The evolution dynamics of electron spin polarization vector P of alkali atomic ensemble in our zero-field OPM can be well-described by the Bloch equation [28]: d P d t = 1 q ( P ) [ γ e B × P + R o p ( s z ^ − P ) − R r e l P ] (1) where q(P) is the nuclear slowing-down factor which is a function of the magnitude of P [29], γ e ≈ 2 π × 28   H z / n T is the gyromagnetic ratio of electron, B is the magnetic field vector, [ B x , B y , B z ] T , R _op is the optical pumping rate along z-axis, R _rel is the spin-relaxation rate, and s is the polarization degree of pump light.

Single-beam zero-field OPMs perform magnetic field measurements by detecting the transmitted light intensity of pump beam, which only reflects the longitudinal polarization P _z [17, 28]. According to Ref. [17], the approximated relationship between the output of the magnetometer R _out and P _z is R o u t = R 0 ∙ exp [ − O D ( ν ) ( 1 − P z ) ] (2) where R ₀ is the original signal before the vapor cell, and O D ( ν ) = n L σ ( ν ) is the optical depth of the vapor cell. And n is the number density of the alkali vapor, σ(ν) is the absorption cross-section as a function of light frequency ν, and L is the length of the vapor cell. Thus, we focus on the response of P _z to magnetic field. The P _z component of the stable-state solution to (1) is P z s t a b l e = P 0 B z 2 + [ ( R o p + R r e l ) / γ e ] 2 B x 2 + B y 2 + B z 2 + [ ( R o p + R r e l ) / γ e ] 2 (3)

For low-frequency system noise suppression and magnetic field sensitivity improvement, single-beam zero-field OPMs are usually modulated with high-frequency magnetic field. Conventionally, only one transverse axis is modulated, in which way only magnetic field measurement of the modulated axis is achieved [10, 30]. For dual-axis measurement, using the transverse rotationally modulated magnetic field results in the output signal being orthogonally demodulated to obtain the in-phase and out-of-phase terms, realizing the simultaneous measurement of both transverse directions [26, 31, 32]. According to Ref. [26], the response of P _z with dual-axis measurement containing both information about B _x0 and B _y0 is P z d u a l ∝ R o p J 0 3 ( u ) J 1 ( u ) [ B x 0 ⁡ sin ( ω 1 t ) + B x 0 ⁡ cos ( ω 1 t ) ] Γ 1 2 + J 0 2 ( u ) B x 0 2 + J 0 2 ( u ) B y 0 2 + J 0 4 ( u ) B z 0 4 (4) where Γ 1 = ( R o p J 0 2 ( u ) + R r e l ) / γ e represents the zero-field resonance linewidth, and J n ( u ) represents the nth order Bessel function of the first kind. Variable u = γ e B x y m / [ q ( P ) ω 1 ] is the modulation index. B_xym is the modulated magnetic field amplitude along the transverse directions. Emitting the quadratic terms in the denominator of (4), the P z d u a l response to B _x0 and B _y0 are orthogonal in phase, which means that orthogonal demodulation can offer dual-axis measurement from P z d u a l .

For measurement scheme mentioned above, optical pumping aligns the atomic spin polarization along the z-axis, resulting in the insensitivity of spin polarization precession to a small magnetic field along the same z-axis. Hence, we made an attempt to apply another modulated magnetic field with different frequency along the longitudinal direction.

In summary, our combined modulated magnetic field are expressed as B x = B x y m ⁡ sin ( ω 1 t ) + B x 0 B y = B x y m ⁡ cos ( ω 1 t ) + B y 0 (5) B z = B z m ⁡ sin ( ω 2 t ) + B z 0 where B _xym and B _zm are the amplitudes of the transverse rotational and longitudinal modulated magnetic field, respectively, and B _×0, B _y0, and B _z0 are the magnetic fields to be detected along the three orthogonal directions. The combined modulated magnetic field results in an amalgamated precession of atomic spin polarization, which can be sensitive to magnetic field along all the three orthogonal directions. From Equation 3, it could be found that P _z has naturally contained information of B _z0. Through employing the modulated magnetic field along the z-axis, the scale factor of the magnetometer to B _z0 is enhanced, and its sensitivity to B _z0 presents. This conclusion is drawn mostly from numerical simulation, for that analytical solution to Bloch Equation 1 under combined modulated magnetic field (5) is difficult to obtain. Specifically, the mathematical operation of cross product B × P in (1) can be transformed into matrix and vector multiplication: B × P = [ B x B y B z ] × [ P x P y P z ] = [ 0 − B z B y B z 0 − B x − B y B x 0 ] ∙ [ P x P y P z ] = B t ∙ P (6)

When the third modulated magnetic field is applied, the time-varying cross terms with trigonometric functions of B _t complicate the analytical solution to (1).

