The SERF atomic co-magnetometers have some properties similar to the SERF atomic magnetometers. As we discussed in our previous work,^[24] we take the alkali metal vapor cell (the shape of the cell is roughly spherical) of the SERF atomic co-magnetometers based on HOP containing two types of alkali metal atoms, ³⁹K–⁸⁵Rb or ¹³³Cs–⁸⁵Rb, we take ³⁹K or ¹³³Cs as atom A, select ⁸⁵Rb as atom B in the SERF regime,^{[24,25] 21}Ne as buffer gas to suppress the spin relaxation and N₂ as quench gas to restrain radiative de-excitation of alkali metal atoms.^[26] The saturated density^[27] of the alkali-metal atoms vapor in units of cm^–3 at cell temperature T in Kelvin is . A₁ and B₁ are phase parameters,^[25] where , , , , and for the temperature higher than 400 K. Because the SERF regime can be reached by operating with sufficiently high alkali metal number density (at higher temperature) and in sufficiently low magnetic field,^[28,29] as we discussed in our previous work,^[24] 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 co-magnetometers be in the SERF regime.

Considering the spin exchange between alkali-metal atoms A and B in the hybrid vapor cell, we assume that the vapor densities obey Raoult’s law,^[30] , where f_B is the mole fraction of atom B in the metal and 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 atoms A and B, , . For K–Rb–²¹Ne atomic co-magnetometer, we suppose that in an alkali-metal vapor cell filled with K, Rb, and ²¹Ne atoms, the density ratio of K to Rb is D_r = n_K/n_Rb (where n_K and n_Rb are the number densities of K and Rb, respectively). K atom spins are directly polarized by the D1 line light and Rb atom spins are polarized through spin exchange with K atom spins. If the frequency rate of the laser beam, ν, is tuned away from the absorption center of the pressure^[31] and the power-broadened^[32] D1 absorption line, there will be an AC Stark shift of K, which is denoted as in the light propagation direction,^[35] where the light propagation direction is in the z direction. For Rb atom spins, the far-off-resonant laser will cause an AC Stark shift on the D1 and D2 line transitions of Rb atoms. K atom spins polarize Rb atom spins by the spin-exchange interaction. The spin transfer rate from Rb to K is given by , where is the spin-exchange cross section between Rb and K^[33] and v_{Rb − K} is the relative velocity between Rb and K. Thus the spin transfer rate from K to Rb is .

The full Bloch equations are given as follows:^[34,35]

We can find that the relaxation rates for diffusion of ⁸⁵Rb in the ²¹Ne gas to the wall of ³⁹K–⁸⁵Rb–²¹Ne co-magnetometer , the total relaxation rates for diffusion of ⁸⁵Rb in the ²¹Ne gas to the wall of ³⁹K–⁸⁵Rb–²¹Ne co-magnetometer , the relaxation rates for diffusion of ⁸⁵Rb in the ²¹Ne and N₂ gas to the wall of ¹³³Cs–⁸⁵Rb–²¹Ne co-magnetometer and the total relaxation rates for diffusion of ⁸⁵Rb in the ²¹Ne and N₂ gas to the wall of ¹³³Cs–⁸⁵Rb–²¹Ne co-magnetometer decrease when ²¹Ne atom number density n_Ne increases in Fig. 1(a), because the buffer gas ²¹Ne atom suppresses the spin relaxation caused by wall collisions. The relaxation rates for diffusion of ⁸⁵Rb in the N₂ gas to the wall of ³⁹K–⁸⁵Rb–²¹Ne co-magnetometer is , the relaxation rates for diffusion of ⁸⁵Rb in the N₂ gas to the wall of ¹³³Cs–⁸⁵Rb–²¹Ne co-magnetometer is . , , and decrease when n_N2 increases for N₂ gas colliding with excited alkali metal atoms and absorbing the energy, which restrains radiative de-excitation of the excited alkali metal atoms, and do not change when n_N2 increases because they are not related with n_N2 in Fig. 1(b).

Fig. 1.

