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Abstract
In the spin-exchange relaxation-free (SERF) magnetometer of a perpendicular pump-probe configuration, the pump and probe beam characteristics significantly affect the performance. In this paper, an efficient evaluation of optical parameters to improve the sensitivity of a miniature magnetometer has been presented. We have determined the pump light’s optimal intensity and wavelength through theoretical analysis and the zero-field resonance experiments. Chirp signals are applied to measure the optical rotations at different probe intensities and frequencies. Through theoretical and experimental analysis of noise source characterization under different beam intensities and wavelengths, we demonstrate that dual-beam magnetometer performance is mainly limited by photon shot noise. Based on the optimum pump and probe beam parameters, we demonstrate magnetic field sensitivity of 6.3 ${\textrm {fT/}}\sqrt {\textrm {Hz}}$ in an 87Rb vapor cell filled with nitrogen gas, with an active measurement volume of 3 × 3 × 3 mm3.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The dominant technology for high-sensitive magnetic field measurements have been liquid-helium cooled superconducting quantum interference device (SQUID) magnetometer for nearly half a century due to their exquisite advantages, i.e., the high sensitivity of approximate 150 ${\textrm {aT/}}\sqrt {\textrm {Hz}}$ in the white noise regime between about 20 ${\textrm {kHz}}$ and a large bandwidth over 2.5 ${\textrm {MHz}}$ [1]. These sensitive magnetometers have already been used in biomagnetic signal detection, such as magnetoencephalography (MEG) [2] and magnetocardiography (MCG) [3]. Although the SQUID technology has been fully developed and SQUID-based magnetometers have been commercially available, the prerequisite cryogenics and the continual supply of expensive liquid helium limit their reconfiguration and increase the operating cost. Over the last 20 years, atomic magnetometers based on the interaction of resonant magneto-optics with atomic vapor show the capacity of high sensitivity in magnetic field measurements. Especially the descendant, spin-exchange relaxation-free (SERF) magnetometers have advantages in the detection of the weak magnetic field with the level of subfemtotesla [4,5], which eliminates the relaxation due to spin-exchange collision and achieves superior sensitivity performance [6,7]. Except for precisely fundamental measurements [8,9], the sensitivity and bandwidth of SERF magnetometer are sufficiently high to observe MCG [10–12] and MEG [13,14] signals. In recent years, a commercial 15 ${\textrm {fT/}}\sqrt {\textrm {Hz}}$ in 3-100 ${\textrm {Hz}}$ band QuSpin Zero-Field Magnetometer (QZFM) was released, which is compact, low cost, and easy to operate [15]. Combined with the inherent advantages of being free from cryogenic cooling and allowing flexible positioning of the sensor, tens to hundreds of microfabricated SERF magnetometers can be easily placed around a subject to create a spatial map of the magnetic field at a low cost [16,17].
The application above requires the magnetometer to be miniaturized enough to obtain sufficient spatial resolution and ultra sensitivity to detect the weak magnetic signals. Considering spatial resolution consideration, we make a microfabricated vapor cell with an inner volume of $3 \times 3 \times 3\;{\textrm {mm}^3}$. But in early articles, the sensitivity of magnetometers is below 8 ${\textrm {fT/}}\sqrt {\textrm {Hz}}$ [15,18] with the same measurement volume, which is much larger than the theoretical limited sensitivity, about 1 ${\textrm {fT/}}\sqrt {\textrm {Hz}}$ [19]. So it is worthwhile to analyze the factor restraining the sensitivity of magnetometers. With respect to improve sensitivity, there are two approaches to be considered. One is narrowing the magnetic linewidth and increasing transverse relaxation time $T_{2}$. In our previous work [20], we studied the single-beam SERF magnetometer with $^{87}$Rb vapor cells under different nitrogen gas pressures, and achieved a high magnetic sensitivity of 8.8 ${\textrm {fT/}}\sqrt {\textrm {Hz}}$. The other is improving the noise-to-signal ratio, maximizing magnetometer response, and suppressing the system noise. A differential method to measure the optical rotation has been employed to reduce the noise from the fluctuation of probe beam. Furthermore, the performance of magnetometer is affected by additional types of noise including electronic and magnetic noise. The pioneering work by Romalis $et al.$ [19,21] analyzed the quantum noise, derived the noise limit, and presented a theoretical optimization of the magnetometer. Krzyzewski $et al.$ [22] proposed the test methods of various types of noise and experimentally evaluated noise arising from the laser intensity, laser frequency, and other sources of single-beam SERF magnetometer, but they did not give the analysis and test conclusions of the optimum intensity. Xing $et al.$ [23] presented a probe noise characteristic model demonstrating the relationship between the intensity and the signal frequency but they did not analyze the combined effect caused by the probe wavelength and intensity. Moreover, the photo-elastic modulator modulated system technique is unsuitable for the microfabricated structure in this paper.
