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Abstract
In this paper, we experimentally study transient dynamics of spin polarized atoms in the spin-exchange-relaxation-free (SERF) regime with a single-beam configuration. We pumped atoms with a weak detuning pumping beam, along with a sequence of magnetic field pulses orthogonal to the pumping beam were applied. The dynamics of atomic spin, which experiences Larmor precession under the perturbation of magnetic field, is detected by the transmitted pumping beam. Benefited from the long coherence time of atomic spin in the SERF regime, the dependence of precession frequency and decay rate, which is equal to the magnetic resonance linewidth of atomic spin, on magnetic fields is studied with the transient dynamics of atomic spin in the limit of low spin polarization. Moreover, we demonstrate that coil constants can be calibrated by analyzing the precession frequency of the transient dynamics of atomic spin. And the experimental results show that the coil constants are 114.25 ± 0.02 nT/mA and 114.12 ± 0.04 nT/mA in x- and y-axis, respectively. This method is particularly applicable to study the atomic spin dynamics and calibrate the coil constant in situ of a miniature single-beam SERF magnetometer.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Ultrasensitive magnetic field measurements based on optically pumped atomic magnetometers (OPMs) have seen rapid development [1,2]. A long spin coherence time has been realized in a spin-exchange-relaxation-free (SERF) magnetometer by operating in a zero field with high alkali-metal density [3–5], achieving the highest sensitivity (experimentally, 160 aT/$\textrm {Hz}^{1/2}$) and surpassing some state-of-the-art superconducting quantum interference devices (SQUIDs) that have dominated ultrasensitive magnetic field precision measurements in recent years [6,7]. Compared with SQUIDs, OPMs have the inherent advantage of noncryogenic cooling, and offer the possibility of flexible configuration [8–11]. With these advantages, they have been applied in various areas, including magnetoencephalography (MEG) [12,13], magnetocardiography (MCG) [14], inertial rotation sensing [15], tests of CPT and Lorentz invariance [16,17], and the exploration of the dynamics of geomagnetic fields [18].
SERF magnetometers require that the relaxation caused by spin-exchange collisions is eliminated so that a longer coherence time can be achieved compared with non-SERF magnetometers [19]. The conventional method to study the spin exchange in the SERF regime is measuring the dependence of the precession frequency and magnetic resonance linewidth of atomic spin on the magnetic field with the Faraday rotation effect of a linear polarized probe beam. In Ref. [20], authors measured the relation between the magnetic resonance linewidth and precession frequency to study spin exchange in the SERF regime with a pump-probe configuration based on the Faraday rotation of the probe beam. The authors of Ref. [21] measured the relation between the magnetic resonance linewidth and magnetic field based on spin noise spectra. Driven by thermal fluctuation, the atoms undergo Larmor spin precession under a finite magnetic field, and the Larmor frequency and magnetic resonance linewidth are obtained by measuring the Faraday rotation of a probe beam. However, these two methods are not suitable for studying the spin dynamics for a miniature integrated single-beam SERF magnetometer.
SERF magnetometers generally operate in an extremely low magnetic field environment, which is realized by multilayer magnetic shields and triaxial magnetic compensation coils, to eliminate the broadening from spin-exchange collisions [22]. The ability to precisely control the magnetic field of an OPM is especially important since the accuracy of the magnetic field is directly related to the accuracy of the calibration field in open-loop mode [23], or limited to the coil constant in closed-loop mode [24]. Therefore, the precise calibration of magnetic coil constants has a substantial effect on the overall performance of an OPM, for both the estimation of sensitivity and the accurate measurement of the magnetic field, and it is necessary to research a high-precision method to calibrate coil constants.
In theory, based on the known distribution of the coil current, the intensity and distribution of the magnetic field can be determined through magnetic simulation. The coupling effect between the high-permeability magnetic shielding material and the coil, however, will distort the magnetic field, which is challenging to account for by quantitative theoretical modeling [25,26]. Therefore, the coil constant of a certain coil is usually calibrated with a fluxgate magnetometer installed in the volume of interest [27]. Unfortunately, the measurement accuracy with fluxgate is limited by the installation position error or sensitivity.
