the measurement precision of $B_0$ is still excellent. For example, the measured value $B'_0$ is $3550.0\pm 47.0~$nT with $n=8$ resonance.
Fig. 5. Behavior of the demodulated $R$ component under different $\omega _1$ for a given $\omega _0~ \approx 2\pi \times 25 \textrm{kHz}$ when slowly sweeping the modulation frequency $\omega$. High-order resonances are observed at $\omega =\gamma B_0/n$ ($n\ge 1$) as $\omega _1$ increases.
Fig. 6. Measurement of the large $B_0$ with high-order resonances. (a) The high-order resonances and its Lorentzian fitting are represented by dots and solid line, respectively. (b) The measured value $B'_0$ and its confidence interval for high-order resonances.
The above discussion is based on the assumption that the resonance order $n$ is known ahead of time. In practical applications, the exact resonance orders are unknown since magnetic field $B_0$ is unknown. However, the unknown $B_0$ can also be determined by two adjacent high-order resonant points $\omega _k$ and $\omega _{k+1}$ since
It worth is pointing out that there is a trade-off between the modulation strength $\omega _1$ and the modulation frequency $\omega$, since $u=\omega _1/\omega$ and the lower-order resonances are more easily observed than the higher-order resonances at the same modulation depth $u$. For example, the $n=8$ resonance can be observed when the $\omega _1$ is increased to $18.83~\textrm{kHz}$ (as shown in Fig. 5). Therefore, the high-order resonance effects provide a new method to measure the strong DC magnetic field and greatly reduce the bandwidth requirements of detectors and current sources, which may find applications in particular magnetic field sensing.
3.2 High-frequency AC magnetic field sensing
Measuring the AC magnetic fields, especially the strong AC magnetic field, is important for practical applications. Due to the bandwidth or dynamic-range limitations, most atomic magnetometers can only measure the low-frequency (usually on the order of a few hundred Hertz) AC magnetic field with small amplitude, which can be measured with the $n=0$ resonance [as shown in Fig. 2(a)]. However, it is difficult to measure the AC magnetic field with a large frequency (> kHz). Here, we show that an AC magnetic field with small or large amplitude can be measured provided its frequency is much larger than the full width at half maximum (FWHM) of high-order resonances. In our system, the values of FWHM of high-order resonances are less than 400 Hz, therefore the AC field with a frequency larger than 400 Hz can be measured.
To measure the AC magnetic field, we focus on the amplitude behavior of the resonances, i.e., the value of the $R$ component at $\omega _0=n\omega$, with respect to modulation depth $u$. According to Eqs. (4) – (5), the amplitude of the $n$th magnetic resonance is proportional to $g_{n}(u)$,
Figure 7 shows the theoretical amplitude behavior of the resonances for $n=0-5$ as a function of the modulation depth $u$. The corresponding experiment measured resonance amplitudes with different modulation depth $u$ are in good agreement with the Eq. (8), demonstrating that the amplitudes can be precisely controlled by the modulation depth $u$ (see Fig. 8).
Fig. 7. Amplitude behavior $g_n(u)$ under the different modulation depths. The linear response regions of the high-order resonances are marked with different colors.
Fig. 8. Amplitude behavior under the different modulation depths. (a)-(f) are the amplitudes of from $n=0$ to $n=5$ resonances, respectively. The dots are the experimental data, and the blue lines are the theoretical results obtained with Eq. (8).
The relation between the resonance amplitudes and the modulation depth $u$ enables us to measure the strength of an AC magnetic field $B_1=u\omega /\gamma$. The measurement of $B_1$ can be divided into two steps. We first determine the frequency of the AC field, which can be obtained by analyzing the spectrum of the transmitted pumping signal or scanning the magnetic field $B_0$ and reference frequency to determine the resonant points of the $R$ component, and use the determined frequency as the reference frequency. Then we determine the value of modulation depth $u$ with the amplitude response. Since the resonance amplitude is a non-monotonic function of the modulation depth $u$, a measured $n$-th order resonance amplitude may correspond to two different $u_1$ and $u_2$ as shown in Fig. 7. Although $g_0(u_1)=g_0(u_2)$, we can determine the true value of $u$ by investigating the first-order resonances amplitude since $g_1(u_1)$ and $g_1(u_2)$ are unequal. By this means, the strength of an unknown AC field, which can be regarded as the modulation field, can be measured by using high-order resonances provided the frequency is much larger than the relaxation rate $\Gamma$.
