Abstract

We demonstrate a novel method of measuring magnetic field based on the transient signal of the K-Rb-21Ne co-magnetometer operating in nuclear spin magnetization self-compensation magnetic field regime. The operation condition for self-compensation magnetic field by nuclear spin magnetization of 21Ne in steady state is presented. We characterize the dynamics of the coupled spin ensembles by a set of Bloch equations, and formulate the expression of transient output signal. After verifying the stability of this method, the measurement range and error are studied. This method is also verified to be valid in various temperature and pumping light power density.

© 2017 Optical Society of America

1. Introduction

Precision measurements of magnetic field have been finding a wide range of applications, from searching for permanent electric dipole moments [1], geomagnetic mapping [2], to biomagnetism measurement [3,4]. The low-temperature super-conducting quantum interference device (SQUID) with sensitivity of about 1 fT/ Hz1/2 [5] has held the magnetic field sensitivity record for many years, hence it is dominant in the ultrasensitive magnetic field measurement. However, with the development of technology, such as the improvement in the performance of diode laser, the refinement of the production of demanding atomic vapor, and the progress on quantum manipulation, the atomic magnetometer has achieved ultrahigh sensitivity record [6], which surpasses the sensitivity of SQUID. Currently, the atomic optical magnetometer operating in spin-exchange relaxation-free (SERF) regime is the most sensitive magnetometer with the sensitivity of 0.16 fT/ Hz1/2 in a gradiometer arrangement [7] and the sensitivity of 0.54 fT/ Hz1/2 as a scalar magnetometer [8].

Co-magnetometers utilizing the precession of nuclear spin rather than electron are used in inertial rotation sensing and the test of fundamental symmetries [9], for example the K-Rb-21Ne co-magnetometer is applied to search for tensor interactions violating the local Lorentz invariance [10] and the K-3He co-magnetometer is a promising nuclear spin gyroscope with rotation sensitivity of 5 × 10−7 rad s−1 Hz−1/2 [11]. The measurement of magnetic field based on co-magnetometers has not been demonstrated, because co-magnetometers operating in nuclear spin magnetization self-compensation magnetic field regime are insensitive to magnetic field in steady state compared to the similar alkali metal magnetometers [12]. However, co-magnetometers operating in self-compensation magnetic field regime are still sensitive to magnetic field in transient process, thus the transient response could be used to measure magnetic field. Moreover, the study of transient signal would have benefit for revealing the dynamic performance of co-magnetometer and measuring optical parameters. In addition, the change of transverse residual magnetic field inside the alkali vapor cell of co-magnetometer would disturb the measurement of inertial rotation and the test of fundamental symmetries [12, 13]. Besides, the longitudinal compensation magnetic field of co-magnetometers would deviate from the self-compensation point due to the drift of co-magnetometer system, resulting in that the co-magnetometers cannot operate normally anymore because the nuclear spin magnetization of noble gas atoms cannot self-compensate small magnetic field [14]. Therefore, in-situ measurement of the change of the transverse residual magnetic fields inside the alkali vapor cell and the deviation from the self-compensation point based on the transient signal of co-magnetometer could be potentially used to rapidly and accurately compensate the change of the magnetic fields and the deviation, and would have significant benefit for the operation of co-magnetometers.

In this work, we study the self-compensation of magnetic field in steady state by nuclear spin magnetization of 21Ne in the K-Rb-21Ne co-magnetometer, formulate the expression of the transient output signal under external excitation, and measure the transient response to step change of the transverse magnetic field. We demonstrate a novel method of measuring magnetic field by fitting the transient signal of the co-magnetometer. Under various operation conditions the performance of the co-magnetometer is considered. The measurement range of this method is from 0.016 nT to 1.760 nT and the measurement error is lower than 0.01 nT, which can be potentially improved by suppressing the magnetic noise and increasing the stability of the pumping light. On one hand, this method could be potentially used to rapidly and accurately compensate the change of residual magnetic field and the derivation in co-magnetometers. On the other hand, it could be potentially used in high precision simultaneous measurement of magnetic field and inertial rotation in miniaturization applications.

2. Basic principle

The K-Rb-21Ne co-magnetometer principally consists of a spherical glass vapor filled with 21Ne gas and a small mixed droplet of K-Rb as well as N2 gas (quenching and buffer gas). K atoms are spin polarized along the z-axis of xyz Cartesian coordinate system by circularly polarized pump light whose wavelength is locked on the D1 line of K atoms. Hybrid optical pumping [15, 16], concretely the spin-exchange collisions with K atoms, polarize Rb atoms along z-axis. The 21Ne nuclear spin is polarized along z-axis by spin-exchange collisions with alkali atoms [17]. The linearly polarized probe light is utilized to measure the precession of Rb spin ensembles in the presence of magnetic field by optical rotation [18].

The K-Rb spin-exchange rate exceeds 106 s−1 at typical densities of about 1014 cm−3 because of sufficiently large spin-exchange cross section, resulting in that the Rb and K atoms are in spin-temperature equilibrium with same spin polarization P [15]. The K-Rb spin-exchange interaction for Rb atoms could be represented by an equivalent polarization effect with an equivalent rate Rp, and the K-Rb spin-destruction interaction for Rb atoms is considered in the spin destruction rate of Rb atoms. The interactions between alkali atoms and 21Ne atoms are determined by the imaginary part of spin-exchange cross section [19], and could be described by an effective magnetic field experienced by one spin species from the average magnetization of the other in a spherical cell, B = λMP, where λ = 8π k0 /3 [20]. The Fermi contact shift enhancement factor of the alkali-noble atoms pair k0 for Rb-21Ne pair and K-21Ne pair are 35.7 ± 3.7 and 30.8 ± 2.7 respectively [21]. M = μn is the magnetization corresponding to fully polarization. Since the density ratio of K and Rb atoms is typically on the order of 10−2, the effective magnetic field of K-21Ne is two orders of magnitude smaller than that of Rb-21Ne. The K-21Ne spin destruction interaction for 21Ne atoms is included in the spin destruction rate of 21Ne atoms, which is two orders of magnitude smaller than that of Rb-21Ne. Therefore, the interaction of 21Ne atoms and alkali atoms are dominated by coupled spin ensembles Rb-21Ne. The magnetic field generated by Rb electron spins and 21Ne nuclear spins are denoted by Be and Bn respectively, and the magnetizations are designated by Me and Mn. Moreover, the spin ensembles experience no effective magnetic field produced by themselves because of the symmetry of spherical vapor [19]. The co-magnetometer can be properly approximated by Bloch equations coupling the Rb ensemble polarization Pe with the 21Ne ensemble polarization Pn [11,13,18]:

Pet=γeQ(B+λMnPn+L)×PeΩ×Pe+RmSm+RpSp+RsenePnQPeQ{T1,T2,T2},Pnt=γn(B+λMePe)×PnΩ×Pn+RseenPeRtotnPn.

