High sensitivity closed-loop Rb optically pumped magnetometer for measuring nuclear magnetization

Siran Li; Siran Li; Siran Li; Siran Li; Danyue Ma; Danyue Ma; Danyue Ma; Danyue Ma; Kun Wang; Kun Wang; Yanan Gao; Yanan Gao; Bozheng Xing; Bozheng Xing; Xiujie Fang; Xiujie Fang; Xiujie Fang; Bangcheng Han; Bangcheng Han; Wei Quan; Wei Quan

doi:10.1364/OE.473654

1. Introduction

Alkali metal atoms can be employed to polarize noble-gas nuclei through spin-exchange optical pumping (SEOP) and simultaneously function as an in situ atomic magnetometer to measure the nuclear magnetization with high sensitivity [1]. Due to the Fermi-contact interaction between the spin-exchange partners, which results in an effective magnetic field arising from the alkali electron spin polarization, the ability for the alkali metal magnetometer to sense the noble gas magnetization can be significantly enhanced [2]. Various configurations of optically pumped magnetometers (OPMs) are thus built to measure the nuclear magnetization for several applications including sensitive tests of fundamental physics [3–5] and rotation sensing [6–8], etc. Among them, OPMs based on electron paramagnetic resonance (EPR) can operate in large-scale magnetic field environments with longitudinal field modulation [9]. Rb and Xe atoms are the common combination of the atomic sources in this configuration. Rb atoms are employed to polarize Xe nuclei through SEOP and measure the transverse nuclear magnetization at the presence of an alterable bias longitudinal magnetic field. These OPMs also have short start-up to maintain real-time sensitivity and two isotopes (¹²⁹Xe and ¹³¹Xe) with different gyromagnetic ratios for elimination of the bias magnetic field dependence [10]. For these reasons, these Rb OPMs are widely succeeded in searching for new physics beyond the standard model [11,12] and serving for inertial measurement as nuclear magnetic resonance gyroscopes (NMRGs) [13,14].

For these applications, it is essential to improve the performances of the Rb OPMs based on EPR to measure the nuclear magnetization. One of key performances is the magnetometry sensitivity, which generally achieves sub-pT/Hz^1/2 in practice [11,13]. The sensitivity level depends on the signal strengths of the polarization and noises. The signal strength can be enhanced by parametrical optimization, while the noises are mainly suppressed by magnetic field or light modulation. Since the magnetic field modulation operates at the Rb Lamour frequency in this configuration, Rb OPMs usually utilize the balanced detection to measure the probe light rotation angle without the probe light modulation [15,16]. However, both frequency and magnitude of the magnetic field modulation are positively related with the longitudinal bias magnetic field [17]. Therefore, the field modulation cannot suppress the probe noises such as low-frequency noises efficiently when the bias magnetic field is weak (several hundreds of nT) in some applications [3,12]. To overcome the limitation, the light modulation for the optical rotation detection can be introduced to suppress the probe noises of the linearly-polarized probe beam. For other types of OPMs without the magnetic field modulation, several light modulation methods have already been applied to realize high probe sensitivity such as the Faraday [18], photoelastic [19] and acousto-optic modulation [20,21]. Among them, acousto-optic modulation can achieve radio-frequency modulation with a high extinction ratio, a small crystal volume and without extra magnetic noises, which can be used for sensitivity improvement in this configuration and is more suitable for integration than other methods.

In this study, we construct a Rb OPM for measuring nuclear magnetization with high performances. To further suppress the probe noise and pursue a higher sensitivity, we modify the conventional combination of the magnetic field modulation and balanced detection of this configuration with acousto-optic modulation. The working principle of the OPM is demonstrated theoretically. In the open-loop mode, a dual-axis transverse measurement based on EPR is achieved simultaneously with 30 fT/Hz^1/2 sensitivity in a 200 Hz bandwidth and a 250 nT linear working range. What’s more, although Rb OPMs realize transverse measurement of the nuclear magnetization with high sensitivities, their tolerant external fields in the transverse direction are much weaker than that in the longitudinal direction. Therefore, we utilize a closed-loop feedback to enlarge the transverse measurement range to µT level and improve the stability effected by the magnetic field fluctuations while maintain the open-loop performances. In the closed-loop mode, a longitudinal quasi-static field measurement can also be obtained based on nuclear magnetic resonance (NMR).

