Optical magnetic combination method for suppressing the Rb polarization-induced magnetic gradient in Rb-Xe NMR co-magnetometers

Wang Tengyue; Wang Tengyue; Peng Jinpeng; Peng Jinpeng; Liu Zhanchao; Liu Zhanchao; Liu Zhanchao; Mao Yunkai; Mao Yunkai; Wang Xuelei; Wang Xuelei

doi:10.1364/OE.489463

1. Introduction

The nuclear magnetic resonance (NMR) magnetometer is widely used in the magnetic field measurement area, such as fundamental physics [1,2], medical imaging [3], and atomic gyroscopes in inertial navigation [4,5]. Based on spin-exchange optical pumping (SEOP) of noble gases [6], the Rb-Xe co-magnetometers can measure the inertial rotation rate as NMR gyros (NMRGs) [7,8]. In the core sensing atomic ensemble, the magnetic moments of Xe nuclei can be hyperpolarized by the polarized-Rb valence electron and sensed as magnetic fields due to the Fermi-contacting [9,10], and vice versa. Therefore, the effective magnetic field gradient induced by optically spin-polarized Rb atoms has a non-negligible effect on the properties of Xe nuclear spin.

In NMR co-magnetometers, the polarization and relaxation of nuclear spins are the key factors limiting their performance. When the Rb-Xe atomic ensemble is illuminated by the resonant $\sigma ^+$ or $\sigma ^-$ laser, the optical power of the pump laser attenuates with increasing propagation distance due to Rb absorptions of spin angular momentum transferred from photons [11]. Then the non-uniformly polarized alkali metal electrons will form a polarization-induced magnetic gradient, which results in the reduction of nuclear spin relaxation time and SNR [12,13]. The Rb polarization-induced magnetic gradient and the residual magnetic gradient from the coil system are the main sources of magnetic field inhomogeneity. In the presence of magnetic field gradient, the nuclear spin precession of the noble gas Xe will dephase and its macroscopic transverse relaxation time will be shortened [14], which can be qualified by the spin echo method [15,16]. Specifically, for the $^{129}$Xe relaxation rate in a cubic vapor cell with an inner length of 5mm, Yim $et$ $al.$ showed that the magnetic gradient relaxation accounts for more than half of the total relaxation rates, in addition to minor cell wall relaxation and minimal spin-exchange collision relaxation in their experiment [17]. Therefore, it is necessary to suppress the magnetic gradient and eliminate its influences. To quantify the polarization-induced magnetic gradient relaxation, Lee $et$ $al.$ measured the relaxation time of $^{129}$Xe by repeatedly switching on/off the pumping light. By optimizing the duty cycle of the pumping time, the transverse relaxation time can approach the longitudinal relaxation time [18]. However, how to suppress the polarization-induced magnetic gradient remains a challenge.

In order to alleviate Rb polarization-induced magnetic gradient in a $^{3}$He-$^{129}$Xe co-magnetometer, Limes $et$ $al.$ applied continuous $\pi$ pulses and alternating $\sigma ^+$/$\sigma ^-$ beam to polarized-Rb atoms with the help of coils and an electro-optic-modulator (EOM) [19]. By continuously flipping its polarization state, the gradient was rapidly reversed, thus reducing the influence on spin relaxation. However, for cm-scale NMRGs [20], EOM is bulky to fit into the compact system, and the method is more suitable for their synchronous SEOP NMR scheme. Zhao $et$ $al.$ used counter-propagating pump beams in a magnetometer to suppress the electron polarization gradient [21], but they didn’t compensate for the magnetic gradient. Moreover, its vapor cell is different from NMR co-magnetometers atomic ensemble, which contains both alkali metal and noble gases for angular rate measurements, while magnetometers only contain alkali metal as working atoms. Therefore, this analysis is not appropriate for our system with Xe noble gases. For analyzing the pump beam influence on spin polarization homogeneity in NMRGs, Jia $et$ $al.$ adopted the finite element method (FEM) to simulate the spatial distribution and drew conclusions that the electronic spin polarization homogeneity is more sensitive to pump power than beam diameter [22]. While increasing the pump power can easily improve the spatial uniformity, excessive power will produce more bias instability error by increasing the alkali field shifts [23]. Therefore, it is necessary to elucidate the effect of the Rb polarization-induced magnetic gradient on nuclear spin properties, and develop a more effective method to suppress the polarization-induced magnetic gradient in NMR co-magnetometers.

