Introduction

In recent years, ultrahigh sensitive magnetic field measurement technology has become a hotspot in research of weak magnetic field. In the field of biomedicine, it is used in magnetoencephalography (MEG) and magnetocardiography (MCG)1,2,3. In physics, it is used to analyze the magnetism of material and measure the symmetry broken of charge conjugation, parity transformation and time reversal (CPT)4,5,6. At present, the sensitivity of the spin exchange relaxation free (SERF) atomic magnetometer is the highest in the ultrahigh sensitive magnetometers3, 7,8,9,10. The shot-noise limit of the K SERF magnetometer11 is estimated to be 2 aT/Hz 1/2 and with more optimization, it should be possible to approach the shot-noise-limited sensitivity in the range 10 − 1 aT/Hz 1/2 for K SERF magnetometer10. The effects of the spin-exchange relaxation can be suppressed in the SERF regime, when the spin-exchange rate is much larger than the Larmor precession frequency12, 13. The SERF regime can be reached by operating with sufficiently high alkali metal number density (at higher temperature) and in sufficiently low magnetic field13, 14.

It was found that hybrid optical pumping can make the SERF magnetometer realize higher experimental detecting sensitivity and more homogeneous atomic spin polarization15 and it is suitable for quantum nondestructive measurement16. Ito et al.17, 18 realized a sensitivity of 3 × 104 aT/Hz 1/2 in magnetic field measurement by SERF atomic magnetometers by hybrid optical pumping of K-Rb. Fang et al.19 obtained a sensitivity of approximately 5 × 103 aT/Hz 1/2 by optimizing the parameters of SERF magnetometer based on K-Rb hybrid optical pumping. Li et al.20 measured the magnetic field sensitivity better than 700 aT/Hz 1/2 by a subfemtotesla atomic magnetometer based on hybrid optical pumping of K-Rb. However, there is almost no work about the systematic analysis of the influence factors on the polarization and the fundamental sensitivity of K (Cs)-Rb-He hybrid optical pumping SERF atomic magnetometers. We need more practical methods to obtain higher fundamental sensitivity of the hybrid optical pumping SERF atomic magnetometer.

In this report, we obtain a general formula on the fundamental sensitivity of the hybrid optical pumping SERF magnetometer, which describes the fundamental sensitivity of the magnetometer varying with the number density of buffer gas and quench gas, pumping rate of pump beam, external magnetic field, cell effective radius (the shape of the cell is roughly spherical), measurement volume, cell temperature and measurement time. We have investigated two types of hybrid optical pumping SERF atomic magnetometers based on 39K (133Cs)-85Rb-4He (39K (133Cs)-85Rb-4He magnetometers), then found the fundamental sensitivity of 133Cs-85Rb-4He magnetometer is lower than the one of 39K-85Rb-4He magnetometer at the same cell temperature and in the SERF regime when the pumping rate of pump beam is bigger than about 1916 s −1 and N2 number density is bigger than about 1.974×1016 cm -3 at our chosen conditions. Optimizing the magnetometer parameters is advantageous to improve the sensitivity of the magnetometer in measuring weak magnetic field. Furthermore, we obtained a higher fundamental sensitivity of 1.8359 × 10−2 aT/Hz 1/2 with 39K-85Rb-4He magnetometer when the polarization of 85Rb atom is 1.3174 × 10−4 and the fundamental sensitivity is higher than the shot-noise-limited sensitivity of K SERF atomic magnetometer10 in the range 10 − 1 aT/Hz 1/2. Among 39K, 85Rb and 133Cs SERF magnetometers, there is a maximum temperature range for 39K to make the magnetometer in the SERF regime with the number density of 39K satisfies the condition of the SERF regime, so the SERF magnetometer based on 39K is suitable for an environment with the temperature varying drastically. These findings not only optimize the parameters for the SERF regime, but also provide an experimental guide for the design of the hybrid optical pumping SERF magnetometer.

Results

The number density of alkali-metal atoms

The alkali metal vapor cell (the shape of the cell is roughly spherical) of the SERF atomic magnetometer based on hybrid optical pumping contains two types of alkali metal atoms, they are 39K-85Rb or 133Cs-85Rb. 133Cs can reach large saturation vapor pressure at lower temperature21 and realize SERF regime at lower temperature, which has more advantages for low temperature conditions. 39K has the highest theoretical sensitivity, so we study the hybrid optical pumping SERF atomic magnetometers based on 39K-85Rb and 133Cs-85Rb respectively. We select 4He as the buffer gas and take N2 as quench gas (that is 39K (133Cs)-85Rb-4He magnetometers). 4He gas suppresses the spin relaxation caused by wall collisions, colliding with excited alkali metal atoms and absorbing the energy, N2 gas restrains radiative deexcitation of the excited alkali metal atoms22. One type of alkali-metal atom which is directly pumped and polarized by a circularly polarized pump beam is called A and the other type of alkali-metal atom which is polarized by the spin-exchange collisions with A is called B in the hybrid optical pumping SERF magnetometer16, 23, we take 39K or 133Cs as A respectively, select 85Rb as B in the SERF regime. The number density of alkali-metal vapor and the polarization of alkali-metal vapor are two most important parameters of the cell24.

The saturated density of the alkali-metal atoms vapor in units of cm -3 at cell temperature T in Kelvin is given by ref. 5

$${n}_{sat}=\frac{1}{T}{10}^{21.866+{A}_{1}-{B}_{1}/T},$$
(1)

where n sat is the saturated density of alkali-metal (39K, 85Rb and 133Cs) atom vapor. When the alkali-metal vapor cell only contains one type of alkali-metal atom (single alkali-metal vapor cell), the number density of the alkali-metal atom equals to the saturated density of the alkali-metal atom, the parameters A 1 and B 1 are phase parameters21, where\({A}_{1}^{K}=4.402,{A}_{1}^{Rb}=4.312,{A}_{1}^{Cs}=4.165,\) \({B}_{1}^{K}=4453,{B}_{1}^{Rb}=4040\,{\rm{and}}\,{B}_{1}^{Cs}=3830\) for the temperature is higher than 400 K.

We can obtain the number density of 39K, 133Cs and 85Rb varying with the cell temperature for the single alkali-metal vapor cell from equation (1) as shown in Fig. 1. When the number density of 39K, 85Rb and 133Cs atom are the same, 39K need the highest temperature. In general, the number density of the alkali metal atoms is 1013  cm −3 to 1014  cm −3 in the SERF regime, we can find that there is a maximum temperature range for 39K to make the magnetometer in the SERF regime with the number density of 39K satisfies the condition of the SERF regime, therefore, the SERF magnetometer based on 39K is suitable for an environment with the temperature varying drastically. If we increase the cell temperature, we can obtain higher number density of alkali-metal atoms, however, the cell glass will be corroded, the laser power and heating equipment will be unable to bear and there will be other problems experimentally when the number density of alkali-metal atoms is greater than or equal to 1015  cm −3. What’s more, the optical depth will be too big so that the laser will be largely absorbed by the atoms. If the vapor cell is made of special glass, the laser power is very big and the volume of the vapor cell is very small, we can appropriately increase the temperature of the vapor cell. Depending on equation (1), when cell temperature T = 457.5 K, for single alkali-metal vapor cell, we obtain the number density of 39K is n K  = 7.4864 × 1013  cm −3, the number density of 85Rb is n Rb  = 9.9776 × 1014  cm −3, the number density of 133Cs is n Cs  = 4.8642 × 1014  cm −3. When T = 457.6 K, we obtain n K  = 7.5216 × 1013  cm −3, n Rb  = 1.0017 × 1015  cm −3, n Cs  = 4.8848 × 1014  cm −3. Because SERF regime can be reached by operating with sufficiently high alkali metal number density (at higher temperature) and in sufficiently low magnetic field13, 14, we choose T = 457.5 K as the highest temperature to reduce the corrosion of alkali metal atoms to the vapor cell and make the magnetometer in the SERF regime.