In this paper, an efficient approach to improve the sensitivity of a miniature magnetometer has been presented. We analyze and measure the different noise sources of a microfabricated vapor cell with the power spectral density and propose a method combining multiple noise sources using the noise floor to optimize the probe beam characteristics. The different noise sources are characterized by converting noise spectrum to magnetic field units using a magnetometer signal response curve. We modify the shot noise and electronic noise equation to analyze the combined effect caused by the probe wavelength and intensity. We demonstrate the optimal pump and probe beam characterization based on the theoretical and experimental results. This method is helpful in noise analysis and could promote the sensitivity of the SERF magnetometer.
2. Theory analysis
The Bloch equation is the theoretical basis of the spin dynamics of the dual-beam SERF magnetometer, describing the fundamental physical processes of optical pumping, spin relaxation, and spin precession [7,19].
The pumping rate for pump and probe beam can be expressed above [19], where $I$ is the intensity of the laser beam, ${{r_e}}$ is the classical electron radius, $f$ is the oscillator strength, ${\Delta v}$ is the frequency detuning from the corresponding transition. ${{\Gamma _{\textrm {D}}}}$ is the Full widths at half maximum of Lorentzian curve due to pressure broadening and natural lifetime. For convenience, we define ${R_{\textrm {PR}}} = \eta {I_{\textrm {probe}}}$ where the $\eta$ is the optical pumping factor which is influenced by the detuning.
We solve the Eq. (1) in steady-state. After the magnetic field in the shield is nulled and only applied small magnetic field of $B_{y}$ in $y$ axis, we can simplify the solution as:
Linearly polarized probe light on ${\textrm {D}_2}$ off-resonance frequency along the $x$ axis is used to detect the optical rotation $\theta$, given by the Eq. (4). After the cell the probe beam passes through a polarizing beam splitter setting at ${45^ \circ }$ to the initial polarization.
The performance of magnetometer is mainly limited by quantum noise, electronic noise, and magnetic noise. With a magnetic field $B$ to be measured, we evaluate those noise sources in magnetic field units for simplification as in Eq. (6).
The fundamental quantum noise mainly contain spin projection noise and photon shot noise which can be expressed as $\delta {B_{{\textrm {spn}}}} = \sqrt {2{\Gamma ^3}} /({R_{{\textrm {OP}}}}{\gamma ^{\textrm {e}}}\sqrt {nV} )$ and $\delta {B_{{\textrm {psn}}}} = {k_{{\textrm {psn}}}}/(G\sqrt {{I_{{\textrm {PR}}}}{e^{ - {\textrm {OD}}}}} )$ respectively [19,21,24], $k_{{\textrm {psn}}}$ is the ratio of the shot noise unrelated with the probe intensity. The electronic noise is mainly from the instruments as PD, PDA and DAQs, according to Eq. (5), the equivalent magnetic noise can rearranged as $\delta {B_{\textrm {e}}} = {k_{\textrm {e}}}/({I_{{\textrm {PR}}}}{e^{ - {\textrm {OD}}}}G)$. Magnetic noise ${\delta {B_{\textrm {m}}}}$ is mainly from the magnetic shields and nulling coils. Combining the equations above, we can calculate the optimum optical parameters to ensure that the magnetometer achieves the highest sensitivity as the Fig. 1 shows. The simulation sensitivity is analyzed by the intensity and detuning of the probe beam, which could be calculated from Eq. (6) for probe beam center wavelength $\lambda$ = 780.255 ${\textrm {nm}}$, relaxation rate of the cell ${R_{\textrm {rel}}}$ = 100 Hz, full width at half maximum (FWHM) of pressure broadening $\Gamma _{{\textrm {D}}}$ = 24.5 GHz, and the rubidium vapor density $n$ = $1 \times {10^{14}}\;{{\textrm {cm}}^{ - 3}}$.
