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Analysis of effects of magnetic field gradient on atomic spin polarization and relaxation in optically pumped atomic magnetometers

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Abstract

The magnetic field gradient within optical pumping magnetometers (OPMs) suppresses sensitivity improvement. We investigated the effects of the magnetic field gradient along the x-, y-, and z-axes on the limiting factors of magnetometers under extremely low magnetic field conditions. We modified the magnetic field gradient relaxation model such that it can be applied to atoms in the spin exchange relaxation free (SERF) regime. The gradient relaxation time and spin polarizations, combined with fast spin-exchange interaction, were determined simultaneously using the oscillating cosine magnetic field excitation and amplitude spectrum analysis method. During the experiments, we eliminated the errors caused by the temperature and pumping power, and considered different isotope spin exchange collisions in naturally abundant Rb during the data analysis to improve the fitting accuracy. The experimental results agreed well with those of theoretical calculations and confirmed the accuracy of the improved model. The contribution of the transverse magnetic field gradient to the relaxation of the magnetic field gradient cannot be ignored in the case of small static magnetic fields. Our study provides a theoretical and experimental basis for eliminating magnetic gradient relaxation in atomic sensors in the SERF region.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Optical pumping atomic magnetometers are used for detecting magnetic fields with extremely high sensitivity. In general, magnetometers are used in a range of applications, including geological exploration [1], biomagnetism [2,3], and tests of fundamental physics, such as electric dipole moment measurements [4]. Based on the underlying physical mechanism, magnetometers can be classified as nuclear magnetic resonance (NMR) magnetometers [5], superconducting quantum interference device magnetometers [6], and spin-exchange-relaxation-free (SERF) magnetometers [7,8]. SERF magnetometers eliminate spin exchange relaxation by operating in a low magnetic field and at a high alkali metal density. A SERF magnetometer with a sensitivity of 0.16 fT/Hz1/2 was reported in 2010 [9]. The polarization and relaxation time of alkali metal atoms are the factors limiting further improvements in the performance of magnetometers. Based on the fundamental sensitivity limit of SERF magnetometers [10], the intrinsic sensitivity of these devices increases significantly with large increases in the transverse relaxation time. A uniform static magnetic field is required to ensure accurate measurements. The magnetic field gradient within magnetometers results in the transversal and longitudinal gradient relaxation of the atoms. The magnetic gradient increased the magnetic linewidth and loss magnetic field measurement resolution owing to a reduction in the spin relaxation time [11]. Therefore, it is essential to elucidate the effects of the magnetic field gradient on the atomic spin polarization and relaxation.

The magnetic field gradient caused by the static magnetic and residual magnetic fields of the magnetic shielding and electrical heating systems used is the main source of magnetic inhomogeneity in magnetometers. When a high-pressure buffer gas is used to fill the interior of the atomic gas cell, the magnetic field gradient causes the decoherence of the spins of the alkali metal atoms. Cates and Happer studied the atomic spin caused by a nonuniform magnetic field under low-pressure conditions [12,13]. In [11], the effects of the first- and second-order magnetic fields on the performance of comagnetometers were elucidated. Subsequently, transverse spin relaxation and diffusion coefficient measurements were performed using polarized nuclei in a magnetic field gradient in an NMR system [14], and the compensation of the magnetic field system was optimized [15]. However, in most studies, the effects of the magnetic field gradient along the x- and y-axes are ignored, and measurements and analysis are performed based only on the magnetic field along the z-axis. Because the sensitivity of SERF magnetometers is of the order of fT, the effects of the magnetic field gradient along the x- and y-axes cannot be ignored. Under extremely low-magnetic-field conditions, the magnetic field gradient along the x-, y-, and z-axes affects the performance of magnetometers. The spin-exchange (SE) collision of atoms should be considered based on the gradient relaxation model, which is considered more suitable for SERF magnetometers. The relaxation of the magnetic field gradient is a key factor limiting further improvements in magnetometer sensitivity. Therefore, the effects of the magnetic field gradient along the three axes on the limiting factors of magnetometers should be investigated.

Several conventional methods have been used to measure the polarization and relaxation of alkali atoms. The commonly used methods include those involving electron paramagnetic resonance (EPR) [16] and pumping decay transient (PDT) measurements [17]. In the case of EPR methods, polarization is achieved by measuring the population distribution of particles corresponding to the different hyperfine Zeeman sublevels using a transverse radiofrequency field with a frequency close to the paramagnetic resonance peak of alkali metals or by adding a large background magnetic field. The normal operation of the sensor in the SERF state is affected by the presence of large magnetic and radio frequency fields. In the case of the PDT method. Either the pump light of the sensor must be attenuated or an instantaneous dynamic response must be employed to achieve the polarization of the alkali metal atoms. Thus, this method requires the addition of a chopper to the optical path to control the on–off state of the pump light, which reduces the accuracy. Moreover, the optical depth of alkali metals is large, and the pump light cannot pass through completely; this attenuates the detection signal.

The transverse relaxation time of an alkali metal atom has been calculated using the magnetic resonance linewidth. The commonly used methods for measuring the magnetic resonance linewidth include synchronous pumping (SP), free induction decay (FID), and EPR. The SP method [7] uses a chopper to modulate the pump rate, similar to the PDT method. Magnetic resonance is achieved when the modulation frequency of the chopper is close to the Larmor precession frequency of the atom. In the FID method [18], a pulsed driving magnetic field is applied; however, it is difficult to precisely control the pulse time.

In this study, we investigated the effects of magnetic field gradients along the x-, y-, and z-axes on the limiting factors of an optically pumped SERF magnetometer under extremely low- magnetic-field conditions. For most magnetometers, the resonance frequency is significantly higher than the linewidth. However, the resonance frequency of SERF magnetometers is very low. Therefore, we initially substituted the rotating magnetic field with a small-amplitude, low-oscillating-cosine magnetic field. Next, the effect of a decrease in the precession frequency of the alkali metal atoms in the SERF regime owing to spin exchange collisions because of the slowing-down factor was considered [7,19]. We simultaneously measured the polarization and relaxation of the magnetometer in the presence of a magnetic field gradient. The transverse and longitudinal relaxations attributable to the magnetic field gradient were determined. The relaxation contributions of the x-, y-, and z-axes magnetic field gradients were found to be different under different primary magnetic fields along the z-axis as well as for different gas cell parameters. Finally, we measured the variations in the slowing-down factor with changes in the temperature and pump-light intensity. Furthermore, during data analysis, the effects of the different isotope SE collisions in naturally abundant Rb (72.2% 85Rb with I = 5/2 and 27.8%87Rb with I = 3/2) on the precession characteristics of the pumping process were investigated to confirm whether the obtained results were more consistent with those of a theoretical analysis performed in the improved SERF regime.

2. Principles and methods

In the presence of a magnetic field, alkali metal atoms at two hyperfine levels precess in opposite directions at the same Larmor frequency, ω0=γeB/(2I+1), where γe is the electron gyromagnetic ratio and I is the nuclear spin. The alkali metal atoms in |F1, mF1> and |F2, mF2> exchange spin angular momentum through SE collisions. Although the projection value, mF1+ mF2, of the total angular momentum is conserved, the angular momentum is redistributed at different hyperfine sublevels. The SE collisions between the alkali atoms redistribute the atoms in the hyperfine states and change their precession direction; however, the total angular momentum is maintained. Under typical SERF conditions, a high alkali metal atom density, and an extremely weak magnetic field, the SE collision rate, Rse, is much higher than ω0, the atoms in the spin-temperature distribution are locked together in the same direction owing to the SE collisions, and the frequency, ωe = γeB/Q(Pz), is small. Q(Pz) is the slowing-down factor, which depends on I and the polarization, Pz, in the pumping direction of the alkali metal atoms. For different values of I, the components of Q(Pz) are as follows [19]:

$${Q_{3/2}} P_z^{} = \frac{{\textrm{6 + }2P_z^2}}{{\textrm{1} + P_z^2}},{Q_{5/2}} P_z^{} = \frac{{6({19 + 26P_z^2 + 3P_z^4} )}}{{9 + 30P_z^2 + 9P_z^4}}.$$
In the SERF regime, the spin behavior can be described completely using the density matrix equation, which expresses the evolution of the electron spin polarization [20]:
$${\frac{d}{{dt}}{\boldsymbol P} = \frac{1}{{Q P_z^{} }}\left[ {{\gamma^e}{\boldsymbol B} \times {\boldsymbol P} + {R_{\textrm P}}({s\overrightarrow {\textrm z} - {\boldsymbol P}})- \frac{{\boldsymbol P}}{{{T_2}}}} \right]}$$
where B is the magnetic field applied to the sensor, and P is the atom spin polarization. In our experimental device, the optically pumped light travels along the z-axis while the probe light is incident along the x-axis. Furthermore, the spins of the alkali metal atoms are polarized along the z-axis. Finally, Rp is the pump rate of the pump light, and s is the photon polarization of the pump beam, where s$\textrm{ = }$+1 is the average photon spin for ${\sigma ^ + }$ light.

Multiple factors affect the transverse relaxation rate of alkali metal atoms, which can be expressed as follows:

$$\frac{1}{{{T_2}}} = \frac{1}{{T_{\textrm{SD}}^{}}} + \frac{1}{{T_\textrm{2}^{\textrm{SE}}}} + \frac{1}{{T_{\Delta \textrm{B}}^{}}} + \frac{1}{{T_\textrm{D}^{}}} + {R_\textrm{p}} + {R_{\textrm{pr}}},$$
where 1/TSD is the spin-destruction relaxation rate, which causes the spin angular momentum of the alkali metal atoms to be transformed into the rotational angular momentum of the colliding atom pair during collision; 1/T2SE is the spin-exchange relaxation rate; 1/TD is the diffusion relaxation rate; and 1/TΔB is the rate of relaxation caused by the inhomogeneity of the magnetic field. Furthermore, Rpr is the depolarization rate resulting from the absorption of the detection light. For the SE rate, Rse is much larger than ω0 [21].
$$\frac{1}{{T_\textrm{2}^{\textrm{SE}}}} = \frac{{\omega _0^2}}{{{R_{se}}}}\left[ {\frac{1}{2} - \frac{{{{(2I + 1)}^2}}}{{2Q{{({P_z})}^2}}}} \right]Q{({P_z})^2}.$$
In this case, the spin-exchange relaxation is proportional to the square of the magnetic field and inversely proportional to Rse. Therefore, as the magnetic field decreases and Rse increases, it decays rapidly. In other words, the SERF regime is realized.

Brownian motion of a spin in the presence of a magnetic-field gradient can significantly influence the spin-lattice relaxation time [22]. When there is a magnetic field gradient in the atomic cell, atomic spins with nonzero magnetic moments precess around the background magnetic field of the magnetic field gradient at different Larmor precession frequencies. In a cell not filled with a buffer gas, the alkali metal atoms move rapidly, and the total magnetic field experienced by them owing to atomic spin during the coherence time is the average magnetic field of the magnetic field gradient in the cell. Once the interior of the cell has been filled with a high-pressure buffer gas, the dynamic range of motion of the atoms within the spin coherence time decreases, leading to the spin decoherence of the alkali metal atoms. The inhomogeneity of the magnetic field causes a precession phase shift, and the incoherent precession causes the attenuation of the total spin polarization of the atoms and an increase in the resonance spectral linewidth. Consider the spin precession characteristics of the atoms in the SERF region. When the density of alkali metals is very high, the rate of spin exchange between them is significantly greater than the Larmor precession frequency of the atoms in a magnetic field. Although the direction of the spin precession of the atoms at different superfine levels is the opposite, a fast spin exchange collision pulls the spin angular momentum of the atoms to the same position [21]. The relationship between the magnetic field gradient and the longitudinal and transverse relaxation rates of the spin of the alkali metal atoms can be expressed as follows [12,13]:

$$\begin{array}{c} \frac{1}{{{T_{1\Delta B}}}} = 2D\frac{{{{|{\vec{\nabla }{B_x}} |}^2} + {{|{\vec{\nabla }{B_y}} |}^2}}}{{B_0^2}} \times \sum\limits_n {\frac{{Q({P_z})}}{{[{x_{1n}^2 - 2} ][{Q({P_z}) + {D^2}x_{1n}^4B_0^{ - 2}{\gamma^e}^2{R^{ - 4}}} ]}}} ,\\ \\ \frac{1}{{{T_{2\Delta B}}}} = \frac{{8{\gamma ^e}^2{R^4}{{|{\vec{\nabla }{B_z}} |}^2}}}{{175DQ({P_z})}} + D\frac{{{{|{\vec{\nabla }{B_x}} |}^2} + {{|{\vec{\nabla }{B_y}} |}^2}}}{{B_0^2}} \times \sum\limits_n {\frac{{Q({P_z})}}{{[{x_{1n}^2 - 2} ][{Q({P_z}) + {D^2}x_{1n}^4B_0^{ - 2}{\gamma^e}^2{R^{ - 4}}} ]}}} . \end{array}$$
where R is the radius of the alkali metal cell, x1n are the spatial frequency coefficients with the zeros of the derivatives of Bessel functions, B0 is the static magnetic field applied along the z axis, and $\vec{\nabla }{B_x}$, $\vec{\nabla }{B_y}$, and $\vec{\nabla }{B_z}$ represent the first-order gradient magnetic field in the x-, y-, and z-axes, respectively. Taking the z-axis as a representative example, the magnetic field gradient can be expressed as follows:
$$\vec{\nabla }{B_z} = \frac{{\partial {B_z}}}{{\partial x}}\vec{x} + \frac{{\partial {B_z}}}{{\partial y}}\vec{y} + \frac{{\partial {B_z}}}{{\partial z}}\vec{z},$$
where D is the diffusion coefficient related to the temperature and pressure in the cell [18]:
$$D = {D_0}\left( {\frac{{\sqrt {1 + T/(273K)} }}{{{p^{}}/(1\textrm{amg})}}} \right).$$
where D0 is the characteristic diffusion constant (D0Rb-He = 0.5 cm2/s [23]). T is the temperature of the cell, and p is the gas pressure inside the atomic cell.