In situ calibration of the coil constant in a high-sensitivity OPM has provided a more attractive method. One method detects the spin precession frequency signal of the hyperpolarized $^3\textrm {He}$ with an ultrasensitive atomic spin comagnetometer, from which the coil constant can be determined by the Larmor precession frequency [28]. Another method, developed by Chen et al., measures the initial amplitude of the free induction decay (FID) of the $^{129}\textrm {Xe}$; the coil constant is calibrated by calculating the $\pi /2$ pulse duration in a nuclear magnetic resonance gyroscope (NMRG) [29]. The magnetic field can be measured accurately using both methods, but noble gases with a long polarization lifetime are essential in the atomic ensemble, which limits their application, especially in an OPM whose sensitive source is alkali-metal atoms. Other methods employed the alkali-metal ensemble were proposed including Bell-Bloom magnetometer and self-sustaining vector atomic magnetometer [30,31]. These two methods are difficult to reduce the package volume in a fully integrated magnetometer.
The dynamics response of spin-polarized alkali-metal atoms is a useful method for precision measurements. Kornack et al. studied the coherent spin-exchange interaction of $^{3}\textrm {He}$ and K by analyzing the transient response of spin-polarized K atoms, useful for tests of the fundamental symmetries of nature [32]. The authors of Ref. [33–35] used the transient dynamics of spin-polarized alkali atoms to determine the coil constant and search the magnetic field produced by hyperpolarized $^{3}\textrm {He}$.
In this paper, we study transient atomic spin dynamics in the SERF regime with a single-beam configuration. We consider the atoms to be optically pumped with a weak detuning pumping beam in the SERF regime. A sequence of square-wave magnetic fields orthogonal to the pumping beam is applied to drive the atomic spin from the equilibrium state. By recording the transient response with the transmitted pumping beam and analyzing the recorded data, the spin exchange and magnetic resonance linewidth are studied with transient atomic spin dynamics. Furthermore, the slowing-down factor $q$ is measured by comparing the transient atomic spin precession frequencies in the non-SERF and SERF regimes. We demonstrate that coil constants can be calibrated with transient atomic spin dynamics in the SERF regime. This method is useful to study the atomic spin dynamics and calibrate the constant of the coils that are integrated of a miniature single-beam SERF magnetometer where traditional methods are difficult to perform.
2. Methods
The behavior of electron spin generally is governed by the evolution of the density matrix equation. With a low magnetic field and high alkali-metal density, in which spin-exchange rate is far large compared to the Larmor precession frequency, the density matrix equation is simplified by assuming a spin-temperature distribution, and the dynamics of the electron spin polarization $\mathbf {P}$ under the magnetic field $\mathbf {B}$ is described by the phenomenological Bloch equation [20]
In the absence of any magnetic field, the equilibrium spin polarization $P_{0}$ of atoms under a circularly polarized pumping beam is $R_\textrm {op}/(R_\textrm {op}+R_\textrm {rel})$. With a magnetic field $B_x$ applied along the $x$-direction, the exact solution of $P_y$ and $P_z$ for the atomic spin polarization $\mathbf {P}$ along the $y$- and $z$-directions are
The steady-states $P_{y},~P_{z}$ contain the information of $B_x$. However, $P_{y}$ can not be detected since only one beam is applied in $z$-direction in our scheme (see experimental setup section). The new steady-state polarization $P_\textrm {ss}$ along the $z$-direction under the effect of magnetic field $B_x$ is $R'_\textrm {op}\Gamma /(\Gamma ^2+\omega ^2_{q})$, which is insensitive to $\omega _{q}$ (corresponding to the magnetic field $B_x$) in a low magnetic field regime. In the following, we use the transient state to study the dynamics of atomic spin in the SERF regime, which provides more information than that use steady-state in our scheme. According to Eq. (2) and Eq. (3), a spiral curve of the precession of atomic spin is obtained under the disturbance of a square-wave magnetic fields [as shown in Fig. 1(a)]. The projection of the spiral curve in the direction of the pumping beam shows a damped oscillation, from which the decay rate $\Gamma$ (corresponding to the coherence time) and oscillating frequency $\omega$ can be extracted [as shown in Fig. 1(b)]. The oscillating frequency $\omega$ and decay rate $\Gamma$ are equal to the precession frequency $\omega _{q}$ in the SERF regime and magnetic resonance linewidth of the atomic spin under the corresponding magnetic field. With the dependence of the precession frequency $\omega _q$ and decay rate $\Gamma$, the basic parameters such as the spin-exchange rate $\Gamma _\textrm {SE}$ and the alkali-metal vapor density $n$ can be obtained. Moreover, by precisely measuring the value of the nuclear slowing-down factor $q$, one could obtain $B_x$ from the precession frequency of the transient atomic spin dynamics since $g_s$ and $\mu _B$ are constants. Combined with the applied current strength $I_x$ in the magnetic field coil of the $x$-axis, the coil constant $k_x$ can be described as follows:
where $B_0$ is the residual field that is incomplete compensation or overcompensation, whose value is comparable to zero. With a similar method, the coil constant in the $y$-direction is obtained by applying a square-wave magnetic field along the $y$-direction.