In this work, we demonstrate an optical magnetic combination method for suppressing Rb polarization-induced magnetic gradient based on the second-order gradient coils under the counter-propagating pump beams condition. To confirm its validity, the spatial distribution of polarization-induced effective magnetic field is theoretically simulated and experiments are conducted simultaneously. Compared with the conventional single pump beam scheme, the compensation of the gradient coils is more effective in the counter-propagating pump beams scheme. Moreover, the transverse relaxation time of the Xe nuclear spin and the SNR of the Rb-Xe atomic ensemble are further improved. This method shed more light on the way to improve the performance of NMR co-magnetometers.

2. Principles and methods

2.1 Formation of polarization-induced magnetic gradient

The heart of NMR co-magnetometers is a mm-scale vapor cell containing alkali-metal Rb atoms, noble gas Xe atoms and N$_2$ et al. When illuminated by the resonant circularly polarized laser, the alkali-metal atoms absorb the spin angular momentum transferred from photons and attain a spin polarization state by the pumping depopulation. In the process of continuous absorption, the optical pumping power attenuates with increasing propagation distance, which results in different Rb electron spin polarizability at different spatial positions [18].

In order to analyze the distribution of optical polarization in 3D space for counter-propagating pump beams, the electronic spin polarization distribution considering the atomic diffusion effect can be calculated by the numerical simulations. In the Rb-Xe atomic ensemble, the evolution of the electronic spin polarization $P_e$ can be described by the Bloch-Torrey equations [24]:

(1)$$\frac{\partial \boldsymbol{P_e}}{\partial t}=D_e \nabla ^2\boldsymbol{P_e}+ \gamma _e \boldsymbol{P_e} \times \boldsymbol{B}+ R_{\text{op}} \left(1-\boldsymbol{P_e}\right) -R_e \boldsymbol{P_e},$$

where the subscript $e$ indicates the electrons of alkali-metal Rb, $D_e$ is the diffusion coefficient, $B$ is the magnetic field, $\gamma {}_e$ is the gyromagnetic ratio, and $R_{\text {op}}$, $R_{\text {e}}$ represent the rate of optical pumping and spin relaxation respectively. The boundary condition is [25]:

(2)$$D_e{\nabla}P_e={-}P_e \sqrt{\frac{D_e R_{\text{op}}}{2}}.$$

Considering the Gaussian distribution of the pump laser, the spatial distribution of the optical pumping rate can be expressed as [26]:

(3)$$R_{\text{op}}(x,y,z)=R_{\text{op}}(z) \exp[\frac{-2 \left(x^2+y^2\right)} {{r_{\text{beam}}^2}}],$$

where $r_{\text {beam}}$ is the radius of the pump beam, and the pump laser is propagating along $z$ -axis. In the counter-propagating pump light path, the right-handed and left-handed circularly polarized beams should be applied simultaneously, propagating top-down and bottom-up respectively. Then the variations of the optical pumping rate $R_{\text {op}}$ can be described in Eq. (4), in which the total optical pumping rate $R_\text {op}$ is the sum of $R_\text {op1}$ and $R_\text {op2}$.