2. Principles and methods

The basic principle of the OPM for measuring nuclear magnetization is represented in Fig. 1, the disordered atoms are contained in a heated alkali metal vapor cell, including alkali metal atoms, nuclei, quenching gas and buffer gas. In presence of a static magnetic field B₀ along z-axis and an oscillating magnetic field with the Larmor frequency ω=ω_a, ω_b of Xe nuclei (corresponding to ¹²⁹Xe or ¹³¹Xe respectively) along x-axis, a circularly-polarized pump beam is turned to the Rb D1 transition to polarize Rb and Xe atoms through SEOP along the z-axis. At this time, all individual nuclear magnetic moments precess at their Larmor frequencies and the net nuclear magnetizations M = M₁, M₂ (corresponding to ¹²⁹Xe or ¹³¹Xe respectively) is tilted away from z-axis and precess in the x-y plane.

Fig. 1. Schematic of the nuclear magnetization measurement. (a)Disordered atoms are contained in a heated vapor cell. (b) In presence of a static magnetic field B₀ along z-axis, all individual nuclear magnetic moments precess about the z-axis at their Larmor frequencies and the net magnetization of Xe nuclei is zero. (c)Through SEOP, the net nuclear magnetization M = M₁, M₂ is aligned along the z-axis but is still zero in the x-y plane due to the out of phase precession of the magnetic moments. (d) By applying an oscillating field B₁ or B₂ with the Larmor frequency ω=ω_a, ω_b of Xe nuclei along the x-axis, M is tilted away from z-axis and precess in the x-y plane because the magnetic moments become phase coherent at this time.

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On this basis, the dynamic model of the nuclear magnetization $\mathbf{M} = {[{M_x},{M_y},{M_z}]^\textrm{T}}$ precessing about the magnetic field can be expressed as [22]:

(1)$$\frac{d}{{dt}}\mathbf{M} = \frac{{{M_0} - {M_z}}}{{{T_1}}}\hat{z} - \frac{{{M_x}}}{{{T_2}}}\hat{x} - \frac{{{M_y}}}{{{T_2}}}\hat{y} + \mathbf{M} \times {\gamma ^{Xe}}\mathbf{B}, $$

where T₁ and T₂ are the longitudinal and transverse relaxation time of Xe nuclei, respectively. M₀ is the z-component magnetization at the equilibrium position along the z-axis that depend on the optical pumping. γ^Xe is the nuclear gyromagnetic ratio of Xe nuclei. $\hat{x},\hat{y},\hat{z}$ are the unit vectors of the cartesian coordinate, and $\hat{z}$ is set to be the pumping direction. Taking ¹²⁹Xe for example, when an oscillating field B₁cos(ω_at) is applied along the x-axis, it is equivalent to apply two identical magnetic fields rotating in opposite directions. For the clockwise magnetic field, the x-axis component is B_x = B₁cos(ω_at) while the y-axis component is B_y = -B₁ sin (ω_at). Accordingly, the anticlockwise magnetic field consists of B_x = B₁cos(ω_at) along the x-axis and B_y =B₁ sin (ω_at) along the y-axis. Since B₁ rotates about the z-axis, a rotating reference coordinate is created for simplification as follow:

(2)$$\begin{array}{l} {M_x} = M_x^{\prime}\cos ({\omega _a}t) + M_y^{\prime}\sin ({\omega _a}t)\\ {M_y} ={-} M_x^{\prime}\sin ({\omega _a}t) + M_y^{\prime}\cos ({\omega _a}t) \end{array}, $$

where Mx’ rotates in-phase with B₁ about the z-axis and M_y’ rotates in-quadrature. Therefore Eq. (1) can be expressed in the rotating coordinate

(3)$$\begin{array}{l} \frac{{d{{M^{\prime}}_x}}}{{dt}} ={-} {\omega _\textrm{a}}{{M^{\prime}}_y} + {\gamma ^{X\textrm{e}}}{{M^{\prime}}_y}{B_0} - \frac{{{{M^{\prime}}_x}}}{{{T_2}}}\\ \frac{{d{M_y}}}{{dt}} = {\gamma ^{X\textrm{e}}}{M_z}{B_1} + {\omega _\textrm{a}}{{M^{\prime}}_y} - {\gamma ^{X\textrm{e}}}{{M^{\prime}}_y}{B_0} - \frac{{{{M^{\prime}}_y}}}{{{T_2}}}\\ \frac{{d{M_z}}}{{dt}} ={-} {\gamma ^{X\textrm{e}}}{B_1}{{M^{\prime}}_y} + \frac{{{M_0} - {M_z}}}{{{T_1}}} \end{array}. $$