Besides, it is worth noticing that with combined modulated magnetic field for triaxial measurement, the response of P _z to transverse magnetic field did not change substantially due to different modulation frequency. We rely on response characteristics at transverse modulation frequency (Figure 1A) through demodulating with lock-in amplifiers, which are not different from that in dual-axis measurement scheme. Therefore, Equation 4 can still be used in our triaxial measurement scheme.

We performed numerical simulation based on 1) and 5) to study the response characteristics of P _z . It is crucial to investigate three independent demodulated observables which only depend on the changes of B _x0, B _y0, or B _z0 from the response signal, respectively. We distinguish the observables by frequency and phase, and in particular, the response of frequency ω ₁ and ω ₂, i.e., P _zω1 (R _zω1) and P _zω2 (R _zω2), are the quantities of interest.

The numerical simulation is executed with typical parameters of R o p = R r e l = 150   s − 1 , B x y m = B z m = 140   n T , ω 1 = 2 π × 1000   H z , ω 2 = 2 π × 1500   H z ,. The simulated 3D trajectory of P subjected to the combined modulated magnetic fields is depicted in Figure 1A, showing an apparent time-domain variation of P _z . This variation could finally be detected to sense the triaxial magnetic fields, on condition that it needs to be transformed into the frequency domain. Therefore, we made a preliminary exploration to observe the dependence of P _zω1 and P _zω2 on B _x0 and B _z0 (B _x0 as an example of transverse directions). As shown in Figure 1B, P _zω1 responses markedly to a change of 0.1 nT in B _x0, and by contrast it hardly responses to a change of B _z0. Moreover, under the condition of zero magnetic field, P _zω2 is not zero but has a remarkable peak value. And it is only sensitive to the variation of B _z0, which is detailed in the inset.

Based on the above investigation, a systematic analysis within the typical dynamic range of zero-field OPMs (especially SERF magnetometers), [-20, 20] nT, was carried out and presented in Figure 2. Figure 2A shows that the in-phase signal of P _zω1 are linearly dependent on B _x0 near B _x0 = 0, and Figure 2B indicates that the out-of-phase signal of P _zω1 linearly response to B _y0 near B _y0 = 0. And these two signals are zero when magnetic field along their corresponding direction is zero, respectively. These features are the same with dual-axis measurement scheme. Besides, in Figure 2A and Figure 2B, the amplitude of P _zω2 presents stable response near B _x0 = 0 or B _y0 = 0. Comparatively, Figure 2C shows that P _zω1 hardly responds to the variation of B _z0, while the amplitude of P _zω2, exhibits an observable approximately linear dependence on B _z0, although P _zω2 has a large offset when B _z0 = 0. Note that the steepness of the distinct response to B _z0 over the offset is about an order of magnitude smaller than that of the in-phase signal of P _zω1 replying to B _x0 (or the out-of-phase signal of P _zω1 replying to B _y0), which implies that the sensitivity of the transverse directions is better than the longitudinal one. Therefore, we can conclude that B _x0, B _y0, and B _z0 can be independently and simultaneously measured in the vicinity of zero magnetic field just by demodulating the in-phase signal of P _zω1, the out-of-phase signal of P _zω1 and the amplitude of P _zω2 respectively from the response signal of P _z . This analysis through numerical simulation has verified our aforementioned investigation.

For zero-field OPMs, the frequency response of the input-output relationship could be described as [1, 9]. R ( f ) = A ( 2 π f ) 2 + B W 2 (7) where A is a constant related to the scale factor of the magnetometer, BW represents the bandwidth of the magnetometer. The transfer function corresponding to (7) is G o p e n ( s ) = A s + B W (8)

The transfer function of PI controllers is G c ( s ) = K p + K i s (9) where K _p and K _i denote the proportional and integral gains, respectively.

When the atomic magnetometer works in the closed-loop mode, the triaxial magnetic information is extracted and fed into three separate proportional and integral controllers (PI controllers). The outputs of three controllers are converted into feedback current and applied to the triaxial coils to keep magnetic field along each direction locked at zero. At the moment, the magnetic field generated by the coil is the magnetic field to be measured. Each single axis closed loop is operated independently, as depicted in Figure 3. The transfer function of the closed-loop magnetometer is G c l o s e d ( s ) = G o p e n ( s ) 1 + G c ( s ) ∙ K c o i l ∙ G o p e n ( s ) = A s s 2 + ( K p K c o i l A + B W ) s + K i K c o i l A (10)

Comparing (8) with (10), it could be found that the system characteristics of the closed-loop magnetometer are different from that of the open-loop magnetometer. For instances, the closed-loop magnetometer is regarded as a second-order system. By comparison, the open-loop magnetometer can be simply considered as a first-order system. Through adjusting the controller parameters, the frequency responses of the closed-loop magnetometer can be altered to enlarge the bandwidth.