(4)$$\frac{{dR}_{\text{op}}(z)}{d_z}= n_{\text{Rb}} \sigma{(\nu)} \left({-}R_{\text{op1}}\right) \left(1-P_{ez}\right) d_z +n_{\text{Rb}} \sigma{(\nu)} R_{\text{op2}} \left(1-P_{ez}\right) d_z,$$

where $n_{\text {Rb}}$ is the Rb atoms density, $\sigma (\nu )$ is the optical pumping cross section that is a function of laser frequency $\nu$ and related to the Voigt profile depending on the nature lifetime, pressure broading and Doppler broadening of the alkali metal. The optical pumping cross section can be expressed by [27]:

(5)$$\sigma(\nu)=\pi r_e cf\sum_{F,F'}^{}A_{F,F'} V(\nu-\nu_{F,F'}),$$

where $r_e$ is the classical electron raidus, $c$ is the light velocity, $f$ is the oscillator strength for the Rb D1 line which is approximately 1/3. $A_{F,F'}$ is the transition strength of the hyperfine structure transition $F$$\rightarrow$$F'$, $V$ is the Voigt profile, and $\nu _{F,F'}$ is the corresponding resonance frequency. The relavant values can be found in [27].

When the spatial distribution of electron spin polarization is viewed along the direction of pumping light propagation (i.e. 1-dimensional), we can have a more intuitive understanding of the physical process that continuous absorption of photons leads to the attenuation of electronic polarization along the propagation distance. The changing rate of the alkali metal electron spin is described as follows [6]:

(6)$$\frac{d{\langle S_z\rangle}}{dt}=\alpha\left(1-2\langle S_z\rangle \right)R_{\text{op}}-\langle S_z\rangle R_{\text{sd}},$$

where the first term on the right is the photon absorption rate, the optical pumping efficiency $\alpha$ is 1/2 under the condition of sufficient quenching buffer gas. $R_{\text {sd}}$ is the electron spin destruction relaxation rate. Considering the Rb polarization $P_{\text {Rb}}= 2\langle S_z\rangle$, a time-dependent Rb polarizability expression is obtained from Eq. (6):

(7)$$P_{\text{Rb}}(t)=\frac{R_{\text{op}}}{R_{\text{op}}+R_{\text{sd}}}\left\{1-\exp\left[ \left(-\left(R_{\text{op}}+R_{\text{sd}}\right)t\right)\right] \right\}.$$

The optical pumping rate $R_{\text {op}}$ is the dominant factor affecting the distribution of electron spin polarizability. When a single pump beam propagates through the vapor cell, the pumping rate in light propagating direction $z$-axis can be expressed as:

(8)$$\frac{\text{d}R_{\text{op}}(z)}{dz}={-}n_{\text{Rb}} \sigma{(\nu)} \left(1-P_{\text{ez}}\right) R_{\text{op}} dz,$$

where $n_{\text {Rb}}$ is Rb atoms density, $\sigma {(\nu )}$ is the optical pumping cross section, which is a function of laser frequency $\nu$. The optical power can be converted into the luminous flux per unit area in Eq. (9), which is the pumping rate $R_{\text {op}}$. Then the steady-state electronic polarizability in the direction of light propagation $P_{\text {ez}}$ is obtained.

(9)$$R_{\text{op}}=\frac{I_{\text{op}}\sigma(\nu)} {Ah\nu}, P_{\text{ez}}=\frac{R_{\text{op}}}{R_{\text{op}}+R_{\text{sd}}},$$

solved from the above equations to get:

(10)$$I_{\text{op}} (z)=\frac{{{A}{h}{}\nu} R_{\text{sd}}}{{\sigma (\nu)}}W\left(\frac{R_{\text{op}}(0)}{R_{\text{sd}}} \exp(\frac{R_{\text{op}}(0)}{R_{\text{sd}}}-n_{\text{Rb}} \sigma(\nu) z) \right),$$

where $A$ is the pumping beam area, $h$ is the Planck constant, $W$ is the Lambert W function, and $R_{\text {op}}(0)$ is the pumping rate at the entrance of the cell. Fig. 1 shows the attenuation of the optical power and the electronic spin polarization $P_e$ as determined by Eq. (10).

Fig. 1. (a). The optical power as a function of light propagation distance. (b). The electronic spin polarization $P_e$ as a function of light propagation distance.