Fig. 1. Schematic depiction of the dynamic evolution of the system under the disturbance of a square-wave magnetic field. (a) A spiral curve showing the precession of atomic spin; and (b) the damped oscillation curve of $P_z$. The initial polarization without an external magnetic field is $P_{0}$, and the new steady-state polarization is $P_\textrm {ss}$. The dynamic evolution including the transient response exhibits the atomic spin dynamics of the system.
3. Experimental setup
Figure 2 shows the experimental setup. A small droplet of $^{87}$Rb was contained in a borosilicate glass cubic cell with an inner length of 3 mm. The cell was filled with another 50 torr $\textrm {N}_2$ as quenching gas and 700 torr $^4\textrm {He}$ as buffer gas. We placed the cell in a boron nitride ceramic oven and heated it to 423 K using a twisted-pair winding resistor driven by a high-frequency AC electronic current. A dense Rb vapor density of approximately $1.07\times 10^{14}~\textrm {cm}^{-3}$ was achieved from a saturated density of alkali vapor at this temperature [19]. The cell and oven were mounted in four-layer cylindrical magnetic shields, including three layers of mu-metal and one layer of coaxial aluminum, with a residual magnetic field about several nT after degaussing. To further reduce the residual magnetic fields, three pairs of Helmholtz coils are integrated outside the cell along the $x$-, $y$-, and $z$-directions to actively compensate the residual magnetic fields and apply the square-wave magnetic fields.
Fig. 2. Schematic of the experiment. A circularly polarized pump beam, blue detuned by 20 GHz from the D1 line optical transition center of $^{87}$Rb propagating along the $z$-axis is used to polarize rubidium atoms, and a photodiode is used to detect the intensity of beams transmitted through the vapor cell. PMF: polarization-maintaining fiber. C: collimating lens. LP: linear polarizer. $\lambda /4$: quarter-wave plate. PD: photodiode. TIA: transimpedance amplifier. LPF: low-pass filter. DAQ: data acquisition.
The optical configuration is simple. The pumping beam is generated by a distributed feedback (DFB) laser, whose frequency is blue detuned by 20 GHz from the $^{87}\textrm {Rb}$ D1 line optical resonance transition center and intensity is about 320 µW/ $\textrm {cm}^2$. Also, the pumping beam is changed into circular polarization by a linear polarizer and a quarter-wave plate before it enters the cell. The transmitted pumping beam is monitored by a photodiode to extract the information of magnetic fields. A Lorentzian absorption curve is observed by sweeping the magnetic field $B_x$, as a result of the magneto-optical interaction with atoms that undergoes a zero-field resonance.
4. Results
The coherence time of atomic spin determines the sensitivity of SERF magnetometers. Therefore it is necessary to study the spin-exchange collisions of a SERF magnetometer. For a single-beam configuration, the methods based on Faraday rotation with pump-probe configurations are limited. Besides, coil constants can not be calibrated with conventional methods, including pump-probe configurations and fluxgate, for a miniature integrated single-beam magnetometer. In the following, we show that the study of transient atomic spin dynamics enables us to investigate spin-exchange collisions and to calibrate the coil constant of an integrated SERF magnetometer whose size is on the order of $\textrm {cm}^{3}$.
4.1 Atomic spin relaxation in the SERF regime
To study transient atomic spin dynamics, we applied a disturbance square-wave current signal in the magnetic field coil in the $x$-direction. The spin coherence time of atoms was on the order of 10 milliseconds, far longer than the rise time of the square-wave magnetic field signal (no more than 1 µs). The transient atomic spin dynamics shows a damped oscillation curve under the disturbance of square-wave magnetic fields (as shown in Fig. 3). Figure 3(a) is the square-wave current signal applied to the coil in the $x$-axis, and the step response of the atomic spin is shown in Fig. 3(b). Figure 3(c) provides a zoom-in of the measured signal, the oscillating frequency $\omega$ and the decay rate $\Gamma$ can be obtained by Fourier transformation of the response signal or by fitting the response signal with Eq. (3). The red line is a fitting curve based on Eq. (3), and it shows that the experimental data are in good agreement with the theoretical Bloch model.