According to Eq. (5), the rate of relaxation caused by the magnetic field gradient is related to the static magnetic field, B0, along the z-axis; the pressure, p, within the atomic cell; and the radius, R, of the cell. Only the magnetic field gradients along the x- and y-axes contribute to the longitudinal gradient relaxation rate. However, the magnetic field gradients along the x-, y-, and z-axes all contribute to the transverse gradient relaxation rate. A comparison of their expressions indicates that, when the z-axis magnetic field gradient is eliminated, the inhomogeneity is purely transversal (TB= 2TB) for all the other conditions [13]. Therefore, we mainly focused on the transverse gradient relaxation rate.

We simulated the changes in the transverse gradient relaxation rate under different conditions and analyzed the contributions of the x- and y-axes magnetic field gradients as well as that of the z-axis magnetic field gradient to the transverse gradient relaxation rate. In most studies on magnetic field gradient relaxation, the static magnetic field along the z-axis is of the order of µT [14,15]. As shown in Fig. 1, the transverse gradient relaxation rate stabilizes for a large static magnetic field (γB0R2/D ≫1), which is related only to the z-axis magnetic field gradient because the change in B0 is independent of the effect of the z-axis magnetic field gradient on the transverse gradient relaxation rate. However, for SERF optical pumping magnetometers (OPMs), the internal static magnetic field is approximately 10 nT. For the static magnetic field range corresponding to the SERF region, 1/TB decreases rapidly as B0 increases and gradually stabilizes after ∼20 nT. Thus, the magnetic field gradients along the x- and y-axes in a small static magnetic field have a significant effect on 1/TB.

 figure: Fig. 1.

Fig. 1. Sum of transverse gradient relaxation rate causes by variations in triaxial magnetic field gradient for static magnetic field, B0, based on Eq. (5). Inset magnified image corresponds to B0 values in 0–30 nT range. Transverse gradient relaxation rate produced by z-axis magnetic field gradient does not change with B0. When B0 > 20 nT, transverse gradient relaxation rate is mainly related to z-axis magnetic field gradient, and effects of x-axis and y-axis magnetic field gradients are negligible. However, when B0 < 20 nT, magnetic field gradients along x- and y-axes also affect transverse gradient relaxation rate.

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We compared the variations in the transverse gradient relaxation rate with the cell pressure, p, and cell radius, R, for different B0 values. The solid line in Fig. 2(a) shows that 1/TBtot increases with increasing p for different B0 values. As p increases, the diffusion coefficient, D, decreases, which affects the relaxation of the magnetic field gradient along the x-, y-, and z-axes. The dashed line shows the gradient relaxation produced by the transverse x- and y-axes magnetic field gradients. Under the condition B0 = 5 nT, the gradient relaxation produced by the transverse magnetic field gradient is larger and first increases and then decreases as the pressure increases. The effect of the magnetic field gradient is greater under low-pressure, low-static-magnetic-field conditions than under high-pressure, high-static-magnetic-field conditions. Figure 2(b) shows the contribution of the transverse and longitudinal magnetic field gradients to the gradient relaxation with changes in R under different static magnetic fields. As the radius of the cell, R, increases, the gradient relaxation generated by both the transverse magnetic field gradient and the longitudinal magnetic field gradient increases. The smaller the value of B0, the greater is the effect of increasing the transverse magnetic field gradient. Although the longitudinal magnetic field gradient has a determining effect on gradient relaxation under different conditions, the influence of the transverse magnetic field gradient cannot be ignored. We determined the gradient relaxation time by applying magnetic field gradients along the different axes and measuring the transverse relaxation time of the SERF OPMs.

 figure: Fig. 2.

Fig. 2. Comparison of changes in transverse gradient relaxation rate; cell pressure, p; and cell radius, R, for different B0 values. Solid lines represent overall transverse gradient relaxation of triaxial magnetic field gradient while dashed lines represent transverse gradient relaxation resulting from transverse magnetic field gradients along x- and y-axes.

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The value of the transverse relaxation time, T2, can be determined by measuring the half-width at half-maximum of the magnetic resonance curve. An oscillating magnetic field, By = B’cos(ωt), is applied in the y-direction perpendicular to the x-z plane, and the magnetic field is decomposed into two counter-propagating rotating fields, such that

$${B_y} = \frac{{i{B^\prime }}}{2}({{e^{i\omega t}} + {e^{ - i\omega t}}} ).$$
When a small bias field, B0, is applied, the output signals of magnetometer to different frequencies of magnetic field are obtained by sweeping frequency of oscillating magnetic field. The response of the SERF OPM is given by Eq. (2) as follows [24]:
$${P_x} = \frac{{{P_0}{\gamma ^e}{B^\prime }}}{{2Q({P_z})}}\left[ {\frac{{\Delta \omega \cos (\omega t) + ({\omega - {\omega_0}} )\sin (\omega t)}}{{{{(\Delta \omega )}^2} + {{({\omega - {\omega_0}} )}^2}}}} \right.\left. { + \frac{{\Delta \omega \cos ( - \omega t) + ({\omega + {\omega_0}} )\sin ( - \omega t)}}{{{{(\Delta \omega )}^2} + {{({\omega + {\omega_0}} )}^2}}}} \right]$$
The amplitude–frequency response function is used to fit the orthogonal sum signals of the in-phase and out-of-phase optical rotations. When the resonance frequency, ω0, is much larger than the linewidth, Δω, Eq. (9) can be expressed as follows:
$${P_x}^{(R)} = \frac{{{P_0}{\gamma ^e}{B^\prime }}}{{2Q({P_z})}}\frac{1}{{\sqrt {{{(\Delta \omega )}^2} + {{({\omega - {\omega_0}} )}^2}} }}.$$
In the above equation, ω0 = γeBz/Q(Pz), the magnetic resonance linewidth Δω = (Q(Pz)T2)-1. The effects of the magnetic field gradient on the relaxation and polarization of SERF OPMs can be determined simultaneously from the SE effect.

3. Experimental setup

Figure 3 shows the experimental setup used in this study, which is similar to that employed in our previous study [25]. The sensing unit of the device was an aluminosilicate glass (GE180) spherical cell with a radius of 8 mm. The cell was filled with naturally abundant Rb, 50 Torr N2, and 350 Torr 4He in a liquid nitrogen environment. N2 was used as the quenching gas to improve the efficiency of optical pumping by eliminating radiation trapping, and 4He was used as the buffer gas to reduce wall collision relaxation. The cell was placed in a boron nitride ceramic oven whose vacuum components were made of polyether ether ketone. The density of alkali metal atoms within the cell was measured using laser absorption spectroscopy and calculated using Raoult’s law and the saturated vapor density equation [26]. To ensure a sufficiently high atomic density, we used twisted-pair wires with an alternating current with a frequency of 300 kHz in an oven to heat the sensing unit and a platinum resistor attached to the cell wall for real-time temperature monitoring. The magnetic shield was composed of five layers of nested cylindrical high-permeability µ-metals. The shielding factor of the magnetic shield was 106, so that the atoms were in a low-magnetic-field environment. The device was equipped with a set of compensating three-axis magnetic coils as well as a set of gradient three-axis magnetic coils, which were used to compensate for the residual magnetic field and generate the magnetic field gradients ($\nabla Bx$= ∂Bx/ ∂x, $\nabla By$= ∂By/ ∂y, $\nabla Bz$= ∂Bz/ ∂z). Two high-precision function generators (Keysight 33500 B) were used to drive the magnetic coils. All the internal components of the device were made of nonmagnetic or weakly magnetic materials to ensure that the atoms were in the SERF region.

 figure: Fig. 3.

Fig. 3. Experimental setup of SERF OMP. Cell of naturally abundant Rb was placed within four-layer magnetic shielding and nonmagnetic vacuum system. Pumping light is incident along z-axis while probe light is incident along x-axis

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The pump light produced by a distributed Bragg reflector (DBR) diode laser was tuned to the D1 resonance of Rb, which was used to polarize the alkali atoms along the z-axis. The pump laser power was stabilized using a noise eater (Thorlabs, NEM03L). The linearly polarized pump light was converted into circularly polarized light using a quarter-wave plate before irradiating the cell. The radius of the pumping light beam was increased to 12 mm using a beam expander to ensure that the cell was fully covered. The optical rotation angle is proportional to the transverse component of the alkali metal atom spin polarization. Another DBR diode laser was used to produce a probe light for detecting the transverse polarization component, Px, along the x-axis. The probe light was transformed into a 3-mm linear-polarized light by being made to pass through the beam expander and quarter-wave plate, whose wavelength was tuned away by 1 nm of the Rb D2 resonance line. A photo elastic modulator (Hinds Instruments PEM100) modulated the detection light to an amplitude of 0.08 rad and frequency of 50 kHz to reduce the low-frequency noise [27]. The signal was demodulated using a lock-in amplifier (Zurich Instruments HF2LI). We used a wavelength meter (High Finesse WS7) with a measurement accuracy of 60 MHz to monitor the wavelengths of the pump and probe lights. Before the measurements, the magnetic shield was demagnetized, and three-axis magnetic field compensation coils were used to eliminate all three components of the magnetic field around the cell. A small transverse oscillating magnetic field was applied along the y-axis at different static magnetic field, B0, values. Next, different magnetic field gradients were applied along the different axes using saddle-gradient coils. Data acquisition was performed using an NI PXI-4461 DAQ card to determine the response of the SERF OPM. The amplitude–frequency response function was fitted after amplitude spectrum analysis to obtain the resonance frequency ω0, and linewidth Δω.

It should be noted that, for alkali metal atoms with different nuclear spins, I, the differences in the strength of the SE collision effect can be determined from the precession characteristics [19], which, in turn, affect their polarization and relaxation. The atom source used during the experiment was naturally abundant Rb with a composition of 72.2% 85Rb (I = 5/2) and 27.8%87Rb (I = 3/2). The precession rates of the two isotopes should be considered when fitting the amplitude–frequency response function of the system. The total amplitude–frequency response function of naturally abundant Rb can be obtained from Eq. (10) as follows:

$$\begin{array}{l} {P_x}^{(R)} = \frac{{{P_{a0}}{\gamma ^e}{B^\prime }}}{{2{Q_a}({P_z})}}\frac{1}{{\sqrt {{{(\Delta {\omega _a})}^2} + {{({\omega - {\omega_a}_0} )}^2}} }}\textrm{ + }\frac{{{P_{b0}}{\gamma ^e}{B^\prime }}}{{2{Q_b}({P_z})}}\frac{1}{{\sqrt {{{(\Delta {\omega _b})}^2} + {{({\omega - {\omega_b}_0} )}^2}} }}.\\ s = \sum\limits_i {{n_i}} \frac{1}{{\sqrt {{{(\Delta {\omega _i})}^2} + {{({\omega - {\omega_i}_0} )}^2}} }} + {d_i},i = a,b. \end{array}$$
In the above equation, a represents isotope 85Rb, and b represents isotope 87Rb. As the temperature of the device increased and the power of the pump light decreased, the double-peak effect became more pronounced. After loading the image, light was made to pass through the cell, and its polarization was detected using a photo elastic modulator (Hinds Instruments PEM100). The fitting results for the experimental and theoretical data based on Eq. (11) are shown in Fig. 4. The triangles represent the amplitude–frequency response of the OPM when there is no magnetic field gradient for a temperature of 150 °C, pumping laser of 0.5 mW, and static magnetic field of 7 nT. 85Rb and 87Rb are represented by the dashed and dashed-dotted lines, respectively, based on the single-peak fitting of Eq. (10). The solid line represents the fitting of the experimental data using bimodal fitting. The effect of bimodal fitting is closer to the experimental results. Based on the fitting results, we decided to focus on 85Rb, which is present in a greater proportion in naturally abundant Rb. Thus, we use the bimodal fitting results for the 85Rb data in the subsequent discussion.