Fig. 1. The simulation sensitivity as a function of probe intensity and detuning are calculated from Eq. (6) for ${R_{\textrm {rel}}}$ = 100 Hz, $\Gamma _{{\textrm {D}}}$ = 24.5 GHz, rubidium vapor density $n$ = $1 \times {10^{14}}\;{{\textrm {cm}}^{ - 3}}$.
3. Experimental setup
A schematic of the experimental setup is shown in Fig. 2, which is similar to the one used in our earlier studies [20]. Selecting borosilicate with the advantages of a low melted point and low cost as cell glass material, we made a microminiature alkali cell with an inner volume $3 \times 3 \times 3 {\textrm {m}}{{\textrm {m}}^3}$, containing a droplet of $^{87}\textrm {Rb}$ metal. In order to meet the needs of long-term testing, this cell is filled with N$_2$ buffer gas with a pressure of 1.4 amg. FWHM $\Gamma _{\textrm {D}}$ of this cell is 24.5 GHz, the center wavelength of ${{\textrm {D}}_1}$ line is 794.9984 nm, and the center wavelength of ${\textrm {D}_2}$ line is 780.2561 nm. Optical depth is 6.38 for pump beam which is measured by spectral profiles at 423 K. The cell is heated by a wire heater using high-frequency (200 kHz) alternating current as high as 150 $^{\circ }$C inside a boron nitride ceramic oven, and the temperature, which is monitored by a PT1000 resistor, is stable to 0.01$^{\circ }$C and vary less than 1$^{\circ }$C across the cell. The rubidium vapor density $n$ is about $1 \times {10^{14}}\;{{\textrm {cm}}^{ - 3}}$. The magnetic shield consists of four layers of $\mu$-metal and an inner layer of MnZn ferrite is constructed to attenuate the earth’s magnetic field and maintain the SERF regime. A triaxial uniform field Coils’ system is established to further compensate the residual magnetic field and provide the external magnetic field [25]. The radial coil is designed with a quasi-elliptic function fitting method in a simple structure with the constant of 74.55 ${\textrm {nT/mA}}$ [26] and the axial coil is designed with a Taylor expansion method with the constant of 92.32 ${\textrm {nT/mA}}$ [27]. As a design result, the absolute maximum relative error in the cubic target region (with a side length of $R$/2, where the radius of the ferrite $R$ = 45.5 mm) is optimized to 0.023$\%$. The coil constant is optimized considering the coupling effect between the coil and the ferrite material.
Fig. 2. Schematic of the experimental setup for measurements. A circularly polarized laser beam at $^{87}\textrm {Rb}\;{\textrm {D}_1}$ resonance line optical pumps the rubidium atoms along the $z$ direction, and the transmitted light intensity is monitored by PD0. And a linearly polarized probe beam which is blue detuned from the ${{\textrm {D}}_2}$ line, propagates along the $x$ direction, passing sequentially through the Rb cell, a polarizing beam splitter, and two subtraction photodiodes (PD+ and PD-). LP denotes linearly polarized plate; QWP is a quarter-wave plate; PD is a photodiode; PBS is a polarized beam splitter; DAQ is the data acquisition system. Shutters are used to turn on/off the probe and pump beam when measuring the system and probe noise.
A circularly polarized laser beam at $^{87}\textrm {Rb}\;{\textrm {D}_1}$ line optical pumps the atoms along the $z$ direction. And a linearly polarized probe beam which is tuned to the ${\textrm {D}_2}$ line, propagates along the $x$ direction, passing sequentially through the Rb cell, a polarizing beam splitter, and two subtraction photodiodes (PD+ and PD-). The diameter $d$ of the pump and probe beam is 2.7 mm, and the linewidth of both is on the order of 1 MHz. The polarization beam splitter and noise eater combination are used to control the probe and pump laser intensity which changes less than 0.1$\%$ during the experiment. The extinction ratio of two beams passing through the fibers is larger than 40 dB. In the experiment, all data are collected by DAQs for 100 s with the sampling rate of 1000 Hz and analyzed with power spectral density of averaging spectral resolution of 1 Hz.