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	Fig. 1. The wall relaxation rates change with the 21Ne number density nNe in (a), the N2 number density nN2 in (b), the cell temperature T in (c), and the cell effective radius a in (d).

The , , , , , and increase with increasing cell temperature T in Fig. 1(c) because the increasing T makes the number density and relative velocity of ⁸⁵Rb atoms greater and more ⁸⁵Rb atoms move towards the wall. The R_wall decreases when the cell effective radius a increases in Fig. 1(d) for the ⁸⁵Rb atoms colliding more ²¹Ne atoms and N₂ atoms with the increasing a when ⁸⁵Rb atoms move towards the wall, which will decrease the wall relaxation rate.

The polarization in the z direction could be calculated by^[35]

The angle between the spins polarization and the z direction is small by applying a small magnetic field B_y, and the polarizations of K, Rb and ²¹Ne are approximately constant in the z direction, then the coupled Bloch equations can be simplified.^[35] In the x direction, the polarizations of K and Rb can be solved and are given by^[35]

For small atomic polarization, the spin precession rate is given by ω₀ = gμ_BB/Q(P^e)ħ, which is the Larmor frequency, T_SE is the spin-exchange time, , where is the spin-exchange cross section between Rb and Rb, and is the relative velocity between Rb and Rb. Q(P^e) is the slow-down factor for polarization and nuclear angular momentum,^[27,50] Q(P^e)_⁸⁵Rb = (38 + 52P² + 6P⁴)/(3 + 10P² + 3P⁴) for the ⁸⁵Rb atom. In this limit spin exchange contributes to transverse relaxation only in second order and vanishes for zero magnetic field,^[17] , for low polarization limit and small magnetic fields, spin-exchange relaxation is quadratic in the magnetic field, I is nuclear spin of the alkali-metal atoms, we consider the relaxation rate due to ⁸⁵Rb–⁸⁵Rb, ⁸⁵Rb–³⁹ K spin-exchange collisions, then

Table 1.

Table 1.

Table 1. Parameters used for the calculation. . Parameter Value Boltzmann’s constant kB 1.38 × 10–23 J/K Atomic mass unit m 1.660539040(20) × 10–27 kg π 3.14 21Ne nuclear gyromagnetic ratio[59] 2.1 × 107 rad/s/T Electron spin g factor 2 × 1.001159657 Planck’s constant ħ 1.054589 × 10–34 J·s Bohr magneton μB 9.27408 × 10–24 J/T [55] 1.8 × 10–14 cm2 [52] 1.9 × 10–14 cm2 [52] 2.1 × 10–14 cm2 [62] 1 × 10−18 cm2 [46] 1.6 × 10–17 cm2 [63] 2 × 10–16 cm2 [36,39] 1 × 10–23 cm2 [56] 5.2 × 10–23 cm2 [57] 1 × 10−22 cm2 [27,58] 7.9 × 10–23 cm2 [27,36] 1 × 10–22 cm2 [27,36] 5.5 × 10–22 cm2

Table 1. Parameters used for the calculation. .

To improve the practicability of the HOPSERF co-magnetometers, it is necessary for us to investigate the fundamental sensitivity of the co-magnetometers to improve the sensitivity, stability of the co-magnetometers and to realize the miniaturization of the co-magnetometers. The fundamental shot-noise-limited sensitivity of an atomic gyroscope based on the co-magnetometer is given by^[17,59]

In the limit of fast spin-exchange and small magnetic field, the spin-exchange relaxation rate vanishes for sufficiently small magnetic field.^[36] In Eq. (12), we can find that the fundamental sensitivity of the co-magnetometers increases when part or all of , R_SD, R_B, , and (the latter two terms are approximately zero in sufficiently low magnetic field and the co-magnetometer is in the SERF regime, which is helpful for us to study how B influence the SERF regime and fundamental sensitivity of the co-magnetometers) decrease, n_B and V increase. For the expressions of , R_SD, R_B, , , n_B, and V, we just need to consider the fundamental sensitivity of the co-magnetometers change with one of the cell effective radius a, n_Ne, n_N2, cell temperature T, wavelength of pump beam (λ_K and λ_Cs), power density of pump beam ( and ), pump beam spot radius a_p, the mole fraction of ⁸⁵Rb f_Rb, external magnetic field B and measurement volume V.