To ensure that the magnetic field gradient relaxation and polarization are measured accurately, the device should be placed such that it is only sensitive to the magnetic field gradient. The slowing-down factor, Q, was measured in the absence of a magnetic field gradient as the static magnetic field, B0, was changed for different temperatures and pumping optical powers, as shown in Fig. 5. For the same temperature and pumping power, Q remained almost constant for different B0 values. As the temperature increased, the effect of the pumping power on the slowing-down factor, Q, weakened such that the values of the slowing-down factor for the different pumping powers were barely distinguishable. Therefore, we set the experimental ambient temperature to 150 °C to eliminate the effect of the pumping power on the measurement results. Figure 6 shows the measured linewidths for different pumping powers at 150 °C as functions of B0; the solid line represents the fitted curve based on Eq. (4) and Eq. (11), which yielded an SE rate of Rse≈ 4.36×104 s-1 at a pumping power of 0.5 mW. The theoretical SE rate, Rse, for the condition n = 7.51×1013 cm-3 is 4.62×104 s-1, which is close to the experimentally determined value. The linewidth increases with the increase in the pumping power because increases in the pumping power result in a larger Rp, and a better fit can be achieved at a small pumping power. In summary, the device was used successfully to measure the magnetic field gradient relaxation at a temperature of 150 °C and pumping power of 0.5 mW.

 figure: Fig. 4.

Fig. 4. Comparison of single-peak fitting using Eq. (10) and bimodal fitting using Eq. (11) to determine amplitude–frequency response function of investigated system.

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 figure: Fig. 5.

Fig. 5. Experimentally measured slowing-down factor, Q(Pz), as function of static magnetic field, B0, at different temperatures and pumping optical powers in absence of magnetic field gradient in investigated OPM.

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 figure: Fig. 6.

Fig. 6. Linewidths at different pumping powers and temperature of 150 °C as functions of static magnetic field, B0. Solid line is fitted curve-based Eq. (4) and Eq. (11), which yielded SE rate of Rse≈ 4.36×104 s-1 at pumping power of 0.5 mW.

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4. Results and discussion

The simultaneously measured polarization and transverse relaxation time of naturally abundant Rb atoms in the SERF regime under the magnetic field gradient generated by the gradient coil along the three axes are shown in Figs. 79. Figure 7 shows the polarization of alkali metal atoms along the z-axis of the OPM under different magnetic field gradients as measured using the slowing down factor, Q(Pz), based on Eq. (1) for SE interactions. The values of the static magnetic field, B0, in Fig. 7(a) and 7(b) are 5 and 20 nT, respectively. The dotted line represents the change trend line. The measurements were performed thrice continuously, and the uncertainties are represented by the error bars in the figures. The high uncertainties at low frequencies are attributable to the weak phase-locked capability at low frequencies. Pz increased with an increase in the magnetic field gradient, indicating that the detection signal of the device, which is proportional to Px, decreased. The effect of the magnetic field gradients applied along the x- and y-axes on the change in Pz became less pronounced with an increase in B0.

 figure: Fig. 7.

Fig. 7. Polarization of alkali metal atoms along z-axis of OPM under different magnetic field gradients (-20 nT/cm to ∼20 nT/cm) measured using slowing-down factor, Q(Pz), based on Eq. (1) for SE interactions. Static magnetic field, B0, in (a) is 5 nT and that in (b) is 20 nT.

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 figure: Fig. 8.

Fig. 8. Measured transverse relaxation rate and transverse gradient relaxation rate under different magnetic field gradients (-20 nT/cm to ∼20 nT/cm) generated by gradient coil along three axes at static magnetic field of B0 = 5 nT. (a) Experimentally measured transverse relaxation rate fitted using quadratic equation (y1 = a1(x + b1)2 + c1) based on Eq. (3) and Eq. (5). (b) Dashed line represents transverse gradient relaxation along x-axis, which changes by 7 s-1 while that along y-axis changes by 6.8 s-1; quadratic coefficients used were a1x = 0.0162 and a1y = 0.0144 based on (a). Solid line represents theoretical values obtained using Eq. (5). (c) Dashed line represents transverse gradient relaxation rate for z-axis changes, which changes by 30.9 s-1; quadratic coefficient used was a1z = 0.0714. Solid line represents theoretical values.

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 figure: Fig. 9.

Fig. 9. Measured transverse relaxation rate and transverse gradient relaxation rate under different magnetic field gradients (-20 nT/cm to ∼20 nT/cm) generated by gradient coil along three axes at static magnetic field of B0 = 20 nT. (a) Experimentally measured transverse relaxation rate fitted using quadratic equation (y2 = a2(x + b2)2 + c2). (b) Dashed line represents transverse gradient relaxation rate along x-axis, which changes by 4.3 s-1 while that along y-axis changes by 5.4 s-1; quadratic coefficients used were a2x = 0.009 and a2y = 0.0112 based on (a). Solid line represents theoretical values. (c) Dashed line represents transverse gradient relaxation along z-axis, which change by 31.2 s-1; quadratic coefficient used was a2z = 0.0719. Solid lines represent theoretical values.

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The polarization Pz along the pumping direction measured is affected by the magnetic field and the transverse relaxation under the same pumping light power. According to the steady-state solution of Bloch equation of SERF magnetometer [28], as magnetic field Bz increases, the polarization Pz rises rapidly and then slowly to a steady state. The polarization Pz changes very little with transverse relaxation rate, the change in the magnetic field is more important for Pz. When we applied a x- or y-axis gradient magnetic field, Pz slowly rises. The reason is that the non-orthogonality angle of the coils used in our experiment is about ∼3° transversely and ∼7° longitudinally [29]. We measured the three-axis coils coupling used in the experiment to be 5%. when we apply a x- or y-axis gradient magnetic field, we also introduce a magnetic change in the z-axis, causing the polarization Pz to slowly increase.

The values of the static magnetic field, B0, in Figs. 8 and 9 are 5 and 20 nT, respectively. We applied magnetic field gradients in the range of -20 nT/cm to ∼20 nT/cm along the three axes separately. The experimental data in Figs. 8(a) and 9(a) were fitted using a quadratic equation, y = a(x + b)2 + c, based on Eq. (3) and Eq. (5). Here, b is the original magnetic field gradient offset within the device, and c is the relaxation excluding the gradient relaxation. The quadratic coefficients of the fitting curves are a1x = 0.0162, a1y = 0.0144, and a1z = 0.0714 for B0 = 5 nT and a2x = 0.009, a2y = 0.0112, and a2z = 0.0719 for B0 = 20 nT. Using the quadratic coefficients, we could accurately determine the transverse gradient relaxation produced by the gradient coil, as indicated by the dashed lines in Figs. 8(b), 8(c), 9(b), and 9(c). The experimental results show that the transverse gradient relaxation time changed by 7 and 4.3 s-1 along the x-axis, 6.8 and 5.4 s-1 along the y-axis, and 30.9 1 and 31.2 s-1 in the z-axis for B0 = 5 and 20 nT, respectively. The solid lines in the figures represent the theoretical values for the SERF OPM obtained using Eq. (5). The transverse relaxation rate in the x-axis and y-axis changed by 2.9 s-1, 0.3 s-1, and 26.3 s-1, 26.3 s-1 in the z-axis for B0 = 5 nT and B0 = 20 nT, respectively.

These results are consistent with those of our previous analysis; the z-axis transverse gradient relaxation rate does not change with the static magnetic field, B0, while the transverse gradient relaxation rates along the x- and y-axes decrease with an increase in B0. The observed differences between the experimental and theoretical values are attributable to the fact that the magnetic field gradient generated by the gradient coils is not a perfect one-order one, and other-order magnetic gradients are also present, which may cause transverse gradient relaxation [15].In the measurement environment, other-order magnetic field gradients are also generated by the electric heating system used as well as the coupling of the coils and mutual coupling between the coils and the magnetic shield [30,31]. In addition, nonorthogonality in the processing and installation of the gradient coils [32] also results in measurement errors. The deviations along the z-axis are smaller than those along the x- and y-axes because the magnetic field gradients along the x- and y-axes are generated by two saddle coils, whereas that along the z-axis is generated by the Helmholtz coil, and the magnetic field gradients generated by saddle gradient coils have low linearity. The experimental data confirmed the validity of the proposed gradient relaxation model for SERF OPMs. It provides reference for the follow-up study of gradient relaxation of SERF OPMs.

The sensitivities measured under the experimental conditions described above are shown in Fig. 10 to allow for a comparison of the effects of the presence and absence of a magnetic field gradient on the performance of the SERF OPM. The orange dotted line represents the probe noise of the device. The green solid line represents the sensitivity in the absence of a magnetic field gradient; the sensitivity is 26.32 fT/Hz1/2 at 30 Hz. The black solid line represents the sensitivity after the application of 10 nT/cm magnetic field gradient along the x-axis; the sensitivity is 62.55 fT/Hz1/2 at 30 Hz. The red solid line represents the sensitivity after the application of a 10 nT/cm magnetic field gradient along the y-axis; the sensitivity is 130 fT/Hz1/2 at 30 Hz. Finally, the blue solid line represents the sensitivity after the application of a 10 nT/cm magnetic field gradient along the z-axis; the sensitivity is 93.51 fT/Hz1/2 at 30 Hz. The y-axis sensitivity decreased the most because the y-axis was the sensitive axis of the magnetometer. Thus, in the case of the SERF magnetometers used for detecting extremely weak magnetic fields, in addition to the effect of the z-axis magnetic field gradient, those of the x- and y-axes magnetic field gradients should also be considered. Therefore, in the future, we intend to evaluate the magnetic field gradient that exists in SERF OPMs and compensate for it to eliminate magnetic gradient relaxation and improve the sensitivity of the sensor.

 figure: Fig. 10.

Fig. 10. Comparison of effects of presence and absence of magnetic field gradient on performance of SERF OPM, and sensitivities measured under experimental conditions described above using 10 nT/cm magnetic field gradients applied independently along all three axes.

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5. Conclusion

In this study, we investigated the effects of magnetic field gradients along the x-, y-, and z-axes on the factors limiting the performance of SERF OPMs under extremely low-magnetic-field conditions and modified the magnetic field gradient relaxation model such that it can be applied to the atoms in the SERF regime. We used the oscillating cosine magnetic field excitation and amplitude spectrum analysis method to determine the magnetic resonance linewidth and slowing-down factor. The transverse relaxation time and z-axis polarization of the alkali metal atoms were determined simultaneously based on the SE collision effect. During the data analysis, we considered the different isotope SE collisions in naturally abundant Rb, which improved the accuracy of the fit. Under the experimental conditions corresponding to a temperature of 150 °C and pumping power of 0.5 mW, the device was sensitive only to the magnetic field gradient, thus confirming the accuracy of the measurement results. When the three axial magnetic field gradients, which ranged from -20 to ∼20 nT/cm, were applied independently, the transverse spin relaxation time and polarization of the alkali metal atoms, Pz, increased with an increase in the magnetic field gradient. A comparison of the experimental results for different static magnetic fields, B0, indicated that the z-axis transverse gradient relaxation does not change with B0 while the transverse gradient relaxations along the x -and y-axes decrease with an increase in B0. In addition, the effect of the x- and y-axes magnetic field gradients on Pz decreases with an increase in B0. The experimental results confirmed the accuracy of the improved model. However, the measurement accuracy can be enhanced further by improving the phase-locked performance at low frequencies and by correcting the problems of the nonorthogonality and coupling of the gradient coils. The sensitivity of the magnetometer is 26.32 fT/Hz1/2 at 30 Hz when there is no magnetic field gradient. However, when a 10 nT/cm magnetic field gradient is applied along all three axes, the sensitivity becomes 62.55, 130, and 93.51 fT/Hz1/2, respectively, at 30 Hz. Therefore, the effect of the magnetic field gradients along the x- and y-axes on the performance of the SERF optical pump magnetometer is not negligible. Precise simultaneous measurements of the transverse gradient relaxation and atomic polarization are necessary to improve the sensitivity of both the magnetometer and the comagnetometer. Our study provides a theoretical and experimental basis for eliminating the gradient relaxation caused by static magnetic and residual magnetic fields in SERF OPMs.