Here γe and γn are the gyromagnetic ratios of electron and 21Ne nucleon. Q is the slowing down factor due to hyperfine interaction between electron and nuclear spins of Rb atoms, and it depends on the nuclear spin I and the polarization rate of the ensemble P [22]. Since the natural-isotopic abundance of Rb (72.2% 85Rb with nuclear spin I = 5/2 and 27.8% 87Rb with nuclear spin I = 3/2) is used in our experiment and the Rb spin ensemble is in the regime of strong spin-exchange, the slowing down factor for the Rb spin ensemble is the hybrid of Q(I = 3/2) and Q(I = 5/2). B and Ω are external magnetic field and inertial rotation. L is the light shift from pump and probe lasers, and can be set to zero as long as the pump light is tuned to optical resonance and the probe light is linearly polarized [12]. Rp and Rm are the pumping rate of pump and probe lights, while Sp and Sm are their photon spins. Rsene is the spin-exchange rate from 21Ne atoms to Rb atoms, and Rseen is the spin-exchange rate from Rb to 21Ne. For Rb spin ensemble, the longitudinal and transverse relaxation rates are (QT1)1=(Rm+Rp+Rsene+Rsde)/Q and (QT2)1=(QT1)1+Rseee respectively, where Rsde is the spin destruction rate of Rb atoms. The spin-exchange relaxation rate of Rb-Rb, Rseee=ω02Tse[Q2(2I1)2]/2 [22, 23], cannot be ignored for large Be, which is negligible in SERF regime [11, 19], where ω0 is the reduced precession frequency, Tse is the spin-exchange time of Rb-Rb, I is the nuclear spin of Rb atoms, and T21 is denoted by Rtote. Rtotn=Rseen+Rquadn+Rsdn is the total spin relaxation rate of 21Ne, where Rquadn is the quadrupole relaxation rate of 21Ne, Rsdn is the spin destruction rate.

With small transverse excitation, the longitudinal polarization Pze and Pzn nearly remain constant, therefore the Bloch equations could be linearized. By solving Eq. (1) we can obtain the complete output signal Sx proportional to the transverse polarization Pxe including transient and steady components in Eq. (2).

Sx=KdPxe=eλ1rt(P1rcosλ1itP1isinλ1it)+eλ2rt(P2rcosλ2itP2isinλ2it)+Sxsteady,λ1=λ1r+iλ1i=Rtote2Q+a2+b2+a22+i[(γe(LzBze)2QγnBzn2)+Sign[b]a2+b2a22],λ2=λ2r+iλ2i=Rtote2Q+a2+b2+a22+i[(γe(LzBze)2QγnBzn2)Sign[b]a2+b2a22],a=(RtoteQ)2(γeLzBzeQ+γnBzn)24γeBzeγnBznQ,b=2RtoteQ(γeLzBzeQ+γnBzn)4γeBzeγnBznQ,
where Kd is the factor that converts the transverse polarization of Rb spin ensemble to output voltage signal by the probe system; λ1r, λ1i, λ2r and λ2i are determined by the system itself; P1r, P1i, P2r, P2i and Sxsteady depend on the input signals.

As inputting transverse magnetic field By, the basic operation of this co-magnetometer is that a bias magnetic field Bz produced by coils is applied along z-axis whose value is set to Bc = − BeBn, which is defined as the compensation point. At this point, the nuclear magnetization of 21Ne could arbitrarily cancel slowly changing By in steady state [11], leading to the steady signal of x-axis no longer depending on By, while the transient signal of x-axis still related to By. Hence, the transient signal can be used to obtain the input magnetic field. The intuitive compensation principle is illustrated in Fig. 1 and will be discussed in more detail below. At the compensation point, the parameters P1r, P1i, P2r, P2i and Sxsteady of Eq. (2) can be specifically rewritten with different types of change in transverse magnetic field, for example step-like change and sinusoidal change. When the change is step-like with amplitude By0, these parameters can be expressed by Eq. (3), and with other types of change they can be similarly rewritten:

P1r=KrBy0+K1r0Ωy,P1i=KiBy0+K1i0Ωy,P2r=KrBy0+K2r0Ωy,P2i=KiBy0+K2i0Ωy,Kr=λ1rλ2r(λ2rλ1r)2+(λ2iλ1i)2KdγePzeQ,Ki=λ2iλ1i(λ2rλ1r)2+(λ2iλ1i)2KdγePzeQ,Sxsteady=Ksteady(δBzBnBy+Ωyγn),Ksteady=KdγePzeRtoteRtote+γe2(Lz+δBz)2.

 figure: Fig. 1

Fig. 1 Intuitive compensation principle of self-compensation co-magnetometer. (a) Without transverse magnetic field By, Pe and Pn direct along z-axis. (b) When applying By, Pe and Pn would precess around the total magnetic field in the transient process. (c) Pn would stabilize at the direction in y–z plane in a short time, resulting in the projection of Bn with respect to y-axis compensating By, hence Pe returns back to z-axis unaffected by By in steady state.

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Here Ωy, projection of earth rotation along y-axis, is invariable equaling to 5.586×10−5 rad/s because the y-axis is directed along the direction of the earth rotation horizontal component, hence K1r0Ωy, K1i0Ωy, K2r0Ωy and K2i0Ωy are constants. Since KdγePze/Q could be obtained by the calibration procedure [13], Kr and Ki can be determined by the system parameters λ1r, λ1i, λ2r and λ2i. Ksteady is the scale factor of steady state value, and Sxsteady is constant only depending on Ωy because δ Bz = BzBc equals to zero at the compensation point. Therefore, after determining the system parameters, the magnetic field By0 can be acquired by fitting the transient signal based on Eq. (2) and Eq. (3).

3. Experimental setup

The schematic of the co-magnetometer is shown in Fig. 2, which is similar to our previous apparatus [14,24]. There is a 14 mm diameter spherical aluminosilicate glass vapor in the centre of the device, which contains a mixture droplet of K-Rb (natural abundance) alkali metals, 2020 Torr 21Ne gas (70% isotope enriched) and 31 Torr N2. The cell is installed in a boron nitride ceramic oven and heated by a 110 kHz AC electrical heater. A PID temperature control system is applied to control the current of the 110 kHz AC electrical heater to reduce temperature fluctuation of the cell. The oven is enclosed by 5-layers cylindrical mu-metal magnetic shields and a ferrite barrel for shielding the ambient magnetic field, and the residual magnetic field is further compensated by a set of three-axis Helmholtz coils. A water-cooling system is used to reduce the cross-ventilation due to the temperature difference between the oven and the room.

 figure: Fig. 2

Fig. 2 The schematic of the co-magnetometer. GT, Glan-Thompson polarizer; PD, photo detector; PBS, polarization beam splitter; ECDL, external cavity diode laser; TA, tapered amplifier; BE, beam expander; DFB, distributed feedback laser; PEM, photo elastic modulator.