4. Results and discussion
In order to evaluate the noise characteristics under optimized condition, we optimize the pump intensity and wavelength first. On-resonance circular wavelength used to pump the alkali atoms is 794.9984 nm, which avoids the creation of a virtual magnetic field by near-resonant light. Although the AC-Stark shift can be compensated by the reverse magnetic field generated by the coil, gradient magnetic field is difficult to cancel due to attenuation of near-resonance light intensity especially in a cell with a large $\textrm { OD }$ parameter [28]. After determining the wavelength of the pump light, we optimize the optimal pump intensity. The fundamental sensitivity limit of the SERF magnetometer is given by the quadratic sum of the spin-projection noise, the photon-shot noise, electronic noise, and magnetic noise. We see that if the quantum limit is dominated by the spin-projection noise, the fundamental sensitivity is minimized by setting ${R_\textrm {OP}} = 2({R_\textrm {rel}}{\textrm { + }}{R_\textrm {PR}})$, where the polarizability is 0.67. If the photon shot noise, electronic noise dominates, then the sensitivity limit is optimized by setting ${R_\textrm {OP}} = {R_\textrm {rel}}{\textrm { + }}{R_\textrm {PR}}$, where the polarizability is 0.5, which in general result in the largest signal strength described in the Eq. (3). Under the experimental conditions of this paper. We simulate $\delta {B _\textrm {psn}}{\textrm { < }} 1\textrm {fT}/\sqrt {\textrm {Hz}}$ and $\delta {B _\textrm {spn}}{\textrm { > }} 4\textrm {fT}/\sqrt {\textrm {Hz}}$. The ${R_\textrm {OP}} = {R_\textrm {rel}}{\textrm { + }}{R_\textrm {PR}}$ is optimal choice for the following experiments. After nulling the magnetic fields, we sweep the amplitude of ${B_y}$ from −80 nT to 80 nT. Figure 3(a) shows the optical rotation signal of the cell at different incident pump beam intensity, while the probe beam intensity remain 8.73 ${\textrm {mW/c}}{{\textrm {m}}^2}$ and about 100 GHz detuning from the center wavelength. Combining the Eq. (3) with Eq. (5), we can simplify the solution as:
Fig. 3. (a): Optical rotation signal of probe beam as a function of $B_{y}$ at different pump beam intensity. The probe intensity is 7${\textrm {mW/c}}{{\textrm {m}}^2}$. (b): The slope $k$ of the center of magnetometer response by sweeping ${B_y}$ amplitude at different pump beam intensity. The black dots are experimental data, and the red solid curve is fitted by Eq. (7).
Using the method above, we can also calculate the optical pumping rate of the probe beam, which will be used in the following calculation. The half-width at half maximum (HWHM) of the Lorentzian curve represents the whole relaxation rate $\Gamma$. When we change the intensity of the probe beam, the slope represents the optical pumping factor $\eta$ as shown in Fig. 4(a). Figure 4(b) shows the optical pumping factor as a function of wavelength, which is discussed in Eq. (2). The red Lorentzian curve fits the black experimental data very well, but the width of the fitted Lorentzian curve is not consistent with that of pressure broadening of the cell. The reason is that the intensity of the probe light decays along the direction of the probe beam in the alkali cell. The smaller the detune is, the higher the attenuation of the probe intensity is. thereby, the actual measured value $\eta$ departs from theoretical value seriously. For reasonability, we use Lorentzian lineshape for fitting in this experiment.
Fig. 4. (a): The optical pumping rate of probe beam as a function of intensity at different wavelengths. The pump intensity is 7.43 ${\textrm {mW/c}}{{\textrm {m}}^2}$. (b): The optical pumping factor of probe beam under different wavelengths.