Funding

Major Scientific Research Project of Zhejiang Lab (2019MB0AE03); Foundation from Beijing Academy of Quantum Information Sciences (Y18G28); National Natural Science Foundation of China (61975005).

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.

References

1. C. Zimmer, K. K. Khurana, and M. G. Kivelson, “Subsurface Oceans on Europa and Callisto: Constraints from Galileo Magnetometer Observations,” Icarus. 147(2), 329–347 (2000). [CrossRef]  

2. E. Boto, N. Holmes, and J. Leggett, “Moving magnetoencephalography towards real-world applications with a wearable system,” Nature 555(7698), 657–674 (2018). [CrossRef]  

3. Y. J. Kim, I. Savukov, and S. Newman, “Magnetocardiography with a 16-channel fiber-coupled single-cell Rb optically pumped magnetometer,” Appl. Phys. Lett . 114(14), 143702 (2019). [CrossRef]  

4. V. V. Flambaum, M. Pospelov, A. Ritz, and Y. V. Stadnik, “Sensitivity of EDM experiments in paramagnetic atoms and molecules to hadronic CP violation,” Phys. Rev. D. 102(3), 035001 (2020). [CrossRef]  

5. S. Begus and D. Fefer, “An absorption-type proton NMR magnetometer for measuring low magnetic fields,” Meas. Sci. Technol. 18(3), 901–906 (2007). [CrossRef]  

6. S. Chatraphorn, E. F. Fleet, and F. C. Wellstood, “Scanning SQUID microscopy of integrated circuits,” Appl. Phys. Lett. 76(16), 2304–2306 (2000). [CrossRef]  

7. J. C. Allred, R. N. Lyman, T. W. Kornak, and M. V. Romalis, “High sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002). [CrossRef]  

8. I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003). [CrossRef]  

9. H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultra-high sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 227 (2010). [CrossRef]  

10. D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002). [CrossRef]  

11. D. Sheng, A. Kabcenell, and M. V. Romalis, “New Classes of Systematic Effects in Gas Spin Comagnetometers,” Phys. Rev. Lett. 113(16), 163002 (2014). [CrossRef]  

12. G. D. Cates, S. R. Schaefer, and W. Happer, “Relaxation of spins due to field inhomogeneities in gaseous samples at low magnetic field and low pressures,” Phys. Rev. A. 37(8), 2877–2885 (1988). [CrossRef]  

13. K. C. Hasson, G. D. Cates, K. Lerman, P. Bogorad, and W. Happer, “Spin relaxation due to magnetic-field inhomogeneities: quartic dependence and diffusion-constant measurements,” Phys. Rev. A. 42(9), 5766 (1990). [CrossRef]  

14. X. Liu, C. Chen, T. Qu, K. Yang, and H. Luo, “Transverse spin relaxation and diffusion-constant measurements of spin-polarized 129xe nuclei in the presence of a magnetic field gradient,” Sci. Rep. 6(1), 24122 (2016). [CrossRef]  

15. X. Zhan, C. Chen, Z. Wang, Q. Jiang, and H. Luo, “Improved compensation and measurement of the magnetic gradients in an atomic vapor cell,” AIP. Adv. 10(4), 045002 (2020). [CrossRef]  

16. Z. C. Ding, X. W. Long, J. Yuan, Z. F. Fan, and H. Luo, “Sensitive determination of the spin polarization of optically pumped alkali-metal atoms using near-resonant light,” Sci. Rep. 6(1), 32605 (2016). [CrossRef]  

17. R. J. Li, W. Quan, and J. C. Fang, “Polarization Measurement of Cs Using the Pump Laser Beam,” IEEE Photonics J. 9(6), 1–8 (2017). [CrossRef]  

18. W. Franzen, “Spin Relaxation of Optically Aligned Rubidium Vapor,” Phys. Rev. 115(4), 850–856 (1959). [CrossRef]  

19. I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A. 71(2), 023405 (2005). [CrossRef]  

20. M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker, and M. V. Romalis, “Spin-exchange relaxation free magnetometry with cs vapor,” Phys. Rev. A. 77(3), 1012–1015 (2007). [CrossRef]  

21. W. Happer and A. C. Tam, “Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors,” Phys. Rev. A. 16(5), 1877–1891 (1977). [CrossRef]  

22. L. D. Schearer and G. K. Walters, “Nuclear Spin-Lattice Relaxation in the Presence of Magnetic-Field Gradients,” Phys. Rev. 139(5A), A1398–A1402 (1965). [CrossRef]  

23. F. A. C. Franz and Volk, “Spin relaxation of rubidium atoms in sudden and quasimolecular collisions with light-noble-gas atoms,” Phys. Rev. A. 14(5), 1711–1728 (1976). [CrossRef]  

24. S. J. Seltzer and M. V. Romalis, “High-temperature alkali vapor cells with antirelaxation surface coatings,” J. Appl. Phys. 106(11), 114905 (2009). [CrossRef]  

25. X. J. Fang, K. Wei, T. Zhao, Y. Y. Zhai, D. Y. Ma, B. Z. Xing, Y. Liu, and Z. S. Xiao, “High spatial resolution multi-channel optically pumped atomic magnetometer based on a spatial light modulator,” Opt. Express. 28(18), 26447–26460 (2020). [CrossRef]  

26. K. Wei, T. Zhao, X. J. Fang, Y. Y. Zhai, H. R. Li, and W. Quan, “In-situ measurement of the density ratio of K-Rb hybrid vapor cell using spin-exchange collision mixing of the K and Rb light shifts,” Opt. Express. 27(11), 16169–16183 (2019). [CrossRef]  

27. B. Xing, C. Sun, Z. Liu, J. Zhao, J. Lu, B. Han, and M. Ding, “Probe noise characteristics of the spin-exchange relaxation-free (SERF) magnetometer,” Opt. Express. 29(4), 5055–5067 (2021). [CrossRef]  

28. S. J. Seltzer, “Developments in alkali-metal atomic magnetometry,” Ph.D. thesis, Dept. Phys., Princeton Univ., Princeton, NJ, USA (2008).

29. L. Xing, W. Quan, W. F. Fan, W. J. Zhang, Y. Fu, and T. X. Song, “The method for measuring the non-orthogonal angle of the magnetic field coils of a K-Rb-21Ne co-magnetometer,” IEEE Access. 7, 63892–63899 (2019). [CrossRef]  

30. D. Y. Ma, J. X. Lu, X. J. Fang, K. Wang, J. Wang, N. Zhang, H. J. Chen, M. Ding, and B. C. Han, “Analysis of coil constant of triaxial uniform coils in Mn-Zn ferrite magnetic shields,” J. Phys. D. 54(27), 275001 (2021). [CrossRef]  

31. D. H. Pan, S. X. Lin, L. Y. Li, J. Li, Y. X. Jin, Z. Y. Sun, and T. H. Liu, “Research on the Design Method of Uniform Magnetic Field Coil Based on the MSR,” IEEE Trans. Ind. Electron. 67(2), 1348–1356 (2020). [CrossRef]  

32. H. Zhang, S. Zou, W. Quan, X. Y. Chen, and J. C. Fang, “On-Site Synchronous Determination of Coil Constant and Nonorthogonal Angle Based on Electron Paramagnetic Resonance,” IEEE Trans. Instrum. Meas. 69(6), 3191–3197 (2020). [CrossRef]  

References

  • View by:

  1. C. Zimmer, K. K. Khurana, and M. G. Kivelson, “Subsurface Oceans on Europa and Callisto: Constraints from Galileo Magnetometer Observations,” Icarus. 147(2), 329–347 (2000).
    [Crossref]
  2. E. Boto, N. Holmes, and J. Leggett, “Moving magnetoencephalography towards real-world applications with a wearable system,” Nature 555(7698), 657–674 (2018).
    [Crossref]
  3. Y. J. Kim, I. Savukov, and S. Newman, “Magnetocardiography with a 16-channel fiber-coupled single-cell Rb optically pumped magnetometer,” Appl. Phys. Lett. 114(14), 143702 (2019).
    [Crossref]
  4. V. V. Flambaum, M. Pospelov, A. Ritz, and Y. V. Stadnik, “Sensitivity of EDM experiments in paramagnetic atoms and molecules to hadronic CP violation,” Phys. Rev. D. 102(3), 035001 (2020).
    [Crossref]
  5. S. Begus and D. Fefer, “An absorption-type proton NMR magnetometer for measuring low magnetic fields,” Meas. Sci. Technol. 18(3), 901–906 (2007).
    [Crossref]
  6. S. Chatraphorn, E. F. Fleet, and F. C. Wellstood, “Scanning SQUID microscopy of integrated circuits,” Appl. Phys. Lett. 76(16), 2304–2306 (2000).
    [Crossref]
  7. J. C. Allred, R. N. Lyman, T. W. Kornak, and M. V. Romalis, “High sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
    [Crossref]
  8. I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
    [Crossref]
  9. H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultra-high sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 227 (2010).
    [Crossref]
  10. D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
    [Crossref]
  11. D. Sheng, A. Kabcenell, and M. V. Romalis, “New Classes of Systematic Effects in Gas Spin Comagnetometers,” Phys. Rev. Lett. 113(16), 163002 (2014).
    [Crossref]
  12. G. D. Cates, S. R. Schaefer, and W. Happer, “Relaxation of spins due to field inhomogeneities in gaseous samples at low magnetic field and low pressures,” Phys. Rev. A. 37(8), 2877–2885 (1988).
    [Crossref]
  13. K. C. Hasson, G. D. Cates, K. Lerman, P. Bogorad, and W. Happer, “Spin relaxation due to magnetic-field inhomogeneities: quartic dependence and diffusion-constant measurements,” Phys. Rev. A. 42(9), 5766 (1990).
    [Crossref]
  14. X. Liu, C. Chen, T. Qu, K. Yang, and H. Luo, “Transverse spin relaxation and diffusion-constant measurements of spin-polarized 129xe nuclei in the presence of a magnetic field gradient,” Sci. Rep. 6(1), 24122 (2016).
    [Crossref]
  15. X. Zhan, C. Chen, Z. Wang, Q. Jiang, and H. Luo, “Improved compensation and measurement of the magnetic gradients in an atomic vapor cell,” AIP. Adv. 10(4), 045002 (2020).
    [Crossref]
  16. Z. C. Ding, X. W. Long, J. Yuan, Z. F. Fan, and H. Luo, “Sensitive determination of the spin polarization of optically pumped alkali-metal atoms using near-resonant light,” Sci. Rep. 6(1), 32605 (2016).
    [Crossref]
  17. R. J. Li, W. Quan, and J. C. Fang, “Polarization Measurement of Cs Using the Pump Laser Beam,” IEEE Photonics J. 9(6), 1–8 (2017).
    [Crossref]
  18. W. Franzen, “Spin Relaxation of Optically Aligned Rubidium Vapor,” Phys. Rev. 115(4), 850–856 (1959).
    [Crossref]
  19. I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A. 71(2), 023405 (2005).
    [Crossref]
  20. M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker, and M. V. Romalis, “Spin-exchange relaxation free magnetometry with cs vapor,” Phys. Rev. A. 77(3), 1012–1015 (2007).
    [Crossref]
  21. W. Happer and A. C. Tam, “Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors,” Phys. Rev. A. 16(5), 1877–1891 (1977).
    [Crossref]
  22. L. D. Schearer and G. K. Walters, “Nuclear Spin-Lattice Relaxation in the Presence of Magnetic-Field Gradients,” Phys. Rev. 139(5A), A1398–A1402 (1965).
    [Crossref]
  23. F. A. C. Franz and Volk, “Spin relaxation of rubidium atoms in sudden and quasimolecular collisions with light-noble-gas atoms,” Phys. Rev. A. 14(5), 1711–1728 (1976).
    [Crossref]
  24. S. J. Seltzer and M. V. Romalis, “High-temperature alkali vapor cells with antirelaxation surface coatings,” J. Appl. Phys. 106(11), 114905 (2009).
    [Crossref]
  25. X. J. Fang, K. Wei, T. Zhao, Y. Y. Zhai, D. Y. Ma, B. Z. Xing, Y. Liu, and Z. S. Xiao, “High spatial resolution multi-channel optically pumped atomic magnetometer based on a spatial light modulator,” Opt. Express. 28(18), 26447–26460 (2020).
    [Crossref]
  26. K. Wei, T. Zhao, X. J. Fang, Y. Y. Zhai, H. R. Li, and W. Quan, “In-situ measurement of the density ratio of K-Rb hybrid vapor cell using spin-exchange collision mixing of the K and Rb light shifts,” Opt. Express. 27(11), 16169–16183 (2019).
    [Crossref]
  27. B. Xing, C. Sun, Z. Liu, J. Zhao, J. Lu, B. Han, and M. Ding, “Probe noise characteristics of the spin-exchange relaxation-free (SERF) magnetometer,” Opt. Express. 29(4), 5055–5067 (2021).
    [Crossref]
  28. S. J. Seltzer, “Developments in alkali-metal atomic magnetometry,” Ph.D. thesis, Dept. Phys., Princeton Univ., Princeton, NJ, USA (2008).
  29. L. Xing, W. Quan, W. F. Fan, W. J. Zhang, Y. Fu, and T. X. Song, “The method for measuring the non-orthogonal angle of the magnetic field coils of a K-Rb-21Ne co-magnetometer,” IEEE Access. 7, 63892–63899 (2019).
    [Crossref]
  30. D. Y. Ma, J. X. Lu, X. J. Fang, K. Wang, J. Wang, N. Zhang, H. J. Chen, M. Ding, and B. C. Han, “Analysis of coil constant of triaxial uniform coils in Mn-Zn ferrite magnetic shields,” J. Phys. D. 54(27), 275001 (2021).
    [Crossref]
  31. D. H. Pan, S. X. Lin, L. Y. Li, J. Li, Y. X. Jin, Z. Y. Sun, and T. H. Liu, “Research on the Design Method of Uniform Magnetic Field Coil Based on the MSR,” IEEE Trans. Ind. Electron. 67(2), 1348–1356 (2020).
    [Crossref]
  32. H. Zhang, S. Zou, W. Quan, X. Y. Chen, and J. C. Fang, “On-Site Synchronous Determination of Coil Constant and Nonorthogonal Angle Based on Electron Paramagnetic Resonance,” IEEE Trans. Instrum. Meas. 69(6), 3191–3197 (2020).
    [Crossref]

