1. Introduction

Ultra-sensitive magnetometers enable the measurements of extremely weak magnetic fields from the human brain (magnetoencephalograms; MEGs) [1–3] and heart (magnetocaridograms; MCGs) [4, 5]. Superconducting quantum interference device (SQUID) sensors are widely used for MEGs, MCGs [6] and other applications [7, 8]. However, SQUIDs requires helium cryogenics, which would hinder low-cost maintenance and compact system design. Atomic magnetometers, therefore, have been considered as alternatives to SQUID sensors, and, indeed, human MCG and MEG measurements via atomic magnetometers have been reported [9–12].

Among various atomic magnetometers developed to date, the type with the most potential for biomagnetism applications is the spin-exchange relaxation free (SERF) magnetometer first introduced by J. C. Allred et al. [13]. It can reach an ultra-high sensitivity of 0.1 fT/Hz^1/2 by completely eliminating relaxations due to spin-exchange collisions [14]. However, its effectively long spin-coherence time limits on the bandwidth of the magnetometer [15–18]. Several methods to expand bandwidth have been reported. Self-oscillating magnetometers possess a bandwidth exceeding 1 kHz as a result of feeding a modulated current back to the light source [19, 20]. With a high density atomic cell, a bandwidth of up to 10 kHz can be obtained by increasing the optical pumping rate [21, 22]. Unfortunately, those systems have mediocre sensitivities as low as 10 fT/Hz^1/2, which may not be capable of detecting weak biomagnetic signals such as the human MEG.

It is well known that negative feedback improves the linearity of an amplifier by exceeding its characteristic bandwidth, and the feedback gain governs the size of the bandwidth extension. Recently, the same principle was applied to a radio frequency (RF) atomic magnetometer, and bandwidths were reported that were broader than a kilohertz and centered at 423 kHz. Therein, the suppression of the spin coherence time under negative feedback was referred to as spin damping. The RF atomic magnetometer with spin damping exhibited a promising sensitivity, 0.3 fT/Hz^1/2 [23]. However, for practical applications in sensing time-varying biomagnetic fields, the center frequency must be close to direct current (DC), which therefore is not compatible with RF atomic magnetometers.

In this work, we applied negative feedback to a SERF atomic magnetometer. Increasing the bandwidth by a factor of nearly three results in a (nearly) flat-response in the frequency domain from DC to 190 Hz. The noise level was measured as 3 fT/Hz^1/2 at 100 Hz, which is comparable to DC-SQUID sensors. Defining the whole system of a SERF atomic magnetometer as an amplifier having a single pole in the frequency response, delivers a simple description of how negative feedback works. It does not require microscopic modeling of the dynamics of atomic systems under interactions with optical fields. We demonstrate that the signal measured under negative feedback shows a higher correlation coefficient and it more in-phase with the test field, which simulates MCG signal of a rat. In order to show the feasibility of our method, we choose an example of a high frequency biomagnetic application, a rat MCG, which gives an exaggerated band-limit-distortion. Once we succeed in obtaining the less distorted waveform from the rat MCG, we could get a more clear and subtle high frequency features in the human MCG and MEG.

2. Experimental setup

The experimental schematic of the atomic magnetometer is shown in Fig. 1. The cell and coil system were placed inside a three-layer set of cylindrical mu-metal chambers, to minimize the effects of the external magnetic field including the Earth’s magnetic field. After degaussing, the residual fields inside the shield were of the order of 0.15 ± 0.05 nT in the directions of x̂ and ẑ and 3±1 nT in ŷ.

 

Fig. 1 Schematic of the setup for a flat-response magnetometer with SERF regime. HWP, half-wave plate; QWP, quarter-wave plate; PD, photodiode; B_t, test field; FG, function generator; V_fg, output of function generator

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The glass cell contained: potassium (K) vapor, the density was measured to be about 1×10¹³ cm⁻³; 600 Torr of He buffer gas, to reduce the rate that atoms in the cell diffuse towards its wall; and 15 Torr of N₂, to improve optical pumping by quenching. The cell consisted of two parts, a main cell and a stem cell. The main cell was borosilicate glass which was connected to the stem cell, acting as a reservoir. The stem was made of aluminosilicate glass that prevents K vapor being absorbed by the surface of the cell. During the experiment, two independent heaters are used to avoid deposition of the vapor. The main cell was heated to 200±0.5 °C and the stem cell was heated to 185±0.5 °C. The temperature difference between main cell and stem cell was maintained automatically. A resistive heater was insulated by insulation panels.

A distributed feedback (DFB) laser was used in the experiment. The optical pumping was accomplished by a circularly polarized laser after passing through the polarization maintaining fiber (single mode TEM₀₀), and exactly tuned to the center of the K D₁ line. The pump beam power was amplified up to 1 W by a tapered amplifier and the beam diameter was expanded to 5 mm by passing through two lenses. The linearly-polarized probe beam propagating in x̂ was detuned from the K D₁ line by several nanometers to minimize the probe absorption. The probe beam was generated by a single mode DFB laser and monitored by a Fabry-Perot interferometer and spectrometer. After passing the half wave plate and the K vapor cell, the laser beam entered the balanced polarimeter that included an analyzer and photodiode. An analyzer divides the laser beam into two paths with orthogonal polarization and then measures the difference between the signals on the photodiodes.