Fig. 3. Transient atomic spin dynamics under a certain magnetic field disturbance. (a) A square-wave excitation signal is applied in the $x$-direction. (b) Transient response of the atomic spin under the disturbance signal. (c) Comparison of the measured data and theoretical fitting model. The blue square represents the experimental results, whereas the red line, overlapping the data points, represents the theoretical prediction based on Eq. (3).
Figure 4 depicts a cluster of damped oscillation curves under different amplitudes of square-wave magnetic fields in the $x$-axis. As shown in Fig. 4, the oscillating frequency increases with increasing current (corresponding to the magnetic field $B_{x}$). Moreover, the decay rate $\Gamma$ also changes under the different amplitudes of the magnetic field $B_{x}$. To study the relation between decay rate $\Gamma$ and magnetic fields $B_{x}$, we obtained the decay rate $\Gamma$ by fitting the response signal with Eq. (3) under different currents (see Fig. 5). Due to the spin-exchange relaxation turning on at a finite magnetic field, the decay rate $\Gamma$ increases (corresponding to a shorter coherence time) as the magnetic field $B_{x}$ grows larger. Considered spin-exchange relaxation, the decay rate $\Gamma$ can be written as
Fig. 4. The measured transient response of the single-beam SERF magnetometer under different square-wave excitation in the $x$-axis. The other components of the magnetic field are zeroed carefully at the beginning of the experiment.
Fig. 5. Decay rate $\Gamma$ under different magnetic fields $B_{x}$. (a) With increasing magnetic field, the decay rate $\Gamma$ also increases. Compared with a low temperature (blue square), spin-exchange relaxation is suppressed for a high temperature (red dot) under the same magnetic field. (b) Relation between precssion frequency $\omega$ and decay rate $\Gamma$ at 423 K is fitted with Eq. (6), which gives $T_\textrm {SE} =12.3\times 10^{-6}~\textrm {s}$.
The relation between $\Gamma ^\textrm {SE}_{2}$ and the spin-exchange rate $\Gamma _\textrm {SE}$ under a small magnetic field in the low polarization limit is
High-sensitivity SERF magnetometers generally work in a low polarization limit since a high atomic spin polarization has a negative effect on the coherence time $1/\Gamma$ [36]. Since spin polarization is sensitive to the frequency and intensity of the pumping beam, we study the relation between the decay rate $\Gamma$ and these two parameters. The precession frequency for Fig. 6 is about 175 Hz, much smaller than the spin-exchange rate $\Gamma ^\textrm {SE}_{2}$. As the frequency of the pumping beam approaches the center of the D1 line resonance transition, the spin polarization becomes larger, resulting in a larger decay rate $\Gamma$ [see Fig. 6(a)]. Similarly, if one increases the pumping intensity under a certain frequency (blue detuned by 20 GHz from the $^{87}$Rb D1 line resonance transition center) of the pumping beam, the decay rate $\Gamma$ also increases with increasing pumping intensity. This is reasonable for an optically thin rubidium vapor [see Fig. 6(b)].
Fig. 6. Relation between $\Gamma$ and spin polarization. (a)The decay rate $\Gamma$ under different frequency detunings of the pumping beam for a given pumping intensity (320 $\mu \textrm {W}/\textrm {cm}^2$). (b) The decay rate $\Gamma$ under different pumping intensities when the pumping beam is blue detuned by 20 GHz from the $^{87}$Rb D1 line resonance transition center.
4.2 Magnetic field calibration
As we mentioned before, the accuracy of the coil constant is vital for magnetometers, especially for high sensitivity magnetometers. For our scheme, conventional methods are limited for the size of the magnetometer and the experimental configuration.
The transient dynamics of atomic spin enables us to calibrate the coil constant of a small size magnetometer based on single-beam configuration. According to Eq. (3), the precession frequency $\omega$ of transient atomic spin dynamics is a linear function of the magnetic field $B_x$ since the parameters $g_{s}$, $\mu _B$ and slowing-down factor $q$ are constants.
To measure the slowing-down factor $q$, we perform the experiment at $\textrm {T}=363 ~{\textrm {K}}$, where the oscillating frequency $\omega$ is equal to the Larmor precession frequency $\omega _{0}$ (as shown in Fig. 7). As we can see from Fig. 7, the decay rate $\Gamma$ is larger than that in the SERF regime (as shown in Fig. 4) mainly due to spin-exchange collisions. The slowing-down factor $q=(2I+1)\omega _{0}/\omega _{q}$, where $I=3/2$ is the nuclear spin of $^{87}\textrm {Rb}$, is obtained by measuring the precession frequency in the non-SERF and SERF regimes.