2021 (2)

B. Xing, C. Sun, Z. Liu, J. Zhao, J. Lu, B. Han, and M. Ding, “Probe noise characteristics of the spin-exchange relaxation-free (SERF) magnetometer,” Opt. Express. 29(4), 5055–5067 (2021).
[Crossref]

D. Y. Ma, J. X. Lu, X. J. Fang, K. Wang, J. Wang, N. Zhang, H. J. Chen, M. Ding, and B. C. Han, “Analysis of coil constant of triaxial uniform coils in Mn-Zn ferrite magnetic shields,” J. Phys. D. 54(27), 275001 (2021).
[Crossref]

2020 (5)

D. H. Pan, S. X. Lin, L. Y. Li, J. Li, Y. X. Jin, Z. Y. Sun, and T. H. Liu, “Research on the Design Method of Uniform Magnetic Field Coil Based on the MSR,” IEEE Trans. Ind. Electron. 67(2), 1348–1356 (2020).
[Crossref]

H. Zhang, S. Zou, W. Quan, X. Y. Chen, and J. C. Fang, “On-Site Synchronous Determination of Coil Constant and Nonorthogonal Angle Based on Electron Paramagnetic Resonance,” IEEE Trans. Instrum. Meas. 69(6), 3191–3197 (2020).
[Crossref]

X. J. Fang, K. Wei, T. Zhao, Y. Y. Zhai, D. Y. Ma, B. Z. Xing, Y. Liu, and Z. S. Xiao, “High spatial resolution multi-channel optically pumped atomic magnetometer based on a spatial light modulator,” Opt. Express. 28(18), 26447–26460 (2020).
[Crossref]

V. V. Flambaum, M. Pospelov, A. Ritz, and Y. V. Stadnik, “Sensitivity of EDM experiments in paramagnetic atoms and molecules to hadronic CP violation,” Phys. Rev. D. 102(3), 035001 (2020).
[Crossref]

X. Zhan, C. Chen, Z. Wang, Q. Jiang, and H. Luo, “Improved compensation and measurement of the magnetic gradients in an atomic vapor cell,” AIP. Adv. 10(4), 045002 (2020).
[Crossref]

2019 (3)

Y. J. Kim, I. Savukov, and S. Newman, “Magnetocardiography with a 16-channel fiber-coupled single-cell Rb optically pumped magnetometer,” Appl. Phys. Lett. 114(14), 143702 (2019).
[Crossref]

K. Wei, T. Zhao, X. J. Fang, Y. Y. Zhai, H. R. Li, and W. Quan, “In-situ measurement of the density ratio of K-Rb hybrid vapor cell using spin-exchange collision mixing of the K and Rb light shifts,” Opt. Express. 27(11), 16169–16183 (2019).
[Crossref]

L. Xing, W. Quan, W. F. Fan, W. J. Zhang, Y. Fu, and T. X. Song, “The method for measuring the non-orthogonal angle of the magnetic field coils of a K-Rb-21Ne co-magnetometer,” IEEE Access. 7, 63892–63899 (2019).
[Crossref]

2018 (1)

E. Boto, N. Holmes, and J. Leggett, “Moving magnetoencephalography towards real-world applications with a wearable system,” Nature 555(7698), 657–674 (2018).
[Crossref]

2017 (1)

R. J. Li, W. Quan, and J. C. Fang, “Polarization Measurement of Cs Using the Pump Laser Beam,” IEEE Photonics J. 9(6), 1–8 (2017).
[Crossref]

2016 (2)

Z. C. Ding, X. W. Long, J. Yuan, Z. F. Fan, and H. Luo, “Sensitive determination of the spin polarization of optically pumped alkali-metal atoms using near-resonant light,” Sci. Rep. 6(1), 32605 (2016).
[Crossref]

X. Liu, C. Chen, T. Qu, K. Yang, and H. Luo, “Transverse spin relaxation and diffusion-constant measurements of spin-polarized 129xe nuclei in the presence of a magnetic field gradient,” Sci. Rep. 6(1), 24122 (2016).
[Crossref]

2014 (1)

D. Sheng, A. Kabcenell, and M. V. Romalis, “New Classes of Systematic Effects in Gas Spin Comagnetometers,” Phys. Rev. Lett. 113(16), 163002 (2014).
[Crossref]

2010 (1)

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultra-high sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 227 (2010).
[Crossref]

2009 (1)

S. J. Seltzer and M. V. Romalis, “High-temperature alkali vapor cells with antirelaxation surface coatings,” J. Appl. Phys. 106(11), 114905 (2009).
[Crossref]

2007 (2)

M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker, and M. V. Romalis, “Spin-exchange relaxation free magnetometry with cs vapor,” Phys. Rev. A. 77(3), 1012–1015 (2007).
[Crossref]

S. Begus and D. Fefer, “An absorption-type proton NMR magnetometer for measuring low magnetic fields,” Meas. Sci. Technol. 18(3), 901–906 (2007).
[Crossref]

2005 (1)

I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A. 71(2), 023405 (2005).
[Crossref]

2003 (1)

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref]

2002 (2)

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

J. C. Allred, R. N. Lyman, T. W. Kornak, and M. V. Romalis, “High sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref]

2000 (2)

S. Chatraphorn, E. F. Fleet, and F. C. Wellstood, “Scanning SQUID microscopy of integrated circuits,” Appl. Phys. Lett. 76(16), 2304–2306 (2000).
[Crossref]

C. Zimmer, K. K. Khurana, and M. G. Kivelson, “Subsurface Oceans on Europa and Callisto: Constraints from Galileo Magnetometer Observations,” Icarus. 147(2), 329–347 (2000).
[Crossref]

1990 (1)

K. C. Hasson, G. D. Cates, K. Lerman, P. Bogorad, and W. Happer, “Spin relaxation due to magnetic-field inhomogeneities: quartic dependence and diffusion-constant measurements,” Phys. Rev. A. 42(9), 5766 (1990).
[Crossref]

1988 (1)

G. D. Cates, S. R. Schaefer, and W. Happer, “Relaxation of spins due to field inhomogeneities in gaseous samples at low magnetic field and low pressures,” Phys. Rev. A. 37(8), 2877–2885 (1988).
[Crossref]

1977 (1)

W. Happer and A. C. Tam, “Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors,” Phys. Rev. A. 16(5), 1877–1891 (1977).
[Crossref]

1976 (1)

F. A. C. Franz and Volk, “Spin relaxation of rubidium atoms in sudden and quasimolecular collisions with light-noble-gas atoms,” Phys. Rev. A. 14(5), 1711–1728 (1976).
[Crossref]

1965 (1)

L. D. Schearer and G. K. Walters, “Nuclear Spin-Lattice Relaxation in the Presence of Magnetic-Field Gradients,” Phys. Rev. 139(5A), A1398–A1402 (1965).
[Crossref]

1959 (1)

W. Franzen, “Spin Relaxation of Optically Aligned Rubidium Vapor,” Phys. Rev. 115(4), 850–856 (1959).
[Crossref]

Acosta, V. M.

M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker, and M. V. Romalis, “Spin-exchange relaxation free magnetometry with cs vapor,” Phys. Rev. A. 77(3), 1012–1015 (2007).
[Crossref]

Allred, J. C.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref]

J. C. Allred, R. N. Lyman, T. W. Kornak, and M. V. Romalis, “High sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref]

Begus, S.

S. Begus and D. Fefer, “An absorption-type proton NMR magnetometer for measuring low magnetic fields,” Meas. Sci. Technol. 18(3), 901–906 (2007).
[Crossref]

Bogorad, P.

K. C. Hasson, G. D. Cates, K. Lerman, P. Bogorad, and W. Happer, “Spin relaxation due to magnetic-field inhomogeneities: quartic dependence and diffusion-constant measurements,” Phys. Rev. A. 42(9), 5766 (1990).
[Crossref]

Boto, E.

E. Boto, N. Holmes, and J. Leggett, “Moving magnetoencephalography towards real-world applications with a wearable system,” Nature 555(7698), 657–674 (2018).
[Crossref]

Budker, D.

M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker, and M. V. Romalis, “Spin-exchange relaxation free magnetometry with cs vapor,” Phys. Rev. A. 77(3), 1012–1015 (2007).
[Crossref]

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

Cates, G. D.

K. C. Hasson, G. D. Cates, K. Lerman, P. Bogorad, and W. Happer, “Spin relaxation due to magnetic-field inhomogeneities: quartic dependence and diffusion-constant measurements,” Phys. Rev. A. 42(9), 5766 (1990).
[Crossref]

G. D. Cates, S. R. Schaefer, and W. Happer, “Relaxation of spins due to field inhomogeneities in gaseous samples at low magnetic field and low pressures,” Phys. Rev. A. 37(8), 2877–2885 (1988).
[Crossref]

Chatraphorn, S.

S. Chatraphorn, E. F. Fleet, and F. C. Wellstood, “Scanning SQUID microscopy of integrated circuits,” Appl. Phys. Lett. 76(16), 2304–2306 (2000).
[Crossref]

Chen, C.

X. Zhan, C. Chen, Z. Wang, Q. Jiang, and H. Luo, “Improved compensation and measurement of the magnetic gradients in an atomic vapor cell,” AIP. Adv. 10(4), 045002 (2020).
[Crossref]

X. Liu, C. Chen, T. Qu, K. Yang, and H. Luo, “Transverse spin relaxation and diffusion-constant measurements of spin-polarized 129xe nuclei in the presence of a magnetic field gradient,” Sci. Rep. 6(1), 24122 (2016).
[Crossref]

Chen, H. J.

D. Y. Ma, J. X. Lu, X. J. Fang, K. Wang, J. Wang, N. Zhang, H. J. Chen, M. Ding, and B. C. Han, “Analysis of coil constant of triaxial uniform coils in Mn-Zn ferrite magnetic shields,” J. Phys. D. 54(27), 275001 (2021).
[Crossref]

Chen, X. Y.