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The electron spin of K atoms is polarized along z-axis by the circularly polarized pumping light, which is produced by a external cavity diode laser (ECDL) and amplified by a tapered amplifier (TA). The wavelength of the pumping light is tuned to D1 resonance of K and the maximum light power is about 1W. The pumping light beam is expanded by beam expander (BE) and selected by aperture, consequently the cell is covered by light with relatively uniform power density. The precession of the electron-nuclear spin ensembles in the presence of magnetic field can be measured by the linearly polarized probe light based on optical rotation. The probe light along x-axis is formed by a distributed feedback (DFB) laser, whose wavelength is 0.3 nm red detuning from D1 line of 85Rb, for detecting the x-component of electron polarization. The power density of the probe light and the pumping light are stabilized by electrically controlled half waveplate respectively. A photo-elastic modulator (Hinds Instrument) modulates the probe light with resonance frequency of about 50 kHz and modulation amplitude about 0.08 rad. And the signal measured by a photodetector is demodulated by a lock-in amplifier (Stanford Research SR830) and the first harmonic of the signal is recorded [25].

This co-magnetometer operates with the cell temperature set to 456K, at which the densities of K and Rb atoms are about 3.8 × 1012 cm−3 and 3 × 1014 cm−3 respectively. And the pumping light power density is 81.3 mW cm−2. Applying a series of procedures for zeroing residual magnetic field [12], the residual magnetic fields are found to be approximate to 1.586 ± 0.001 nT and 3.626 ± 0.002 nT in x and y axis respectively, while the compensation point is found to be Bc = −312.7 nT. Utilizing the calibration procedure [11,13], the light shift Lz is measured to be 0.127 nT, Rtote/γe is 31.84 nT, and the steady state scale factor Ksteady is 22.1 V/nT. A sensitivity of 2.1 × 10−8 rad s−1 Hz−1/2 for the K-Rb-21Ne co-magnetometer has been demonstrated in our previous work [14], corresponding to a magnetic field sensitivity of 1.0 fT Hz−1/2.

4. Results and discussions

The responses to transverse input magnetic field with various bias magnetic field are shown in Fig. 3. The output signals Sx, following a small step magnetic field By0 = 0.32 nT formed by the three-axis Helmholtz coils, are measured when the bias magnetic field Bz is set to compensation point or non-compensation point, and are fitted with Eq. (2). As shown in Fig. 3(a), when Bz is set to Bc, the response signal would damp to the initial offset value Sx0=0.1286V after a short-time oscillation, that means By completely compensated by the nuclear magnetization of 21Ne in steady state. In concrete term, the offset value Sx0=0.1286V is steady response to the projection of earth rotation along y-axis, and the characteristic time determining the time of transient process is approximately equals to 0.45 second. In Fig. 3(b), when Bz tuned to non-compensation point, the transient oscillation signal attenuates to the steady state value unequal to the initial offset value, and the time of transient process is longer than the one at the compensation point. The ratio of the difference between the initial value and the steady value ΔSx0 to the initial value is −0.493, indicating By incompletely compensated in steady state. Specifically, the difference values ΔSx0 with same input magnetic field By0 = 0.32 nT and various bias magnetic field are illustrated in Fig. 3(c). At the compensation point δ Bz =0, the steady response to magnetic field is entirely suppressed, and ΔSx0 increases along with the growth of deviation of compensation point δ Bz with the increase rate declining as expected with Eq. (3).

 figure: Fig. 3

Fig. 3 The responses to transverse magnetic field By0 = 0.32 nT with various bias magnetic field. (a) The bias magnetic field Bz is set to the compensation point Bc = −312.7 nT. (b) Bz is set to the non-compensation point Bc = −362.7 nT. (c) The difference between the initial offset value and the steady state value ΔSx0 is measured with same input magnetic field By0 = 0.32 nT and various bias magnetic field δ Bz.

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At the compensation point, the determined system parameters cannot be utilized to fit the transient signal to acquire the input magnetic field, unless the system parameters are stable over time and unaffected by inputting signals. The stability of the system parameters is studied and shown in Fig. 4. A series of response signals to the same input step magnetic field By0 = 0.32 nT are recorded every half hour, and these 2-second transient signals are fitted by Eq. (2) as shown in Fig. 4(a). The coefficients of determination R-square (the goodness of fit) are superior to 0.99.

 figure: Fig. 4

Fig. 4 The stability of the system parameters. (a) The variation of the four fitted system parameters over time. The response signals to By0 = 0.32nT at every half hour are continuously measured five times, of which the standard deviations of the fitted system parameters are plotted as the error bars. (b) The R-square of the fitting curves with different input magnetic field. The response signals to each input magnetic field are measured five times, of which the standard deviations of the fitted R-square are regarded as the error bars.

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The average values of the system parameters (λ1i, λ1r , λ2i, λ2r) are −1.78, −2.24, −8.3, −45.5 respectively, and the standard deviations of the system parameters are less than 0.03, 0.02, 0.1 and 0.8 respectively. The corresponding relative uncertainty (percent uncertainty) of the system parameters is less than 1.5%, 0.8%, 1.5%, 1.8% respectively, proving that the system parameters are constant immune to time. To verify that different input signals have no effect on the system parameters, λ1i0=1.78, λ1r0=2.24, λ2i0=8.3, λ2r0=45.5, obtained from the fitting of the response signal to step magnetic field By0 = 0.32 nT, are applied to fit the responses of magnetic fields with various amplitudes. In Fig. 4(b), the corresponding R-square are all better than 0.99, indicating that the fitting curves are in good agreement with the measured signals. The goodness of fit of the signals with smaller input amplitudes are slightly poorer than those with larger input amplitudes, probably because their response amplitudes are relatively small that would be influenced by the fluctuation of the system more easily.