Because the slow-down factors are different in the polarization of ³⁹K–⁸⁵Rb–²¹Ne and ¹³³Cs–⁸⁵Rb–²¹Ne co-magnetometers, we take the slow-down factors at a low polarization limit for convenience of the theoretical analysis. The number density of the ²¹Ne, the power density of pump beam and the pump beam wavelength will affect the polarization greatly by affecting the pumping rate of pump beam. We systematically studied the variations of the pumping rate of the pump beam, frequency shift, Bⁿ, B^e, the relaxation rate for the alkali-alkali spin-exchange collisions. We obtain the following results by MATLAB and take several points to plot with Origin 8.

We choose one of ²¹Ne number density n_Ne, N₂ number density n_N2, cell temperature T, wavelength of pump beam λ_K(λ_Cs), mole fraction of ⁸⁵Rb f_Rb, cell effective radius a, and power density of pump beam (), the input rotation velocity in the x direction Ω_x and y direction Ω_y, external magnetic field in the y direction B_y and z direction B_z by Eq. (4) as a variable (other parameters are invariable) to obtain the results that the polarization of the ³⁹K(¹³³Cs)–⁸⁵Rb–²¹Ne HOPSERF co-magnetometer changes with the variable. From the formula of R_p, we can find that R_p, the frequency shift and B^e do not change with the increasing a_p due to a_p in () and A_t are canceled out. Depending on suggestions and the typical conditions of the experiment group,^[34–54] in order to facilitate the theoretical analysis, we take the mole fraction of ⁸⁵Rb atom as f_Rb = 0.97, n_{N e} = 2 × 10¹⁹ cm^–3, n_N2 = 2 × 10¹⁷ cm^–3, W/cm², Ω_x = 7.292 × 10^–6 rad/s, Ω_y = 7.292 × 10^–5 rad/s, B_y = 1 × 10^–12 T, B_z = 7 × 10^–11 T, a = 1 cm, a_p = 1 cm, λ_K = 769.808 nm, λ_Cs = 894.3 nm, and T = 457.5 K, at the moment, ³⁹K, ⁸⁵Rb, and ¹³³Cs are in the SERF regime.

To ensure the validity of the low polarization limit and considering that the noble gas has enough magnetic moment which can compensate for the external magnetic field and the system can be a co-magnetometer, we make the ⁸⁵Rb polarization smaller than about 0.08 and larger than 10^–4. We discuss the variations of pumping rate of the pump beam, the frequency shift, the Bⁿ, the B^e, the alkali-alkali spin-exchange collision relaxation rate related with ⁸⁵Rb atoms. Figure 2 demonstrates the ⁸⁵Rb polarization of ³⁹K (¹³³Cs)–⁸⁵Rb–²¹Ne co-magnetometers changes with n_Ne, n_N2, λ_K(λ_Cs) and T, respectively.

Fig. 2.

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	Fig. 2. The 85Rb polarization of 39K (133Cs)–85Rb–21Ne co-magnetometers versus the number density of the buffer gas 21Ne in (a), the number density of the quench gas N2 in (b), the wavelength of pump beam in (c), and the cell temperature in (d).

Figure 2(a) shows that ⁸⁵Rb polarizations decrease with the increasing n_Ne in ³⁹K–⁸⁵Rb–²¹Ne and ¹³³Cs–⁸⁵Rb–²¹Ne co-magnetometers. The ⁸⁵Rb polarization of ¹³³Cs–⁸⁵Rb–²¹Ne co-magnetometer is larger than the one of ³⁹K–⁸⁵Rb–²¹Ne co-magnetometer.