As shown in Eq. (6), the fundamental noise are related to the factor $G$ except for the constant value of magnetic noise ${\delta {B_{\textrm {m}}}}$. At the condition ${R_{{\textrm {OP}}}} = {R_{{\textrm {rel}}}}{\textrm { + }}{R_{{\textrm {PR}}}}$ when magnetometer has the best sensitivity, the magnetometer response $G$ can be simplified as:
where the factor $\kappa = l{r_e}nc{f_{{\textrm {D2}}}}{\gamma ^{\textrm {e}}}/16$, $D(v)$=$\Delta v/(\Delta {v^2} + {({\Gamma _D}/2)^2})$. Obviously, the coefficient $G$ is affected by the intensity of the probe beam due to the change of the pumping rate of probe beam ${R_{{\textrm {PR}}}}$, it is also affected by the wavelength of probe beam as a result of absorption cross-sectional area changed. The frequency response is used to calculate the sensitivity of the magnetometer. A chirp signal in the $x$ direction, varied from 1 Hz to 200 Hz in 50 s period with the amplitude of 100 ${{\textrm {pT}}_{{\textrm {rms}}}}$, is used to calibrate the frequency response under different probe beam intensities [29]. The frequency reponse can be described by a first-order low-pass filter with a fit to $A/{({(f - B)^2} + f_{3{\textrm {dB}}}^2)^{1/2}}$ where $f$ is the frequency and ${f_{3{\textrm {dB}}}}$ is the 3 - dB bandwidth [30]. As Fig. 5(a) shows, the magnetometer response is inversely proportional to the probe intensity, which is due to the pumping rate of probe beam ${R_{{\textrm {PR}}}}$. Then, we apply a 100 ${{\textrm {pT}}_{{\textrm {rms}}}}$ magnetic field with a frequency of 173.7 Hz to the magnetometer during the process of scanning the wavelength. The demodulation signal through the lock-in amplifier divided by incident intensity of probe beam is the magnetometer response signal $G{e^{ - {\textrm {OD}}}}$, taking into account absorption of the probe beam. These signals recorded as a function of probe wavelength at different probe beam intensities are shown in Fig. 5(b).Fig. 5. (a): The magnetometer frequency response $G$ under different probe beam intensity at the wavelength of 780.183 nm. (b): The magnetometer response as a function of probe wavelength at different probe beam intensities, taking into account absorption of the probe beam.
As shown by the photon shot noise, $\delta {B_{{\textrm {psn}}}} = {k_{{\textrm {psn}}}}/(G\sqrt {{I_{{\textrm {PR}}}}{e^{ - {\textrm {OD}}}}} )$, although increasing the intensity of probe beam can reduce the noise, as discussed above, it will cause the factor $G$ to become smaller so as to offset the noise reduction effect of increasing the intensity. Combined with the Eq. (8), the $\delta {B_{{\textrm {psn}}}}$ expands to:
Fig. 6. The simulation results of photon shot noise as a function of probe intensity at different wavelengths from Eq. (9)
Through the formulation of electronic noise, it can be seen that the electronic noise is inversely proportional to the intensity of the probe beam. This attenuation coefficient of inverse proportional function is mainly affected by the detuning of the probe wavelength. As Fig. 7(a) shows, we measure the parameter $2K{I_{{\textrm {PR}}}}{e^{ - \textrm {OD}}}G$ as a function of probe wavelength at different probe beam. This quasi dispersion curve, in which the center wavelength is 780.255 nm, has two peaks, about 780.12 nm and 780.42 nm. The center zero area is due to the thick optical depth. As shown in Fig. 7(b), the total noise will be affected by electronic noise in the condition of slight detuning and weak intensity of the probe intensity. For example, the electronic noise is significant when the wavelength is 780.20 nm due to the small signal amplitude. When the probe intensity increases, the electronic noise approaches a small constant below 0.5 ${\textrm {fT/}}\sqrt {{\textrm {Hz}}}$ and can be neglected.
Fig. 7. (a): The signal amplitude of magnetometer as a function of probe wavelength at different probe intensities. (b): The electronic noise as a function of probe intensity at different wavelength.