Assuming Δω=γ^XeΒ₀ - ω_a is the difference value between the Xe Larmor frequency and the oscillating frequency of the applied magnetic field, the static solutions of the above equations can be obtained. Therefore, the net magnetization in the stationary coordinate are

(4)$$\begin{array}{l} {M_x} = {M_0}\frac{{{\gamma ^{X\textrm{e}}}{B_1}{T_2}^2\Delta \omega }}{{1\textrm{ + }( {T_2}\Delta \omega {) ^2}\textrm{ + }( {\gamma ^{X\textrm{e}}}{B_1}{) ^2}{T_1}{T_2}}}\cos ({\omega _a}t) + {M_0}\frac{{{\gamma ^{X\textrm{e}}}{B_1}{T_2}}}{{1\textrm{ + }( {T_2}\Delta \omega {) ^2}\textrm{ + }( {\gamma ^{X\textrm{e}}}{B_1}{) ^2}{T_1}{T_2}}}\sin ({\omega _a}t)\\ {M_y} ={-} {M_0}\frac{{{\gamma ^{X\textrm{e}}}{B_1}{T_2}^2\Delta \omega }}{{1\textrm{ + }( {T_2}\Delta \omega {) ^2}\textrm{ + }( {\gamma ^{X\textrm{e}}}{B_1}{) ^2}{T_1}{T_2}}}\sin ({\omega _a}t) + {M_0}\frac{{{\gamma ^{X\textrm{e}}}{B_1}{T_2}}}{{1\textrm{ + }( {T_2}\Delta \omega {) ^2}\textrm{ + }( {\gamma ^{X\textrm{e}}}{B_1}{) ^2}{T_1}{T_2}}}\cos ({\omega _a}t), \end{array}$$

which can be detected by the Rb in situ magnetometer. Based on EPR, the dynamics of Rb polarization $\mathbf{P} = {[{P_x},{P_y},{P_z}]^\textrm{T}}$ in the presence of the magnetic field $\mathbf{B} = {[{B_x},{B_y},{B_z}]^\textrm{T}}$ is described by the Bloch equation [2].

(5)$$\frac{d}{{dt}}\mathbf{P} = \frac{1}{q}[\mathbf{P} \times {\gamma ^e}\mathbf{B} + {R_{OP}}(s\hat{z} - \mathbf{P}) - \mathbf{RP}],\mathbf{R} = diag\left[ {\begin{array}{ccc} {\frac{1}{{{T_2}^e}}}&{\frac{1}{{{T_2}^e}}}&{\frac{1}{{{T_1}^e}}} \end{array}} \right], $$

where q is the slowing factor, which depends on the polarization and is considered as a constant. γ^e is the electron gyromagnetic ratio and R_op is the pumping rate. s is the photon polarization and s = 1 when the pump beam is circularly polarized. The transverse magnetic field to be measured is incapable to create any orientation when the applied B₀ is much larger, and thus an oscillating carrier field B_ccos(ω_ct) is also applied in the z-axis to force Rb magnetization to oscillate at various harmonics pω_c of ω_c, where ω_c = γ^eB₀ is the electron resonance frequency. On this basis, a first-order differential equation is derived from Eq. (5) and simplified by Jacobi–Anger expansions. Assuming ${\omega _0} = {\gamma ^e}{B_\textrm{0}}$, ${\omega _1} = {\gamma ^e}{B_\textrm{c}}$, ${R_{\textrm{op}}} + 1/{T_2}^e = 1/\tau$ and ${P_x}$ can be expressed as