The coil system was composed of 3-axial Helmholtz coils (two pairs in ŷ and one pair in each of x̂ and ẑ) and a circular coil in ŷ. The 3-axial Helmholtz coils, wound around a circular frame with symmetry axes orthogonal to each other, were used to eliminate the residual fields in the magnetic shield. The circular coil generated a test field, B_t, contained oscillating signals with multiple frequencies. The second set of Helmholtz coil in the ŷ direction was connected to the output of the balanced polarimeter, to produce the negative feedback field, B_fb, which antiparallel to B_t

3. Experimental results

The resonance curve of a dispersion type in Fig. 2(a) shows the polarization rotation of a probe beam with respect to the transverse magnetic field B_y. The sensitivity of an atomic magnetometer is proportional to the slope of the dispersive signal near zero field. To retain the best sensitivity, the peak-to-peak amplitude and spectral width of the polarization rotation signal were observed during the experiment by adjusting the tunable parameters of the experiment to maximize the dispersion curve slope. In our conditions, the laser intensities of pump and probe beam were estimated to be 33 mW/cm² and 5 mW/cm², respectively. The measured rotation curve agrees well to the expression given by Ledbetter et al. [24], as indicated by the red line in Fig. 2(a), from which the slope on resonance, amplitude, and width can be determined.

 

Fig. 2 (a) Typical optical rotation signal as a function of B_y in the SERF regime. The red solid line represents the theoretical results given by the steady state solution of the Bloch equation. The inset shows the optical rotation in a near zero field and linear fit. (b) Signal response of the amplifier in Fig. 1 as a function of frequency for four different amplifier gains.

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The principle of the negative feedback amplifier can be applied to atomic magnetometers, in order to improve linearity of output, namely to extend the characteristic bandwidth. This is based on the fact that the system of an atomic magnetometer can be regarded as a single-pole amplifier, in which the input is external field and the output is the signal voltage. The unit of the open-loop gain, G, is V/nT. Like other systems possessing resonances, SERF atomic magnetometers exhibit a Lorentzian-type response in the frequency domain, which has a center frequency of nearly zero and a cut-off frequency, f_C. Therefore, a phenomenological approach allows G to inherit a Lorentzian form, as $G=\frac{G_0}{1+jf/f_C}$, where G₀ represents the DC gain. Under negative feedback, the gain G_fb is expressed in terms of G and the feedback parameter, β in Eq. (1), as below,

To verify the validity of the Eq. (2) in a SERF atomic magnetometer, the open-loop and feedback gains at DC, G₀ and β, must be measured beforehand. The insert in Fig. 2(a) shows a narrow scanning range near resonance. In DC B_y fields, the slope of the signal determines G₀, and we measured it to be 0.6 V/nT. In order to estimate the feedback parameter, β, we applied a sinusoidally oscillating electric signal of 350 mV_pp to the input of the amplifier connected to the feedback coil. The magnetic field generated by the feedback coil was detected by the atomic magnetometer with a frequency range from 15 Hz to 1 kHz. Then the measured V_out was converted to the magnetic field using G₀, which forms the vertical axis in Fig. 2(b). The power bandwidth of the feedback amplifier in our system has a limited range from 10 Hz to 40 kHz, therefore, the data fitted to the Lorentzian function was centered on 0 Hz to obtain the output voltage at zero frequency. Consequently, the amplifier gains −25, −15 and −8 dB were converted to 0.5, 2, and 3.5 nT/V, respectively.

To investigate the frequency response of the atomic magnetometer under negative feedback, closed-loop signal versus frequency data were measured for the β values 0, 0.5, 2, and 3.5 nT/V (Fig. 3). Fitting the curves to Lorentzian forms, allowed estimates of G_fb(0) and f_C to be made, which are shown in Fig. 3. Without the negative feedback, the data for the frequencies lower than 50 Hz cannot be measured due to a photodiode output saturation. When β = 3.5 nT/V, the curve of the frequency response became nearly flat up to 195 Hz, which is the half width a half maximum (HWHM) of the curve. According to Eq. (2), f_C should be multiplied by the factor (1 + βG₀), and G_fb(0) divided by the same factor. The calculated f_C values for the three β values, 0.5, 2, and 3.5 nT/V, are given by 93.6, 158 and 223 Hz, respectively. Although the experimentally obtained bandwidths f_C are less than these values, overall, considering an atomic magnetometer as a single-pole amplifier, provides a reasonable phenomenological description of the whole system.