As discussed above, we should analyze the different noise sources ${\delta {B_{{\textrm {spn}}}}}$, ${\delta {B_{{\textrm {psn}}}}}$, ${\delta {B_{\textrm {e}}}}$ and ${\delta {B_{\textrm {m}}}}$. Assuming the magnetometer working under optimal conditions, the spin-projection noise ${\delta {B_{{\textrm {spn}}}}}$ is the level of 0.73 ${\textrm {fT/}}\sqrt {{\textrm {Hz}}}$. For the magnetic noise ${\delta {B_{\textrm {m}}}}$ consists of the coils’ noise and magnetic shield’s noise in our experiment condition, the magnetic coils’ noise floor which caused by the circuit voltage uncertainty is measured by applying a calibrating sine signal of 100 ${{\textrm {pT}}_{{\textrm {rms}}}}$ from the function generator and calculated using coil calibration constant. The magnetic shield consists of four layers of $\mu$-metal and an inner layer of MnZn ferrite with a calculated noise floor of 2 ${\textrm {fT/}}\sqrt {{\textrm {Hz}}}$ at 1 Hz [31,32]. The magnetic noise is a constant no matter what the performance of the magnetometer is. Due to the magnetometer frequency response, the electronic noise ${\delta {B_{\textrm {e}}}}$ measured by blocking off the pump and probe beam is different under different probe beam intensity. In our experiment, optical rotation signal is measured by a pair of difference PDs, which can estimate the noise from the fluctuation of the laser power. However, there are some offsets that remain between the two PDs, the optical rotation signal might be slightly influenced. We monitor the probe incident intensity to evaluate the probe intensity noise, including noises from the probe beam, fiber, quantum, and electronic devices. By blocking the pump beam, we can get the probe shot noise which contains probe intensity noise and photon shot noise, and define the photon shot noise by comparing the probe intensity noise with probe shot noise. Figure 8 shows the magnetic noise spectrum of probe beam, coils, electronic system, when the pump beam intensity is 7 ${\textrm {mW/c}}{{\textrm {m}}^2}$ and probe beam intensity is 11 ${\textrm {mW/c}}{{\textrm {m}}^2}$ with the wavelength of 780.12 nm. As we can see, the electronic and coils noise is about 0.1-0.2 ${\textrm {fT/}}\sqrt {{\textrm {Hz}}}$, and the probe intensity noise is about 1.5 ${\textrm {fT/}}\sqrt {{\textrm {Hz}}}$, while the probe shot noise is about 6 ${\textrm {fT/}}\sqrt {{\textrm {Hz}}}$. Compared to other noise sources, the performance of the magnetometer is mainly limited by the photon shot noise.
Fig. 8. Magnetic noise spectrum of probe photon shot noise, probe intensity noise, electronic noise, coils noise, and theoretical ferrite noise.
To measure the sensitivity of the magnetometer and analyze the influence of probe beam, a small sine calibrating magnetic field at 30.5 Hz with an amplitude of 100 ${{\textrm {pT}}_{{\textrm {rms}}}}$ is applied at $x$-direction. We choose several probe beam wavelengths (780.02 nm, 780.065 nm, 780.12 nm, 780.165 nm, and 780.2 nm) to evaluate the magnetometer’s noise density spectrum under several different probe beam intensities as Fig. 9 shows. The five noise floors are too close to tell the difference under the relatively small probe intensity. For brief verification, we average the noise floor from 10 Hz to 100 Hz noise density spectrum except the calibrating and power frequency while the average range contains the magnetometer bandwidth.
Fig. 9. The noise density spectrum of the magnetometer at several different probe intensities while the probe beam wavelength is 780.165 nm. Calibrating magnetic field is 30.5 Hz, and the power frequency is 50 Hz.