H. Zhang, S. Zou, W. Quan, X. Y. Chen, and J. C. Fang, “On-Site Synchronous Determination of Coil Constant and Nonorthogonal Angle Based on Electron Paramagnetic Resonance,” IEEE Trans. Instrum. Meas. 69(6), 3191–3197 (2020).
[Crossref]

Dang, H. B.

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultra-high sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 227 (2010).
[Crossref]

Ding, M.

D. Y. Ma, J. X. Lu, X. J. Fang, K. Wang, J. Wang, N. Zhang, H. J. Chen, M. Ding, and B. C. Han, “Analysis of coil constant of triaxial uniform coils in Mn-Zn ferrite magnetic shields,” J. Phys. D. 54(27), 275001 (2021).
[Crossref]

B. Xing, C. Sun, Z. Liu, J. Zhao, J. Lu, B. Han, and M. Ding, “Probe noise characteristics of the spin-exchange relaxation-free (SERF) magnetometer,” Opt. Express. 29(4), 5055–5067 (2021).
[Crossref]

Ding, Z. C.

Z. C. Ding, X. W. Long, J. Yuan, Z. F. Fan, and H. Luo, “Sensitive determination of the spin polarization of optically pumped alkali-metal atoms using near-resonant light,” Sci. Rep. 6(1), 32605 (2016).
[Crossref]

Fan, W. F.

L. Xing, W. Quan, W. F. Fan, W. J. Zhang, Y. Fu, and T. X. Song, “The method for measuring the non-orthogonal angle of the magnetic field coils of a K-Rb-21Ne co-magnetometer,” IEEE Access. 7, 63892–63899 (2019).
[Crossref]

Fan, Z. F.

Z. C. Ding, X. W. Long, J. Yuan, Z. F. Fan, and H. Luo, “Sensitive determination of the spin polarization of optically pumped alkali-metal atoms using near-resonant light,” Sci. Rep. 6(1), 32605 (2016).
[Crossref]

Fang, J. C.

H. Zhang, S. Zou, W. Quan, X. Y. Chen, and J. C. Fang, “On-Site Synchronous Determination of Coil Constant and Nonorthogonal Angle Based on Electron Paramagnetic Resonance,” IEEE Trans. Instrum. Meas. 69(6), 3191–3197 (2020).
[Crossref]

R. J. Li, W. Quan, and J. C. Fang, “Polarization Measurement of Cs Using the Pump Laser Beam,” IEEE Photonics J. 9(6), 1–8 (2017).
[Crossref]

Fang, X. J.

D. Y. Ma, J. X. Lu, X. J. Fang, K. Wang, J. Wang, N. Zhang, H. J. Chen, M. Ding, and B. C. Han, “Analysis of coil constant of triaxial uniform coils in Mn-Zn ferrite magnetic shields,” J. Phys. D. 54(27), 275001 (2021).
[Crossref]

X. J. Fang, K. Wei, T. Zhao, Y. Y. Zhai, D. Y. Ma, B. Z. Xing, Y. Liu, and Z. S. Xiao, “High spatial resolution multi-channel optically pumped atomic magnetometer based on a spatial light modulator,” Opt. Express. 28(18), 26447–26460 (2020).
[Crossref]

K. Wei, T. Zhao, X. J. Fang, Y. Y. Zhai, H. R. Li, and W. Quan, “In-situ measurement of the density ratio of K-Rb hybrid vapor cell using spin-exchange collision mixing of the K and Rb light shifts,” Opt. Express. 27(11), 16169–16183 (2019).
[Crossref]

Fefer, D.

S. Begus and D. Fefer, “An absorption-type proton NMR magnetometer for measuring low magnetic fields,” Meas. Sci. Technol. 18(3), 901–906 (2007).
[Crossref]

Flambaum, V. V.

V. V. Flambaum, M. Pospelov, A. Ritz, and Y. V. Stadnik, “Sensitivity of EDM experiments in paramagnetic atoms and molecules to hadronic CP violation,” Phys. Rev. D. 102(3), 035001 (2020).
[Crossref]

Fleet, E. F.

S. Chatraphorn, E. F. Fleet, and F. C. Wellstood, “Scanning SQUID microscopy of integrated circuits,” Appl. Phys. Lett. 76(16), 2304–2306 (2000).
[Crossref]

Franz, F. A. C.

F. A. C. Franz and Volk, “Spin relaxation of rubidium atoms in sudden and quasimolecular collisions with light-noble-gas atoms,” Phys. Rev. A. 14(5), 1711–1728 (1976).
[Crossref]

Franzen, W.

W. Franzen, “Spin Relaxation of Optically Aligned Rubidium Vapor,” Phys. Rev. 115(4), 850–856 (1959).
[Crossref]

Fu, Y.

L. Xing, W. Quan, W. F. Fan, W. J. Zhang, Y. Fu, and T. X. Song, “The method for measuring the non-orthogonal angle of the magnetic field coils of a K-Rb-21Ne co-magnetometer,” IEEE Access. 7, 63892–63899 (2019).
[Crossref]

Gawlik, W.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

Han, B.

B. Xing, C. Sun, Z. Liu, J. Zhao, J. Lu, B. Han, and M. Ding, “Probe noise characteristics of the spin-exchange relaxation-free (SERF) magnetometer,” Opt. Express. 29(4), 5055–5067 (2021).
[Crossref]

Han, B. C.

D. Y. Ma, J. X. Lu, X. J. Fang, K. Wang, J. Wang, N. Zhang, H. J. Chen, M. Ding, and B. C. Han, “Analysis of coil constant of triaxial uniform coils in Mn-Zn ferrite magnetic shields,” J. Phys. D. 54(27), 275001 (2021).
[Crossref]

Happer, W.

K. C. Hasson, G. D. Cates, K. Lerman, P. Bogorad, and W. Happer, “Spin relaxation due to magnetic-field inhomogeneities: quartic dependence and diffusion-constant measurements,” Phys. Rev. A. 42(9), 5766 (1990).
[Crossref]

G. D. Cates, S. R. Schaefer, and W. Happer, “Relaxation of spins due to field inhomogeneities in gaseous samples at low magnetic field and low pressures,” Phys. Rev. A. 37(8), 2877–2885 (1988).
[Crossref]

W. Happer and A. C. Tam, “Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors,” Phys. Rev. A. 16(5), 1877–1891 (1977).
[Crossref]

Hasson, K. C.

K. C. Hasson, G. D. Cates, K. Lerman, P. Bogorad, and W. Happer, “Spin relaxation due to magnetic-field inhomogeneities: quartic dependence and diffusion-constant measurements,” Phys. Rev. A. 42(9), 5766 (1990).
[Crossref]

Holmes, N.

E. Boto, N. Holmes, and J. Leggett, “Moving magnetoencephalography towards real-world applications with a wearable system,” Nature 555(7698), 657–674 (2018).
[Crossref]

Jiang, Q.

X. Zhan, C. Chen, Z. Wang, Q. Jiang, and H. Luo, “Improved compensation and measurement of the magnetic gradients in an atomic vapor cell,” AIP. Adv. 10(4), 045002 (2020).
[Crossref]

Jin, Y. X.

D. H. Pan, S. X. Lin, L. Y. Li, J. Li, Y. X. Jin, Z. Y. Sun, and T. H. Liu, “Research on the Design Method of Uniform Magnetic Field Coil Based on the MSR,” IEEE Trans. Ind. Electron. 67(2), 1348–1356 (2020).
[Crossref]

Kabcenell, A.

D. Sheng, A. Kabcenell, and M. V. Romalis, “New Classes of Systematic Effects in Gas Spin Comagnetometers,” Phys. Rev. Lett. 113(16), 163002 (2014).
[Crossref]

Khurana, K. K.

C. Zimmer, K. K. Khurana, and M. G. Kivelson, “Subsurface Oceans on Europa and Callisto: Constraints from Galileo Magnetometer Observations,” Icarus. 147(2), 329–347 (2000).
[Crossref]

Kim, Y. J.

Y. J. Kim, I. Savukov, and S. Newman, “Magnetocardiography with a 16-channel fiber-coupled single-cell Rb optically pumped magnetometer,” Appl. Phys. Lett. 114(14), 143702 (2019).
[Crossref]

Kimball, D. F.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

Kivelson, M. G.

C. Zimmer, K. K. Khurana, and M. G. Kivelson, “Subsurface Oceans on Europa and Callisto: Constraints from Galileo Magnetometer Observations,” Icarus. 147(2), 329–347 (2000).
[Crossref]

Kominis, I. K.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref]

Kornack, T. W.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref]

Kornak, T. W.

J. C. Allred, R. N. Lyman, T. W. Kornak, and M. V. Romalis, “High sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref]

Ledbetter, M. P.

M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker, and M. V. Romalis, “Spin-exchange relaxation free magnetometry with cs vapor,” Phys. Rev. A. 77(3), 1012–1015 (2007).
[Crossref]

Leggett, J.

E. Boto, N. Holmes, and J. Leggett, “Moving magnetoencephalography towards real-world applications with a wearable system,” Nature 555(7698), 657–674 (2018).
[Crossref]

Lerman, K.

K. C. Hasson, G. D. Cates, K. Lerman, P. Bogorad, and W. Happer, “Spin relaxation due to magnetic-field inhomogeneities: quartic dependence and diffusion-constant measurements,” Phys. Rev. A. 42(9), 5766 (1990).
[Crossref]

Li, H. R.

K. Wei, T. Zhao, X. J. Fang, Y. Y. Zhai, H. R. Li, and W. Quan, “In-situ measurement of the density ratio of K-Rb hybrid vapor cell using spin-exchange collision mixing of the K and Rb light shifts,” Opt. Express. 27(11), 16169–16183 (2019).
[Crossref]

Li, J.

D. H. Pan, S. X. Lin, L. Y. Li, J. Li, Y. X. Jin, Z. Y. Sun, and T. H. Liu, “Research on the Design Method of Uniform Magnetic Field Coil Based on the MSR,” IEEE Trans. Ind. Electron. 67(2), 1348–1356 (2020).
[Crossref]

Li, L. Y.

D. H. Pan, S. X. Lin, L. Y. Li, J. Li, Y. X. Jin, Z. Y. Sun, and T. H. Liu, “Research on the Design Method of Uniform Magnetic Field Coil Based on the MSR,” IEEE Trans. Ind. Electron. 67(2), 1348–1356 (2020).
[Crossref]

Li, R. J.

R. J. Li, W. Quan, and J. C. Fang, “Polarization Measurement of Cs Using the Pump Laser Beam,” IEEE Photonics J. 9(6), 1–8 (2017).
[Crossref]

Lin, S. X.

D. H. Pan, S. X. Lin, L. Y. Li, J. Li, Y. X. Jin, Z. Y. Sun, and T. H. Liu, “Research on the Design Method of Uniform Magnetic Field Coil Based on the MSR,” IEEE Trans. Ind. Electron. 67(2), 1348–1356 (2020).
[Crossref]

Liu, T. H.

D. H. Pan, S. X. Lin, L. Y. Li, J. Li, Y. X. Jin, Z. Y. Sun, and T. H. Liu, “Research on the Design Method of Uniform Magnetic Field Coil Based on the MSR,” IEEE Trans. Ind. Electron. 67(2), 1348–1356 (2020).
[Crossref]

Liu, X.

X. Liu, C. Chen, T. Qu, K. Yang, and H. Luo, “Transverse spin relaxation and diffusion-constant measurements of spin-polarized 129xe nuclei in the presence of a magnetic field gradient,” Sci. Rep. 6(1), 24122 (2016).
[Crossref]

Liu, Y.

X. J. Fang, K. Wei, T. Zhao, Y. Y. Zhai, D. Y. Ma, B. Z. Xing, Y. Liu, and Z. S. Xiao, “High spatial resolution multi-channel optically pumped atomic magnetometer based on a spatial light modulator,” Opt. Express. 28(18), 26447–26460 (2020).
[Crossref]

Liu, Z.

B. Xing, C. Sun, Z. Liu, J. Zhao, J. Lu, B. Han, and M. Ding, “Probe noise characteristics of the spin-exchange relaxation-free (SERF) magnetometer,” Opt. Express. 29(4), 5055–5067 (2021).
[Crossref]

Long, X. W.