After determining the system parameters ( λ1r0, λ1i0, λ2r0, λ2i0), the input magnetic field By can be acquired by fitting the response signal based on Eq. (2) and Eq. (3) with Kr = 5.96 and K1r0Ωy=0.034. The responses to input signals varying from 0.008 nT to 2.080 nT are measured. As shown in Fig. 5(a), the fitted By with R-square superior to 0.99 is proportional to the input magnetic field, except for By =1.920 nT and By =2.080 nT whose fitted By are smaller than the input ones respectively with their responses exceeding the measurement range of the lock-in amplifier. In Fig. 5(b), the corresponding residuals are smaller than the relevant input By by at least two orders of magnitude, excluding the one of By = 0.008 nT whose residual is only one order smaller than the relevant input By. The exception of By = 0.008 nT probably results from the response is relative small and affected by fluctuation of the signal significantly. Therefore, the measurement range of this method is from 0.016 nT to 1.760 nT, and the reverse input By can also be fitted, and the measurement error is less than 0.01 nT. To further examine the transient response to small magnetic field, a high precision function generator (Agilent 33522A, minimum output 1 mV), which is attenuated by a 100 kΩ resistor and drives the Helmholtz coils directly (generating a minimum magnetic field about 1.6 pT), is utilized. The 0.5-second responses to small input magnetic fields (16 pT, 8 pT and 1.6 pT) are measured and fitted. As shown in Fig. 5(c), the measured signals are fitted with λ1r0, λ1i0, λ2r0, λ2i0, Kr = 5.96 and K1r0Ωy=0.034 based on Eq. (2) and Eq. (3), and the fitted curves are consistent with the measured signals with R-square superior to 0.9. The transient response model could still effectively describe the response to small magnetic field on the order of pT. The 0.5-second probe signal without input signal is denoted by black solid curve in Fig. 5(c), and the noise peak-to-peak value of this probe signal, which is primarily caused by magnetic field noise inside the magnetic shields, pumping light and probe light power density noises, operation temperature noise and electronic noise, is an order of magnitude smaller than the peak value of response to By0 = 1.6 pT, hence the minimum measurement range could be potentially reduced to approach the noise level of probe signal without input signal. In addition, the transient signal model can also be applied to fit the short time transient signal to acquire the input signal as long as there is sufficient data to characterize the characteristic of the transient process. And the characteristic time of the transient process approximates to 0.45 second in this K-Rb-21Ne co-magnetometer, that means the required transient signal could be shorter than the 2-second signal in Fig. 5(a). The characteristic time of this K-Rb-21Ne co-magnetometer could be potentially reduced by optimizing the operation condition. And the time of transient process could be significant reduced by using K-3He co-magnetometer, whose characteristic time approximates to 0.05 second [19]. Therefore, the measurement time can be potentially diminished by reducing the characteristic time of co-magnetometer and fitting a short time transient signal.

 figure: Fig. 5

Fig. 5 (a) The fitted By as function of the input By. The red dots denote the measured data, while the function of the black line is y = x. The response signals to each input signal are measured five times, of which the standard deviations of the fitted magnetic fields are illustrated as the error bars. (b) The corresponding residuals between the fitted By and the input By. The horizontal axis is in logarithmic coordinates. (c) The measured and fitted responses to small magnetic field. The black solid curve is the measured probe signal without input magnetic field. The circle point, asterisk point and upward-pointing triangle point are the measured signals to 1.6 pT, 8 pT, and 16 pT respectively, while the red solid curve, blue solid curve and green solid curve are the fitted curves to 1.6 pT, 8 pT, and 16 pT respectively.

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Finally, the validity of this method in different operation temperature and pumping light power density is also studied. In Fig. 6(a) and 6(b), the pumping light power density changes from 58.2 mW cm−2, 81.3 mW cm−2 to 97.3 mW cm−2, while the temperature holds at 459 K. In Fig. 6(c) and 6(d), the temperature ranges within typical experimental conditions from 437 K, 448 K to 459 K with the pumping light power density fixed on 81.3 mW cm−2. Under each of these conditions, the magnetic field zeroing procedure is executed to eliminate the residual magnetic field and find the new compensation point after the system reaching equilibrium state. The system parameters, determined by the fitting of the response signal to step magnetic field By0 = 0.32nT under each of these conditions, are used to fit the responses of various input magnetic fields. As shown in Fig. 6, the responses to various input magnetic field are recorded and fitted with Eq. (2) and Eq. (3). The fitted curves are consistent with the measured signals because the corresponding goodness of fit are superior to 0.99 in Fig. 6(a) and (c). The reason why the goodness of fit of signals with smaller input magnetic field are slightly poor is identical to the one described above. In Fig. 6(b) and 6(d), the function of the black solid lines is y = x, and the lines are plotted to indicate the deviation of the fitted magnetic field and the input magnetic field. The fitted magnetic field values are all close to the solid lines, which indicate that the fitted results are approximate to the input signals. And for clarity, these fitted magnetic fields are fitted by y = ax + b. The slopes of the fitting lines for the results in Fig. 6(b) with pumping light power density 58.2 mW cm−2, 81.3 mW cm−2 and 97.3 mW cm−2 are 1.048, 1.013 and 0.9677 respectively, and the corresponding R-square are all better than 0.99. The slopes of the fitting lines for the results in Fig. 6(d) with temperature 437 K, 448 K and 459 K are 1.093, 1.081 and 1.044 respectively, and the corresponding R-square are also all better than 0.99. Therefore, this method is still feasible in different typical temperature and pumping light power density.

 figure: Fig. 6

Fig. 6 The fitting results of response signals to various input magnetic field in different conditions. (a) and (b) denote the fitted R-square and By with different pumping light power density respectively. (c) and (d) designate the fitted R-square and By with different operation temperature respectively. The function of the black line in (b), identical to the one in (d), is y = x. The insets in (b) and (d) show the magnification of the fitted By with the input magnetic field of about 0.288 nT and 0.672 nT respectively. In (a) and (b), the response signals to each input magnetic field are measured five times under different pumping light power density, of which the standard deviations of the R-square and the fitted magnetic fields are plotted as the error bars respectively. While in (c) and (d), the definitions of the error bars are the same as (a) and (b) except for the response signals are measured under different operation temperature.

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5. Conclusion

In conclusion, we have characterized the dynamic performances of coupled spin ensembles in K-Rb-21Ne co-magnetometer using Bloch equations. We have also studied the operation of the system as a self-compensation co-magnetometer that can be used to measure magnetic field in transient process, as well as the theoretical expression of the transient signal. The stability of this model to different input signals over time and the linear relationship between input signals and fitting results have been verified. Meanwhile, this method has been proved to be valid in a set of typical temperature and pumping light power density. The measurement range of this method is from 0.016 nT to 1.760 nT, and the measurement error is lower than 0.01 nT, which can be further possibly improved by reducing the noise of the magnetic field and the fluctuation of the pumping light power. This method can be utilized in small magnetic field measurement situation demanding high precision and miniaturization. The parameter λ1r, acquired by fitting the transient signal to step transverse magnetic field, is a function of δ Bz and could be potentially utilized to zero δ Bz to find a better compensation condition for co-magnetometers. And it is capable of improving the sensitivity of inertial rotation measurement based on this co-magnetometer by accurately eliminating the disturbance of magnetic field. Moreover, this method could be possibly developed to measure magnetic field and inertial rotation simultaneously.

Appendix

Here detailed solutions to the Bloch equations are provided. With small transverse excitation, the longitudinal polarization Pze and Pzn nearly remain constant, therefore the resulting 4 × 4 system of Bloch equations for the transverse components Pxe, Pye, Pxn and Pyn, becomes linear [11, 13]. Utilizing P˜e=Pxe+iPye and P˜n=Pxn+iPyn, the 4 × 4 system of equations can be written in the 2 × 2 system of equations:

P˜et={RtoteQ+i[γeQ(Bz+λMnPzn+Lz)Ωz]}P˜e+(RseneQiγeλMnPzeQ)P˜n+Pzeγe(ByiBx)/Q,
P˜nt=(RseeniγnλMePzn)P˜e+{Rtotn+i[γn(Bz+λMePze)Ωz]}P˜n+Pznγn(ByΩyiBx+iΩx).

