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Abstract
In optical atomic magnetometers (AMs), the light-shift caused by the circularly polarized pumping beam have a significant impact on the response and is also one of the non-negligible sources of the noise. In this paper, we develop a novel method whereby utilizing the symmetry of the frequency response in an AM to measure and cancel the light-shift. Furthermore, we theoretically analyze and experimentally verify a rapid method of magnetic field compensation and the approach is convenient to measure and cancel of the light-shift. Moreover, the influence of intensity and frequency of the pumping beam is also investigated. The proposed method of in − situ measurement and cancellation of light-shift will be particularly profitable to other optical systems based on AMs.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Spin-exchange relaxation-free (SERF) atomic magnetometers (AMs) have been found in widely applications, such as the survey of geomagnetism [1], the field of fundamental physics [2–7], biomagnetism [8–12], medicine imaging [13,14], and other areas [15–18]. Typically, SERF magnetometers are pumped by the absorption of an on- or near-resonance circularly or elliptically polarized light in the ambient magnetic fields on the order of 10 nT or less. However, the light-shift induced by the virtual transitions and real transitions is equivalent to the actual ambient field felt by the alkali-atoms, including the vector and tensor light-shift generated by circularly and linearly polarized light, respectively [19]. This effect can be important when such fields are comparable to the ambient field. The imperfect optical conditions force the atoms to be interfered by the light-shift and the fluctuations of light-shift are also uncorrelated, which will limit the common mode noise suppression of the multi-channel gradiometer magnetometers [20]. In addition, the long-term stability is also limited by the light-shift in other optical systems, such as the atomic clock [21,22], atomic gyroscope [23,24], ion-trap quantum information processing [25], and coherent population trapping [26,27]. Therefore, the measurement and cancellation of the light-shift is of great significance.
The light-shift due to the virtual absorption of circularly polarized light was first discovered by Arditi and Carver in the 1960s [28]. Since then, substantial advances have been developed to measure and cancel the light-shift in optically-pumped atomic devices. The AC-Stark shift could be suppressed by reducing the dependence of the light-shift on the frequency and intensity [29] of the pump. Several classical methods covered by relying on diffusion to transport polarized atoms in a small subvolume [30], using two separate detuned pumping lasers in the opposite direction to obtain a more uniform spatial pumping profile [31,32], and using hybrid optical pumping to improve the polarization efficiency [33] were adopted to minimize the light-shift. However, these approaches are required high-power laser, limited volume, two consistent laser beams, and hybrid optical pumping, respectively, which could limit their applications, especially in a miniaturized integrated AMs. In addition, the light-shift can also be set to zero at a certain zero-shift laser frequency point [23,34,35] or by using a feedback loop to lock the laser at a certain frequency [36–38]. Although the two methods can cancel the light-shift, it is quite difficult to maximize the sensitivity of the AMs due to the effects of large pump rates and pressure broadening in cell, respectively.
In this paper, we present a novel approach for the $\textit {in-situ}$ measurement and the cancellation of light-shift in fiber-coupled AMs, realized by utilizing the frequency response around the resonance frequency point in the magnetometer. This simple method of measuring and cancelling the light-shift is quite efficient and the proposed approach can also be used for the $\textit {in-situ}$ three-axis magnetic compensation of an AM based on the orthogonal pump-probe configuration. Furthermore, we also verify that the intensity and frequency of the circularly polarized pump light have a significant impact on the longitudinal residual magnetic field, and a preliminary analysis is also performed in the proposed fiber-coupled AMs. The developed method provides a convenient approach for the measurement and cancellation of the light-shift in the optically-pumped AM.
2. Theory
In the SERF regime, the spin behavior described by a density matrix equation can be assumed by the spin temperature distribution. In this case, the spin-exchange rate is much faster than the Larmor precession in the magnetic field $\mathbf {B}$ and the overall evolution of the alkali-metal atomic spin polarization $\mathbf {P}$ can be well-described by the Bloch equations [39,40]:
Fig. 1. Values for the fictious magnetic field given by Eq.(3) due to the vector light-shift caused by light with complete circular polarization $s=1$ and $\Gamma$=50 GHz. The Values is symmetric around frequency detuning and increases with pump intensity.