Figure 1
figure 1

The number density of 39K, 133Cs and 85Rb vary with temperature. Among 39K, 85Rb and 133Cs SERF magnetometers, there is a maximum temperature range for 39K to make the magnetometer in the SERF regime with the number density of 39K (black line in squares) satisfies the condition of the SERF regime, so the SERF magnetometer based on 39K is applicable to the working environment with the temperature varies drastically.

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The polarization of alkali-metal atom

Considering the spin-exchange between two types of alkali-metal atoms A and B in the hybrid vapor cell, here, we assume that the vapor densities obey the Raoult’s law25, n B ≈f B n sat B, where f B is the mole fraction of atom B in the metal and n sat B is the saturated vapor density for pure atom B metal. When the mole fraction of atom B is 0.97, we can obtain the number density of atom A and B, n A  ≈ 0.03n sat A, n B ≈ 0.97n sat B, the spin polarization P of each type of atoms in zero magnetic field can be described as19,

$${P}_{A}=\frac{{R}_{SE}^{B}{P}_{B}+{R}_{p}}{{R}_{p}+{R}_{SD}^{A}+{R}_{SE}^{B}},$$
(2)
$${P}_{B}=\frac{{R}_{SE}^{A}{P}_{A}}{{R}_{SD}^{B}+{R}_{SE}^{A}},$$
(3)
$${R}_{SD}^{A}={n}_{A}{\sigma }_{SD}^{A-A}{\bar{v}}_{A-A}+{n}_{He}{\sigma }_{SD}^{A-He}{\bar{v}}_{A-He}+{n}_{{N}_{2}}{\sigma }_{SD}^{A-{N}_{2}}{\bar{v}}_{A-{N}_{2}}+{n}_{A}{\sigma }_{SD}^{A-B}{\bar{v}}_{A-B},$$
(4)
$${R}_{SD}^{B}={n}_{B}{\sigma }_{SD}^{B-B}{\bar{v}}_{B-B}+{n}_{He}{\sigma }_{SD}^{B-He}{\bar{v}}_{B-He}+{n}_{{N}_{2}}{\sigma }_{SD}^{B-{N}_{2}}{\bar{v}}_{B-{N}_{2}}+{n}_{B}{\sigma }_{SD}^{B-A}{\bar{v}}_{B-A},$$
(5)

where A represents alkali-metal atom 39K or 133Cs, B represents alkali-metal atom 85Rb. \({R}_{SE}^{A}\) is the spin-exchange rate of atom B with atom A, \({R}_{SE}^{A}={k}_{SE}{n}_{A}={n}_{A}{\sigma }_{SE}^{B-A}{\bar{v}}_{B-A},{R}_{SE}^{B}\) is the spin-exchange rate of atom A with atom B, \({R}_{SE}^{B}={k}_{SE}{n}_{B}={n}_{B}{\sigma }_{SE}^{A-B}{\bar{v}}_{A-B}\), R p is the pumping rate of pump beam, which is mainly determined by pumping laser parameters6, R B  = k SE n A P A is the pumping rate of atom B17, R SD is the spin destruction relaxation rate, k SE is the spin-exchange rate constant, n is the number density of atoms, \({\bar{v}}_{alkali-alkali}\) is the relative velocity between the alkali atoms, \({\bar{v}}_{alkali-alkali}=\sqrt{\frac{8{\kappa }_{B}T}{\pi {m}_{alkali-alkali}}}\), κ B is Boltzmann’s constant, T is cell temperature, \({\bar{v}}_{alkali-{N}_{2}}=\sqrt{\frac{8{\kappa }_{B}T}{\pi {m}_{alkali-{N}_{2}}}}\) is the relative velocity between the alkali atoms and quench gas N2 respectively, the reduced mass of alkali atoms and quench gas N2 is \({m}_{alkali-{N}_{2}}=\frac{{m}_{alkali}{m}_{{N}_{2}}}{{m}_{alkali}+{m}_{{N}_{2}}}\), the relative velocity between the alkali atoms and buffer gas 4He is \({\bar{v}}_{alkali-He}=\sqrt{\frac{8{\kappa }_{B}T}{\pi {m}_{alkali-He}}}\), the reduced mass of alkali atoms and buffer gas 4He is \({m}_{alkali-He}=\frac{{m}_{alkali}{m}_{He}}{{m}_{alkali}+{m}_{He}}\), k SE for different alkali metal atoms are nearly the same26,27,28,29,30, k SE  = 〈σ SE v〉, \({k}_{SE}^{Rb-Cs}={k}_{SE}^{Cs-Rb},{k}_{SE}^{Rb-K}={k}_{SE}^{K-Rb}\), the spin-exchange cross section of 39K, 85Rb and 133Cs is \({\sigma }_{SE}^{K}=1.8\times {10}^{-14}\)  cm 2, \({\sigma }_{SE}^{Rb}=1.9\times {10}^{-14}\)  cm 2 and \({\sigma }_{SE}^{Cs}=\) \(2.1\times {10}^{-14}\)  cm 2 respectively27, 31, \({k}_{SE}^{Cs-Rb}={\sigma }_{SE}^{Cs-Rb}\sqrt{\frac{{m}_{Rb}+{m}_{Cs}}{{m}_{Rb}{m}_{Cs}}}\sqrt{\frac{8{\kappa }_{B}T}{\pi }}\,{\rm{and}}\,{k}_{SE}^{K-Rb}={\sigma }_{SE}^{K-Rb}\sqrt{\frac{{m}_{Rb}+{m}_{K}}{{m}_{Rb}{m}_{K}}}\sqrt{\frac{8{\kappa }_{B}T}{\pi }}\), where \({\sigma }_{SE}^{K-Rb}\) is the spin-exchange cross section of 39K and 85Rb by spin-exchange collisions with each other, \({\sigma }_{SE}^{Cs-Rb}\) is the spin-exchange cross section of 133Cs and 85Rb by spin-exchange collisions with each other, for \({\sigma }_{SE}^{{}^{87}Rb-{}^{87}Rb}=\) \(\mathrm{(1.9}\pm \mathrm{0.2)}\times {10}^{-14}\)  cm 2, \({\sigma }_{SE}^{{}^{87}Rb-{}^{133}Cs}=\mathrm{(2.3}\pm \mathrm{0.2)}\times {10}^{-14}\)  cm 2, however, we don’t find the values of \({\sigma }_{SE}^{{}^{39}K-{}^{85}Rb}\) and \({\sigma }_{SE}^{{}^{133}Cs-{}^{85}Rb}\), we take \({\sigma }_{SE}^{{}^{39}K-{}^{85}Rb}={\sigma }_{SE}^{{}^{85}Rb-{}^{39}K}\approx {\sigma }_{SE}^{K}\frac{{n}_{K}}{{n}_{K}+{n}_{Rb}}+{\sigma }_{SE}^{Rb}\frac{{n}_{Rb}}{{n}_{K}+{n}_{Rb}}\), \({\sigma }_{SE}^{{}^{133}Cs-{}^{85}Rb}={\sigma }_{SE}^{{}^{85}Rb-{}^{{\rm{133}}}Cs}\,\approx \) \({\sigma }_{SE}^{Cs}\frac{{n}_{Cs}}{{n}_{Cs}+{n}_{Rb}}+\) \({\sigma }_{SE}^{Rb}\frac{{n}_{Rb}}{{n}_{Cs}+{n}_{Rb}}\) by considering the weight of 39K, 85Rb and 133Cs.