From the formula of , R_p, T_1n, frequency shift, Bⁿ, and B^e, we can find that increases with the increasing n_Ne, , , , decrease when ²¹Ne atom number density n_Ne increases in Fig. 1(a), R_p, Bⁿ and B^e increase with increasing n_Ne, T_1n, , , and decrease with increasing n_Ne.The change value of ⁸⁵Rb polarization, which is mainly determined by the , , and R_p.

Figure 2(b) demonstrates that ⁸⁵Rb polarization decreases with increasing n_N2. From the formulae of , R_Wall and B^e, we can find that increases with the increasing n_N2, , , , and decrease when n_N2 increases. and do not change when n_N2 increases in Fig. 1(b), B^e decreases slowly with increasing n_N2. Here ⁸⁵Rb polarization is mainly determined by the and .

Figure 2(c) shows that ⁸⁵Rb polarization of ³⁹K (¹³³Cs)–⁸⁵Rb–²¹Ne co-magnetometer increases when λ_K(λ_Cs) increases but is smaller than the value of about 770.085 nm (894.5032 nm), decreases when λ_K(λ_Cs) is larger than the value of about 770.1309 nm (894.696 nm) (⁸⁵Rb polarization is about 0.08). From the formulae of the ⁸⁵Rb polarization, R_p, L_z, and B^e, we can find that R_p, and B^e increase and decreases with increasing λ_K(λ_Cs) when the λ_K(λ_Cs) is smaller than the value of about 770.085 nm (894.5032 nm). R_p and B^e decrease with increasing λ_K(λ_Cs) when the λ_K(λ_Cs) is larger than the value of about 770.1309 nm (894.696 nm). Therefore, the change of ⁸⁵Rb polarization with λ_K(λ_Cs) is mainly determined by the pumping rate of the pump beam, the frequency shift and B^e.

Figure 2(d) demonstrates that ⁸⁵Rb polarization decreases with the increasing T. The ⁸⁵Rb polarization of ¹³³Cs–⁸⁵Rb–²¹Ne co-magnetometer is larger than the one of ³⁹K–⁸⁵Rb–²¹Ne co-magnetometer. From the formula of the and Fig. 1(c), R_p, frequency shift, Bⁿ, B^e, alkali-alkali spin exchange collision relaxation rate, we can find that increases with increasing T. , , , , , and increase with increasing T in Fig. 1(c) because the increasing T makes the number density and relative velocity of ⁸⁵Rb atoms larger and more ⁸⁵Rb atoms move towards the wall. and increase with increasing T, due to the fact that the increasing T can increase the linewidth of the K–Ne and Cs–Ne pressure broadening, which make and increase. , , , and decrease slowly with increasing T, due to the fact that the increasing T increases the linewidth of the K–Ne and Cs-Ne pressure broadening, which make , , and decrease. However, and do not change with T because and have nothing to do with the T. Bⁿ increases with increasing T due to the fact that changes with T from the equation of Bⁿ and increases with the increasing T. B^e increases with increasing T. , , , , , decrease with the increasing T because the increasing T makes , and increase, which is inversely proportional to the , , , in Eq. (5), respectively. Therefore, the total effect of the , the pumping rate of the pump beam ( and ), the frequency shift (, , and ), Bⁿ, B^e and the alkali-alkali spin-exchange collision relaxation rate (, , , , , ) make the ⁸⁵Rb polarization decrease with the increasing T.

Figure 3 shows that the ⁸⁵Rb polarization of ³⁹K (¹³³Cs)–⁸⁵Rb–²¹Ne co-magnetometers changes with the mole fraction of ⁸⁵Rb f_Rb, cell effective radius a and power density of pump beam I_PI.

Fig. 3.

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	Fig. 3. The 85Rb polarization of 39K (133Cs)–85Rb–21Ne co-magnetometers versus the mole fraction of 85Rb (a), the cell effective radius (b), and the power density of pump beam (c).

Figure 3(a) shows that ⁸⁵Rb polarizations decrease with increasing mole fraction of ⁸⁵Rb f_Rb. From the formulae of the D_r, , B^e and alkali-alkali spin exchange collision relaxation rate, we can find that D_r, B^e and alkali-alkali spin exchange collision relaxation rate decrease and increases with the increasing f_Rb. Therefore, the decrease process of the ⁸⁵Rb polarization is determined by D_r, , B^e and , , , , , , where D_r and play important roles.