(6)$$\begin{aligned} {P_x} &= \frac{{{\gamma ^e}\tau {P_z}{J_1}(\frac{{{\omega _1}}}{{{\omega _c}}})}}{{1 - {{[({\omega _0} - {\omega _\textrm{c}}\textrm{)}\tau ]}^2}}} \{ [{B_x}({\omega _0} - {\omega _c})\tau - {B_y}]\{ {J_1}(\frac{{{\omega _1}}}{{{\omega _c}}}) + [{J_{1 + p}}(\frac{{{\omega _1}}}{{{\omega _c}}})\textrm{ + }{J_{1 - p}}(\frac{{{\omega _1}}}{{{\omega _c}}})]\cos (p{\omega _\textrm{c}}t)\} \\ &- [{B_x} + {B_y}({\omega _0} - {\omega _c})\tau ]\mathrm{\{ }[{J_{1 + p}}(\frac{{{\omega _1}}}{{{\omega _c}}}) - {J_{1 - p}}(\frac{{{\omega _1}}}{{{\omega _c}}})]\sin (p{\omega _\textrm{c}}t)\mathrm{\} }\} \end{aligned}, $$

where J_p is the p-th Bessel function. When ω₀=ω_c and p = 1, B_x and B_y which reflect the transverse of the Xe magnetization are thus obtained by a lock-in amplifier (LIA) demodulating the output signal at ω_c with a 90° phase difference as

(7)$$\begin{array}{l} {P_{x - Bx}} \propto {B_x}{\gamma ^e}\tau {P_z}{J_1}(\frac{{{\omega _1}}}{{{\omega _c}}})[{J_2}(\frac{{{\omega _1}}}{{{\omega _c}}}) - {J_0}(\frac{{{\omega _1}}}{{{\omega _c}}})]\sin ({\omega _\textrm{c}}t)\\ {P_{x - By}} \propto {B_\textrm{y}}{\gamma ^e}\tau {P_z}{J_1}(\frac{{{\omega _1}}}{{{\omega _c}}})[{J_2}(\frac{{{\omega _1}}}{{{\omega _c}}}) + {J_0}(\frac{{{\omega _1}}}{{{\omega _c}}})]\cos ({\omega _\textrm{c}}t) \end{array}. $$

For the single-axis magnetic field measurement, the B_x is expected while the B_y is interferential. As to the dual-axis magnetic field measurement, the B_x and the B_y signal are expected to be equal. According to Eq. (7), the B_x and B_y signals have different output gains and thus the amplitude B_c and frequency ω_c of the carrier field should be optimized to maximize the output signals. In this study, we mainly evacuate the simultaneous dual-axis measurement of the Rb magnetometer and thus γ^eB_c/ω_c is fixed at 2.405 where the B_x and B_y signals are equal according to the signal simulation as shown in Fig. 2.

Fig. 2. Signal strengths under different carrier field parameter γ^eB_c/ω_c. The black and red solid lines denote the normalized signal outputs of B_x and B_y according to Eq.(7), respectively. The red star denotes the optimum of the normalized signal outputs for simultaneous dual-axis measurement. The corresponding carrier field parameter γ^eB_c/ω_c should be fixed at 2.405 where the B_x and B_y signals are equal, as denoted by the red dashed line. As for single-axis measurement, the black block denotes the maximum of the B_x signal, where γ^eB_c/ω_c is 1.357 but the output contains the B_y coupling. The black star denotes the optimum for single-axis measurement, where γ^eB_c/ω_c = 1.841. At this point, though the B_x signal slightly reduces but the B_y signal is completely suppressed. The black dashed lines denote the corresponding carrier field parameter γ^eB_c/ω_c, respectively.

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On this basis, in order to observe the magnetization by detecting the polarization P_x, a probe beam is emitted to propagate along the x-axis and linearly polarized to interact with Rb atoms near the D2 line. This results in a tiny optical rotation angle

(8)$$\theta = \frac{\pi }{2}ln{r_e}c{P_x}{f_{D2}}\frac{{{\nu _{\textrm{pr}}} - {\nu _{\textrm{D2}}}}}{{{{({\nu _{\textrm{pr}}} - {\nu _{\textrm{D2}}})}^2} - {{({\Gamma _{\textrm{D2}}}/2)}^2}}}, $$

where l is the vapor cell length along the probe direction, n is the atomic density number, r_e is the electron radium, c is the velocity of light, f_D2 represents the oscillating strength of the D2 line of Rb atoms. ν_pr is the probe frequency, ν_D2 is the resonance frequency of D2 line. Γ_D2 represents the pressure broadening of buffer and quenching gas. Since the magnetic field modulation operates at the Lamour frequency ω_c according to Eq.(7), noises can be suppressed by the LIA demodulation and Rb OPMs rarely use the light modulation. The conventional optical rotation angle detection is the balanced detection which suppress the common-mode noises without the light modulation for convenience and integration [23]. The probe beam is directly divided into two branches with equal intensities by setting the probe polarization to 45°, and then the differential signal of both branches is obtained by a balanced amplified photodetector to reflect the θ change.