The spin-destruction cross section of 39K-39K, 85Rb-85Rb and 133Cs-133Cs are \({\sigma }_{SD}^{K}=1\times {10}^{-18}\)  cm 2, \({\sigma }_{SD}^{Rb}=1.6\times {10}^{-17}\)  cm 2 and \({\sigma }_{SD}^{Cs}=2\times {10}^{-16}\)  cm 2 respectively32,33,34, \({\sigma }_{SD}^{K-Rb}\), \({\sigma }_{SD}^{Rb-K}\), \({\sigma }_{SD}^{Cs-Rb}\) and \({\sigma }_{SD}^{Rb-Cs}\) are the spin-destruction cross section of 39K and 85Rb, 85Rb and 39K, 133Cs and 85Rb, 85Rb and 133Cs by collisions with each other respectively. However, we do not find the values of \({\sigma }_{SD}^{K-Rb}\), \({\sigma }_{SD}^{Rb-K}\), \({\sigma }_{SD}^{Cs-Rb}\) and \({\sigma }_{SD}^{Rb-Cs}\) in any references, depending on the spin-destruction cross section of 39K-39K, 85Rb-85Rb and 133Cs-133Cs, we take \({\sigma }_{SD}^{K-Rb}={\sigma }_{SD}^{Rb-K}\approx {\sigma }_{SD}^{K}\frac{{n}_{K}}{{n}_{K}+{n}_{Rb}}+{\sigma }_{SD}^{Rb}\frac{{n}_{Rb}}{{n}_{K}+{n}_{Rb}},{\sigma }_{SD}^{Cs-Rb}={\sigma }_{SD}^{Rb-Cs}\approx {\sigma }_{SD}^{Cs}\frac{{n}_{Cs}}{{n}_{Cs}+{n}_{Rb}}+{\sigma }_{SD}^{Rb}\frac{{n}_{Rb}}{{n}_{Cs}+{n}_{Rb}}\) by considering the weight of 39K, 85Rb and 133Cs. The spin-destruction cross section of 4He and 39K, 4He and 85Rb, 4He and 133Cs are \({\sigma }_{SD}^{K-He}=8\times {10}^{-25}\)  cm 2, \({\sigma }_{SD}^{Rb-He}=9\times {10}^{-24}\)  cm 2 and \({\sigma }_{SD}^{Cs-He}=2.8\times {10}^{-23}\)  cm 2 respectively11. The spin-destruction cross section of N2 and 39K, N2 and 85Rb, N2 and 133Cs is \({\sigma }_{SD}^{K-{N}_{2}}=7.9\times {10}^{-23}\)  cm 2, \({\sigma }_{SD}^{Rb-{N}_{2}}=\) \(1\times {10}^{-22}\)  cm 2 and \({\sigma }_{SD}^{Cs-{N}_{2}}=5.5\times {10}^{-22}\)  cm 2 respectively11, 35.

Substitute equation (2) into equation (3), we obtain

$${P}_{B}=\frac{{R}_{p}{R}_{SE}^{A}}{({R}_{SD}^{B}+{R}_{SE}^{A})({R}_{p}+{R}_{SD}^{A}+{R}_{SE}^{B})-{R}_{SE}^{A}{R}_{SE}^{B}},$$
(6)

We take one of 4He number density n He , N2 number density n N2, cell temperature T and pumping rate of pump beam R p by equation (6) as a variable (other parameters are invariable) to obtain the results that the polarization of the hybrid optical pumping SERF magnetometer based on 39K (133Cs)-85Rb-4He respectively vary with the variable. Depending on suggestions and the typical conditions of the experiment group19, 30, 36, in order to facilitate the theoretical analysis, we take the mole fraction of 85Rb f Rb = 0.97, n He  = 1019  cm −3, n N2 = 2 × 1017  cm −3, \({R}_{p}={R}_{p}^{K}={R}_{p}^{Cs}=200000\,{s}^{-1}\) and T = 457.5 K, at the moment, 39K, 85Rb and 133Cs are in the SERF regime.

The number density of 39K, 85Rb and 133Cs vary with cell temperature and the mole fraction for the hybrid alkali-metal vapor cell, so the relation among the polarization of 85Rb, the number density of alkali metal atom (39K, 85Rb or 133Cs) and cell temperature is nonlinear in groups 39K-85Rb and 133Cs-85Rb when the mole fraction of the alkali-metal atoms are fixed. At the same cell temperature, the number density of 39K, 85Rb and 133Cs are different. When the number density of 39K, 85Rb and 133Cs are equivalent, the cell temperature of 39K-85Rb and 133Cs-85Rb magnetometers are different, and the 85Rb polarization of 39K-85Rb and 133Cs-85Rb magnetometers are also different.

Figure 2 demonstrates the 85Rb polarization of 39K (133Cs)-85Rb-4He magnetometers as a function of 39K, 133Cs, 85Rb, 4He, and N2 number density, the pumping rate of pump beam and cell temperature, including the effects of spin exchange due to 39K-85Rb, 85Rb-39K, 133Cs-85Rb, 85Rb-133Cs and spin relaxation due to 39K-39K, 39K-85Rb, 85Rb-85Rb, 133Cs-85Rb, 133Cs-133Cs, 39K-4He, 85Rb-4He, 133Cs-4He collisons, 39K-N2, 85Rb-N2, 133Cs-N2 destructions. The 85Rb polarization almost does not vary with 4He number density when 4He number density is smaller than a critical value about 1020 cm −3 in 39K-85Rb-4He (black line in squares) and 133Cs-85Rb-4He (red line in dots) magnetometers, otherwise, the 85Rb polarization decreases rapidly in Fig. 2(a). The 85Rb polarization almost does not vary with N2 number density when N2 number density is smaller than about 2 × 1019 cm −3 for 39K-85Rb-4He magnetometer and 133Cs-85Rb-4He magnetometer. Otherwise, the 85Rb polarization decreases rapidly in Fig. 2(b). The 85Rb polarization increases with the increasing pumping rate of pump beam \({R}_{p}^{K}\) and \({R}_{p}^{Cs}\) respectively in Fig. 2(c). The polarization of 85Rb decreases with the cell temperature increasing in Fig. 2(d). The 85Rb polarization of 133Cs-85Rb-4He magnetometer is bigger than the one of 39K-85Rb-4He magnetometer in (a)–(d).

Figure 2
figure 2

The 85Rb polarization of 39K (133Cs)-85Rb-4He magnetometers varies with the number density of the buffer gas (4He), the pumping rate of pump beam, number density of the quench gas N2 and the cell temperature respectively. (a) 85Rb polarization almost does not vary with the increasing 4He number density when 4He number density is smaller than a critical value about 1020 cm −3 in 39K -85Rb-4He (black line in squares) and 133Cs-85Rb-4He (red line in dots) magnetometers, otherwise, 85Rb polarization decreases rapidly. (b) 85Rb polarization almost does not vary with increasing N2 number density when N2 number density is smaller than about 2 × 1019 cm −3 in 39K-85Rb-4He and 133Cs-85Rb-4He magnetometers, otherwise, the 85Rb polarization decreases rapidly. (c) 85Rb polarization increases with increasing \({R}_{p}^{K}\) and \({R}_{p}^{Cs}\) respectively. (d) 85Rb polarization decreases with the increasing cell temperature. The 85Rb polarization of 133Cs-85Rb-4He magnetometer is bigger than the one of 39K-85Rb-4He magnetometer in (a)–(d).