To demonstrate the validity of our method, we modulate the amplitude $B_1$ of the modulation field with a small oscillating field $B_{\textrm{m}}\cos (\omega _{\textrm{m}}t+\varphi )$, i.e.,
such that the resulting modulation depth $u$ lies in the linear regions of $g_n(u)$ (see the solid lines in Fig. 7). In the experiment, a modulation field with $\omega =2\pi \times 5~\textrm{kHz}$ is applied for $t<5~\textrm{s}$, and then amplitude $B_1$ is modulated with an oscillating field with frequency $\omega _{\textrm{m}}=2\pi \times 20~\textrm{Hz}$ [as shown in Fig. 9(a)]. The corresponding response cure $R$ oscillates with $\omega _{\textrm{m}}$ [see Fig. 9(b)], meaning this method is feasible. By analyzing the behavior of the $R$ component under the action of modulation field in the frequency regime, we can obtain the sensitivity of the magnetometer based on the $n=1-4$ resonances, and they are shown in Fig. 10. The sensitivity of the magnetometer based on the $n=1$ resonance is $7~\textrm{pT}/\sqrt{\textrm{Hz}}$, and the sensitivities are about $100~\textrm{pT}/\sqrt{\textrm{Hz}}$ for $n=~2,~3,~4$ resonances. In principle, the frequency $\omega$ of the AC magnetic can be measured provided $B_0=\omega /\gamma$ in the linear regimes of energy shift of Zeeman levels of the ground states of $^{87}\textrm{Rb}$. However, in the real experiment, the maximum frequency of the AC magnetic field is determined by the bandwidth of the detector and lock-in amplifier, and the maximum DC magnetic field generated by the coils. Due to these limitations, the maximum frequency of the AC magnetic field measured with this method is about 150 kHz in our experiment.Fig. 9. Method to measure the AC magnetic field. (a) The amplitude of the modulation field is modulated with $20~\textrm{Hz}$ after $t\ge 0.5~$s. (b) Response of the $R$ component of $n=1$ resonance under the action of modulation field in (a).
Fig. 10. Sensitivity of the magnetometer based on the $n=1-4$ resonances. They are obtained by analyzing the spectrum of the $R$ component of $n$th resonance under the amplitude modulation, respectively.
3.3 Coil constants calibration
The calibration of coil constants is a challenging issue for a small-volume single-beam configuration magnetometry based on atomic spins. Since the magnetic field $B_0$ generated by the coil is linearly dependent on the current $I$ in the coil, $B_0=kI$, the coil constant $k$ could be determined by measuring the magnetic field $B_0$ for a given current $I$.
There are several commonly used methods in coil constant calibrations, and most of them are based on the measurement of the Larmor frequencies under different magnetic fields, and the coil constant is calculated from the average value. In our previous work [40], the transient dynamics response of atomic spins under the different magnetic fields $B_0$ (i.e., free induction decay) was employed to calibrate the coil constants in the $x$ and $y$ axes for a single-beam configuration. In the parametric modulation atomic magnetometers, the first-order resonance can be used to calibrate coil constant by sweeping the frequency $\omega$ of the modulation field under a finite current $I$ and observing the resonant point.
To eliminate stochastic errors, these experiments need to be repeated many times by varying the current $I$. This labored work can be avoided by taking advantage of the high-order resonance effects. By increasing the modulation depth $u$, high-order resonances can be well resolved. Under a fixed bias current $I$, the observed first $m$ resonant points $(\nu _1,\nu _2,\ldots,\nu _m)$ give a characterization of the coil constant by
The coil constants of the $x$ and $y$ directions calibrated via this method agree well with the transient dynamics response method, thus providing an efficient way of calibrating the coil constant since one measurement is sufficient.
4. Conclusion
In conclusion, we demonstrated a single-beam $M_x$ scheme atomic magnetometer with a broad dynamic range in an open-loop operation. The magnetometer is based on the multi-resonance phenomenon of atomic spins and can be used to measure the DC and AC fields of small or large amplitudes. The atomic magnetometer based on zeroth-order resonance obtains a sensitivity of $1.5~\textrm{pT}/\sqrt{\textrm{Hz}}$, and those based on the other high-order resonances are below $3~\textrm{pT}/\sqrt{\textrm{Hz}}$. By using the high-order resonance, a large DC magnetic field $B_0$ can be measured with a lower modulation frequency that is $n$ times smaller than the Larmor precession frequency $\omega _0=\gamma B_0$. Furthermore, the strength of the AC magnetic field can also be measured with the amplitude response of the high-order resonances, and the sensitivity based on first-order resonance is $7~\textrm{pT}/\sqrt{\textrm{Hz}}$, and those based on $n=2-4$ resonances are on the level of $100~\textrm{pT}/\sqrt{\textrm{Hz}}$. The method presented in this paper expands the dynamic range of the magnetometer and largely reduces the bandwidth requirement of the detectors and the current sources. This method provides a new path for high magnetic fields sensing with single-beam miniaturized atomic magnetometries in a practical environment.