2The experimental system of our single-beam closed-loop triaxial zero-field OPM is depicted in Figure 4. The cubic vapor cell, made of borosilicate glass, has an internal size of 8 mm × 8 mm × 8 mm, which is filled with a droplet of ⁸⁷Rb and about 600 torr N₂. The vapor cell is heated to 433 K for working using 200 kHz alternating current generated by an electric heater. The laser beam generated by a distributed-feedback (DFB) laser is first transmitted to the zero-field OPM via a polarization-maintaining fiber, and then is transformed into circularly polarized light by a quarter-wave plate before illuminating the vapor cell. It plays the dual roles as both optical pumping and probing, whose wavelength is set to about 794.98 nm, near the center of the D1 line of ⁸⁷Rb. The transmitted light through the vapor cell is sensed and converted into current signal by the photodiode and finally transformed to voltage signal by the trans-impedance amplifier (PDA200C, Thorlabs). Then, the two channels of one lock-in amplifier (MFLI, Zurich Instruments) are used as quadrature demodulator for the in-phase and the out-of-phase signal of R _zω1. Another lock-in amplifier (MFLI, Zurich Instruments) is used to demodulate the amplitude of R _zω2. Afterward, all the three different signals are sent into built-in PI controllers of LIAs to generate negative feedback signals. The feedback signals are utilized to control the waveform generators (33522B, Keysight) which drive the coil via selected resistors, keeping the triaxial magnetic field sensed by the zero-field OPM locked at zero. Finally, all signals are acquired by the data-acquisition system (PXIe-4464, National Instruments).

The near-zero magnetic field environment of the atomic magnetometer is created by four-layer μ-metal magnetic shield. Then, inside the shield a group of triaxial coils is used for active magnetic field compensation and generation of modulated magnetic fields. The coil group is composed of a nested saddle coil [33] for radial magnetic field and a Lee-Whiting coil [34] for axial magnetic field.

Our experiments are conducted by two steps. At the first step, the triaxial measurement is conducted, including the magnetic field sweeps along the three axes as a verification of the simulation results. And we offer the triaxial sensitivity test result. Then, negative feedbacks are adopted to build a closed-loop magnetometer. Plus, comparison is made between sensitivity, bandwidth, and dynamic range of open-loop mode and those of closed-loop mode.

To confirm the validity of triaxial measurement with the proposed method, we sweep the DC magnetic fields B _×0, B _y0 and B _z0 around the zero magnetic field respectively, and simultaneously record the variation of the in-phase and the out-of-phase signal of R _zω1, and the amplitude of R _zω2. The sweep range of B _x0, B _y0 and B _z0are all set to [-20, 20] nT.

As shown in Figures 5A–C, the experimental results are in good agreement with the simulation results. By observing and comparing the dependence of the three observables on B _x0, B _y0 and B _z0, respectively, major conclusions can be drawn below. Within the range of [-20, 20] nT, the in-phase and the out-of-phase signal of R _zω1 are orthogonally dependent on B _x0 and B _y0, whose dependence features show a dispersion curve. And the relationship between the amplitude of R _zω2 and B _x0 (B _y0) presents an absorption curve. Specifically, in the vicinity of zero field ([-2, 2] nT), as shown in Figure 5D inset, the in-phase signal of R _zω1 is linearly dependent on B _x0, while the out-of-phase signal of R _zω1 and the amplitude of R _zω2 are not sensitive to B _x0. Similar features reproduce in Figure 5E inset. As shown in Figure 5C, only the amplitude of R _zω2 shows a linear dependence on B _z0 within the range of [-20, 20] nT, while the signals of R _zω1 do not change substantially when B _z0 varies. These characteristics are more pronounced in the vicinity of zero field ([-2, 2] nT), which is shown in Figure 5F inset. The interference between the transverse and longitudinal signals is probably caused by non-orthogonal angles of the beams and triaxial coils [35].

Moreover, triaxial scale factors, which is defined as k=ΔR/ΔB, are also evaluated. ΔR is the magnetometer response in voltage when sensing magnetic field variation ΔB. The results show that x and y directions have scale factors of 0.128 V/nT and 0.136 V/nT, respectively. In contrast, the scale factor of longitudinal direction is -0.0148 V/nT, whose absolute value is about one-ninth of that of transverse directions. This scale factor difference among transverse directions and the longitudinal direction is basically consistent with that in numerical simulation mentioned before in Section 2.1. The response of R _zω2 to magnetic field along the longitudinal direction is over its offset, and the response is about one order smaller than that of R _zω1 to magnetic field along transverse directions.