3. Experimental setup
The experimental setup is shown in Fig. 2. The cell contained a droplet of enriched $^{87}$Rb atoms and the buffer gas (50 torr of N$_2$ and 700 torr of $^{4}$He) was heated to 418 K by a boron-nitride oven. The cube-shaped glass cell with 4 mm side long and 3 mm windows was enclosed within five-layer $\mu$-metal shield. To further reduce the residual ambient magnetic field, the three pairs of orthogonal Helmholtz coils were installed inside the shield. These coils were connected to three independent function generators, from which the magnetic field in arbitrary direction can be generated to meet the experimental demand.
Fig. 2. Schematic of the experimental setup. PBS: polarized beam splitter, ${\lambda }$/2: half wave plate, ${\lambda }$/4: quarter-wave plate, PMF: polarization-maintaining fiber, MMF: multi-mode optical fiber, C: collimating lens, FG: function generator, PD: photodiode, TIA: transimpedance amplifier, DAQ: data acquisition.
The frequency of the pump and probe beam were synchronously monitored using a wavelength meter (HighFinesse WS-7-60) with an absolute accuracy of 60 MHz. The Rb atoms were directly polarized along the $z$ axis by the pump beam emitted by a distributed feedback laser (DFB) that was tuned to the center of $^{87}$Rb D1 transition at 795 nm. The beam was coupled to a polarization-maintaining optical fiber (PMF) and collimated with the 1/$e^{2}$ diameter of approximately 2.7 mm, then became the circularly polarized light after passing through a quarter-wave plate (${\lambda }$/4) to create the spin polarization in the vapor cell. The linear polarized probe beam that used to detect the spin polarization was emitted by an external cavity diode laser (ECDL) that was tuned near the $^{87}$Rb D2 line. After the cell, the Faraday rotation of the D2 line linear polarized light was detected by a balanced detector that was composed by a half-wave plate (${\lambda }$/2), a polarization beam splitter (PBS) and two photodetectors (PD). The initial differential current was nulled carefully before applying the pump beam and magnetic field. Thereafter, the detected signal was amplified by a transimpedance amplifier (Thorlabs, PDA200C), demodulated by a lock-in amplifier (Zurich Instruments, MFLI 5MHz) and then collected by a data acquisition system.
4. Experimental results and discussion
As mentioned in Section 2, a weak oscillating field $B_1\cos \omega t$ with the amplitude $B_1$=1 nT and frequency $\omega$=2$\pi \times$80 Hz was applied along the $y$-direction. The alkali-metal atoms were polarized by the on-resonant circularly polarized pumping beam of the 795nm with the light intensity of 1 mW$/$cm$^2$, and the residual magnetic field was compensated to the zero-field by three-axis Helmholtz coils. As shown in Fig. 3, when the magnetic field is applied along $x$-, $y$-, and $z$-axis, a symmetric response including the in-phase and the quadrature component could be obtained. Both the in-phase and the quadrature signals were measured using the lock-in amplifier and the dependence of the obtained signals as a function of an arbitrary component of the magnetic field were fitted by the combination of the Lorentzian- absorption and Lorentzian- dispersion form. As shown in Fig. 3(a), the two signals were fitted by a simple Lorentzian with a half width of about 1.7 nT and both of them can be used to measure the tensor light-shift. Figure 3(b) shows the multiple resonance peaks, among which the zero-field resonance peak can be exploit to monitor the residual magnetic field along the $y$-axis and the two subsymmetry peaks can be used to measure the gyromagnetic ratio precisely. Furthermore, small changes of the gyromagnetic ratio under different pump intensity can also benefit from this property, as they are not affected by the light-shift. Figure 3(c) shows the response of the in-phase and quantrature component obtained by analytic solving the time-dependent Bloch equation described in Eq. (2). This measurement matches the theoretical analysis represented by Eq. (4). For the purpose of verifying that the change of these two resonance peak points $B_0^{left}$ and $B_0^{right}$ can be utilized to measure the vector light-shift, we firstly demonstrated that the response of the AM could be used to compensate the magentic field effectively.
Fig. 3. The response of the AM for the experimental data (dots) and the fitting result (solid lines), including the in-phase and the quadrature signal. The fitting results are a Lorentzian-absorption or a Lorentzian-dispersion or a combination of both. (a) The obtained response signal when the magnetic field is scanned around zero-field in $x$-axis. (b) The multiple resonance peaks are observed by sweeping the magnetic field along the $y$-axis. (c) When a magnetic field is applied on the $z$-axis around zero, an absorptive in-phase and dispersive quadrature signal is obtained.