Z. C. Ding, X. W. Long, J. Yuan, Z. F. Fan, and H. Luo, “Sensitive determination of the spin polarization of optically pumped alkali-metal atoms using near-resonant light,” Sci. Rep. 6(1), 32605 (2016).
[Crossref]

Lu, J.

B. Xing, C. Sun, Z. Liu, J. Zhao, J. Lu, B. Han, and M. Ding, “Probe noise characteristics of the spin-exchange relaxation-free (SERF) magnetometer,” Opt. Express. 29(4), 5055–5067 (2021).
[Crossref]

Lu, J. X.

D. Y. Ma, J. X. Lu, X. J. Fang, K. Wang, J. Wang, N. Zhang, H. J. Chen, M. Ding, and B. C. Han, “Analysis of coil constant of triaxial uniform coils in Mn-Zn ferrite magnetic shields,” J. Phys. D. 54(27), 275001 (2021).
[Crossref]

Luo, H.

X. Zhan, C. Chen, Z. Wang, Q. Jiang, and H. Luo, “Improved compensation and measurement of the magnetic gradients in an atomic vapor cell,” AIP. Adv. 10(4), 045002 (2020).
[Crossref]

Z. C. Ding, X. W. Long, J. Yuan, Z. F. Fan, and H. Luo, “Sensitive determination of the spin polarization of optically pumped alkali-metal atoms using near-resonant light,” Sci. Rep. 6(1), 32605 (2016).
[Crossref]

X. Liu, C. Chen, T. Qu, K. Yang, and H. Luo, “Transverse spin relaxation and diffusion-constant measurements of spin-polarized 129xe nuclei in the presence of a magnetic field gradient,” Sci. Rep. 6(1), 24122 (2016).
[Crossref]

Lyman, R. N.

J. C. Allred, R. N. Lyman, T. W. Kornak, and M. V. Romalis, “High sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref]

Ma, D. Y.

D. Y. Ma, J. X. Lu, X. J. Fang, K. Wang, J. Wang, N. Zhang, H. J. Chen, M. Ding, and B. C. Han, “Analysis of coil constant of triaxial uniform coils in Mn-Zn ferrite magnetic shields,” J. Phys. D. 54(27), 275001 (2021).
[Crossref]

X. J. Fang, K. Wei, T. Zhao, Y. Y. Zhai, D. Y. Ma, B. Z. Xing, Y. Liu, and Z. S. Xiao, “High spatial resolution multi-channel optically pumped atomic magnetometer based on a spatial light modulator,” Opt. Express. 28(18), 26447–26460 (2020).
[Crossref]

Maloof, A. C.

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultra-high sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 227 (2010).
[Crossref]

Newman, S.

Y. J. Kim, I. Savukov, and S. Newman, “Magnetocardiography with a 16-channel fiber-coupled single-cell Rb optically pumped magnetometer,” Appl. Phys. Lett. 114(14), 143702 (2019).
[Crossref]

Pan, D. H.

D. H. Pan, S. X. Lin, L. Y. Li, J. Li, Y. X. Jin, Z. Y. Sun, and T. H. Liu, “Research on the Design Method of Uniform Magnetic Field Coil Based on the MSR,” IEEE Trans. Ind. Electron. 67(2), 1348–1356 (2020).
[Crossref]

Pospelov, M.

V. V. Flambaum, M. Pospelov, A. Ritz, and Y. V. Stadnik, “Sensitivity of EDM experiments in paramagnetic atoms and molecules to hadronic CP violation,” Phys. Rev. D. 102(3), 035001 (2020).
[Crossref]

Qu, T.

X. Liu, C. Chen, T. Qu, K. Yang, and H. Luo, “Transverse spin relaxation and diffusion-constant measurements of spin-polarized 129xe nuclei in the presence of a magnetic field gradient,” Sci. Rep. 6(1), 24122 (2016).
[Crossref]

Quan, W.

H. Zhang, S. Zou, W. Quan, X. Y. Chen, and J. C. Fang, “On-Site Synchronous Determination of Coil Constant and Nonorthogonal Angle Based on Electron Paramagnetic Resonance,” IEEE Trans. Instrum. Meas. 69(6), 3191–3197 (2020).
[Crossref]

K. Wei, T. Zhao, X. J. Fang, Y. Y. Zhai, H. R. Li, and W. Quan, “In-situ measurement of the density ratio of K-Rb hybrid vapor cell using spin-exchange collision mixing of the K and Rb light shifts,” Opt. Express. 27(11), 16169–16183 (2019).
[Crossref]

L. Xing, W. Quan, W. F. Fan, W. J. Zhang, Y. Fu, and T. X. Song, “The method for measuring the non-orthogonal angle of the magnetic field coils of a K-Rb-21Ne co-magnetometer,” IEEE Access. 7, 63892–63899 (2019).
[Crossref]

R. J. Li, W. Quan, and J. C. Fang, “Polarization Measurement of Cs Using the Pump Laser Beam,” IEEE Photonics J. 9(6), 1–8 (2017).
[Crossref]

Ritz, A.

V. V. Flambaum, M. Pospelov, A. Ritz, and Y. V. Stadnik, “Sensitivity of EDM experiments in paramagnetic atoms and molecules to hadronic CP violation,” Phys. Rev. D. 102(3), 035001 (2020).
[Crossref]

Rochester, S. M.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

Romalis, M. V.

D. Sheng, A. Kabcenell, and M. V. Romalis, “New Classes of Systematic Effects in Gas Spin Comagnetometers,” Phys. Rev. Lett. 113(16), 163002 (2014).
[Crossref]

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultra-high sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 227 (2010).
[Crossref]

S. J. Seltzer and M. V. Romalis, “High-temperature alkali vapor cells with antirelaxation surface coatings,” J. Appl. Phys. 106(11), 114905 (2009).
[Crossref]

M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker, and M. V. Romalis, “Spin-exchange relaxation free magnetometry with cs vapor,” Phys. Rev. A. 77(3), 1012–1015 (2007).
[Crossref]

I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A. 71(2), 023405 (2005).
[Crossref]

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref]

J. C. Allred, R. N. Lyman, T. W. Kornak, and M. V. Romalis, “High sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref]

Savukov, I.

Y. J. Kim, I. Savukov, and S. Newman, “Magnetocardiography with a 16-channel fiber-coupled single-cell Rb optically pumped magnetometer,” Appl. Phys. Lett. 114(14), 143702 (2019).
[Crossref]

Savukov, I. M.

M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker, and M. V. Romalis, “Spin-exchange relaxation free magnetometry with cs vapor,” Phys. Rev. A. 77(3), 1012–1015 (2007).
[Crossref]

I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A. 71(2), 023405 (2005).
[Crossref]

Schaefer, S. R.

G. D. Cates, S. R. Schaefer, and W. Happer, “Relaxation of spins due to field inhomogeneities in gaseous samples at low magnetic field and low pressures,” Phys. Rev. A. 37(8), 2877–2885 (1988).
[Crossref]

Schearer, L. D.

L. D. Schearer and G. K. Walters, “Nuclear Spin-Lattice Relaxation in the Presence of Magnetic-Field Gradients,” Phys. Rev. 139(5A), A1398–A1402 (1965).
[Crossref]

Seltzer, S. J.

S. J. Seltzer and M. V. Romalis, “High-temperature alkali vapor cells with antirelaxation surface coatings,” J. Appl. Phys. 106(11), 114905 (2009).
[Crossref]

S. J. Seltzer, “Developments in alkali-metal atomic magnetometry,” Ph.D. thesis, Dept. Phys., Princeton Univ., Princeton, NJ, USA (2008).

Sheng, D.

D. Sheng, A. Kabcenell, and M. V. Romalis, “New Classes of Systematic Effects in Gas Spin Comagnetometers,” Phys. Rev. Lett. 113(16), 163002 (2014).
[Crossref]

Song, T. X.

L. Xing, W. Quan, W. F. Fan, W. J. Zhang, Y. Fu, and T. X. Song, “The method for measuring the non-orthogonal angle of the magnetic field coils of a K-Rb-21Ne co-magnetometer,” IEEE Access. 7, 63892–63899 (2019).
[Crossref]

Stadnik, Y. V.

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B. Xing, C. Sun, Z. Liu, J. Zhao, J. Lu, B. Han, and M. Ding, “Probe noise characteristics of the spin-exchange relaxation-free (SERF) magnetometer,” Opt. Express. 29(4), 5055–5067 (2021).
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D. H. Pan, S. X. Lin, L. Y. Li, J. Li, Y. X. Jin, Z. Y. Sun, and T. H. Liu, “Research on the Design Method of Uniform Magnetic Field Coil Based on the MSR,” IEEE Trans. Ind. Electron. 67(2), 1348–1356 (2020).
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W. Happer and A. C. Tam, “Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors,” Phys. Rev. A. 16(5), 1877–1891 (1977).
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Volk,

F. A. C. Franz and Volk, “Spin relaxation of rubidium atoms in sudden and quasimolecular collisions with light-noble-gas atoms,” Phys. Rev. A. 14(5), 1711–1728 (1976).
[Crossref]

Walters, G. K.

L. D. Schearer and G. K. Walters, “Nuclear Spin-Lattice Relaxation in the Presence of Magnetic-Field Gradients,” Phys. Rev. 139(5A), A1398–A1402 (1965).
[Crossref]

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D. Y. Ma, J. X. Lu, X. J. Fang, K. Wang, J. Wang, N. Zhang, H. J. Chen, M. Ding, and B. C. Han, “Analysis of coil constant of triaxial uniform coils in Mn-Zn ferrite magnetic shields,” J. Phys. D. 54(27), 275001 (2021).
[Crossref]

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D. Y. Ma, J. X. Lu, X. J. Fang, K. Wang, J. Wang, N. Zhang, H. J. Chen, M. Ding, and B. C. Han, “Analysis of coil constant of triaxial uniform coils in Mn-Zn ferrite magnetic shields,” J. Phys. D. 54(27), 275001 (2021).
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X. Zhan, C. Chen, Z. Wang, Q. Jiang, and H. Luo, “Improved compensation and measurement of the magnetic gradients in an atomic vapor cell,” AIP. Adv. 10(4), 045002 (2020).
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X. J. Fang, K. Wei, T. Zhao, Y. Y. Zhai, D. Y. Ma, B. Z. Xing, Y. Liu, and Z. S. Xiao, “High spatial resolution multi-channel optically pumped atomic magnetometer based on a spatial light modulator,” Opt. Express. 28(18), 26447–26460 (2020).
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K. Wei, T. Zhao, X. J. Fang, Y. Y. Zhai, H. R. Li, and W. Quan, “In-situ measurement of the density ratio of K-Rb hybrid vapor cell using spin-exchange collision mixing of the K and Rb light shifts,” Opt. Express. 27(11), 16169–16183 (2019).
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D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
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X. J. Fang, K. Wei, T. Zhao, Y. Y. Zhai, D. Y. Ma, B. Z. Xing, Y. Liu, and Z. S. Xiao, “High spatial resolution multi-channel optically pumped atomic magnetometer based on a spatial light modulator,” Opt. Express. 28(18), 26447–26460 (2020).
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B. Xing, C. Sun, Z. Liu, J. Zhao, J. Lu, B. Han, and M. Ding, “Probe noise characteristics of the spin-exchange relaxation-free (SERF) magnetometer,” Opt. Express. 29(4), 5055–5067 (2021).
[Crossref]

Xing, B. Z.

X. J. Fang, K. Wei, T. Zhao, Y. Y. Zhai, D. Y. Ma, B. Z. Xing, Y. Liu, and Z. S. Xiao, “High spatial resolution multi-channel optically pumped atomic magnetometer based on a spatial light modulator,” Opt. Express. 28(18), 26447–26460 (2020).
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L. Xing, W. Quan, W. F. Fan, W. J. Zhang, Y. Fu, and T. X. Song, “The method for measuring the non-orthogonal angle of the magnetic field coils of a K-Rb-21Ne co-magnetometer,” IEEE Access. 7, 63892–63899 (2019).
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X. Liu, C. Chen, T. Qu, K. Yang, and H. Luo, “Transverse spin relaxation and diffusion-constant measurements of spin-polarized 129xe nuclei in the presence of a magnetic field gradient,” Sci. Rep. 6(1), 24122 (2016).
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D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
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Z. C. Ding, X. W. Long, J. Yuan, Z. F. Fan, and H. Luo, “Sensitive determination of the spin polarization of optically pumped alkali-metal atoms using near-resonant light,” Sci. Rep. 6(1), 32605 (2016).
[Crossref]

Zhai, Y. Y.