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The effective magnetic field due to the optically polarized-Rb electron spin is denoted as:

(11)$$B_{\text{e}}=\frac{2}{3} \mu _0 \mu _B g_s {{\kappa}n}_{\text{Rb}}\langle S_z\rangle =\frac{1}{3} \mu _0 \mu _B g_s{{\kappa}n}_{\text{Rb}}P_{\text{e}},$$

where $\mu _0$ is the permeability of the free space, $\mu _B$ is the Bohr magneton, $g_s$ is the g-factor of electrons and $\kappa {}\sim$493 is the enhancement factor between Rb-Xe [10]. Due to the absorption of photons by Rb atoms, the light intensity at different positions will induce different magnitudes of the electron spin polarization. Since each electron can be regarded as a small magnetic moment, according to Eq. (11), an effective induced magnetic gradient will come into being by the inhomogeneous Rb polarization. Moreover, the strength of the polarization-induced magnetic gradient increases with the temperature.

The transverse relaxation rate $\Gamma _2$ of nuclear spin, which is the inverse of the relaxation time $T_2$, can be expressed as:

(12)$$\Gamma_2= {\Gamma}_{{coll}}+\Gamma_{{wall}}+\Gamma _{{\Delta B}}.$$

where $\Gamma _{coll}$ is the spin exchange collision relaxation, $\Gamma _{wall}$ is the relaxation due to collisions with the inner walls of the cell, and $\Gamma _{\Delta B}$ is the relaxation caused by the magnetic gradient, among which the magnetic gradient relaxation has a dominant impact [14,28]. Besides the residual magnetic gradient of the coil system, the effective magnetic gradient induced by the polarization gradient in the device is a crucial source of the total magnetic gradient.

In the case of the Xe nucleus, $^{129}$Xe is more appropriate for measurement than $^{131}$Xe due to the higher sensitivity of the transverse relaxation rate of $^{129}$Xe to the magnetic gradient, which is proportional to the square of the nuclear gyromagnetic ratio and thus contributes to a stronger gradient relaxation. As for a cubic cell of side length $L$, the transverse relaxation rate can be given by [29]:

(13)$$\Gamma _{{\Delta}B}=\frac{\gamma ^2 L^4}{120 D}|{{\nabla}B}_z|^2,$$

where $\gamma$ is the gyromagnetic ratio, $D$ is the diffusion coefficient, and $\nabla B_z$ is the magnetic field gradient of B$_z$ along the $z$-axis. In practical applications, conventional magnetic field gradient coils, which are optimized in terms of size and structure, are usually employed for gradient compensation to reduce the influence of magnetic field inhomogeneity on spin relaxation properties. Through the multi-order Taylor expansion of the magnetic field, various gradient coils are designed and have proven to be applicable in the magnetism detecting area [30,31].

2.2 Measurement of Xe transverse relaxation time

In the NMR regime, the nuclear spin relaxation time of Xe noble gas atoms can be measured by the free induction decay (FID) method [32]. Two types of relaxation time including the transverse relaxation time ($T_2$) and the longitudinal relaxation time ($T_1$) are important indicators of evaluating the nuclear spin relaxation properties. $T_2$ characterizes the weakening of nuclear spin phase coherence, while $T_1$ characterizes the process of the atomic magnetic moment reaching the thermal equilibrium state under the influence of the surrounding environment. The difference between their reciprocals is the relaxation rate caused by magnetic gradient and the spin-exchange collisions influenced by pumping light [33]. In particular, longer $T_2$ improves the long-term stability of NMR comagnetometers, especially of NMRGs [23]. Therefore, we focus on the $T_2$ compensation under the polarization-induced effective magnetic gradient.

The measurement schematic and the data fitting are shown in Fig. 2. When measuring the transverse relaxation time $T_2$, we firstly applied a $\pi$/2 pulse along the $y$-axis to flip the $z$-polarized Xe nuclear spins precessing into the $xy$ plane, as shown in Fig. 2(a). After removing the pulse, the nuclear spins started to spiral precess and decay gradually under the optical pumping along $z$ -axis and the spin-exchange collisions with other atoms. Finally, we could fit the FID signal in an exponential curve and acquire the transverse spin relaxation time.