The standard methods of solving differential equations can be used, and the formal solution is given by:

P˜e(t)=P1eλ1t+P2eλ2t+Pesteady,
Pxe(t)=Re[P˜e]=Re[P1eλ1t+P2eλ2t]+Pxesteady,
where P1 = P1r +i P1i, P2 = P2r +i P2i, λ1 = λ1r +i λ1i and λ2 = λ2r +i λ2i.

For the experimentally realized K-Rb-21Ne co-magnetometer, RseenγnλMePzn, RseneγeλMnPze and γe(Bz+λMn+Pzn+Lz)/QΩz, hence Rseen, Rsene and Ωz are set to zero. Since Rtote/QRtotn and Bz=λMnPznλMePzn at compensation point, the λ1 and λ2 are represented by:

λ1=λ1r+iλ1i=Rtote2Q+a2+b2+a22+i[(γe(LzBze)2QγnBzn2)+Sign[b]a2+b2a22],
λ2=λ2r+iλ2i=Rtote2Q+a2+b2+a22+i[(γe(LzBze)2QγnBzn2)Sign[b]a2+b2a22],
a=(RtoteQ)2(γeLzBzeQ+γnBzn)24γeBzeγnBznQ,
b=2RtoteQ(γeLzBzeQ+γnBzn)4γeBzeγnBznQ,

The general form of output signal Sx, which is proportional to the transverse polarization Pxe including transient and steady components, is given by:

Sx=KdPxe=eλ1rt(P1rcosλ1itP1isinλ1it)+eλ2rt(P2rcosλ2itP2isinλ2it)+Sxsteady.
Here Kd is the factor that converts the transverse polarization of Rb spin ensemble to output voltage signal by the probe system ( Sy=KdPye); λ1r, λ1i, λ2r and λ2i are determined by the system itself; P1r, P1i, P2r , P2i and Sxsteady depend on the input signals.

At the compensation point, the parameters P1r, P1i, P2r , P2i and Sxsteady can be specifically rewritten with different types of change in transverse magnetic field, for example step-like change and sinusoidal change. When the change is step-like with amplitude By0 (here we primarily consider the change of By and ignore the changes of Bx, Ωx and Ωy for the benefit of simplicity) and the horizontal component of earth rotation Ωy is along the y-axis, these parameters can be specifically rewritten.

By setting ∂ P̃e /∂t = 0 and ∂ P̃n /∂t = 0, the steady signal is:

Sxsteady=KdPxesteady=KdγePzeRtoteRtote2+γe2(Lz+δBz)2(δBzBnBy+Ωyγn),
where δ Bz = BzBc.

And P1r , P1i, P2r , P2i can be found by substituting the initial values ( Sx(0)=Sx0, Sy(0)=Sy0, Sxt|t=0=(γeBy0QΩy)KdPze and Syt|t=0=0), and are given by:

P1r=KrBy0+K1r0Ωy,P1i=KiBy0+K1i0Ωy,
P2r=KrBy0+K2r0Ωy,P2i=KiBy0+K2i0Ωy,
Kr=λ1rλ2r(λ2rλ1r)2+(λ2iλ1i)2KdγePzeQ,Ki=λ2iλ1i(λ2rλ1r)2+(λ2iλ1i)2KdγePzeQ.
The expressions for parameters K1r0, K1i0, K2r0 and K2i0 are similar to Kr and Ki.

Funding

National Natural Science Foundation of China (NSFC) (61227902, 61374210); National Key R&D Program of China (2016YFB0501601).

References and links

1. P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009). [CrossRef]  

2. M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

3. G. Bison, R. Wynands, and A. Weis, “Dynamical mapping of the human cardiomagnetic field with a room-temperature, laser-optical sensor,” Opt. Express 11(8), 904 (2003). [CrossRef]   [PubMed]  

4. H. J. Lee, J. H. Shim, H. S. Moon, and K. Kim, “Flat-response spin-exchange relaxation free atomic magnetometer under negative feedback,” Opt. Express 22(17), 19887 (2014). [CrossRef]   [PubMed]  

5. D. Drung, “High-performance DC SQUID read-out electronics,” Physica C 368(1–4), 134–140 (2002). [CrossRef]  

6. I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003). [CrossRef]   [PubMed]  

7. H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010). [CrossRef]  

8. D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013). [CrossRef]   [PubMed]  

9. D. Budker and M. V. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007). [CrossRef]  

10. M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011). [CrossRef]   [PubMed]  

11. T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95(23), 230801 (2005). [CrossRef]   [PubMed]  

12. G. Vasilakis, “Precision measurements of spin interactions with high density atomic vapors,” Ph.D. dissertation, Princeton Univ., Princeton, NJ, USA, (2011).

13. J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016). [CrossRef]  

14. Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016). [CrossRef]  

15. E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003). [CrossRef]   [PubMed]  

16. M. V. Romalis, “Hybrid optical pumping of optically dense alkali-metal vapor without quenching gas,” Phys. Rev. Lett. 105(24), 243001 (2010). [CrossRef]  

17. T. G. Walker and W. Happer, “Spin-exchange optical pumping of noble-gas nuclei,” Rev. Mod. Phys. 69(2), 629–642 (1997). [CrossRef]  

18. R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016). [CrossRef]  

19. T. W. Kornack and M. V. Romalis, “Dynamics of two overlapping spin ensembles interacting by spin exchange,” Phys. Rev. Lett. 89(25), 253002 (2002). [CrossRef]   [PubMed]  

20. S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989). [CrossRef]  

21. R. K. Ghosh and M. V. Romalis, “Measurement of spin-exchange and relaxation parameters for polarizing 21Ne with K and Rb,” Phys. Rev. A 81(4), 043415 (2010). [CrossRef]  

22. I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A 71(2), 023405 (2005). [CrossRef]  

23. J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002). [CrossRef]   [PubMed]  

24. Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016). [CrossRef]  

25. L. Duan, J. Fang, R. Li, L. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express , 23(25), 32481 (2015). [CrossRef]   [PubMed]  

References

  • View by:

  1. P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
    [Crossref]
  2. M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).
  3. G. Bison, R. Wynands, and A. Weis, “Dynamical mapping of the human cardiomagnetic field with a room-temperature, laser-optical sensor,” Opt. Express 11(8), 904 (2003).
    [Crossref] [PubMed]
  4. H. J. Lee, J. H. Shim, H. S. Moon, and K. Kim, “Flat-response spin-exchange relaxation free atomic magnetometer under negative feedback,” Opt. Express 22(17), 19887 (2014).
    [Crossref] [PubMed]
  5. D. Drung, “High-performance DC SQUID read-out electronics,” Physica C 368(1–4), 134–140 (2002).
    [Crossref]
  6. I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
    [Crossref] [PubMed]
  7. H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010).
    [Crossref]
  8. D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
    [Crossref] [PubMed]
  9. D. Budker and M. V. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007).
    [Crossref]
  10. M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
    [Crossref] [PubMed]
  11. T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95(23), 230801 (2005).
    [Crossref] [PubMed]
  12. G. Vasilakis, “Precision measurements of spin interactions with high density atomic vapors,” Ph.D. dissertation, Princeton Univ., Princeton, NJ, USA, (2011).
  13. J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
    [Crossref]
  14. Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
    [Crossref]
  15. E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
    [Crossref] [PubMed]
  16. M. V. Romalis, “Hybrid optical pumping of optically dense alkali-metal vapor without quenching gas,” Phys. Rev. Lett. 105(24), 243001 (2010).
    [Crossref]
  17. T. G. Walker and W. Happer, “Spin-exchange optical pumping of noble-gas nuclei,” Rev. Mod. Phys. 69(2), 629–642 (1997).
    [Crossref]
  18. R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
    [Crossref]
  19. T. W. Kornack and M. V. Romalis, “Dynamics of two overlapping spin ensembles interacting by spin exchange,” Phys. Rev. Lett. 89(25), 253002 (2002).
    [Crossref] [PubMed]
  20. S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
    [Crossref]
  21. R. K. Ghosh and M. V. Romalis, “Measurement of spin-exchange and relaxation parameters for polarizing 21Ne with K and Rb,” Phys. Rev. A 81(4), 043415 (2010).
    [Crossref]
  22. I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A 71(2), 023405 (2005).
    [Crossref]
  23. J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
    [Crossref] [PubMed]
  24. Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
    [Crossref]
  25. L. Duan, J. Fang, R. Li, L. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express,  23(25), 32481 (2015).
    [Crossref] [PubMed]

2016 (4)

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

2015 (1)

2014 (1)

2013 (1)

D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
[Crossref] [PubMed]

2011 (1)

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

2010 (3)

M. V. Romalis, “Hybrid optical pumping of optically dense alkali-metal vapor without quenching gas,” Phys. Rev. Lett. 105(24), 243001 (2010).
[Crossref]

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010).
[Crossref]

R. K. Ghosh and M. V. Romalis, “Measurement of spin-exchange and relaxation parameters for polarizing 21Ne with K and Rb,” Phys. Rev. A 81(4), 043415 (2010).
[Crossref]

2009 (1)

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

2007 (1)

D. Budker and M. V. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007).
[Crossref]

2005 (3)

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95(23), 230801 (2005).
[Crossref] [PubMed]

I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A 71(2), 023405 (2005).
[Crossref]

2003 (3)

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

G. Bison, R. Wynands, and A. Weis, “Dynamical mapping of the human cardiomagnetic field with a room-temperature, laser-optical sensor,” Opt. Express 11(8), 904 (2003).
[Crossref] [PubMed]

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref] [PubMed]

2002 (3)

D. Drung, “High-performance DC SQUID read-out electronics,” Physica C 368(1–4), 134–140 (2002).
[Crossref]

T. W. Kornack and M. V. Romalis, “Dynamics of two overlapping spin ensembles interacting by spin exchange,” Phys. Rev. Lett. 89(25), 253002 (2002).
[Crossref] [PubMed]

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref] [PubMed]

1997 (1)

T. G. Walker and W. Happer, “Spin-exchange optical pumping of noble-gas nuclei,” Rev. Mod. Phys. 69(2), 629–642 (1997).
[Crossref]

1989 (1)

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Allred, J. C.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref] [PubMed]

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref] [PubMed]

Anderson, L. W.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

Babcock, E.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

Bison, G.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

G. Bison, R. Wynands, and A. Weis, “Dynamical mapping of the human cardiomagnetic field with a room-temperature, laser-optical sensor,” Opt. Express 11(8), 904 (2003).
[Crossref] [PubMed]

Brown, J. M.

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

Budker, D.

D. Budker and M. V. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007).
[Crossref]

Castagna, N.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Cates, G. D.

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Chen, Y.

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

Cheuk, L. W.

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

Chien, T. R.

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Dang, H. B.

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010).
[Crossref]

Ding, M.

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

L. Duan, J. Fang, R. Li, L. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express,  23(25), 32481 (2015).
[Crossref] [PubMed]

Driehuys, B.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

Drung, D.

D. Drung, “High-performance DC SQUID read-out electronics,” Physica C 368(1–4), 134–140 (2002).
[Crossref]

Duan, L.

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

L. Duan, J. Fang, R. Li, L. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express,  23(25), 32481 (2015).
[Crossref] [PubMed]

Dural, N.

D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
[Crossref] [PubMed]

Fan, W.

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

Fang, J.

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

L. Duan, J. Fang, R. Li, L. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express,  23(25), 32481 (2015).
[Crossref] [PubMed]

Ghosh, R. K.

R. K. Ghosh and M. V. Romalis, “Measurement of spin-exchange and relaxation parameters for polarizing 21Ne with K and Rb,” Phys. Rev. A 81(4), 043415 (2010).
[Crossref]

T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95(23), 230801 (2005).
[Crossref] [PubMed]

Gonatas, D.

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Grauch, V. J. S.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Hansen, R. O.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Happer, W.

T. G. Walker and W. Happer, “Spin-exchange optical pumping of noble-gas nuclei,” Rev. Mod. Phys. 69(2), 629–642 (1997).
[Crossref]

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Hersman, F. W.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

Hofer, A.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Hu, Z.

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Jiang, L.

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

L. Duan, J. Fang, R. Li, L. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express,  23(25), 32481 (2015).
[Crossref] [PubMed]

Kadlecek, S.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

Kim, K.

Knowles, P.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Kominis, I. K.

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref] [PubMed]

Kornack, T. W.

T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95(23), 230801 (2005).
[Crossref] [PubMed]

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref] [PubMed]

T. W. Kornack and M. V. Romalis, “Dynamics of two overlapping spin ensembles interacting by spin exchange,” Phys. Rev. Lett. 89(25), 253002 (2002).
[Crossref] [PubMed]

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref] [PubMed]

LaFehr, T. R.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Lee, H. J.

Li, R.

Li, S.

D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
[Crossref] [PubMed]

Li, Y.

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Liu, X.

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Lu, Y.

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

Lyman, R. N.

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref] [PubMed]

Maloof, A. C.

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010).
[Crossref]

Moon, H. S.

Mtchedlishvili, A.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Nabighian, M. N.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Nelson, I.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

Pazgalev, A.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Peirce, J. W.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Phillips, J. D.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Quan, W.

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

Romalis, M. V.