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The fundamental sensitivity of the hybrid optical pumping SERF atomic magnetometer

To improve the practicability of the hybrid optical pumping SERF atomic magnetometer, it is necessary for us to investigate the fundamental sensitivity of the magnetometer to improve the sensitivity and stability of the magnetometer and realize the miniaturization of the magnetometer. The fundamental, shot-noise-limited sensitivity of an atomic magnetometer is given by ref. 37

$$\delta B=\frac{1}{\gamma \sqrt{n{T}_{2}Vt}},$$
(7)

it is also the ultimate sensibility of the atomic magnetometer11, where n is the number density of alkali-metal atoms38, γ is their gyromagnetic ratio and the effective γ for sensitivity estimates is \(\gamma =\frac{g{\mu }_{B}}{\hslash }\) (equation (7) of ref. 11) in our magnetometer operating at zero field, we replace it by electron gyromagnetic ratio γ e \(({\gamma }_{e}=\frac{g{\mu }_{B}}{\hslash })\) 5, 38, g is the electron g-factor, μ B is the Bohr magneton, V is the measurement volume, t is the measurement time, T 2 is the transverse spin relaxation time8, \(\frac{1}{{T}_{2}}={R}_{SD}+{R}_{wall}+{R}_{SE}^{ee}\). For the transverse spin relaxation time of the hybrid optical pumping SERF atomic magnetometer, we need consider the spin destruction relaxation R SD caused by He, N2, alkali metal atom A and B, the relaxation rates due to diffusion of alkali metal atoms A and B to the wall6 \({R}_{wall}^{A}\) and \({R}_{wall}^{B}\), the relaxation rate due to alkali-alkali spin-exchange collisions39 \({R}_{SE}^{ee}\), \({R}_{SE}^{ee}={R}_{SE}^{AA}+{R}_{SE}^{AB}+{R}_{SE}^{BA}+{R}_{SE}^{BB}\), which cannot be ignored for large external magnetic field B and is negligible in SERF regime (when T is higher than 418.3 K, B is smaller than 10−10 T, \({R}_{SE}^{ee}\approx 0\)), the pumping rate of pump beam R p and the pumping rate of atom B R B (R B is a function of R p ), buffer gas is 4He, quench gas is N2, therefore, \(\frac{1}{{T}_{2}}={R}_{SD}^{A}+{R}_{SD}^{B}+\) \({R}_{wall}^{A}+{R}_{wall}^{B}+\) \({R}_{SE}^{AA}+{R}_{SE}^{AB}\,+\) \({R}_{SE}^{BA}+{R}_{SE}^{BB}+{R}_{p}+{R}_{B}\), we substitute this term into equation (7) and obtain

$$\delta B=\frac{\sqrt{{R}_{SD}^{A}+{R}_{SD}^{B}+{R}_{wall}^{A}+{R}_{wall}^{B}+{R}_{SE}^{AA}+{R}_{SE}^{AB}+{R}_{SE}^{BA}+{R}_{SE}^{BB}+{R}_{p}+{R}_{B}}}{{\gamma }_{e}\sqrt{nVt}}{\rm{.}}$$
(8)

However, because alkali metal atom B is probed atom, only these items associated with atom B will be considered in the experiments, we don’t consider those items irrelevant to atom B and acquire the fundamental sensitivity of the hybrid optical pumping SERF atomic magnetometer due to the shot-noise as following

$$\delta B^{\prime} =\frac{\sqrt{{R}_{wall}^{B}+{R}_{SD}+{R}_{B}+{R}_{SE}^{BB}+{R}_{SE}^{AB}+{R}_{SE}^{BA}}}{{\gamma }_{e}\sqrt{{n}_{B}Vt}}{\rm{.}}$$
(9)

where \({R}_{wall}=q(P){D}_{buffer}^{alkali}(\frac{\sqrt{1+T\mathrm{/273.15}}}{{P}_{buffer}/1amg}){(\frac{\pi }{a})}^{2}+q(P){D}_{quench}^{alkali}(\frac{\sqrt{1+T\mathrm{/273.15}}}{{P}_{quench}/1amg}){(\frac{\pi }{a})}^{2}\), the second term that alkali-metal atoms diffuse in the quench gas sometimes is ignored in the experiment, but we find that the harmonic mean of the diffusion coefficients in He and N2 are used for D (diffusion constant of the alkali atom within the gas) in the calculations of ref. 22, hence we also consider the second term, \({R}_{wall}^{B}={R}_{wall}^{B-He}+{R}_{wall}^{B-N2}\). q(P) is the nuclear slowing-down factor of alkali-metal atom40, \(q{(P)}_{K}=\frac{6+2{P}^{2}}{1+{P}^{2}}\) for 39 K atom, \(q{(P)}_{Rb}=\frac{38+52{P}^{2}+6{P}^{4}}{3+10{P}^{2}+3{P}^{4}}\) for 85 Rb atom, \(q{(P)}_{Cs}=\frac{22+70{P}^{2}+34{P}^{4}+2{P}^{6}}{1+7{P}^{2}+7{P}^{4}+{P}^{6}}\) for 133 Cs atom, \({D}_{buffer}^{alkali}\) is the diffusion constant of the alkali atom within the buffer gas41,42,43 in units of cm 2/s and is given at 1 amg and 273 K, \({D}_{quench}^{alkali}\) is the diffusion constant of the alkali atom within the quench gas42,43,44 in units of cm 2/s and is given at 1 amg and 273 K, 1amg = 2.69 × 1019  cm −3, \({D}_{He}^{K}=0.35\)  cm 2/s, \({D}_{He}^{Rb}=0.5\)  cm 2/s, \({D}_{He}^{Cs}=0.29\)  cm 2/s, \({D}_{N2}^{K}=0.2\)  cm 2/s, \({D}_{N2}^{Rb}=0.19\)  cm 2/s, \({D}_{N2}^{Cs}=0.098\)  cm 2/s, P buffer is the pressure intensity of buffer gas in amg, P quench is the pressure intensity of quench gas in amg, a is the equivalent radius of vapor cell, \({R}_{SD}={n}_{He}{\sigma }_{SD}^{B-He}{\bar{v}}_{B-He}+{n}_{{N}_{2}}{\sigma }_{SD}^{B-{N}_{2}}{\bar{v}}_{B-{N}_{2}}+{n}_{B}{\sigma }_{SD}^{B-B}{\bar{v}}_{B-B}+{n}_{B}{\sigma }_{SD}^{B-A}{\bar{v}}_{B-A}+{n}_{A}{\sigma }_{SD}^{A-B}{\bar{v}}_{A-B}\), \({R}_{SE}^{BB}\,=\) \({(\frac{g{\mu }_{B}B}{q{\mathrm{(0)}}_{B}\hslash })}^{2}\frac{q{\mathrm{(0)}}_{B}^{2}-{\mathrm{(2}{I}_{B}+\mathrm{1)}}^{2}}{2{k}_{SE}^{B-B}{n}_{B}}\), B is the external magnetic field, q(0) is the low polarization limit of the slowing-down factor, I is nuclear spin of the alkali-metal atoms, for 39K, 85Rb and 133Cs, I K  = 1.5, I Rb  = 2.5, I Cs  = 3.5, the relaxation rate due to alkali-alkali spin-exchange collisions \({R}_{SE}^{BB}={(\frac{g{\mu }_{B}B}{q{\mathrm{(0)}}_{B}\hslash })}^{2}\frac{q{\mathrm{(0)}}_{B}^{2}-{\mathrm{(2}{I}_{B}+\mathrm{1)}}^{2}}{2{k}_{SE}^{B-B}{n}_{B}},{R}_{SE}^{BA}={(\frac{g{\mu }_{B}B}{q{\mathrm{(0)}}_{A}\hslash })}^{2}\frac{q{\mathrm{(0)}}_{A}^{2}-{\mathrm{(2}{I}_{A}+\mathrm{1)}}^{2}}{2{k}_{SE}^{B-A}{n}_{A}}\), \({R}_{SE}^{AB}={(\frac{g{\mu }_{B}B}{q{\mathrm{(0)}}_{B}\hslash })}^{2}\frac{q{\mathrm{(0)}}_{B}^{2}-{\mathrm{(2}{I}_{B}+\mathrm{1)}}^{2}}{2{k}_{SE}^{A-B}{n}_{B}}\). With sufficiently high alkali metal number density (at higher temperature) and in sufficiently low magnetic field, \({R}_{SE}^{BB}\approx {R}_{SE}^{BA}\approx {R}_{SE}^{AB}\approx 0\). The spin precession rate is \({\omega }_{0}=\frac{g{\mu }_{B}B}{q\mathrm{(0)}\hslash }\).