However, as analyzed above, both modulation frequency and magnitude of the carrier field B_ccos(ω_ct) vary positively with the longitudinal field B₀ to maximize the signal strengths. Therefore, the carrier field modulation is insufficient to suppress some low-frequency noises such as the 1/f noise when the bias magnetic field is set relatively weak (such as several hundreds of nT) in practice. To overcome the limitation, an acousto-optic modulator (AOM) is added to the conventional balanced detection. AOM has been widely applied in laser switch, light intensity stabilization and modulation with a high extinction ratio, a small crystal volume and without extra magnetic noises [24,25]. The AOM can modulate the probe beam with adjustable high frequency up to MHz level to efficiently suppress the low-frequency noises. The light modulation frequency can alternate conveniently to skip the resonance frequency of electrons or nuclei and thus the acousto-optic modulation is non-interfering with the magnetic field modulation at ω_c. When the probe beam propagates through the AOM electroacoustic transducer, its light intensity I_probe is manipulated by the drive voltage with the Bragg-diffraction efficiency η [26]. On this basis, the probe light intensity is modulated when a square-wave signal is applied to drive the AOM with a high modulation frequency ω_m and a duty cycle D. The square waveform is approximate to a first harmonic wave based on Fourier expansions

(9)$$f(t) = \frac{4}{\pi }\sin ({\pi D} )\cos ({\omega _m}t). $$

Then the first-order light is divided into two separate probe beams with equal intensities by setting the probe polarization to 45°.

(10)$$\begin{array}{l} {I_1} = \eta {I_{probe}}f(t){\sin ^2}\left( {\theta - \frac{\pi }{4}} \right)\\ {I_2} = \eta {I_{probe}}f(t){\cos ^2}\left( {\theta - \frac{\pi }{4}} \right) \end{array}. $$

Therefore, the differential signal of the two beams can be detected by a balanced amplified photodetector,

(11)$${I_{out}} = {I_2} - {I_1} = \frac{4}{\pi }\sin ({\pi D} )\eta {I_{probe}}\sin ({2\theta } )\cdot \cos ({\omega _m}t). $$

The output signal V_out which is proportional to I_out can be demodulated by a LIA at ω_m

(12)$${V_{out - {\omega _m}}} \propto \frac{4}{\pi }\sin ({\pi D} )\eta {I_{probe}}\sin ({2\theta } )\approx \frac{8}{\pi }\sin ({\pi D} )\eta {I_{probe}}\theta. $$

and thus θ is obtained with a low probe noise. Eventually, the θ signal can be further demodulated at ω_c to reflect the responses of the polarization to the magnetic fields.

Furthermore, compared with the large-scale bias magnetic field in the longitudinal direction, the tolerant bias field for transverse measurement are much weaker [2]. To further enlarged the transverse compensation range, a closed-loop control based on the negative feedback is applied to lock the magnetic field detected by the Rb polarization [27,28]. In the open-loop mode, demodulated responses of the Rb OPM can be modeled by a low-pass filter with a cutoff frequency ω_b and described by a second-order transfer function

(13)$$G(s) = \frac{A}{{1 + s/{\omega _b}}},$$

\begin{aligned} A &\propto {\gamma ^e}\tau {P_z}{J_1}(2.405)[{J_2}(2.405) - {J_0}(2.405)]\\ &\approx {\gamma ^e}\tau {P_z}{J_1}(2.405)[{J_2}(2.405) + {J_0}(2.405)] \end{aligned}

where s = jω and A is the DC gain proportional to the scale factor in the linear range of the modulated part according to Eq. (7). The feedback system is construct by proportion-integration (PI) controllers and the coil compensation, and the schemtic diagram is shown in Fig. 3. The transfer function can be described as

(14)$${G_c}(s) = \frac{{{k_{coil}}G(s)({k_p} + {k_i}/s)}}{{1 + {k_{coil}}G(s)({k_p} + {k_i}/s)}} = \frac{{{k_{coil}}{k_p}A{\omega _b}s + {k_{coil}}{k_i}A{\omega _b}}}{{{s^2} + ({\omega _b} + {k_{coil}}{k_p}A{\omega _b})s + {k_{coil}}{k_i}A{\omega _b}}}, $$

where k_p and k_i represent parameters of the proportional and integral gains, respectively. k_coil is decided by the coil constant.