The triaxial magnetic field sensitivities are analyzed by acquiring the noise spectra of the output signals of magnetometer. A 100 pT_rms magnetic calibration signal at 30.5 Hz is employed along the three axes in turn. At each turn, the voltage output signal was firstly acquired and collected for 60s. We conduct noise spectra analysis towards the voltage output signal. And then the voltage noise spectra are divided by the amplitude frequency responses (Figure 7) to obtain the sensitivities of our magnetometer.

The triaxial sensitivities in our experiments is given in Figure 6A, in which the maximum peak in each noise spectrum represents the calibration signal. The triaxial magnetic field sensitivities are 18 fT/Hz^1/2 along the x-axis, 18 fT/Hz^1/2 along the y-axis, and 140 fT/Hz^1/2 along the z-axis. The transverse sensitivities are excellent, in line with expectations. And we achieve measurement of magnetic field along the longitudinal direction without modifying conventional single-beam configuration. The fact that the sensitivity of longitudinal direction is inferior to that of transverse directions is mainly due to small response amplitude over the large offset of R _zω2 when B _z0 varies around zero-field, while the response amplitudes of R _zω1 to magnetic field along transverse directions are strong, which have zero offset at zero field. A qualitative description is given below: the spin polarization vector P is basically along the z-axis, and the angle between B _z0 and P is small, so the magnetometer signal representing precession variation of P is small as well. By comparison, B _x0 ( B _y0 ) and P approaches perpendicular, so the precession variation of P is distinct leading to the strong magnetometer signal.

When the triaxial measurement is completed, as depicted in Figure 3, we adopt the closed-loop operation based on PI controllers to enlarge the bandwidth of the magnetometer and the dynamic range of the magnetometer, and to keep the triaxial magnetic field sensed by the magnetometer locked at zero. The closed loops of three axes are operating independently of each other, with three separate PI controller.

We firstly alter the open-loop magnetic modulation parameters for extending bandwidth, as a basis for bandwidth expansion in closed-loop mode. With operating parameters same as above simulation conditions, the bandwidths of triaxial magnetic detection are 92 Hz, 92 Hz, and 100 Hz, as Figure 7 shows. According to our previous research [14, 25, 26], high-frequency and large amplitude modulated magnetic field would bring about high bandwidth of the zero-field OPM. Thus, we set transverse rotationally modulated magnetic field at 6 kHz, with 2000 nT_pp amplitude, and longitudinal modulated magnetic field at 9 kHz, with the same 2000 nT_pp amplitude. The ratio of frequencies in this group is the same as that in simulation parameters for better performance. In this condition, the bandwidths of open-loop triaxial magnetic detection are 809 Hz, 757 Hz, and 270 Hz.

Finally, the closed-loop operation is conducted with optimized parameters as in Figure 3 to further expand the triaxial detection bandwidths to 1.86 kHz, 1.90 kHz, and 1.61 kHz. The feedback bandwidth is 2.5 kHz, which is larger than the current triaxial bandwidths of the closed-loop magnetometer, and would not affects the bandwidths.

The response curves in Figure 8 indicate that the triaxial dynamic ranges of the open-loop magnetometer with operating parameters same as simulation ones are restricted to below 20 nT for the transverse directions, and less than 40 nT for the longitudinal direction. Operating in closed-loop mode, the triaxial dynamic ranges of the magnetometer are enhanced to ±150 nT. Furthermore, closed-loop operation with increased triaxial coil constants enlarges the triaxial dynamic ranges of the magnetometer to over ±1,000 nT.

We achieve the high-bandwidth and large dynamic range of the magnetometer, and meanwhile keep its ultra-high triaxial sensitivities, which are presented below in Figure 6B. The triaxial sensitivities are 22 fT/Hz^1/2 along x-axis, 22 fT/Hz^1/2 along y-axis, and 230 fT/Hz^1/2 along z-axis. The sensitivities degrade performance may result from two reasons: the first reason is the large relaxation rate caused by large modulation amplitude, and the second reason is stronger system noise from closed-loop controller and actuators.

In addition, to evaluate the stability of the triaxial closed loop, we lock the triaxial magnetic field sensed by the magnetometer at zero. By contrast, working in open loop, the magnetometer signal may drift along with either the drift of environmental magnetic field or the drift of magnetometer itself (including the drift of light, cell temperature, etc.). We conduct experiment to record the variation of the triaxial sensed magnetic field, and make comparison between the magnetic field variation record with magnetometer operating in open loop and that with magnetometer operating in closed loop. Records in Figure 9 indicate that in 30 min triaxial magnetic field variations with closed loop are less than 1% of that with open loop. This result has confirmed that with triaxial closed loop, we could keep the magnetometer operating in zero field effectively.