Spin-exchange collisions preserve total angular momentum of a colliding pair of atoms but can scramble the hyperfine state of the atoms. Atoms in different hyperfine states do not precess coherently and thereby limit the coherence lifetime of the atoms. However, decoherence due to spin-exchange collisions can be nearly eliminated if the spin-exchange collisions occur much faster than the precession frequency of the atoms. In this regime of fast spin-exchange, all atoms in an ensemble rapidly change hyperfine states, spending the same amounts of time in each hyperfine state and causing the spin ensemble to precess more slowly but remain coherent13. In the limit of fast spin-exchange and small magnetic field, the spin-exchange relaxation rate vanishes for sufficiently small magnetic field11. In equation (9), we can find that the fundamental sensitivity of the hybrid optical pumping SERF atomic magnetometer increases when part or all of \({R}_{wall}^{B}\), R SD , R B , \({R}_{SE}^{BB}\), \({R}_{SE}^{AB}\) and \({R}_{SE}^{BA}\) (the later three terms are approximately zero in sufficiently low magnetic field and the magnetometer is in the SERF regime, which is helpful for us to study how B influence the SERF regime and fundamental sensitivity of the magnetometer) decrease, n B , V and t increase. For the expressions of \({R}_{wall}^{B}\), R SD , R B , \({R}_{SE}^{BB}\), \({R}_{SE}^{AB}\), \({R}_{SE}^{BA}\), n B , V and t, we just need to consider the fundamental sensitivity of the magnetometer change with one of the cell effective radius a, n He , n N2, t, cell temperature T, pumping rate of pump beam (\({R}_{p}^{K}\) and \({R}_{p}^{Cs}\)), external magnetic field B and measurement volume V. Diffusion of alkali metal atoms A and B to the wall will corrode the vapor cell and decrease the fundamental sensitivity of the magnetometer. Sufficiently many buffer gas will reduce diffusion of alkali metal atoms A and B to the wall. The probed alkali-metal atoms have a large absorption effect on the pumping beam, it’s an additional relaxation item for the alkali-metal atoms of the hybrid optical pumping SERF magnetometer. The spin exchange rate between alkali metal atoms A and alkali metal atoms B play a similar “pumping beam” action. Atom B is polarized by the spin exchange collisions between alkali metal atoms A and B. The pumping effect of probe beam means circularly polarized light in the probe beam pumps alkali-metal atoms. The outer electrons of the alkali metal atoms are polarized by the pumping beam, the polarized electrons undergo Larmor precession under the external magnetic field.

If we consider the influence of the light shift noise5, 45 B LS , photon shot noise8 B psn , spin-projection noise5 B spn , magnetic field noise46, 47 B mag , technology noise B tech and other noise B other on the SERF atomic magnetometer. Using the method of superposition of power spectral density, we can obtain the sensitivity of the hybrid optical pumping SERF atomic magnetometer as following

$$Sen=\sqrt{{(\delta B)}^{2}+{B}_{LS}^{2}+{B}_{psn}^{2}+{B}_{spn}^{2}+{B}_{mag}^{2}+{B}_{tech}^{2}+{B}_{other}^{2}}{\rm{.}}$$
(10)

If the noises above are optimized by technology means, the sensitivity of the hybrid optical pumping SERF atomic magnetometer approaches to the ultimate sensitivity, which is also helpful to study the atomic spin gyroscope48,49,50,51,52.

We take one of the cell effective radius a, n He , n N2, t, cell temperature T, pumping rate of pump beam (\({R}_{p}^{K}\) and \({R}_{p}^{Cs}\)), B (it is helpful for us to study how B influence the SERF regime and fundamental sensitivity of the magnetometer) and measurement volume V in equation (9) as a variable (other parameters are invariable) to obtain the results that the fundamental sensitivity of the hybrid optical pumping SERF magnetometer based 39K-85Rb-4He and 133Cs-85Rb-4He vary with the variable. Depending on suggestions and the typical conditions of the experiment group19, 30, 36, in order to facilitate the theoretical analysis, we take the mole fraction of 85Rb f Rb  = 0.97, n He  = 1019  cm −3, n N2 = 2 × 1017  cm −3, T = 457.5 K, \({R}_{p}^{K}={R}_{p}^{Cs}=200000\,{s}^{-1}\), a = 1 cm, V = 1 cm 3, t = 100 s, B = 10−15 T (\({R}_{SE}^{ee}\approx 0\)). Because n K , n Rb and n Cs vary with T, the relation between the fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers and the number density of alkali-metal atoms (n K , n Rb or n Cs ) is nonlinear. At the same T (the mole fraction of 85 Rb f Rb is fixed), n K , n Rb and n Cs are different, the fundamental sensitivity are also different. We will study the vapor cell by the characteristics and properties of the microcavities53,54,55 in the future work.

Figure 3 shows the relaxation rates due to diffusion of 85Rb in the 4He gas to the wall of 39K-85Rb-4He magnetometer \({R}_{wall-K-Rb-He}^{Rb-He}\), the total relaxation rates due to diffusion of 85Rb in the 4He and N2 gas to the wall of 39K-85Rb-4He magnetometer \({R}_{wall-K-Rb-He}^{Rb-He-N2}\), the relaxation rates due to diffusion of 85Rb in the 4He gas to the wall of 133Cs-85Rb-4He magnetometer \({R}_{wall-Cs-Rb-He}^{Rb-He}\) and the total relaxation rates due to diffusion of 85Rb in the 4He and N2 gas to the wall of 133Cs -85Rb-4He magnetometer \({R}_{wall-Cs-Rb-He}^{Rb-He-N2}\) decrease when 4He atom number density n He increases, the relaxation rates due to diffusion of 85Rb in the N2 gas to the wall of 39K-85Rb-4He magnetometer \({R}_{wall-K-Rb-He}^{Rb-N2}\) and the relaxation rates due to diffusion of 85Rb in the N2 gas to the wall of 133Cs-85Rb-4He magnetometer \({R}_{wall-Cs-Rb-He}^{Rb-N2}\) increase slowly when n He increases in Fig. 3(a). \({R}_{wall-K-Rb-He}^{Rb-N2},{R}_{wall-K-Rb-He}^{Rb-He-N2},\) \({R}_{wall-Cs-Rb-He}^{Rb-N2}\,{\rm{and}}\,{R}_{wall-Cs-Rb-He}^{Rb-He-N2}\) decrease when n N2 increases, \({R}_{wall-K-Rb-He}^{Rb-He}\,{\rm{and}}\,{R}_{wall-Cs-Rb-He}^{Rb-He}\) increase slowly when n N2 increases in Fig. 3(b). \({R}_{wall-K-Rb-He}^{Rb-He},{R}_{wall-K-Rb-He}^{Rb-N2},{R}_{wall-K-Rb-He}^{Rb-He-N2},{R}_{wall-Cs-Rb-He}^{Rb-He},\) \({R}_{wall-Cs-Rb-He}^{Rb-N2}\) \({\rm{and}}\,{R}_{wall-Cs-Rb-He}^{Rb-He-N2}\) decreases slowly when pumping rate of pump beam increases in Fig. 3(c). \({R}_{wall-K-Rb-He}^{Rb-N2},{R}_{wall-K-Rb-He}^{Rb-He-N2},{R}_{wall-Cs-Rb-He}^{Rb-N2}\,{\rm{and}}\,{R}_{wall-Cs-Rb-He}^{Rb-He-N2}\) increase, \({R}_{wall-K-Rb-He}^{Rb-He}\,{\rm{and}}\,{R}_{wall-Cs-Rb-He}^{Rb-He}\) increase slowly when cell temperature increases in Fig. 3(d). \({R}_{wall-K-Rb-He}^{Rb-He},{R}_{wall-K-Rb-He}^{Rb-N2},{R}_{wall-K-Rb-He}^{Rb-He-N2},\) \({R}_{wall-Cs-Rb-He}^{Rb-He},{R}_{wall-Cs-Rb-He}^{Rb-N2}\,{\rm{and}}\,{R}_{wall-Cs-Rb-He}^{Rb-He-N2}\) decreases rapidly when the cell effective radius increases in Fig. 3(e). The R wall of 39K-85Rb-4He magnetometer is bigger than the one of 133Cs-85Rb-4He magnetometer.