Funding
Beijing Natural Science Foundation (4191002); Key Reseach and Development Program of Zhejiang (2020C01037).
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.
References
1. C. W. Helstrom, “Quantum detection and estimation theory,” J. Stat. Phys. 1(2), 231–252 (1969). [CrossRef]
2. D. Budker and M. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007). [CrossRef]
3. C. L. Degen, F. Reinhard, and P. Cappellaro, “Quantum sensing,” Rev. Mod. Phys. 89(3), 035002 (2017). [CrossRef]
4. M. Khanahmadi and K. Mølmer, “Time-dependent atomic magnetometry with a recurrent neural network,” Phys. Rev. A 103(3), 032406 (2021). [CrossRef]
5. C. Tsuei and J. Kirtley, “Pairing symmetry in cuprate superconductors,” Rev. Mod. Phys. 72(4), 969–1016 (2000). [CrossRef]
6. H. Mamin, M. Poggio, C. Degen, and D. Rugar, “Nuclear magnetic resonance imaging with 90-nm resolution,” Nat. Nanotechnol. 2(5), 301–306 (2007). [CrossRef]
7. R. Wyllie, M. Kauer, G. Smetana, R. Wakai, and T. Walker, “Magnetocardiography with a modular spin-exchange relaxation-free atomic magnetometer array,” Phys. Med. Biol. 57(9), 2619–2632 (2012). [CrossRef]
8. R. Wyllie, M. Kauer, R. T. Wakai, and T. G. Walker, “Optical magnetometer array for fetal magnetocardiography,” Opt. Lett. 37(12), 2247–2249 (2012). [CrossRef]
9. V. K. Shah and R. T. Wakai, “A compact, high performance atomic magnetometer for biomedical applications,” Phys. Med. Biol. 58(22), 8153–8161 (2013). [CrossRef]
10. A. P. Colombo, T. R. Carter, A. Borna, Y.-Y. Jau, C. N. Johnson, A. L. Dagel, and P. D. Schwindt, “Four-channel optically pumped atomic magnetometer for magnetoencephalography,” Opt. Express 24(14), 15403–15416 (2016). [CrossRef]
11. R. T. Wakai, “The atomic magnetometer: A new era in biomagnetism,” in AIP Conference Proceedings, vol. 1626 (American Institute of Physics, 2014), pp. 46–54.
12. I. Kominis, T. Kornack, J. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003). [CrossRef]
13. J. Osborne, J. Orton, O. Alem, and V. Shah, “Fully integrated standalone zero field optically pumped magnetometer for biomagnetism,” in Steep Dispersion Engineering and Opto-Atomic Precision Metrology XI, vol. 10548 (International Society for Optics and Photonics, 2018), p. 105481G.
14. S. M. Rochester, Modeling nonlinear magneto-optical effects in atomic vapors (University of California, Berkeley, 2010).
15. V. Gerginov, S. Krzyzewski, and S. Knappe, “Pulsed operation of a miniature scalar optically pumped magnetometer,” J. Opt. Soc. Am. B 34(7), 1429–1434 (2017). [CrossRef]
16. V. Lucivero, W. Lee, N. Dural, and M. Romalis, “Femtotesla direct magnetic gradiometer using a single multipass cell,” Phys. Rev. Appl. 15(1), 014004 (2021). [CrossRef]
17. D. Sheng, A. R. Perry, S. P. Krzyzewski, S. Geller, J. Kitching, and S. Knappe, “A microfabricated optically-pumped magnetic gradiometer,” Appl. Phys. Lett. 110(3), 031106 (2017). [CrossRef]
18. E. Boto, N. Holmes, J. Leggett, G. Roberts, V. Shah, S. S. Meyer, L. D. Mu noz, K. J. Mullinger, T. M. Tierney, S. Bestmann, G. R. Barne, R. Bowtell, and M. Brookes, “Moving magnetoencephalography towards real-world applications with a wearable system,” Nature 555(7698), 657–661 (2018). [CrossRef]
19. R. Zhang, R. Mhaskar, K. Smith, and M. Prouty, “Portable intrinsic gradiometer for ultra-sensitive detection of magnetic gradient in unshielded environment,” Appl. Phys. Lett. 116(14), 143501 (2020). [CrossRef]
20. J. Lu, Z. Qian, J. Fang, and W. Quan, “Suppression of vapor cell temperature error for spin-exchange-relaxation-free magnetometer,” Rev. Sci. Instrum. 86(8), 083103 (2015). [CrossRef]