Figure 3(b) demonstrates that ⁸⁵Rb polarizations decrease with increasing cell effective radius a. From the formula of the ⁸⁵Rb polarization, R_Wall and B^e, we can find that the wall relaxation rate , , , , and decrease when a increases in Fig. 1(c) and B^e increases with increasing a. Therefore, the decrease process of the ⁸⁵Rb polarization is determined by R_Wall and B^e, where the wall relaxation rate plays a decisive role.

Figure 3(c) shows that ⁸⁵Rb polarization increases when increases. From the formula of , R_p, frequency shift and B^e, we can find that increases with the increasing , which lead to R_p, frequency shift and B^e increase with increasing . R_p plays a decisive role for the change of ⁸⁵Rb polarization with . The ⁸⁵Rb polarization in the ³⁹K–⁸⁵Rb–²¹Ne co-magnetometer is smaller than the one in the ¹³³Cs–⁸⁵Rb–²¹Ne co-magnetometer.

Figures 4(a) and 4(b) demonstrate that the ⁸⁵Rb polarizations increase with increasing input rotation velocity in the x direction Ω_x and the y direction Ω_y, respectively. Ω/γⁿ is equivalent to a magnetic field, which can be compensated by the compensation magnetic field δ B_z. When L_z is some negative value, , which can help to eliminate the effect of the input rotation velocity Ω_x and Ω_y on the ⁸⁵Rb polarization. Figure 4(c) shows that ⁸⁵Rb polarizations decrease with increasing external magnetic field in the y direction B_y. A B_y modulation method could be utilized to measure the mixed AC Stark shifts.^[43] Figure 4(d) demonstrates that ⁸⁵Rb polarizations increase with increasing external magnetic field in the z direction B_z. The change of B_z will make the compensation magnetic field δ B_z change, which will affect the ⁸⁵Rb polarization.

Fig. 4.

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	Fig. 4. The 85Rb polarization of 39K (133Cs)–85Rb–21Ne co-magnetometers versus the input rotation velocity in the x direction (a), and y-direction (b), the external magnetic field in the y-direction By (c), and z-direction Bz (d).

We take ³⁹K (¹³³Cs) as the A atom, ⁸⁵Rb as the B atom, one of the f_Rb, n_Ne, n_N2, T, λ_K (λ_Cs), , a, B (it is helpful for us to study how B influence the SERF regime and fundamental sensitivity of the co-magnetometers) and V in Eq. (12) as a variable (other parameters are invariable) to obtain the results that the fundamental sensitivity of the co-magnetometers based on ³⁹K–⁸⁵Rb–²¹Ne and ¹³³Cs–⁸⁵Rb–²¹Ne change with the variable. Depending on suggestions and the typical conditions of the experiment group,^[34,35,54] in order to facilitate the theoretical analysis, we take n_Ne = 2 × 10¹⁹ cm^–3, n_N2 = 2 × 10¹⁷ cm^–3, λ_K = 769.808 nm, λ_Cs = 894.3 nm, W/cm², B = 10^–12 T ( 0), T = 457.5 K, a = 1 cm, a_p = 1 cm, f_Rb = 0.97, V = 1 cm³, and t = 1 s.

Figure 5(a) represents that the fundamental sensitivity of ³⁹K (¹³³Cs)–⁸⁵Rb–²¹Ne co-magnetometer augments with the increasing number density of ²¹Ne n_Ne when n_Ne is smaller than a critical value about 1.6230 × 10¹⁹ cm^–3 and decreases when n_Ne is larger than the value. For this phenomenon, just as we discussed in our previous work,^[24] 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 n_Ne 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 n_Ne is larger than the value and increase. Hence, if we take the critical value as n_Ne, spin exchange collisions in alkali-metal atoms are the most, we can obtain the highest fundamental sensitivity of the co-magnetometer. From Eq. (12), Fig. 1(a) and R_p, we can find that R_SD increases with the increasing n_Ne, and decrease with increasing n_Ne, and increase with increasing n_Ne when n_Ne increases, other systemic parameters in Eq. (12) do not change with n_Ne, hence there is a smallest n_Ne to make the fundamental sensitivity highest, which is determined by R_SD, () and .