Fig. 7. The measured transient response of atomic spin in the non-SERF regime under different amplitudes magnetic square-wave excitation along the $x$-direction.
Figure 8 shows the dependence of the precession frequency $\omega$ on the current of $x$-axis (corresponding to the magnetic field $B_{x}$) in the non-SERF (blue squares in Fig. 8) and SERF regimes (red dots in Fig. 8). By comparing the precession frequencies under these two regimes, the slowing-down factor $q$ is obtained. In our experiment, the parameter $q=5.9710\pm 0.0070$, which indicates that the atomic spin in a low polarization limit. Therefore, the coil constant $k_{x}$ can be calibrated with Eq. (4) for a known slowing-down factor $q$. In our experiment, the calibrated coil constants were $114.25\pm 0.02$ nT/mA and $114.12\pm 0.04$ nT/mA in $x$- and $y$-axis.
Fig. 8. Dependence of precession frequency on the applied current in the $x$-axis (corresponding to magnetic field $B_{x}$) in the non-SERF and SERF regimes. The precession frequency is linearly proportional to the current since the precession frequency is determined by $\omega _{0} =g_s \mu _B B_{x}/(2I+1)$ in the non-SERF regime and by $\omega _{q}=g_s\mu _B B_{x}/q$ in the SERF regime. The precession frequency in the non-SERF regime is larger than that in the SERF under the same current since the slowing-down factor $q>4$ in the SERF regime.
Moreover, the slowing-down factor $q$ can also be obtained from Eq. (6) since the spin-exchange rate $\Gamma _\textrm {SE}$ is determined by $\Gamma _\textrm {SE}=n\sigma _\textrm {se}\bar {\upsilon }$. For the temperature at 423K, the vapor density of rubidium is $n=1.07\times 10^{14}~\textrm {cm}^{-3}$, thermal velocity $\bar {\upsilon }=454~\textrm {m/s}$, which gives $T_\textrm {SE}=1/\Gamma _\textrm {SE}=11.4\times 10^{-6}~\textrm {s}$. Combined with the theoretical spin-exchange rate $\Gamma _\textrm {SE}$, the slowing-down factor $q=6.04\pm 0.02$ is attained by fitting the data in Fig. 5(b) with Eq. (6). However, we find this value is a bit larger than that obtained from Fig. 8, and the value exceeds the normal range. This may be caused by the actual temperature of the vapor in the cell is smaller than the temperature we measured since the temperature sensor PT1000 is placed outside the cell, and the slowing-down factor $q$ obtained by this method is strongly dependent on the measured temperature. Therefore, the slowing-down factor $q$ measured by the method above has more advantages than that measured by this method.
For single-beam SERF magnetometer, the calibration of coil constant based on transient atomic spin dynamics in the SERF regime has advantages over that based on transient atomic spin dynamics in the non-SERF regime. On the one hand, it is a challenging task to zero the ambient magnetic field in the non-SERF regime, which can be easily realized in the SERF regime. On the other hand, due to the spin-exchange collisions, the coherence time of atomic spin in the SERF regime is longer than that in the non-SERF regime, which is helpful for measuring the precession frequency (corresponding to magnetic fields) of transient atomic spin.
5. Conclusion
In conclusion, we have studied transient atomic spin dynamics in the SERF regime for a single-beam configuration. By measuring the dependence of the precession frequency and decay rate of atomic spin on magnetic fields in the SERF regime, spin exchange was studied. The relation between the decay rate and spin polarization was also studied by varying the frequency and intensity of the pumping beam. Moreover, the slowing-down factor $q$ was measured by comparing the precession frequency of atomic spin in the non-SERF and SERF regimes. Based on this method, coil constants can be calibrated in situ, and the calibrated coil constants were $114.25\pm 0.02$ nT/mA and $114.12\pm 0.04$ nT/mA in $x$- and $y$-axis in our system. The method presented here is particularly suitable to study the transient dynamics of atomic spin and to calibrate the coil constant in situ in a miniature single-beam OPM operating in the SERF regime.
Funding
Natural Science Foundation of Beijing Municipality (4191002); National Natural Science Foundation of China (61627806, 61703025); National Key Research and Development Program of China (2018YFB2002405); Major Scientific Research Project of Zhejiang Lab (2019MB0AE03).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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