X. J. Fang, K. Wei, T. Zhao, Y. Y. Zhai, D. Y. Ma, B. Z. Xing, Y. Liu, and Z. S. Xiao, “High spatial resolution multi-channel optically pumped atomic magnetometer based on a spatial light modulator,” Opt. Express. 28(18), 26447–26460 (2020).
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K. Wei, T. Zhao, X. J. Fang, Y. Y. Zhai, H. R. Li, and W. Quan, “In-situ measurement of the density ratio of K-Rb hybrid vapor cell using spin-exchange collision mixing of the K and Rb light shifts,” Opt. Express. 27(11), 16169–16183 (2019).
[Crossref]

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X. Zhan, C. Chen, Z. Wang, Q. Jiang, and H. Luo, “Improved compensation and measurement of the magnetic gradients in an atomic vapor cell,” AIP. Adv. 10(4), 045002 (2020).
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H. Zhang, S. Zou, W. Quan, X. Y. Chen, and J. C. Fang, “On-Site Synchronous Determination of Coil Constant and Nonorthogonal Angle Based on Electron Paramagnetic Resonance,” IEEE Trans. Instrum. Meas. 69(6), 3191–3197 (2020).
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D. Y. Ma, J. X. Lu, X. J. Fang, K. Wang, J. Wang, N. Zhang, H. J. Chen, M. Ding, and B. C. Han, “Analysis of coil constant of triaxial uniform coils in Mn-Zn ferrite magnetic shields,” J. Phys. D. 54(27), 275001 (2021).
[Crossref]

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L. Xing, W. Quan, W. F. Fan, W. J. Zhang, Y. Fu, and T. X. Song, “The method for measuring the non-orthogonal angle of the magnetic field coils of a K-Rb-21Ne co-magnetometer,” IEEE Access. 7, 63892–63899 (2019).
[Crossref]

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B. Xing, C. Sun, Z. Liu, J. Zhao, J. Lu, B. Han, and M. Ding, “Probe noise characteristics of the spin-exchange relaxation-free (SERF) magnetometer,” Opt. Express. 29(4), 5055–5067 (2021).
[Crossref]

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X. J. Fang, K. Wei, T. Zhao, Y. Y. Zhai, D. Y. Ma, B. Z. Xing, Y. Liu, and Z. S. Xiao, “High spatial resolution multi-channel optically pumped atomic magnetometer based on a spatial light modulator,” Opt. Express. 28(18), 26447–26460 (2020).
[Crossref]

K. Wei, T. Zhao, X. J. Fang, Y. Y. Zhai, H. R. Li, and W. Quan, “In-situ measurement of the density ratio of K-Rb hybrid vapor cell using spin-exchange collision mixing of the K and Rb light shifts,” Opt. Express. 27(11), 16169–16183 (2019).
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H. Zhang, S. Zou, W. Quan, X. Y. Chen, and J. C. Fang, “On-Site Synchronous Determination of Coil Constant and Nonorthogonal Angle Based on Electron Paramagnetic Resonance,” IEEE Trans. Instrum. Meas. 69(6), 3191–3197 (2020).
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X. Zhan, C. Chen, Z. Wang, Q. Jiang, and H. Luo, “Improved compensation and measurement of the magnetic gradients in an atomic vapor cell,” AIP. Adv. 10(4), 045002 (2020).
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S. Chatraphorn, E. F. Fleet, and F. C. Wellstood, “Scanning SQUID microscopy of integrated circuits,” Appl. Phys. Lett. 76(16), 2304–2306 (2000).
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C. Zimmer, K. K. Khurana, and M. G. Kivelson, “Subsurface Oceans on Europa and Callisto: Constraints from Galileo Magnetometer Observations,” Icarus. 147(2), 329–347 (2000).
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IEEE Trans. Instrum. Meas. (1)

H. Zhang, S. Zou, W. Quan, X. Y. Chen, and J. C. Fang, “On-Site Synchronous Determination of Coil Constant and Nonorthogonal Angle Based on Electron Paramagnetic Resonance,” IEEE Trans. Instrum. Meas. 69(6), 3191–3197 (2020).
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K. Wei, T. Zhao, X. J. Fang, Y. Y. Zhai, H. R. Li, and W. Quan, “In-situ measurement of the density ratio of K-Rb hybrid vapor cell using spin-exchange collision mixing of the K and Rb light shifts,” Opt. Express. 27(11), 16169–16183 (2019).
[Crossref]

B. Xing, C. Sun, Z. Liu, J. Zhao, J. Lu, B. Han, and M. Ding, “Probe noise characteristics of the spin-exchange relaxation-free (SERF) magnetometer,” Opt. Express. 29(4), 5055–5067 (2021).
[Crossref]

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L. D. Schearer and G. K. Walters, “Nuclear Spin-Lattice Relaxation in the Presence of Magnetic-Field Gradients,” Phys. Rev. 139(5A), A1398–A1402 (1965).
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W. Franzen, “Spin Relaxation of Optically Aligned Rubidium Vapor,” Phys. Rev. 115(4), 850–856 (1959).
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Phys. Rev. D. (1)

V. V. Flambaum, M. Pospelov, A. Ritz, and Y. V. Stadnik, “Sensitivity of EDM experiments in paramagnetic atoms and molecules to hadronic CP violation,” Phys. Rev. D. 102(3), 035001 (2020).
[Crossref]

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Rev. Mod. Phys. (1)

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys. 74(4), 1153–1201 (2002).
[Crossref]

Sci. Rep. (2)

X. Liu, C. Chen, T. Qu, K. Yang, and H. Luo, “Transverse spin relaxation and diffusion-constant measurements of spin-polarized 129xe nuclei in the presence of a magnetic field gradient,” Sci. Rep. 6(1), 24122 (2016).
[Crossref]

Z. C. Ding, X. W. Long, J. Yuan, Z. F. Fan, and H. Luo, “Sensitive determination of the spin polarization of optically pumped alkali-metal atoms using near-resonant light,” Sci. Rep. 6(1), 32605 (2016).
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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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Figures (10)

Fig. 1.
Fig. 1. Sum of transverse gradient relaxation rate causes by variations in triaxial magnetic field gradient for static magnetic field, B0, based on Eq. (5). Inset magnified image corresponds to B0 values in 0–30 nT range. Transverse gradient relaxation rate produced by z-axis magnetic field gradient does not change with B0. When B0 > 20 nT, transverse gradient relaxation rate is mainly related to z-axis magnetic field gradient, and effects of x-axis and y-axis magnetic field gradients are negligible. However, when B0 < 20 nT, magnetic field gradients along x- and y-axes also affect transverse gradient relaxation rate.
Fig. 2.
Fig. 2. Comparison of changes in transverse gradient relaxation rate; cell pressure, p; and cell radius, R, for different B0 values. Solid lines represent overall transverse gradient relaxation of triaxial magnetic field gradient while dashed lines represent transverse gradient relaxation resulting from transverse magnetic field gradients along x- and y-axes.
Fig. 3.
Fig. 3. Experimental setup of SERF OMP. Cell of naturally abundant Rb was placed within four-layer magnetic shielding and nonmagnetic vacuum system. Pumping light is incident along z-axis while probe light is incident along x-axis
Fig. 4.
Fig. 4. Comparison of single-peak fitting using Eq. (10) and bimodal fitting using Eq. (11) to determine amplitude–frequency response function of investigated system.
Fig. 5.
Fig. 5. Experimentally measured slowing-down factor, Q(Pz), as function of static magnetic field, B0, at different temperatures and pumping optical powers in absence of magnetic field gradient in investigated OPM.
Fig. 6.
Fig. 6. Linewidths at different pumping powers and temperature of 150 °C as functions of static magnetic field, B0. Solid line is fitted curve-based Eq. (4) and Eq. (11), which yielded SE rate of Rse≈ 4.36×104 s-1 at pumping power of 0.5 mW.
Fig. 7.
Fig. 7. Polarization of alkali metal atoms along z-axis of OPM under different magnetic field gradients (-20 nT/cm to ∼20 nT/cm) measured using slowing-down factor, Q(Pz), based on Eq. (1) for SE interactions. Static magnetic field, B0, in (a) is 5 nT and that in (b) is 20 nT.
Fig. 8.
Fig. 8. Measured transverse relaxation rate and transverse gradient relaxation rate under different magnetic field gradients (-20 nT/cm to ∼20 nT/cm) generated by gradient coil along three axes at static magnetic field of B0 = 5 nT. (a) Experimentally measured transverse relaxation rate fitted using quadratic equation (y1 = a1(x + b1)2 + c1) based on Eq. (3) and Eq. (5). (b) Dashed line represents transverse gradient relaxation along x-axis, which changes by 7 s-1 while that along y-axis changes by 6.8 s-1; quadratic coefficients used were a1x = 0.0162 and a1y = 0.0144 based on (a). Solid line represents theoretical values obtained using Eq. (5). (c) Dashed line represents transverse gradient relaxation rate for z-axis changes, which changes by 30.9 s-1; quadratic coefficient used was a1z = 0.0714. Solid line represents theoretical values.
Fig. 9.
Fig. 9. Measured transverse relaxation rate and transverse gradient relaxation rate under different magnetic field gradients (-20 nT/cm to ∼20 nT/cm) generated by gradient coil along three axes at static magnetic field of B0 = 20 nT. (a) Experimentally measured transverse relaxation rate fitted using quadratic equation (y2 = a2(x + b2)2 + c2). (b) Dashed line represents transverse gradient relaxation rate along x-axis, which changes by 4.3 s-1 while that along y-axis changes by 5.4 s-1; quadratic coefficients used were a2x = 0.009 and a2y = 0.0112 based on (a). Solid line represents theoretical values. (c) Dashed line represents transverse gradient relaxation along z-axis, which change by 31.2 s-1; quadratic coefficient used was a2z = 0.0719. Solid lines represent theoretical values.
Fig. 10.
Fig. 10. Comparison of effects of presence and absence of magnetic field gradient on performance of SERF OPM, and sensitivities measured under experimental conditions described above using 10 nT/cm magnetic field gradients applied independently along all three axes.

Equations (11)

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Q 3 / 2 P z = 6 +  2 P z 2 1 + P z 2 , Q 5 / 2 P z = 6 ( 19 + 26 P z 2 + 3 P z 4 ) 9 + 30 P z 2 + 9 P z 4 .
d d t P = 1 Q P z [ γ e B × P + R P ( s z P ) P T 2 ]
1 T 2 = 1 T SD + 1 T 2 SE + 1 T Δ B + 1 T D + R p + R pr ,
1 T 2 SE = ω 0 2 R s e [ 1 2 ( 2 I + 1 ) 2 2 Q ( P z ) 2 ] Q ( P z ) 2 .
1 T 1 Δ B = 2 D | B x | 2 + | B y | 2 B 0 2 × n Q ( P z ) [ x 1 n 2 2 ] [ Q ( P z ) + D 2 x 1 n 4 B 0 2 γ e 2 R 4 ] , 1 T 2 Δ B = 8 γ e 2 R 4 | B z | 2 175 D Q ( P z ) + D | B x | 2 + | B y | 2 B 0 2 × n Q ( P z ) [ x 1 n 2 2 ] [ Q ( P z ) + D 2 x 1 n 4 B 0 2 γ e 2 R 4 ] .
B z = B z x x + B z y y + B z z z ,
D = D 0 ( 1 + T / ( 273 K ) p / ( 1 amg ) ) .
B y = i B 2 ( e i ω t + e i ω t ) .
P x = P 0 γ e B 2 Q ( P z ) [ Δ ω cos ( ω t ) + ( ω ω 0 ) sin ( ω t ) ( Δ ω ) 2 + ( ω ω 0 ) 2 + Δ ω cos ( ω t ) + ( ω + ω 0 ) sin ( ω t ) ( Δ ω ) 2 + ( ω + ω 0 ) 2 ]
P x ( R ) = P 0 γ e B 2 Q ( P z ) 1 ( Δ ω ) 2 + ( ω ω 0 ) 2 .
P x ( R ) = P a 0 γ e B 2 Q a ( P z ) 1 ( Δ ω a ) 2 + ( ω ω a 0 ) 2  +  P b 0 γ e B 2 Q b ( P z ) 1 ( Δ ω b ) 2 + ( ω ω b 0 ) 2 . s = i n i 1 ( Δ ω i ) 2 + ( ω ω i 0 ) 2 + d i , i = a , b .

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