Figure 3
figure 3

The R wall of 39K-85Rb-4He and 133Cs-85Rb-4He magnetometers vary with the number density of buffer gas 4He and quench gas N2, pumping rate of pump beam, cell temperature and cell effective radius. (a) \({R}_{wall-K-Rb-He}^{Rb-He},{R}_{wall-K-Rb-He}^{Rb-He-N2},{R}_{wall-Cs-Rb-He}^{Rb-He}\,{\rm{and}}\,{R}_{wall-Cs-Rb-He}^{Rb-He-N2}\) decrease when 4He atom number density n He increases, \({R}_{wall-K-Rb-He}^{Rb-N2}\,{\rm{and}}\,{R}_{wall-Cs-Rb-He}^{Rb-N2}\) increase slowly with increasing 4He atom number density. (b) \({R}_{wall-K-Rb-He}^{Rb-N2},{R}_{wall-K-Rb-He}^{Rb-He-N2},{R}_{wall-Cs-Rb-He}^{Rb-N2}\,{\rm{and}}\,{R}_{wall-Cs-Rb-He}^{Rb-He-N2}\) decrease as n N2 increases, \({R}_{wall-K-Rb-He}^{Rb-He}\,and\,{R}_{wall-Cs-Rb-He}^{Rb-He}\) increase slowly when n N2 increases. (c) \({R}_{wall-K-Rb-He}^{Rb-He},{R}_{wall-K-Rb-He}^{Rb-N2},\) \({R}_{wall-K-Rb-He}^{Rb-He-N2},{R}_{wall-Cs-Rb-He}^{Rb-He},{R}_{wall-Cs-Rb-He}^{Rb-N2}\,{\rm{and}}\,{R}_{wall-Cs-Rb-He}^{Rb-He-N2}\) decreases slowly when pumping rate of pump beam increases. (d) \({R}_{wall-K-Rb-He}^{Rb-N2},{R}_{wall-K-Rb-He}^{Rb-He-N2},{R}_{wall-Cs-Rb-He}^{Rb-N2}\,{\rm{and}}\) \({R}_{wall-Cs-Rb-He}^{Rb-He-N2}\) increase rapidly, \({R}_{wall-K-Rb-He}^{Rb-He}\,{\rm{and}}\,{R}_{wall-Cs-Rb-He}^{Rb-He}\) increase slowly when T increases. (e) \({R}_{wall-K-Rb-He}^{Rb-He},{R}_{wall-K-Rb-He}^{Rb-N2},\) \({R}_{wall-K-Rb-He}^{Rb-He-N2},{R}_{wall-Cs-Rb-He}^{Rb-He},{R}_{wall-Cs-Rb-He}^{Rb-N2}\,{\rm{and}}\,{R}_{wall-Cs-Rb-He}^{Rb-He-N2}\) decreases rapidly when the cell effective radius increases. The R wall of 39K-85Rb-4He magnetometer is bigger than the one of 133Cs-85Rb-4He magnetometer.

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Figures 4 and 5 show the fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers as a function of 39K, 133Cs, 85Rb, 4He and N2 number density, \({R}_{p}^{K}\) and \({R}_{p}^{Cs}\), external magnetic field, cell temperature and measurement time, cell effective radius, measurement volume, includes the effects of spin exchange due to 39K-85Rb, 85Rb-85Rb, 85Rb-39K, 133Cs-85Rb, 85Rb-133Cs and spin relaxation due to 39K-85Rb, 85Rb-85Rb, 85Rb-39K, 133Cs-85Rb, 85Rb-133Cs, 85Rb-4He collisions, 85Rb-N2 destructions.

Figure 4
figure 4

The fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers varies with the number density of buffer gas 4He and quench gas N2, pumping rate of pump beam, measurement time. (a) The fundamental sensitivity of 39K-85Rb-4He magnetometer (black line in squares) increases with the increasing number density of 4He when 4He number density is smaller than a critical value about 4.22 × 1019  cm −3 and decreases when 4He number density is bigger than the value. The fundamental sensitivity of 133Cs-85Rb-4He magnetometer (red line in dots) increases with the increasing number density of 4He when 4He number density is smaller than a critical value about 4.15 × 1019  cm −3 and decreases when 4He number density is bigger than the value. (b) The fundamental sensitivity of 39K-85Rb-4He magnetometer (black line in squares) increases with the increasing N2 number density when N2 number density is smaller than a critical value about 1.22 × 1019  cm −3 and decreases when N2 number density is higher than the value. The fundamental sensitivity of 133Cs-85Rb-4He magnetometer(red line in dots) increases with the increasing N2 number density when N2 number density is smaller than a critical value about 1.21 × 1019  cm −3 and decreases when N2 number density is higher than the value. The fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers decrease with the increasing pumping rate of pump beam in (c). The fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers increase with the increasing measurement time. The fundamental sensitivity of 133Cs-85Rb-4He magnetometer is lower than the one of 39K-85Rb-4He magnetometer in (d).

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Figure 5
figure 5

The fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers varies with the external magnetic field, cell temperature, cell effective radius and measurement volume. (a) When the external magnetic field is smaller than about 10−8 T, the fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers almost do not vary with the increasing external magnetic field respectively. (b) The fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers increase with the increasing cell temperature respectively. (c) The fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers increase with the increasing cell effective radius respectively. (d) The fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers with a = 5 cm increase with increasing measurement volume respectively.

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Figure 4(a) represents that the fundamental sensitivity of 39K-85Rb-4He magnetometer (black line in squares) increases with the increasing 4He number density when 4He number density is smaller than a critical value about 4.22 × 1019  cm −3, it decreases when 4He number density is higher than the value. The fundamental sensitivity of 133Cs-85Rb-4He magnetometer (red line in dots) increases with the increasing 4He number density when 4He number density is smaller than a critical value about 4.15 × 1019  cm −3 and it decreases when 4He number density is higher than the value. For this phenomenon, we think that more alkali-metal atoms diffuse to the cell wall and less spin exchange collisions between alkali-metal atoms A and B when 4He number density is smaller than the value and decrease. Less alkali-metal atoms diffuse to the cell wall and more spin exchange collisions between alkali-metal atoms and buffer gas so that there are less spin exchange collisions in alkali-metal atoms when the number density of 4He is bigger than the value and increase. Therefore, if we take the critical value as 4He number density, spin exchange collisions in alkali-metal atoms are the most, we can obtain the highest fundamental sensitivity of the magnetometer.

Figure 4(b) shows that the fundamental sensitivity of 39K-85Rb-4He magnetometer (black line in squares) increases with the increasing N2 number density when N2 number density is smaller than a critical value about 1.22 × 1019  cm −3, it decreases when N2 number density is higher than the value. The fundamental sensitivity of 133Cs-85Rb-4He magnetometer (red line in dots) increases with the increasing N2 number density when N2 number density is smaller than a critical value about 1.21 × 1019  cm −3, it decreases when N2 number density is higher than the value. Therefore, if we take the critical value as N2 number density, we can obtain the highest fundamental sensitivity of the magnetometer. The fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers decrease with the increasing pumping rate of pump beam respectively in Fig. 4(c). When the pumping rate of pump beam is bigger than about 1916 s -1 and N2 number density is bigger than about 1.974×1016 cm −3, the fundamental sensitivity of 39K-85Rb-4He magnetometer is higher than the one of 133Cs-85Rb-4He magnetometer. The fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers (black line in squares and red line in dots) increase with the increasing measurement time. The fundamental sensitivity of 133Cs-85Rb-4He magnetometer is lower than the one of 39K-85Rb-4He magnetometer in Fig. 4(d).