Fig. 5.

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	Fig. 5. The fundamental sensitivity of 39K (133Cs)–85Rb–21Ne co-magnetometers versus the number density of the buffer gas 21Ne in (a), number density of the quench gas N2 in (b), the wavelength of pump beam in (c), and the power density of pump beam in (d).

Figure 5(b) shows that the fundamental sensitivity of ³⁹K(¹³³Cs)–⁸⁵Rb–²¹Ne co-magnetometer increases with the increasing N₂ number density n_N2 when n_N2 is smaller than a critical value about 1.2407 × 10¹⁹ cm^–3 and decreases when n_N2 is higher than the value. From Eq. (12) and Fig. 1(b), we can find that R_SD increases with the increasing n_N2, and decrease with increasing n_N2, other systemic parameters in Eq. (12) do not change with n_N2, hence there is a smallest n_N2 to make the fundamental sensitivity highest, which is determined by R_SD and ().

Figure 5(c) demonstrates that the fundamental sensitivity of ³⁹K (¹³³Cs)–⁸⁵Rb–²¹Ne co-magnetometers decrease when the wavelength of pump beam λ_K(λ_Cs) increases but is smaller than the value of about 770.085 nm (894.5032 nm) and increase when λ_K(λ_Cs) is larger than the value of about 770.1309 nm (894.696 nm) (⁸⁵Rb polarization is about 0.08).

Figure 5(d) shows that the fundamental sensitivity decreases with the increasing power density of pump beam . In the experiment, I_PI is usually a few hundreds of mW/cm². Here λ_K(λ_Cs) and affect the fundamental sensitivity by changing .

Figure 6(a) describes that the fundamental sensitivity of ³⁹K (¹³³Cs)–⁸⁵Rb–²¹Ne co-magnetometers decrease with the increasing external magnetic field B. The increasing B will make the relaxation rate due to alkali-alkali spin-exchange collisions increase, which makes the fundamental sensitivity decrease. The fundamental sensitivity of ³⁹K (¹³³Cs)–⁸⁵Rb–²¹Ne co-magnetometers increases with the increasing cell temperature T in Fig. 6(b) and we can know that the fundamental sensitivity of ³⁹K–⁸⁵Rb–²¹Ne co-magnetometer is lower than the one of ¹³³Cs–⁸⁵Rb–²¹Ne co-magnetometer from the illustration. From Eq. (12), Fig. 1(c), the formula of R_p and alkali-alkali spin exchange collision relaxation rate, we can find that R_SD and n_B (n_Rb) increase with the increasing T, R_wall increases rapidly when the T increases and R_p increases slowly with the increasing T. However, there is a larger increase for the ⁸⁵Rb number density and a decrease for when the T increases. Hence, the fundamental sensitivity increases with the increasing T, which is mainly determined by the ⁸⁵Rb number density.

Fig. 6.

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	Fig. 6. The fundamental sensitivity of 39K (133Cs)–85Rb–21Ne co-magnetometers change with the external magnetic field in (a), the cell temperature in (b), the cell effective radius in 6(c), and mole fraction of 85Rb in (d).

The fundamental sensitivity of ³⁹K (¹³³Cs)–⁸⁵Rb–²¹Ne co-magnetometers increase with the increasing cell effective radius a, respectively in Fig. 6(c) due to the fact that the increasing a makes the R_wall decrease, which will increase the fundamental sensitivity. The fundamental sensitivity of ³⁹K (¹³³Cs)–⁸⁵Rb–²¹Ne co-magnetometers does not change with increasing a_p. From the formula of R_p, we can find that R_p does not change with the increasing a_p due to the fact that a_p in () and A_t are canceled out.