3. Experimental setup

Figure 4 shows the measurement apparatus of the Rb OPM. The sensitive unit of the apparatus is a 25-mm-diameter spherical vapor cell containing a droplet of naturally abundant Rb, 5 Torr ¹²⁹Xe, 25 Torr ¹³¹Xe, 50 Torr N₂ as the quenching gas, and 300 Torr ⁴He as the buffer gas. The vapor cell is placed in a boron nitride ceramic oven inside a vacuum chamber. To maintain a high atomic density, nonmagnetic heaters made by twisted pair wires are attached to outside surfaces of the oven with the heating temperature around 120°C. The home-made heaters are driven by 75 kHz alternating current, together with a platinum resistor attached to the vapor cell for temperature measurement, to perform real-time temperature control of the sensitive unit. The magnetic shields consist of four-layer cylindrical µ-metal to ensure a low magnetic field environment with a shielding factor of 10⁵. The residual magnetic fields are further reduced by internal three-axis orthogonal magnetic coils which is also required to generate desired magnetic fields for modulation or calibration. The triaxial coils are calibrated to ensure orthogonality and accurate coil constants [29]. Function generators (Keysight 33522B) and auxiliary outputs of the lock-in amplifier (LIA, Zurich Instruments HF2LI) are applied to control the magnetic coils.

Fig. 3. Schematic diagram of the closed-loop control.

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Fig. 4. Experimental setup of the Rb OPM. A naturally abundant Rb cell is heated inside magnetic shields and non-magnetic vacuum chamber. The pump beam propagates along the z-axis while the probe beam propagates along the x-axis. PBS: polarization beam splitter, BPD: balanced amplified photodetector.

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The pump beam is emitted from a distributed Bragg reflector diode laser and then tuned to Rb D1 resonance line for polarization of Rb atoms along the z-axis. The beam is expanded to ensure full coverage of the vapor cell and then converted into circularly polarized beam before propagating through the vapor cell. The probe light is emitted from a distributed feedback laser to detect the optical rotation which is proportional to the transverse polarization component P_x along the x-axis. The probe and pump beams are set orthogonally and their propagation directions are aligned with the x- and y-axis coils, respectively [30]. The linear-polarized probe beam is tuned 0.09 nm away from the Rb D2 resonance line and modulated by an AOM (Gooch & Housego AOM 3080-125). The AOM is controlled by a square-wave signal generated by the function generator with a frequency of 100 kHz. According to Eq. (12), the duty cycle D were set to be 50% to obtain the maximum of the output signal. The output signal of the probe beam is detected by a balanced amplified photodetector (New Focus Model 2307) to further suppress common-mode noises and then demodulated using a LIA at the light modulation frequency of AOM.

To pursue high sensitivity of the Rb OPM, residual magnetic fields around the vapor cell are eliminated to almost zero by magnetic shield and coil compensation before the experiment. By sweeping the magnetic field along the transverse direction, the resonance frequency is measured which reflects the atomic Lamour frequency and proportional to the static component of the magnetic field along the z-axis. The Rb OPM can thus realize longitudinal quasi-static measurement with the known gyromagnetic ratio based on magnetic resonance. To realize a dual-axis transverse measurement, the Rb resonance frequency ω_c is determined under a static magnetic field B₀ in a similar way. Then an oscillating carrier field B_ccos(ω_ct) is applied to the z-axis along with B₀, where ${B_\textrm{c}} = 2.405{\omega _\textrm{c}}/{\gamma ^e}$ according to the analysis in Section 2. On this basis, an oscillating field B_x, whose frequency is different from ω_c, is applied along the x-axis to help the adjustment of the phase difference between the x- and y-axis demodulation at ω_c with the LIA. According to Eq.(7), the B_x demodulation signal is adjusted to maximal while the B_y demodulation signal is almost zero. Their offset signals should be zeroed by compensating the transverse residual magnetic fields with the coils. At this time, the two transverse sensitive axes are set in-phase and out-phase respectively which maintain a 90° phase difference and reach an open-looped measurement condition. As an achievement of measuring nuclear magnetization shown in Fig. 5, the nuclear free induction decay (FID) signals are detected in the open-loop mode experimentally to evacuate the nuclear transverse relaxation time and polarization. According to the FID fitting, the nuclear transverse relaxation times are 10.4s of ¹³¹Xe and 8.2s of ¹²⁹Xe, respectively. The nuclear polarizations are 14% of ¹³¹Xe and 11% of ¹²⁹Xe, respectively. The FID signal also indicated the nuclei is polarized effectively by the Rb atoms through SEOP.