D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
[Crossref] [PubMed]

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

M. V. Romalis, “Hybrid optical pumping of optically dense alkali-metal vapor without quenching gas,” Phys. Rev. Lett. 105(24), 243001 (2010).
[Crossref]

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010).
[Crossref]

R. K. Ghosh and M. V. Romalis, “Measurement of spin-exchange and relaxation parameters for polarizing 21Ne with K and Rb,” Phys. Rev. A 81(4), 043415 (2010).
[Crossref]

D. Budker and M. V. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007).
[Crossref]

T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95(23), 230801 (2005).
[Crossref] [PubMed]

I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A 71(2), 023405 (2005).
[Crossref]

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref] [PubMed]

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref] [PubMed]

T. W. Kornack and M. V. Romalis, “Dynamics of two overlapping spin ensembles interacting by spin exchange,” Phys. Rev. Lett. 89(25), 253002 (2002).
[Crossref] [PubMed]

Ruder, M. E.

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

Savukov, I. M.

I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A 71(2), 023405 (2005).
[Crossref]

Schaefer, S. R.

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Sheng, D.

D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
[Crossref] [PubMed]

Shim, J. H.

Smiciklas, M.

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

Smullin, S. J.

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

Vasilakis, G.

G. Vasilakis, “Precision measurements of spin interactions with high density atomic vapors,” Ph.D. dissertation, Princeton Univ., Princeton, NJ, USA, (2011).

Walker, T. G.

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

T. G. Walker and W. Happer, “Spin-exchange optical pumping of noble-gas nuclei,” Rev. Mod. Phys. 69(2), 629–642 (1997).
[Crossref]

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

Wang, W.

Weis, A.

Weisa, A.

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Wynands, R.

Yuan, H.

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Zhang, H.

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

Zou, S.

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Appl. Phys. Lett. (1)

H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010).
[Crossref]

Geophysics (1)

M. N. Nabighian, V. J. S. Grauch, R. O. Hansen, T. R. LaFehr, Y. Li, J. W. Peirce, J. D. Phillips, and M. E. Ruder, “The historical development of the magnetic method in exploration,” Geophysics 70(6), 33ND–61ND (2005).

J. Phys. B (1)

J. Fang, Y. Chen, S. Zou, X. Liu, Z. Hu, W. Quan, H. Yuan, and M. Ding, “Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field,” J. Phys. B 49(6), 065006 (2016).
[Crossref]

Nat. Phys. (1)

D. Budker and M. V. Romalis, “Optical magnetometry,” Nat. Phys. 3(4), 227–234 (2007).
[Crossref]

Nature (1)

I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003).
[Crossref] [PubMed]

Nucl. Instrum. Methods Phys. Res., Sect. A (1)

P. Knowles, G. Bison, N. Castagna, A. Hofer, A. Mtchedlishvili, A. Pazgalev, and A. Weisa, “Laser-driven Cs magnetometer arrays for magnetic field measurement and control,” Nucl. Instrum. Methods Phys. Res., Sect. A 611(2–3), 306–309 (2009).
[Crossref]

Opt. Express (3)

Phys. Rev. A (5)

Y. Chen, W. Quan, L. Duan, Y. Lu, L. Jiang, and J. Fang, “Spin-exchange collision mixing of the K and Rb ac Stark shifts,” Phys. Rev. A 94(5), 052705 (2016).
[Crossref]

R. Li, W. Fan, L. Jiang, L. Duan, W. Quan, and J. Fang, “Rotation sensing using a K-Rb-21Ne comagnetometer,” Phys. Rev. A 94(3), 032109 (2016).
[Crossref]

S. R. Schaefer, G. D. Cates, T. R. Chien, D. Gonatas, W. Happer, and T. G. Walker, “Frequency shifts of the magnetic-resonance spectrum of mixtures of nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal atoms,” Phys. Rev. A 39(11), 5613–5623 (1989).
[Crossref]

R. K. Ghosh and M. V. Romalis, “Measurement of spin-exchange and relaxation parameters for polarizing 21Ne with K and Rb,” Phys. Rev. A 81(4), 043415 (2010).
[Crossref]

I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A 71(2), 023405 (2005).
[Crossref]

Phys. Rev. Lett. (7)

J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002).
[Crossref] [PubMed]

T. W. Kornack and M. V. Romalis, “Dynamics of two overlapping spin ensembles interacting by spin exchange,” Phys. Rev. Lett. 89(25), 253002 (2002).
[Crossref] [PubMed]

D. Sheng, S. Li, N. Dural, and M. V. Romalis, “Subfemtotesla scalar atomic magnetometry using multipass cells,” Phys. Rev. Lett. 110(16), 160802 (2013).
[Crossref] [PubMed]

M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21Ne-Rb-K comagnetometer,” Phys. Rev. Lett. 107(17), 171604 (2011).
[Crossref] [PubMed]

T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95(23), 230801 (2005).
[Crossref] [PubMed]

E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Anderson, F. W. Hersman, and T. G. Walker, “Hybrid spin-exchange optical pumping of 3He,” Phys. Rev. Lett. 91(12), 123003 (2003).
[Crossref] [PubMed]

M. V. Romalis, “Hybrid optical pumping of optically dense alkali-metal vapor without quenching gas,” Phys. Rev. Lett. 105(24), 243001 (2010).
[Crossref]

Physica C (1)

D. Drung, “High-performance DC SQUID read-out electronics,” Physica C 368(1–4), 134–140 (2002).
[Crossref]

Rev. Mod. Phys. (1)

T. G. Walker and W. Happer, “Spin-exchange optical pumping of noble-gas nuclei,” Rev. Mod. Phys. 69(2), 629–642 (1997).
[Crossref]

Sci. Rep. (1)

Y. Chen, W. Quan, S. Zou, Y. Lu, L. Duan, Y. Li, H. Zhang, M. Ding, and J. Fang, “Spin exchange broadening of magnetic resonance lines in a high-sensitivity rotating K-Rb-21Ne co-magnetometer,” Sci. Rep. 6, 36547 (2016).
[Crossref]

Other (1)

G. Vasilakis, “Precision measurements of spin interactions with high density atomic vapors,” Ph.D. dissertation, Princeton Univ., Princeton, NJ, USA, (2011).