Fig. 2. FID method to measure the Xe transverse relaxation time. (a) The process of nuclear spin precession when applying a $\pi$/2 pulse in measurement. (b) The exponential curve fitting of the FID signal.

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3. Experimental setup

The experimental setup is shown in Fig. 3. Two distributed bragger reflector (DBR) lasers were used as the light source (Vescent D2-100). The pump beam was propagated along the $z$-axis, whose frequency was locked near the Rb D1 line at 377.109 THz with an optical power of 2mW, while the frequency of the probe beam was near the Rb D2 line at 384.245 THz with an optical power of 1mW. The beam diameter kept at 2.5mm and both beams were introduced to the light path by the polarization maintaining fiber (PMF).

Fig. 3. Schematic of the experimental setup. PBS: polarized beam splitter, WP: Wollaston prism, HWP: half wave plate, QWP: quarter wave plate, BE: beam expander. The atoms wearing green in the vapor cell are Rb, and the blue are Xe atoms.

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On the light path of the pump beam, there was a half-wave plate (HWP) combined with a polarized beam splitter (PBS) to separate the light into two portions. One portion of the light continued propagating along the positive $z$-axis, while another portion was deflected and propagated along the negative $z$-axis. Before entering the vapor cell, the counter-propagating pump beams passed the beam expander (BE) to expand the beam radius and the quarter-wave plate (QWP) to transform into circularly polarized beams respectively. The probe beam was propagated along the $x$-axis. After passing the Wollaston prism (WP), it was separated into two beams and then detected by a balanced detector (Thorlabs PDB210A). Finally, the output signal was demodulated in a lock-in amplifier (ZI HF2LI) to obtain the atomic spin information. In order to reduce the influence of light source fluctuations on atomic ensemble, we used the noise eater (Thorlabs NEM03L) to stabilize the optical power and the saturated absorption module to stabilize the laser frequency. Moreover, the non-magnetic double-layered polyimide film heater was utilized to avoid possible magnetic field interference, and the temperature was stabilized at 393K. Considering the beam diameter and the ellipticity of the pump beam have a great influence on the electron polarization properties, we ensured the quality and circular polarization properties of the two beams was the same as possible. Furthermore, the $^{129}$Xe nuclear spin relaxation time should be almost equivalent when two pump beams were applied separately.

Additionally, the core sensing unit, a cubic glass vapor cell with a 3mm internal length, was encircled by multiple sets of coils and a four-layer magnetic shielding. The vapor cell was filled with a droplet of enriched-$^{87}$Rb, 2 Torr of $^{129}$Xe, 8 Torr of $^{131}$Xe, 300 Torr of N$_2$ as quenching gas as well as buffer gas, and 10 Torr of H$_2$ used to generate a layer of RbH anti-relaxation coatings [34].

4. Results and discussions

4.1 Simulations of optical polarization spatial distribution

The finite element method with the commercial software COMSOL Multiphysics was utilized to simulate the electron spin polarization spatial distribution in the counter-propagating pump beams condition. By converting it to the polarization-induced magnetic field, an effective gradient magnetic field was obtained and shown in Fig. 4. From the figure, it can be observed that the homogeneity of the electron spin polarization-induced magnetic field gradually improves as the optical intensity increases. The simulation results are obtained according to Eq. (1) to (5) and several key simulation parameters are as follows, including the Rb atom density is 2$\times$10$^{13}$ cm$^3$, the velocity average of the spin exchange cross section is $\sigma _{se}v$=3.7$\times$10$^{-16}$ cm$^3$ s$^{-1}$, and the relaxation rate of the electrons is 2.1$\times$10$^4$ s$^{-1}$. Other simulation parameters can be found in our previous work [22].