Fig. 5. Nuclear FID signal measured by the Rb OPM. According to the FID fitting, experimentally measured nuclear transverse relaxation times T₂ are 10.4s of ¹³¹Xe and 8.2s of ¹²⁹Xe, respectively. The corresponding nuclear polarizations P are 14% of ¹³¹Xe and 11% of ¹²⁹Xe, respectively.

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To increase the compensation range and stability of the Rb OPM, the response signals is locked at the open-loop working point using PI and phase locked loop (PLL) controllers and the triaxial coils. In the closed-loop mode, two PI modules embedded in the LIA is set to control the transverse coils to maintain the optical rotation angle by suppressing the x- and y-axis components of the detected magnetic field, respectively. What’s more, to maintain a stable resonance frequency of the nuclei or electrons which are proportional to B₀, the PLL module embedded in the LIA is also set to control the longitude coil to stable the longitude magnetic field fluctuations. The output signals are acquired for OPM performances, including amplitude–frequency responses, sensitivities, measurement ranges, bandwidth analyses and the nuclear magnetization. The performances are compared between the open-loop and closed-mode loop modes in the next section.

4. Results and discussion

Performances of the Rb OPM, i.e., the measurement ranges, the bandwidths and the sensitivities, are evacuated and analyzed experimentally. Since the two transverse measurement axes are set to be basically equivalent, we take the y-axis results for example hereinafter. Under different offset magnetic fields along the transverse direction, the responses of the Rb OPM to a DC and an AC magnetic field are both acquired as shown in Fig. 6(a) and Fig. 6(b), respectively. Transverse measurement ranges are compared between the open-loop and closed-loop mode. The measurement ranges can be much enlarged by increasing the compensation ranges of the triaxial coils with the closed-loop control. Limited by the voltage output range of the coil excitation and the coil constants of our experimental apparatus, the measurement ranges increased approximately ±10,125 nT compared with the open-loop mode. As shown in the gray region in Fig. 6(a), the sizes of linear ranges remain basically unchanged to be approximately 250nT in the two modes. The scale factor in the linear range changes 3.8% in Fig. 6(a) and the attenuation trends of AC response signal in Fig. 6(b) are also essentially unchanged. These results indicate the Rb OPM with the closed-loop control obtains a large measurement range without loss of the signal strengths. The slight attenuation occurs mainly because the increasing electrical noise with the compensation voltage in the closed-loop mode.

Fig. 6. Transverse measurement ranges of (a) DC and (b) AC magnetic fields, respectively. Limited by the voltage output range of coil excitation and the coil constants of the experimental apparatus, ± 10,125 nT measurement range is increased in the closed-loop mode approximately.

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Results of the amplitude–frequency responses of the simultaneous dual-axis transverse measurement in the open-loop and the closed-mode are compared in Fig. 7. The frequency of the calibration magnetic field ranges from 5 Hz to 500 Hz and is set to skip the resonance frequencies of the Rb or Xe atoms in the experiment. The corresponding demodulation strengths of the output signals are recorded. The bandwidths are both approximately 200 Hz at -3 dB and the dynamic responses of the two modes are basically equal and exhibit Lorentzian-types fitting curves. Therefore, the closed-loop control can enlarge the measurement ranges to more than 10µT order of magnitude with suitable coil compensation and maintain original measurement abilities at the same time. With a large-scale measurement ranges, it also shows the potential for attractive applications such as the magnetic anonymous detection and deep space detection. What’s more, in the closed-loop mode, the resonance frequency along the transverse direction is proportional to the static magnetic field in the z-axis. Therefore, the Rb OPM can thus realize longitudinal quasi-static measurement precisely based on NMR.

Fig. 7. Frequency responses of the simultaneous dual-axis transverse measurement. The bandwidth in the closed-loop and open-loop mode are basically equal to be 200 Hz at -3 dB.