Figure 6(d) describes that the fundamental sensitivity increases with the increasing mole fraction of ⁸⁵Rb f_Rb, due to the alkali-alkali spin exchange collision relaxation rate decreases and ⁸⁵Rb number density increases with the increasing f_Rb. When the f_Rb is larger than about 0.9662, the fundamental sensitivity of ¹³³Cs–⁸⁵Rb–²¹Ne co-magnetometer is higher than the one of ³⁹K–⁸⁵Rb–²¹Ne co-magnetometer. From Eq. (12), the fundamental sensitivity of ³⁹K (¹³³Cs)–⁸⁵Rb–²¹Ne co-magnetometers increases with increasing measurement volume.

As a result, the polarization of the ⁸⁵Rb atom of the co-magnetometer based on ¹³³Cs–⁸⁵Rb–²¹Ne is larger than the one based on ³⁹K–⁸⁵Rb–²¹Ne in Figs. 2–4. The fundamental sensitivity of ¹³³Cs–⁸⁵Rb–²¹Ne co-magnetometer is higher than the ³⁹K–⁸⁵Rb–²¹Ne co-magnetometer at the same cell temperature in the SERF regime with the same frequency detuning of the pump beam when (1) the external magnetic field is smaller than about 1.7884 × 10^–8 T, (2) the mole fraction of ⁸⁵Rb is larger than about 0.9662, or (3) the power density of pump beam is smaller than about 0.229 W/cm² in Figs. 5 and 6 under our chosen conditions.

We obtain a fundamental sensitivity of about 7.542 × 10^–11 rad·s^–1·Hz^–1/2 with ³⁹K–⁸⁵Rb–²¹Ne co-magnetometer with ⁸⁵Rb polarization is about 4.4892 × 10^–5 and n_K/n_Rb ≈ 5.1228 × 10^–4, a fundamental sensitivity of about 7.5355 × 10^–11 rad·s^–1·Hz^–1/2 with ¹³³Cs–⁸⁵Rb–²¹Ne co-magnetometer with ⁸⁵Rb polarization is about 8.193 × 10^–4 and n_Cs/n_Rb ≈ 0.0043 when n_Ne = 6 × 10¹⁹ cm^–3, n_N2 = 6.3 × 10¹⁸ cm^–3, T = 406.696 K, f_Rb = 0.99, W/cm², λ_K = 769.938 nm, λ_Cs = 894.43 nm, a = 2 cm, a_p = 1 cm, V = 10 cm³, B = 10^–12 T, and t = 1 s. The fundamental sensitivity of ³⁹K(¹³³Cs)–⁸⁵Rb–²¹Ne co-magnetometer is higher than the angular velocity sensitivity of 2.1 × 10^–8 rad·s^–1·Hz^–1/2 of K–Rb–²¹Ne comagnetometer.^[61] By optimizing the above parameters, we obtain a fundamental sensitivity of about 3.9993 × 10^–11 rad·s^–1·Hz^–1/2 with ³⁹K–⁸⁵Rb–²¹Ne co-magnetometer for the polarization of the ⁸⁵Rb atom is about 0.0021 and n_K/n_Rb ≈ 7.5791 × 10^–4, a fundamental sensitivity of 3.8221 × 10^–11 rad·s^–1·Hz^–1/2 with ¹³³Cs–⁸⁵Rb–²¹Ne co-magnetometer for the polarization of the ⁸⁵Rb atom is about 0.0224 and n_Cs/n_Rb ≈ 0.0049 with n_Ne = 8.3407 × 10¹⁸ cm^–3, n_N2 = 6.2035 × 10¹⁸ cm^–3, T = 457.5 K, f_Rb = 0.99, W/cm², λ_K = 770.038 nm, λ_Cs = 894.53 nm, a = 2 cm, a_p = 1 cm, V = 10 cm³, B = 10^–12 T and t = 1 s, with higher fundamental sensitivity possible at larger measurement volume, proper amount of buffer gas and quench gas, smaller pumping rate of pump beam and higher temperature.