Fig. 4. Simulations of the polarization-induced effective magnetic field distribution under counter-propagating pump beams with light intensities of (a) 8.2mW/cm$^2$*2, (b) 14.3mW/cm$^2$*2, (c) 20.4mW/cm$^2$*2, and (d) 61.2mW/cm$^2$*2. The unit of the effective magnetic field is "nT".

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4.2 Simulations of magnetic field distribution generated by gradient coils

By expanding the multi-order Taylor series of the magnetic field, the spatial gradient magnetic field can be obtained to compensate the inhomogeneity of magnetic field. As for a pair of circular coils, the magnetic field generated at the origin can be expressed by:

(14)$$\left|\frac{\partial^2 {B_z}}{\partial z^2} |_{(0,0)}\right|= \frac{3\mu_0 IR(R-4d^2)}{(d^2+R^2)^{7/2}},$$

where $\mu _0$ is the permeability of the free space, $I$ is the current in the windings, $R$ is the radius of the coil, and $d$ is the distance between the coil and the central plane. And the magnetic field generated by a standard gradient coil in the target region is expressed by:

(15)$$B_z\left(z,r\right)=a{\cdot}r^2-2a{\cdot}z^2,$$

where $a$ is the coil constant, $r$ is the radius in the transverse cross-section, and $z$ is the longitudinal distance from the center of the coils in the $z$-$r$ coordinates.

As is shown in Fig. 5, by optimizing the coil size and the structures , we simulated the spatial distribution of the magnetic field generated by the second-order magnetic gradient coils. Fig. 5(a) shows the schematic structure of the second-order gradient coils, and the current direction of the two inner coils is opposite to that of the outer. By setting appropriate coils spacing and current, the spatial distribution of the net gradient field can be obtained in Fig.5(b). Comparing Fig. 4 to Fig. 5, we can see that the effective polarization-induced magnetic field and the gradient field generated by the coils are complementary to each other. After quantitively assessing the uniformity of the polarization-induced magnetic field before and after gradient compensation, we found that there is an improvement of 38.3% at the optical intensity of 20.4mW/cm$^2$. Therefore, we anticipate that a better compensation effect for the polarization-induced gradient field can be achieved under the counter-propagating pump beams scheme. Accordingly, we designed and fabricated the gradient coils to verify the experimental feasibility after full simulation.

Fig. 5. Simulations of magnetic field distribution. (a). Schematic structure of the second-order magnetic field gradient coils. $R$ is the radius of the coils. (b). The spatial net gradient field generated by the second-order gradient coils. $z$ is the longitudinal distance from the central plane, $r$ is the radial distance in the transverse cross-section.

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4.3 Experiments of electronic polarization under different optical pumping

In order to study the influence of dual beams on the electronic spin polarizability, the Rb electronic spin polarization under counter-propagating pump beams and the conventional single beam is measured and shown in Fig. 6 respectively. Relying on the Rb in-situ magnetometer in Rb-Xe atomic ensemble, different magnitudes of polarizability correspond to different optical rotation angles. Then the estimation of electron spin polarization $P_\text {e}$ can be obtained by measuring Faraday optical rotation angles [35].

Fig. 6. The Rb electronic spin polarization at the temperature of 393K, as a function of the pump beam intensity under counter-propagating pump beams and the single beam respectively. The intensity of the single beam is equal to the sum of the intensities of the dual beams.

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It can be seen that the electronic spin polarizability between counter-propagating pump beams and the single beam scheme have little difference, which is consistent with our expected results due to the constant pump power. In the experiment, the electronic polarizability of the top-down pumping beam is a little higher than that from bottom to top. This phenomenon can be attributed to slight differences in the circular polarization properties of the two beams, as well as to the light transmittance of the optical devices.