Figure 5 describes that when the external magnetic field B is smaller than about 10−8 T, the fundamental sensitivity of 39K-85Rb-4He magnetometer is higher than the one of 133Cs-85Rb-4He magnetometer and they almost do not vary with the increasing external magnetic field respectively, the fundamental sensitivity decreases rapidly when B is bigger than about 10−8 T and the fundamental sensitivity of 39K-85Rb-4He magnetometer is lower than the one of 133Cs-85Rb-4He magnetometer when B is bigger than about 2.845×10−8 T in Fig. 5(a). The fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers increase with the increasing cell temperature respectively and there are more spin exchange collisions in alkali-metal atoms in Fig. 5(b). The fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers increase with the increasing cell effective radius respectively in Fig. 5(c). The fundamental sensitivity of 39K (133Cs)-85Rb-4He magnetometers with a = 5cm increase with increasing measurement volume respectively in Fig. 5(d). The fundamental sensitivity of 133Cs-85Rb-4He magnetometer is lower than the one of 39K-85Rb-4He magnetometer in Fig. 5(a)–(d).

As a result, the polarization of 85Rb atom of the hybrid optical pumping SERF magnetometer based on 133Cs-85Rb-4He is bigger than the one based on 39K-85Rb-4He in Fig. 2. However, the fundamental sensitivity of 39K-85Rb-4He magnetometer is higher than the one of 133Cs-85Rb-4He magnetometer when the pumping rate of pump beam is bigger than about 1916 s -1 in figures 4 and 5. For another buffer gas 21Ne, a large 85Rb magnetization field due to spin interaction between 85Rb atom and 21Ne atoms causes a large spin exchange relaxation rate of 85Rb atom56 and 85Rb atom can make 21Ne atoms hyperpolarized, which will affect the magnetic field measurement, it is a better choice to take 4He as the buffer gas of the SERF magnetometer to measure the magnetic field and take 21Ne as the buffer gas of the SERF magnetometer to measure inertia. The fundamental sensitivity of the magnetometers based on 133Cs-85Rb-4He is lower than the one based on 39K-85Rb-4He when the pumping rate of pump beam is bigger than about 1916 s −1 and B is bigger than about 2.845 × 10−8 T (the magnetometers are not in the SERF regime for this external magnetic field).

The polarization of 85Rb atom of the magnetometer is uniform when the mole fraction of 85Rb f Rb  = 0.97, n He  = 1019  cm −3, n N2 = 2 × 1017  cm −3, T = 457.5 K, \({R}_{p}^{K}={R}_{p}^{Cs}=200000\,{s}^{-1}\), a = 1 cm, V = 1 cm 3, B = 10−15 T and t = 100 s. Under above condition, we obtain a fundamental sensitivity of 1.5073 aT/Hz 1/2 with 39K-85Rb-4He magnetometer with 85Rb polarization is 0.1116 and n K /n Rb  = 0.0023, a fundamental sensitivity of 1.6949 aT/Hz 1/2 with 133Cs-85Rb-4He magnetometer with 85Rb polarization is 0.1864 and n Cs /n Rb  = 0.0151. By optimizing above parameters, we obtain a fundamental sensitivity of 1.8359 × 10−2 aT/Hz 1/2 with 39K-85Rb-4He with the polarization of 85Rb atom is 1.3174 × 10−4 and n K /n Rb  = 0.0023, a fundamental sensitivity of 1.8181 × 10−2 aT/Hz 1/2 with 133Cs-85Rb-4He with the polarization of 85Rb atom is 2.3316 × 10−4 and n Cs /n Rb  = 0.0151 with the mole fraction of 85Rb f Rb = 0.97, n He  = 1019  cm −3, n N2 = 2 × 1017  cm −3, T = 457.5 K, \({R}_{p}^{K}={R}_{p}^{Cs}=200\,{s}^{-1}\), a = 10 cm, B = 10−15 T, V = 1000 cm 3, t = 100 s, with higher fundamental sensitivity possible at bigger measurement volume, proper amount of buffer gas and quench gas, smaller pumping rate of pump beam, higher temperature and longer measurement time.

Discussion

In conclusion, we find that 85Rb polarization increases with the increasing pumping rate of pump beam. The 85Rb polarization of 133Cs-85Rb-4He magnetometer is bigger than the one of 39K-85Rb-4He magnetometer. The polarization of 85Rb atom of 39K (133Cs)-85Rb-4He magnetometers almost do not vary when the number density of 4He and N2 increase and the number density of 4He and N2 are smaller than some critical values and decrease rapidly when the number density of buffer gas and quench gas are bigger than the values respectively. The fundamental sensitivity increases with the increasing number density of buffer gas and quench gas when the number density of buffer gas and quench gas are smaller than corresponding critical values respectively and decreases when the number density of buffer gas and quench gas are bigger than the values. The fundamental sensitivity increases with the increasing cell effective radius, measurement volume, cell temperature and measurement time respectively. The fundamental sensitivity of the magnetometers decrease with increasing \({R}_{p}^{K}\) and \({R}_{p}^{Cs}\). At the same cell temperature, the polarization of 85Rb atom of 133Cs-85Rb-4He magnetometer is bigger than the one of 39K-85Rb-4He magnetometer and the fundamental sensitivity of 133Cs-85Rb-4He magnetometer is lower than the one of 39K-85Rb-4He magnetometer when the pumping rate of pump beam is bigger than about 1916 s −1 and B is smaller than about 2.845×10−8 T.

From the formula of the relative velocity, R wall , \({R}_{SE}^{ee}\), n A , n B and equation (1), we can find that increasing the cell temperature will increase R wall , R SD and the number density of alkali-metal atoms, reduce \({R}_{SE}^{ee}\). In general, \({R}_{SE}^{ee}\) is smaller than R wall and R SD , raising the cell temperature resulting in an increase in the fundamental sensitivity is mainly due to the great improvement of the probed alkali-metal atomic number density when the cell temperature inceases, which has a greater influence on the fundamental sensitivity than R wall and R SD . If the number density of alkali-metal atoms and cell volume are fixed, in other words, when the alkali-metal atoms in the vapor cell are fully in the vapor regime, if we continue to raise the cell temperature, the alkali-metal atoms number density will not change, \({R}_{SE}^{ee}\) will decrease, R wall and R SD will increase. What’s more, the decreased value of \({R}_{SE}^{ee}\) is smaller than the increased value of R wall and R SD , which will decrease the fundamental sensitivity. For example, there are certain amount of alkali-metal atoims 39K(133Cs) and 85Rb with the mole fraction of 85Rb is 0.97 in the hybrid vapor cell, when T = 418.3 K, all of the alkali-metal atoims become vapor, n K  = 3.0072×1011 cm −3, n Rb  = 1.7385 × 1014  cm −3, n Cs  = 2.3741×1012 cm −3 with a = 1 cm, if we continue to increase T, we will find that n K , n Cs and n Rb do not change. When n He  = 1019  cm −3, n N2 = 2 × 1017  cm −3, \({R}_{p}^{K}={R}_{p}^{Cs}=200000\,{s}^{-1}\), a = 1 cm, V = 1 cm 3, B = 10−15 T, t = 100 s, the fundamental sensitivity of 39K-85Rb-4He and 133Cs-85Rb-4He magnetometers are 3.0191 aT/Hz 1/2 and 3.1630 aT/Hz 1/2 at T = 418.3 K, 2.4204 aT/Hz 1/2 and 2.6163 aT/Hz 1/2 at T = 430 K.

In practical applications, we should consider some questions, one very essential question is the minimum total number of atoms necessary for the operation of the magnetometer with the desired accuracy, as well as the geometric size of the setup - how small can it be made? How does the fundamental sensitivity of the elaborated setup depend on the number of atoms?

Firstly, we can find that when the number density of alkali-metal atom (which is determined by the mole fraction and cell temperature for the hybrid vapor cell with two types of alkali-metal atoms), buffer gas and quench gas are certain, if we also know the effective radius of vapor cell, pumping rate of pump beam, external magnetic field, measurement volume and measurement time, we can obtain the corresponding total number of the atoms and the fundamental sensitivity of the magnetometer. Because the number density of alkali-metal atom is determined by the mole fraction and cell temperature for the hybrid vapor cell with two types of alkali-metal atoms, we can find that how the fundamental sensitivity of the magnetometer depend on the number density of alkali-metal atom, buffer gas and quench gas from Figs 4(a),(b) and 5(b) (the mole fraction and cell temperature corresponds to the number density of alkali metal atoms). The less number density of buffer gas and quench gas, the bigger R wall , it’s hard to say the minimum number density of buffer gas and quench gas, but we find that there are critical values for the number density of buffer gas and quench gas to make the fundamental sensitivity of the magnetometers highest.