21. J. Wang, W. Fan, K. Yin, Y. Yan, B. Zhou, and X. Song, “Combined effect of pump-light intensity and modulation field on the performance of optically pumped magnetometers under zero-field parametric modulation,” Phys. Rev. A 101(5), 053427 (2020). [CrossRef]
22. T. G. Walker and M. S. Larsen, “Spin-exchange-pumped nmr gyros,” Adv. At., Mol., Opt. Phys. 65, 373–401 (2016). [CrossRef]
23. K. Zhang, N. Zhao, and Y. Wang, “Closed-loop nuclear magnetic resonance gyroscope based on rb-xe,” Sci. Rep. 10(1), 1–7 (2020). [CrossRef]
24. C. Hao, Q. Yu, C. Yuan, S. Liu, and D. Sheng, “Herriott-cavity-assisted closed-loop xe isotope comagnetometer,” Phys. Rev. A 103(5), 053523 (2021). [CrossRef]
25. H. Yang, K. Zhang, Y. Wang, and N. Zhao, “High bandwidth three-axis magnetometer based on optically polarized 85rb under unshielded environment,” J. Phys. D: Appl. Phys. 53(6), 065002 (2020). [CrossRef]
26. T. Karaulanov, I. Savukov, and Y. J. Kim, “Spin-exchange relaxation-free magnetometer with nearly parallel pump and probe beams,” Meas. Sci. Technol. 27(5), 055002 (2016). [CrossRef]
27. I. Savukov, Y. Kim, V. Shah, and M. Boshier, “High-sensitivity operation of single-beam optically pumped magnetometer in a khz frequency range,” Meas. Sci. Technol. 28(3), 035104 (2017). [CrossRef]
28. G. Le Gal, G. Lieb, F. Beato, T. Jager, H. Gilles, and A. Palacios-Laloy, “Dual-axis hanle magnetometer based on atomic alignment with a single optical access,” Phys. Rev. Appl. 12(6), 064010 (2019). [CrossRef]
29. N. Castagna and A. Weis, “Measurement of longitudinal and transverse spin relaxation rates using the ground-state hanle effect,” Phys. Rev. A 84(5), 053421 (2011). [CrossRef]
30. E. Breschi and A. Weis, “Ground-state hanle effect based on atomic alignment,” Phys. Rev. A 86(5), 053427 (2012). [CrossRef]
31. W. Xiao, T. Wu, X. Peng, and H. Guo, “Atomic spin-exchange collisions in magnetic fields,” Phys. Rev. A 103(4), 043116 (2021). [CrossRef]
32. X. Qiu, Z. Xu, X. Peng, L. Li, Y. Zhou, M. Wei, M. Zhou, and X. Xu, “Three-axis atomic magnetometer for nuclear magnetic resonance gyroscopes,” Appl. Phys. Lett. 116(3), 034001 (2020). [CrossRef]
33. Z. Li, R. T. Wakai, and T. G. Walker, “Parametric modulation of an atomic magnetometer,” Appl. Phys. Lett. 89(13), 134105 (2006). [CrossRef]
34. J. Lu, Z. Qian, J. Fang, and W. Quan, “Effects of ac magnetic field on spin-exchange relaxation of atomic magnetometer,” Appl. Phys. B 122(3), 59 (2016). [CrossRef]
35. A. Perry, M. Bulatowicz, M. Larsen, T. Walker, and R. Wyllie, “All-optical intrinsic atomic gradiometer with sub20 ft/cm/sqrt[hz] sensitivity in a 22 ut earth-scale magnetic field,” Opt. Express 28(24), 36696–36705 (2020). [CrossRef]
36. V. Gerginov, M. Pomponio, and S. Knappe, “Scalar magnetometry below 100 ft/sqrt[hz] in a microfabricated cell,” IEEE Sens. J. 20(21), 12684–12690 (2020). [CrossRef]
37. R. Jiménez-Martínez, W. C. Griffith, Y.-J. Wang, S. Knappe, J. Kitching, K. Smith, and M. D. Prouty, “Sensitivity comparison of mx and frequency-modulated bell-bloom cs magnetometers in a microfabricated cell,” IEEE Trans. Instrum. Meas. 59(2), 372–378 (2010). [CrossRef]
38. F. Bloch, “Nuclear induction,” Phys. Rev. 70(7-8), 460–474 (1946). [CrossRef]
39. S. J. Seltzer, Developments in alkali-metal atomic magnetometry (Princeton University, 2008).
40. J. Tang, Y. Yin, Y. Zhai, B. Zhou, B. Han, H. Yang, and G. Liu, “Transient dynamics of atomic spin in the spin-exchange-relaxation-free regime,” Opt. Express 29(6), 8333–8343 (2021). [CrossRef]