Figure 4 shows experimental results of the demodulated first harmonic including the in-phase and quadrature signals with a lock-in amplifier at different bias magnetic field ranging from −4 nT to 4 nT in the $x$-, $y$-, and $z$-directions, respectively. During the magnetic compensation procedure, in order to reduce the influence of the light-shift on the magnetic compensation, on one hand, the frequency of the pump light is changed by tuning the temperature of the laser, so that the light is at the absorption point, i.e., the optimal resonance frequency. On the other hand, reduce the output power of the pump laser. As can be seen from Fig. 4(a) and 4(b), the peaks shift with the different bias magnetic field when the transverse magnetic field is swept while the peaks are located at the zero field when the bias ambient magnetic field is nulled. Similarly, Fig. 4(c) and 4(d) show the relationship between the resonance peaks and the bias magnetic field in $y$-direction, and two symmetry peaks illustrated near the zero field indicates the corresponding oscillating field frequency $\omega$. As shown in Fig. 4(e) and 4(f), the curves have a component with an absorption and a dispersive characteristic that are in-phase and quadrature with the oscillating field, respectively. When the resonance peaks of the in-phase signal are symmetry about zero field or the dispersive curve of the quadrature signal are antisymmetric about the zero-crossing, the residual magnetic field along the $z$-axis is nulled, which agrees with the theoretical prediction. Therefore, this method is especially suitable for the application in the magnetic field compensation of an AM with the orthogonal-beam or nearly parallel pump and probe beams configuration. In our system, the technical limits of the compensation scheme are dependent on the triaxial magnetic coils and the output voltage generated by the function generator. Furthermore, the compensation scheme can realize about $\pm 46.2$, $\pm 192.2$ and $\pm 80.2$ nT in the $x$-, $y$-, and $z$-directions, respectively. Therefore, the developed approach is convenient to measure and cancel of the light-shift. Due to the probe beam is linear polarized, the light-shift is not considered in this research and we mainly focus on light-shift caused by the circularly polarized pump beam.
Fig. 4. Experimental in-phase and quadrature signals acquired by scanning the $x$-, $y$- and $z$-axis magnetic field around zero-field. To demonstrate the compensation process, a sequence of the bias field with the magnitude from 2 to 4 nT are applied in $x$-, $y$- and $z$-axis respectively. When the remanence is nulled, the in-phase and quadrature signal curve would be symmetric about the zero field.
Figures 5(a) and 5(b) show the in-phase and quadrature signals at different pump beam frequencies as a function of the magnetic field variation when an oscillatory field with $\omega$=2$\pi \times$80 Hz is applied on the $y$-axis and a DC magnetic field is swept along the $z$-axis. In the experiments shown in Fig. 5, the light intensities of the 795 nm pumping beam and the 780 nm probe beam are 1 mW$/$cm$^2$ and 0.3 mW$/$cm$^2$, respectively. Since the direction of the scanning magnetic field is parallel to the direction of the pump beam and perpendicular to the oscillating field, the coherent precession of the atomic spin induced by the oscillating field in the plane is perpendicular. These measurements were performed by scanning the longitudinal magnetic field along the $z$-direction. As explained above, the in-phase signal shows two symmetry resonance peaks when the ambient magnetic field is nulled. Figure 5(c) shows the measured resonant magnetic field $B_0$ at different frequency of the pump beam with the static magnetic field of $B_z$=18.57 nT and the data are extracted from Fig. 5(a). According to Eq. (3), the relationship between the magnitude of the fictitious magnetic field due to the AC Stack effect and the frequency of the pump beam are shown in Fig. 5(e), and the data are extracted from Fig. 5(c). Compared with Refs. [31] and [33], we provide a fast and convenient way to realize the decoupling of the remanence of the pump light direction and the equivalent fictitious magnetic field of the light-shift, which can be applied to the integrated SERF AMs. Figure 5(d) shows the dependence of the extracted amplitude and full-width at the half-maximum (FWHM) from the in-phase signal resonance peaks [see Fig. 5(a)] on the frequency of the light. Due to the rapid quenching of the excited state by ${\rm N_2}$, the absorption cross section $\sigma$ for light of frequency $\nu$ has the form of a Lorentzian curve [42]. The measured in-phase signal amplitude and full width at half maximum (FWHM) data at different optical detuning frequencies are fitted with the form of $aL$($\nu )/$($L$($\nu$)+b)$^2$ and $a_1L$($\nu$)+$b_1$ , respectively, and both the amplitude and FWHM reach its maximum value when the pumping laser is blue detuned by 6 GHz from the $^{87}$Rb D1 line resonance transition center due to the pressure broadening caused by the presence of buffer gases in the cell [42]. In addition, for highly alkali dense cell, the optical depth is quite large, and the signal amplitude is also affected by absorption. Therefore, the traditional method of eliminating the light-shift by tuning the frequency to the absorption peaks is not always effective.