Secondly, the smallest geometric size of the setup and volume of vapor cell depends on the processing method and materials. For example, Griffith et al. studied a miniature atomic magnetometer integrated with flux concentrators, the magnetometer uses a millimeter scale 87Rb vapor cell (3 × 2 × 1 mm 3) and either mu-metal or Mn-Zn ferrite flux concentrators. They found that the minimum separation of the concentrators is limited to 2 mm by the external dimensions of the vapor cell57 and reached a sensitivity of 104 aT/Hz 1/2.

Thirdly, if the amount of 39K (133Cs) is little (all of 39K (133Cs) atoms become vapor when T = 418.3 K) and there is enough 85Rb in the vapor cell, when T is bigger than 418.3 K. We continue to increase T, we will find that n K /n Rb (n Cs /n Rb ) gets bigger and bigger, the fundamental sensitivity becomes higher and higher. For instance, Fang et al.19 obtained a sensitivity of approximately 5 × 103 aT/Hz 1/2 by optimizing the parameters of SERF magnetometer based on K-Rb hybrid optical pumping when the mole fraction of K atoms is approximately 0.03. Ito et al. studied optimal densities of alkali metal atoms in an optically pumped K-Rb hybrid atomic magnetometer considering the spatial distribution of spin polarization, calculated the spatial distribution of the spin polarization and found that the optimal density of K atoms is 3 × 1013  cm −3 and the optimal density ratio is n K /n Rb  ~ 400 (Rb as pump atoms and K as probe atoms) to maximize the output signal and enhance spatial homogeneity of the sensor property22.

Fourthly, the alkali-metal atoms in the vapor cell are operated in the “hot-gas” regime. If the atomic gas is cooled to the state of a Bose-Einstein condensation (BEC), the operation may be essentially improved. For example, Wildermuth et al. experimentally sensed electric and magnetic fields with BEC and found this field sensor simultaneously features high spatial resolution and high field sensitivity, reached a sensitivity of ~109 aT at 3 μm spatial resolution58. Therefore, we can use BEC magnetometer to obtain higher sensitivity of magnetic field and spatial resolution, which is very important for the application of the magnetometer in the field of biomedicine.

If we take \({\sigma }_{SD}^{K-Rb}={\sigma }_{SD}^{Rb-K}\approx {\sigma }_{SD}^{Rb}=1.6\times {10}^{-17}\)  cm 2, \({\sigma }_{SD}^{Cs-Rb}={\sigma }_{SD}^{Rb-Cs}\approx {\sigma }_{SD}^{Cs}=2\times {10}^{-16}\)  cm 2, the polarization and fundamental sensitivity of the magnetometer will decrease slightly, but it will not affect the change rule of the polarization and fundamental sensitivity discussed above. To obtain a higher fundamental sensitivity between 39K-85Rb-4He and 133Cs-85Rb-4He magnetometers, it is better to choose 39K-85Rb-4He magnetometer (when the pumping rate of pump beam is bigger than about 1916 s −1, N2 number density is bigger than about 1.974 × 1016 cm-3 and B is smaller than about 2.845 × 10−8 T), with 4He as the buffer gas and take the critical value of 4He number density and quench gas, increase a, V, T (when the quantity of alkali-metal atoms are enough), t, then reduce B and the pumping rate of pump beam based on actual demand of the fundamental sensitivity and spatial resolution. We estimate the fundamental sensitivity limit of the hybrid optical pumping SERF magnetometer due to the shot-noise superior to 1.8359 × 10−2 aT/Hz 1/2, which is higher than the shot-noise-limited sensitivity of 1 aT/Hz 1/2 of K SERF atomic magnetometer. We could choose suitable conditions on the basis of the experiment requirements to gain a higher sensitivity of the SERF magnetometer, keep the costs down and carry forward the miniaturization and practical application of the hybrid optical pumping SERF atomic magnetometers. The influences of the mole fraction of 85Rb f Rb as a variable on the polarization and fundamental sensitivity of the hybrid optical pumping SERF atomic magnetometer will be investigated in the future work.

Methods

The fundamental sensitivity calculation details

We obtain the above calculation results by MATLAB and chose some special points to plot with Origin 8. The fundamental sensitivity of the hybrid optical pumping SERF atomic magnetometer was obtained by equation (9) and relevant parameters used listed in Table 1 and taking one of the cell effective radius a, n He , n N2, measurement time t, cell temperature T, pumping rate of pump beam (\({R}_{p}^{K}\) and \({R}_{p}^{Cs}\)), external magnetic field B and measurement volume V by equation (9) as a variable (other parameters are invariable) and the fundamental sensitivity of 39K-85Rb-4He and 133Cs-85Rb-4He magnetometer vary with the variable in Figs 4(a)–(d) and 5(a)–(d), where the mole fraction of 85Rb f Rb = 0.97, n He  = 1019  cm −3, n N2 = 2 × 1017  cm −3, T = 457.5 K, \({R}_{p}^{K}={R}_{p}^{Cs}=200000\,{s}^{-1}\), a = 1 cm, V = 1 cm 3, t = 100 s, B  = 10−15 T and the magnetometer polarization is obtained by equation (6) and its relevant parameters are chosen as above. For \({R}_{SE}^{KK},{R}_{SE}^{RbRb},{R}_{SE}^{KRb},{R}_{SE}^{RbK}\) and their total spin-exchange collisions relaxation rate \({R}_{SE}^{KRbHe}\) increase, \({R}_{SE}^{CsCs},{R}_{SE}^{RbRb},{R}_{SE}^{CsRb},{R}_{SE}^{RbCs}\) and their total spin-exchange collisions relaxation rate \({R}_{SE}^{CsRbHe}\) increase when B increases and decrease when T increases in Fig. 6. We find that \({R}_{SE}^{AA}({R}_{SE}^{AA}={(\frac{g{\mu }_{B}B}{q{\mathrm{(0)}}_{A}\hslash })}^{2}\frac{q{\mathrm{(0)}}_{A}^{2}-{\mathrm{(2}{I}_{A}+\mathrm{1)}}^{2}}{2{k}_{SE}^{A-A}{n}_{A}})\), \({R}_{SE}^{BB},{R}_{SE}^{AB},{R}_{SE}^{BA}\,{\rm{and}}\,{R}_{SE}^{ee}\) \(({R}_{SE}^{ee}={R}_{SE}^{AA}+{R}_{SE}^{BB}+{R}_{SE}^{AB}+{R}_{SE}^{BA})\) increase when B increases and decrease when T increases. When T is higher than 418.3 K, B is smaller than 10−10 T, \({R}_{SE}^{ee}\approx 0\). When B is bigger than 10−9 T and T is lower than 400 K, we can not ignore the effect of the alkali-alkali spin-exchange collisions relaxation rate. Therefore, there need to reduce the external magnetic field to 10−10 T below and make the cell temperature higher than 418.3 K to reduce the effect of alkali-alkali spin-exchange collisions relaxation rate on the SERF regime and weak magnetic field measurement in the experiments.

Table 1 Parameters used for the calculation.
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Figure 6
figure 6

The alkali-alkali spin-exchange collisions relaxation rate of 39K (133Cs)-85Rb-4He magnetometers varies with the external magnetic field and cell temperature. \({R}_{SE}^{KK},{R}_{SE}^{RbRb},{R}_{SE}^{KRb},{R}_{SE}^{RbK}\) and their total spin-exchange collisions relaxation rate \({R}_{SE}^{KRbHe}\) increase, \({R}_{SE}^{CsCs},{R}_{SE}^{RbRb},{R}_{SE}^{CsRb},{R}_{SE}^{RbCs}\) and their total spin-exchange collisions relaxation rate \({R}_{SE}^{CsRbHe}\) increase when B increases and decrease when T increases in (a) and (b).

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