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The simultaneously measured sensitivities of x- and y-axes in the closed-loop mode are shown in Fig. 8. Two 100pTrms 30 Hz sinusoidal magnetic fields are generated for calibration along the x- and y-axis at the same time, respectively. The corresponding demodulation output signals of the Rb OPM are collected for 100s and then their power spectral densities are analyzed and averaged for 1 Hz bin by fast Fourier transform to obtain dual-axis voltage noise spectrum, respectively. The frequency response fitting curves are used to further convert the voltage noises to the sensitivities. The blue and red lines represent the x- and y-axis sensitivities during the simultaneous dual-axis measurement, respectively. Their peaks occurred at 30 Hz in the frequency spectrum with 100 pT/Hz^1/2 peak values, which correspond to the calibration frequency 30 Hz and the calibration strength 100pTrms. In the 200 Hz bandwidth, the sensitivities of the Rb OPM with AOM are measured unprecedentedly to be 30 fT/Hz^1/2 as denoted by the black dashed line. Some small peaks are ignored which are originated from the power frequency interferences at 50 Hz, frequency multiplication and extremely low frequency, etc. The sensitivity may further improve in single-axis measurement with higher signal strengths as shown in Fig. 2 and less coupling in theory. We also detected the probe noise floors with and without AOM light modulation by blocking the pump light while other experimental conditions maintain the same. The noise spectrum with AOM shows a lower and flatter probe noise level compared with the regular balanced detection. Meantime, the dual-axis sensitivities exceed the noise floor of the regular balanced detection in some frequency bands and approach that of the AOM detection. In our experiment, the sensitivity without AOM slightly decreases to around 33 fT/Hz^1/2. Therefore, it can be concluded that the present sensitivities of the Rb OPM are improved by taking the AOM light modulation. The performances of the OPM are summarized in Table 1.

Fig. 8. Measured sensitivities of the Rb magnetometer with AOM light modulation and the measured probe noises with and without AOM. The blue and red lines denote the x- and y-axis sensitivities during the simultaneous dual-axis measurement with a 30 Hz 100pTrms sinusoidal calibration, respectively. The sensitivities are 30 fT/Hz^1/2 under the 200 Hz bandwidth as marked with the black dashed line. The grey line denotes the probe noise with AOM detection, while the green line denotes the probe noise with normal balanced detection. The probe noise with AOM is suppressed more efficiently and flatter especially in low frequencies compared with the balanced detection. In some frequency bands the probe noise without AOM is beyond the measured dual-axis sensitivities with AOM, which indicates higher sensitivities of the Rb magnetometer is restricted by the balanced detection without AOM and can be improved by using AOM detection.

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Table 1. Performances of the OPM in two modes

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5. Conclusion

In this study, a high-sensitive Rb OPM for measuring nuclear magnetization has been realized. This configuration realizes a simultaneous dual-axis transverse measurement based on EPR as well as a longitudinal quasi-static field measurement based on NMR in the closed-loop mode. For the sake of a higher sensitivity, we suppress the probe noise with an acousto-optic modulation detection and enhance the signal strengths by the parametrical optimization. In addition, we utilize a closed-loop feedback to enlarge the transverse measurement range and improve the stability effected by the magnetic field fluctuations. On this basis, we evacuate the performances of the Rb OPM with and without the closed-loop control experimentally. The measurement ranges are increased to 10µT order of magnitude without sacrificing the original measurement abilities. What’s more, the measurement ranges still have potential to be further enlarged (e.g. geomagnetic field) with more proper coil constants and voltage sources. Meantime, the probe noise floors with and without the AOM detection are compared and analyzed experimentally. The probe noise with AOM is suppressed significantly especially in low frequencies. Eventually, a simultaneous dual-axis transverse measurement with 30 fT/Hz^1/2 sensitivity is achieved in a 200 Hz bandwidth and a 250nT linear working range. This Rb OPM can serve for sensitive tests of fundamental physics and NMRGs that requires nuclear magnetization with flexible magnetic field adjustment and high sensitivity. It also shows the potential for attractive applications such as the magnetic anonymous detection and deep space detection with a large-scale measurement range.

Funding

National Natural Science Foundation of China (62203028); China Postdoctoral Science Foundation (2022M710321).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

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

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High sensitivity closed-loop Rb optically pumped magnetometer for measuring nuclear magnetization

Abstract

1. Introduction

2. Principles and methods

3. Experimental setup

4. Results and discussion

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (1)

Equations (15)

Optics Express

	Open-loop	Closed-loop
Sensitivity (fT/Hz^1/2)	30	30
Linear Range (nT)	250	250
Measurement Range (nT)	±125	±10,125
Bandwidth (Hz)	200	200