 

Fig. 3 Signal response to the input oscillating fields with amplitude of 600 pT for multiple frequencies. The incident beam intensities of the pump and probe beams were estimated to be 33 mW/cm² and 5 mW/cm², respectively. The dashed line is a fit to Lorentzians and f_C indicates the half width at half maximum (HWHM) of the Lorentzian. It is possible to regard HWHM as the cut-off frequency of the single pole amplifier. On the black line, open-loop, f_C = 72 Hz, and increased to 195 Hz on the blue line, closed-loop.

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The performance of the feedback system can be evaluated by the correlation between measured signal and a test field that simulates a rat’s MCG. The test field had a maximum peak-to-peak intensity of 500 pT and a period of 10 Hz, as shown in inset of Fig. 4(a). In ref. [5], the measured MCGs of a rat shows a peak-to-peak magnetic field strength of 50 pT. The rat was placed on the stage in the supine position with a hung down heart to the back, and the total distance between the heart surface and the SQUID sensor was about 14 mm. With an atomic magnetometer, we suppose that the distance between the heart and the beam can be reduced to 4.5 mm by using a thin thermal insulation and by placing the rat in the prone position. For these reasons, we applied a test field with the peak-to-peak amplitude of 500 pT. Its FFT spectrum shows that the MCG signal consists of multiple frequency components in the range from zero to over 200 Hz. A circular coil in the ŷ axis, with a diameter of 5 mm located 25 mm above the vapor cell, produced the test field. Figure 4(b) shows output signals in the time domain for several different values of β. A higher feedback gain, β, resulted in a higher correlation co-efficient, corresponding to less distortion. When β =3.5 nT/V, the bandwidth of 195 Hz covers approximately 70 % of the total spectrum in Fig. 4(a). The obtained correlation of 0.75 can plausibly be explained by the consequence of the extended bandwidth. The amplitude of the measured signal was calibrated to the scale of the magnetic field. As β increased, not only were the distortions from the low frequency components suppressed, but also the phase differences were reduced, because of the signal lagging beyond the HWHM point, as shown in Fig 4(c).

 

Fig. 4 The test field, a MCG-like signal with a peak-to-peak amplitude of 500 pT and 0.1 s period between R peaks, was measured with several values of β. (a) The test field is composed of multiple frequency component. (b) When β is zero, open-loop, the low frequency response causes the distortion, and a higher frequency than the HWHM point causes the signal delay. As β continues to increase up to 3.5 nT/V, the distortion is reduced (c) signals become into-phase.

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We calculated the linear correlation between the measured signals and the test field, to evaluate the degree of distortions in the measured signals. The calculated correlation coefficients for varying values of β is listed in Fig. 4(b). When β =3.5 nT/V the correlation coefficient was 0.76, which approaches the ideal case of an output without distortions.

The system’s noise level is also an important factor, determining the practical upper limit of the extension factor, (1 + βG₀). If this is exceeded, the measured signal will be buried in the noise. To measure the noise level of the magnetometer, we use calibration peaks of a small oscillating field by using the Helmholtz coil in Fig. 1. The total noise can be considered with an environmental magnetic noise, light shift noise, spin-projection noise and photon shot-noise. Obviously, the environmental magnetic noise and light shift noise can be reduced by the negative feedback scheme. The SNR changes from the other noises, spin-projection noise and photon-shot noise, are insignificant in our system due to the small extension factor about three and the sufficient optical power about 5 mW/cm², respectively [23]. Figure 5 shows the noise spectral densities of our system with and without the negative feedback, respectively. The noise level in a low frequency range under a negative feedback gain of 3.5 nT/V has been clearly suppressed compared to the noise level without the negative feedback. In the range of zero to 190 Hz, we are able to suppress the noise level by ten times. Therefore, the negative feedback scheme can widen the detection bandwidth without sacrificing SNR at the low frequency range. The feedback gain could not be increased higher than 3.5 nT/V because the signal-to-noise ratio became obviously worse. We believe further study, investigating the optimal feedback gain for a given system noise, will be necessary to maximize the potential of negative feedback in a SERF atomic magnetometer.

 

Fig. 5 The noise spectrum of the atomic magnetometer with (orange line) and without (blue line) negative controlled feedback.

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4. Conclusions

We demonstrated that, analogous to feedback amplifiers, the bandwidth of a SERF atomic magnetometer can be increased by negative feedback. Increased bandwidth reduces distortions in the output signals, which was verified by comparison with a rat’s MCG test field. The linear correlation coefficient was enhanced by a factor of more than three as the feedback gain was increased. Furthermore, the sensitivity remained at 3 fT/Hz^1/2 at 100 Hz. This result leads to the conclusion that a SERF atomic magnetometer with negative feedback is practical for sensing biomagnetic signals containing multiple frequency components, for instance, in pharmaceutical developments that require the Q-T interval of rat MCGs to test drug efficiencies. Consequently, further research would accelerate the development of a compact, portable and cryogenic-free system based on atomic magnetometers.