4.4 Transverse relaxation time improvement using the combination method

In the counter-propagating pump beams scheme, the transverse relaxation rate of Xe nuclear spin under the compensation of second-order magnetic gradient coils is demonstrated in Fig. 7. By applying an appropriate magnetic field gradient, the transverse relaxation rate of the nuclear spin reaches the minimum. The amount of the relaxation rate decrease caused by the magnetic field compensation (compensable gradient relaxation) are marked out with soild vertical lines and labeled with $\Delta {}\Gamma _{yi}$. Note in Fig. 7 that the compensable gradient relaxation $\Delta {}\Gamma _{yi}$ increases with the increase of light intensity. And the different quadratic minima are related to the improvement of spin exchange relaxation. Moreover, when we take the $\Delta {}\Gamma _{yi}$ as the dependent variable and the pump beam intensity as the independent variable, the light intensity dependence of the compensable gradient relaxation $\Delta {}\Gamma _{yi}$ is shown in Fig. 8. It can be seen that the compensable gradient relaxation $\Delta {}\Gamma _{yi}$ increases gradually with the increasing of pump intensity. And with the improvement of transverse relaxation time slowing down, the spatial distribution of polarized-Rb electrons tends to be saturated with the increasing pump intensity. Furthermore, the compensable gradient relaxation $\Delta {}\Gamma _{yi}$ increases with the increase of the temperature, which is consistent with the theoretical analysis of the attenuation of optical power and electronic spin polarizability in Section 2.1.

Fig. 7. The transverse relaxation rate of $^{129}$Xe versus the applied second-order gradient field under the counter-propagating pump beams. The black and red solid lines are the quadratic fitting curves of the measured data, and $\Delta {}\Gamma _{yi}$ is the maximum amount of magnetic gradient relaxation that can be compensated by the gradient coils.

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Fig. 8. Compensable gradient relaxation $\Delta {}\Gamma _{yi}$ as a function of the pump beam intensity. (a) Compensable gradient relaxation $\Delta {}\Gamma _{yi}$ at different temperatures under the counter-propagating pump beams. (b) Compensable gradient relaxation $\Delta {}\Gamma _{yi}$ under counter-propagating pump beams and the single beam.

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As shown in Fig. 8(b), under the same total pump intensity, the compensable gradient relaxation $\Delta {}\Gamma _{yi}$ in dual beams is higher than that in a single beam. It can be indicated that using second-order gradient coils to compensate for the polarization-induced magnetic gradient under the counter-propagating pump beams is more effective than that under a single pump beam. Specifically, the transverse relaxation time measured at 393K is shown in Table 1. When the optical intensity is 20.4mW/cm$^2$*2, the compensation of gradient coils increases $^{129}$Xe transverse relaxation time by 21.3% and 31.7% under the single and the dual beams respectively. When the optical intensity is 40.8mW/cm$^2$*2, the transverse relaxation time increases by 35.8% and 47.4% under the single beam and the dual beams respectively.

Table 1. Transverse relaxation time of $^{129}$Xe under different optical pumping schemes.

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4.5 SNR improvement of NMR co-magnetometers

The SNR of NMR co-magnetometers can be assessed by $B_n$/ $\delta _B$, where $B_n$ is the Xe nuclear spin magnetic field sensed by Rb in-situ magnetometer, and $\delta _B$ is the sensitivity of the magnetometer [27,36]. We adopted the method in Ref. [37] which used the initial value of the nuclear spin FID signal to obtain the magnetic field generated by the hyperpolarized Xe nuclei. As is shown in Eq. (16), the polarizability $P_{^{129}\text {Xe}}$ can be calculated by:

(16)$$B_{\text{FID}}=\frac{2}{3} {\kappa}\mu _0\mu _{^{129}\text{Xe}}n_{ ^{129}\text{Xe}}P_{^{129}\text{Xe}},$$

where $\mu _{^{129}\text {Xe}}$ is the magnetic moment of $^{129}$Xe, $n_{^{129}\text {Xe}}$ is the atoms denstiy of $^{129}$Xe.