To analyze the noise of a dual-beam magnetometer, we have tested the sensitivity of the magnetometer under different probe intensities at different probe wavelengths plotted in Fig. 10. By comprehensively considering the several noises, we have fitted the data using Eq. (6). As shown in Fig. 10, the dots are experimental data and the solid line are fitted with good by the parameters ${R_{{\textrm {rel}}}}{\textrm { = 103 Hz}}$, ${k_{{\textrm {psn}}}} = 0.65 \pm 0.05\;{\textrm {fT/}}\sqrt {{\textrm {Hz}}} \sqrt {{\textrm {mW/c}}{{\textrm {m}}^{\textrm {2}}}}$, and ${k_{\textrm {e}}} = 0.025 \pm 0.005\;{\textrm {fT/}}\sqrt {{\textrm {Hz}}} {\textrm {(mW/c}}{{\textrm {m}}^{\textrm {2}}})$. The five fitting curves at different probe wavelengths are similar to the simulation in Fig. 1. Under different wavelengths, the sensitivities all decrease at first and increase subsequently. Meanwhile, when the detuning of the probe beam is small, the sensitivity is poor and the suitable light intensity range is also very small. When the detuning is bigger, the sensitivity gets better and the suitable light intensity range becomes bigger. For small detuning, e.g., 780.20 nm, the total noise is mainly affected by electronic and photon shot noise under low intensity. Although the electronic noise decreases with the probe intensity increasing, the photon shot noise increases rapidly due to the large factor $\eta$. The total noise floor is always large at different probe intensities. When the detuning gets larger, e.g., 780.12 nm and 780.165 nm, the electronic noise gradually decreases and does not affect total noise. The factor $\eta$ is relatively small, so the photon shot noise grows gradually with probe intensity. When the detuning is large, e.g., 780.02 nm and 780.065 nm, the trend of total noise with probe intensity is almost the same as the photon shot noise.
Fig. 10. The sensitivities under different probe intensities at different probe wavelengths. The results averaged by 10 Hz to 100 Hz noise density spectrum except the calibrating and power frequency. The dots are experimental data, and the solid lines are calculated from Eq. (6).
In the case of large detuning, optimum sensitivity can be achieved at large probe intensity, so the transimpedance amplifier is required to have the ability of a large measuring range. Sometimes the magnetometer output has bias voltage due to the position of the polarization axis of the half-wave plate, and the transimpedance amplifier will overload with the probe intensity increasing. In the case of the larger probe intensity, the photodiode’s photoelectric conversion coefficient will turn in the nonlinear region, resulting in output error. Through the above analysis, the optimal parameter is about 50 - 70 GHz detuning from center wavelength and about 10 - 15 ${\textrm {mW/c}}{{\textrm {m}}^2}$ probe beam intensity, the best sensitivity is about 6.3 ${\textrm {fT/}}\sqrt {{\textrm {Hz}}}$.
5. Conclusion
In this paper, we present a theoretical and experimental analysis of noise sources characterized from a microfabricated cell with an active measurement volume of $3 \times 3 \times 3\;{{\textrm {mm}}^3}$ in a perpendicular pump-probe configuration. Then we use several noises of the magnetometer, including spin projection noise, photon shot noise, electronic noise, and magnetic noise to evaluate the probe beam characteristics and find that the optical pumping rate of the probe beam significantly affects the magnetometer performance. Compared to other noise sources, the performance of the magnetometer is mainly limited by the probe noise at the optimal pumping condition. The optimal probe intensity range increases with more detuning from the resonance frequency. However, limited by the performance of transimpedance amplifiers and photodetectors, the optimal probe intensity is not suitable for large detuning. The experimental results show that the optimal parameter is about 50 - 70 GHz detuning from center wavelength and about 10 - 15 ${\textrm {mW/c}}{{\textrm {m}}^2}$ probe beam intensity with a noise floor of 6.3 ${\textrm {fT/}}\sqrt {{\textrm {Hz}}}$. In order to further improve the sensitivity, the photon shot noise can be suppressed by reducing relaxation ${R_{{\textrm {rel}}}}$, such as replacing rubidium with potassium or filling the cell with Helium gas in the same volume. Fabricating a big size cell can also settle this matter, but this is not friendly to miniaturized integration of magnetometers. There is another sophisticated method of spin squeezing to enhance the magnetometer by reducing the photon noise [33,34]. We theoretically calculate the total noise and fit it with the experimental data in a good arrangement. Based on the results, we demonstrate the optimal pump and probe beam characterizations, which could promote the sensitivity in the SERF magnetometer.
Funding
Beijing Municipal Natural Science Foundation (1222025); Key Research and Development Program of Zhejiang, China under Grant (2020C01037); National Natural Science Foundation of China (62003022); National Key Research and Development Program of China under Grant (2018YFB2002405).
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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