Fig. 5. The response around the resonance magnetic field $B_0$ when an oscillating field with frequency of 80 Hz is applied, consisting of an in-phase component (a) and quadrature component (b). The experimentally measured resonance point, amplitudes, FWHM at different detuning frequency, and light-shift are shown in Fig. (c), (d) and (e), respectively.
Figure 6(a) and 6(b) show the in-phase response and the quadrature response of the magnetometer under different pump intensity. Figure 6(c) is the amplitude and FWHM of the resonance, obtained from Fig. 6(a). The overlaying line is a fit by $\eta P/(\eta P+\Gamma _{SD}$) with $R_{\rm {op}}={\eta }I$, which represents the relationship between the polarization and the pump intensity. Overlaying the FWHM data is a fit to $\eta I+\Gamma _{SD}$, and from which the electron spin-destruction rate $\Gamma _{SD}=880\pm 35\; s^{-1}$ can be obtained, which is in reasonable agreement with the theoretical prediction. Comparing the measurement data shown in Fig. 6(d), we can find that the change of the measured resonance point $B_0$ shows an opposite trend with the increase of the pump intensity due to the light-shift caused by the pump beam. Based on the fitting result, we can obtain the static magnetic field $B_z^{left}=18.82$ nT and $B_z^{right}=18.58$ nT, and from which the $B_z=(B_z^{left}+B_z^{right})/2$ at the zero light-shift can be also observed. Finally, the fitting curve gives the light-shift as a linear function of the pump intensity with slope of 0.65 nT/(mW/cm$^2$). As of yet, we have successfully demonstrated the approach of in-situ measurement and cancellation of light-shift in a fiber-coupled AM.
Fig. 6. Magnetometer frequency response under different pumping intensity around the resonance magnetic field $B_0$ to an oscillating field with the frequency of 80 Hz, consisting of an in-phase component (a) and a quadrature component (b). Relationship between pump beam intensities and the amplitude, FWHM, and resonance point magnetic field, as shown in Fig. 5(c) and Fig. 5(d), respectively.
Figure 7 shows the experimentally observed optical rotation with and without compensation of the light-shift for a frequency detuning of 20 GHz and light intensity of 1 mW$/$cm$^2$. As described in [41] and [43], light-shift acts as if it were a real magnetic field along the axis of the pump light propagation. Therefore, the existence of the light-shift is the same as the existence of the bias magnetic field $B_z$ in the direction of the pump [44]. Obviously, we see that a larger response signal after compensation light-shift than without compensation.
Fig. 7. Experimentally measured optical rotation as a function of $B_y$ with and without compensation light-shift for a frequency detuning of 20 GHz and light intensity of 1 mW$/$cm$^2$.
5. Conclusion
In conclusion, we have developed a novel method that can be used for the in-situ three-axis magnetic compensation of the AM with the orthogonal pump-probe configuration. This method can be also utilized to distinguish the residual magnetic $B_z$ and light-shift $B_{\rm {LS}}$. More importantly, the proposed approach provides a more effective method for the in-situ measurement and cancellation of the light-shift by utilizing the symmetry of the response around the resonance frequency point of a fiber-coupled AMs. Meanwhile, we have also verified that the intensity and frequency of the circularly polarized pumping beam have a significant impact on the longitudinal residual magnetic field, which could limit the accuracy of the magnetometer. Although our method is implemented and validated in the $^{87}$Rb fiber-coupled AM, however, the proposed method can still be extended to the in-situ cancellation of the light-shifts in other types of atomic devices, including but not limited to the atomic clock and the atomic gyroscope. Due to the potentially higher sensitivity of the pump-probe scheme of magnetometers, we will design a miniaturized pump-probe fiber-coupled magnetometer in future work. We believe the proposed technique will be beneficial to developing magnetocardiography [45] and magnetoencephalography instrumentation [46]. Further improvement will be focused on the suppression of the tensor light-shifts of the linearly polarized probe light.
Funding
National Natural Science Foundation of China (62003020); Research on theory and method of gravity disturbance real time compensation for precision inertial measurement (62173019); Key Research and Development Program of Zhejiang (2020C01037).
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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