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Figures (6)

Fig. 1
Fig. 1 Intuitive compensation principle of self-compensation co-magnetometer. (a) Without transverse magnetic field By, P e and P n direct along z-axis. (b) When applying By, P e and P n would precess around the total magnetic field in the transient process. (c) P n would stabilize at the direction in y–z plane in a short time, resulting in the projection of B n with respect to y-axis compensating By, hence P e returns back to z-axis unaffected by By in steady state.
Fig. 2
Fig. 2 The schematic of the co-magnetometer. GT, Glan-Thompson polarizer; PD, photo detector; PBS, polarization beam splitter; ECDL, external cavity diode laser; TA, tapered amplifier; BE, beam expander; DFB, distributed feedback laser; PEM, photo elastic modulator.
Fig. 3
Fig. 3 The responses to transverse magnetic field By0 = 0.32 nT with various bias magnetic field. (a) The bias magnetic field Bz is set to the compensation point Bc = −312.7 nT. (b) Bz is set to the non-compensation point Bc = −362.7 nT. (c) The difference between the initial offset value and the steady state value Δ S x 0 is measured with same input magnetic field By0 = 0.32 nT and various bias magnetic field δ Bz.
Fig. 4
Fig. 4 The stability of the system parameters. (a) The variation of the four fitted system parameters over time. The response signals to By0 = 0.32nT at every half hour are continuously measured five times, of which the standard deviations of the fitted system parameters are plotted as the error bars. (b) The R-square of the fitting curves with different input magnetic field. The response signals to each input magnetic field are measured five times, of which the standard deviations of the fitted R-square are regarded as the error bars.
Fig. 5
Fig. 5 (a) The fitted By as function of the input By. The red dots denote the measured data, while the function of the black line is y = x. The response signals to each input signal are measured five times, of which the standard deviations of the fitted magnetic fields are illustrated as the error bars. (b) The corresponding residuals between the fitted By and the input By. The horizontal axis is in logarithmic coordinates. (c) The measured and fitted responses to small magnetic field. The black solid curve is the measured probe signal without input magnetic field. The circle point, asterisk point and upward-pointing triangle point are the measured signals to 1.6 pT, 8 pT, and 16 pT respectively, while the red solid curve, blue solid curve and green solid curve are the fitted curves to 1.6 pT, 8 pT, and 16 pT respectively.
Fig. 6
Fig. 6 The fitting results of response signals to various input magnetic field in different conditions. (a) and (b) denote the fitted R-square and By with different pumping light power density respectively. (c) and (d) designate the fitted R-square and By with different operation temperature respectively. The function of the black line in (b), identical to the one in (d), is y = x. The insets in (b) and (d) show the magnification of the fitted By with the input magnetic field of about 0.288 nT and 0.672 nT respectively. In (a) and (b), the response signals to each input magnetic field are measured five times under different pumping light power density, of which the standard deviations of the R-square and the fitted magnetic fields are plotted as the error bars respectively. While in (c) and (d), the definitions of the error bars are the same as (a) and (b) except for the response signals are measured under different operation temperature.

Equations (16)

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P e t = γ e Q ( B + λ M n P n + L ) × P e Ω × P e + R m S m + R p S p + R se ne P n Q P e Q { T 1 , T 2 , T 2 } , P n t = γ n ( B + λ M e P e ) × P n Ω × P n + R se en P e R tot n P n .
S x = K d P x e = e λ 1 r t ( P 1 r cos λ 1 i t P 1 i sin λ 1 i t ) + e λ 2 r t ( P 2 r cos λ 2 i t P 2 i sin λ 2 i t ) + S x steady , λ 1 = λ 1 r + i λ 1 i = R tot e 2 Q + a 2 + b 2 + a 2 2 + i [ ( γ e ( L z B z e ) 2 Q γ n B z n 2 ) + Sign [ b ] a 2 + b 2 a 2 2 ] , λ 2 = λ 2 r + i λ 2 i = R tot e 2 Q + a 2 + b 2 + a 2 2 + i [ ( γ e ( L z B z e ) 2 Q γ n B z n 2 ) Sign [ b ] a 2 + b 2 a 2 2 ] , a = ( R tot e Q ) 2 ( γ e L z B z e Q + γ n B z n ) 2 4 γ e B z e γ n B z n Q , b = 2 R tot e Q ( γ e L z B z e Q + γ n B z n ) 4 γ e B z e γ n B z n Q ,
P 1 r = K r B y 0 + K 1 r 0 Ω y , P 1 i = K i B y 0 + K 1 i 0 Ω y , P 2 r = K r B y 0 + K 2 r 0 Ω y , P 2 i = K i B y 0 + K 2 i 0 Ω y , K r = λ 1 r λ 2 r ( λ 2 r λ 1 r ) 2 + ( λ 2 i λ 1 i ) 2 K d γ e P z e Q , K i = λ 2 i λ 1 i ( λ 2 r λ 1 r ) 2 + ( λ 2 i λ 1 i ) 2 K d γ e P z e Q , S x steady = K steady ( δ B z B n B y + Ω y γ n ) , K steady = K d γ e P z e R tot e R tot e + γ e 2 ( L z + δ B z ) 2 .
P ˜ e t = { R tot e Q + i [ γ e Q ( B z + λ M n P z n + L z ) Ω z ] } P ˜ e + ( R se ne Q i γ e λ M n P z e Q ) P ˜ n + P z e γ e ( B y i B x ) / Q ,
P ˜ n t = ( R se en i γ n λ M e P z n ) P ˜ e + { R tot n + i [ γ n ( B z + λ M e P z e ) Ω z ] } P ˜ n + P z n γ n ( B y Ω y i B x + i Ω x ) .
P ˜ e ( t ) = P 1 e λ 1 t + P 2 e λ 2 t + P e steady ,
P x e ( t ) = Re [ P ˜ e ] = Re [ P 1 e λ 1 t + P 2 e λ 2 t ] + P x e steady ,
λ 1 = λ 1 r + i λ 1 i = R tot e 2 Q + a 2 + b 2 + a 2 2 + i [ ( γ e ( L z B z e ) 2 Q γ n B z n 2 ) + Sign [ b ] a 2 + b 2 a 2 2 ] ,
λ 2 = λ 2 r + i λ 2 i = R tot e 2 Q + a 2 + b 2 + a 2 2 + i [ ( γ e ( L z B z e ) 2 Q γ n B z n 2 ) Sign [ b ] a 2 + b 2 a 2 2 ] ,
a = ( R tot e Q ) 2 ( γ e L z B z e Q + γ n B z n ) 2 4 γ e B z e γ n B z n Q ,
b = 2 R tot e Q ( γ e L z B z e Q + γ n B z n ) 4 γ e B z e γ n B z n Q ,
S x = K d P x e = e λ 1 r t ( P 1 r cos λ 1 i t P 1 i sin λ 1 i t ) + e λ 2 r t ( P 2 r cos λ 2 i t P 2 i sin λ 2 i t ) + S x steady .
S x steady = K d P x e steady = K d γ e P z e R tot e R tot e 2 + γ e 2 ( L z + δ B z ) 2 ( δ B z B n B y + Ω y γ n ) ,
P 1 r = K r B y 0 + K 1 r 0 Ω y , P 1 i = K i B y 0 + K 1 i 0 Ω y ,
P 2 r = K r B y 0 + K 2 r 0 Ω y , P 2 i = K i B y 0 + K 2 i 0 Ω y ,
K r = λ 1 r λ 2 r ( λ 2 r λ 1 r ) 2 + ( λ 2 i λ 1 i ) 2 K d γ e P z e Q , K i = λ 2 i λ 1 i ( λ 2 r λ 1 r ) 2 + ( λ 2 i λ 1 i ) 2 K d γ e P z e Q .

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