The $^{129}$Xe nuclear spin polarizability $P_{^{129}Xe}$ versus the pump beam intensity is shown in Fig. 9. The experimental result shows that, compared with the single beam including the top-down pumping or the bottom-up pumping conditions, the counter-propagating pumping scheme can effectively improve the nuclear spin polarizability. This effect is attributed to the improvement of electronic spin polarization homogeneity in the vapor cell, which reduces the nuclear spin magnetic gradient relaxation rate. That is, under the same optical pump intensity, the electronic spin polarization amplitude changes little, but a more homogeneous Rb polarization increases the nuclear spin polarization by suppressing the magnetic gradient relaxation of $^{129}$Xe.

Fig. 9. The pump beam intensity dependence of $^{129}$Xe nuclear spin polarization under the counter-propagating pump beams and the single beam conditions respectively.

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The sensitivity measurements under the counter-propagating pump beams and the single beam conditions are shown in Fig. 10. It can be seen that there is little difference between them, and both of the sensitivities are about 600 fT/Hz$^{1/2}$. Since the sensitivity $\delta _B$ is almost not affected by the pump beam intensity, adopting counter-propagating pump beams will increase the nuclear spin polarizability and then improve the SNR of NMR co-magnetometers.

Fig. 10. Sensitivity of the NMR co-magnetometers when applying a calibrated magnetic field at 80Hz.

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Among the NMR co-magnetometers, NMR Gyros is an angular rate sensor that relies on the Xe isotopes to serve as the working atoms. According to the angle-random-walk (ARW) expression as 1/(T$_2$SNR$\sqrt {\Delta f}$) [23], when the linewidth of the co-magnetometers $\Delta f$ is constant, improving the Xe nuclear spin transverse relaxation time and the SNR is significant for the NMRGs system. Our new scheme simultaneously enhances the relaxation time and the SNR, making it of significance to the ARW and the long-term stability of NMRGs.

5. Conclusion

In conclusion, a novelty optical magnetic combination method that aims to suppress the optically Rb polarization-induced magnetic gradient in NMR co-magnetometers is proposed and investigated theoretically and experimentally. The simulation shows that the spatial distribution of the effective magnetic field caused by the Rb-polarization attenuation in counter-propagating pump beams scheme is complementary with the magnetic field distribution generated by the customized gradient coils. The experimental results indicate that the gradient compensation effect is 10% higher under the counter propagating pump beams condition compared to the compensation effect under the single-beam condition. Besides, owing to the improvement of Rb electron polarization homogeneity in the counter-propagating pump beams, the nuclear spin polarizability and the SNR of NMR co-magnetometers are enhanced in turn, which posts a vital application to NMRGs in inertial navigation. Finally, the effective utilization of optical power could also be improved in the counter-propagating pump beams scheme and practical applications are foreseen for NMRGs.

Funding

Innovation Program for Quantum Science and Technology (2021ZD0300403); National Science Fund for Distinguished Young Scholars (62225102).

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Optical pump intensity(mW/cm $^{2}$ )	20.4mW/cm $^{2}$ *2		40.8mW/cm $^{2}$ *2
Pump scheme	Single-beam	Dual-beams	Single-beam	Dual-beams
Without gradient compensation(s)	10.8	10.4	8.1	7.8
With gradient compensation(s)	13.1	13.7	11.0	11.5
Improvement(%)	21.3%	31.7%	35.8%	47.4%

Optical magnetic combination method for suppressing the Rb polarization-induced magnetic gradient in Rb-Xe NMR co-magnetometers

Abstract

1. Introduction

2. Principles and methods

2.1 Formation of polarization-induced magnetic gradient

2.2 Measurement of Xe transverse relaxation time

3. Experimental setup

4. Results and discussions

4.1 Simulations of optical polarization spatial distribution

4.2 Simulations of magnetic field distribution generated by gradient coils

4.3 Experiments of electronic polarization under different optical pumping

4.4 Transverse relaxation time improvement using the combination method

4.5 SNR improvement of NMR co-magnetometers

5. Conclusion

Funding

Disclosures

Data Availability

References

Data Availability

Cited By

Figures (10)

Tables (1)

Equations (16)

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