Temperature characteristics of K-Rb hybrid optically pumped magnetometers with different density ratios

Sho Ito; Yosuke Ito; Tetsuo Kobayashi

doi:10.1364/OE.27.008037

1. Introduction

Optically pumped magnetometers (OPMs) are employed for performing biomagnetic field measurements such as magnetoencephalograms (MEGs) [1] and magnetocardiograms (MCGs) [2]; they have rapidly advanced in recent years. Magnetometers exhibit the highest sensitivity (theoretically, $10 aT / \sqrt{Hz}$ ) [3,4] in the spin-exchange-relaxation-free (SERF) regime. Furthermore, they require no cryogenic cooling; hence, their installation and maintenance costs are lower than those of the superconductive quantum interference devices.

Multichannel sensors have to be developed for employing OPMs to perform biomagnetic field measurements; further, the sensitivity of the OPMs should be improved. OPMs with a single alkali metal has been extensively studied, and Dang et al. achieved a sensitivity of $0.16 fT / \sqrt{Hz}$ at 40 Hz using a pure potassium (K) cell [5]. However, in such OPMs, the sensor properties are observed to fluctuate along the pump beam with the alkali metal atoms because of the attenuation of the pump beam. The sensor properties should be spatially homogeneous to achieve several sensing regions in a single cell [6]. Therefore, hybrid spin exchange optical pumping (SEOP) is used. The OPMs that use two kinds of alkali metal atoms reduce the optical depth (OD) of the pumped atoms, which makes it possible to improve the spatial homogeneity of the sensor properties [7,8]. Furthermore, the fundamental sensitivity limit of the hybrid OPMs is expected to be higher than that of the single OPMs [9], and the spin polarization of the K obtained in the SEOP is 4.5 times that achieved by the direct optical pumping of K without any quenching gas [10].

The densities of the two alkali metal atoms and their density ratio are especially important for improving the sensitivity of hybrid OPMs [11]. The density ratio changes little as the temperature of the hybrid cell after the hybrid cell is sealed. However, the densities of the two alkali metal atoms change with the temperature of the hybrid cell. The output signal intensity of hybrid OPMs is expected to be temperature dependent because the spin exchange rate of different alkali metal atoms is affected by their densities. Fang et al. investigated the temperature dependence of the output signal intensity [12]. However, their numerical results do not agree well with the experimental results. In addition, they measured the temperature dependence of output signal intensity for one hybrid cell, so that the output signal intensity with respect to the density ratios of two types of alkali metal atoms have not been elaborated. Furthermore, they considered only the case where the density of the atom for probing reached to the saturated density. Therefore, in this study, we fabricated hybrid cells with different densities and density ratios. We measured the output signal intensities of the hybrid OPMs with the hybrid cells to investigate their temperature dependence. Furthermore, the densities of K and Rb were estimated based on Raoult’s law, and we compared the experimental results with the results of the numerical calculations to validate the Bloch equations for the hybrid OPMs.

2. Principle

The behavior of the spin polarization of the alkali atoms is described using the Bloch equation. In hybrid OPMs, the atoms of one type of alkali metal are directly pumped using a pump beam, whereas the other alkali metal atom is polarized by spin exchange collisions using the pumped atoms. In this study, the measurements were performed using a pump and probe configuration [6]. We used Rb for optical pumping and K for probing because K exhibited the highest theoretical sensitivity owing to its lowest spin destruction collision cross section [9]. The Bloch equation for a hybrid cell can be given as follows:

\begin{array}{l} \frac{d}{d t} S^{Rb} = D^{Rb} \nabla^{2} S^{Rb} + \frac{γ^{e}}{q^{Rb}} \times B \times \frac{1}{q^{Rb}} R_{SE}^{RbK} S^{K} + \frac{R_{OP}}{2 q^{Rb}} z \\ - \frac{1}{q^{Rb}} (R_{OP} + R_{SD}^{Rb} + R_{SE}^{RbK} + R_{SE}^{RbRb}) S^{Rb}, \end{array}

\frac{d}{d t} S^{K} = D^{K} \nabla^{2} S^{K} + \frac{γ^{e}}{q^{K}} S^{K} \times B + \frac{1}{q^{K}} R_{SE}^{KRb} S^{Rb} - \frac{1}{q^{K}} (R_{SD}^{K} + R_{SE}^{KRb} + R_{SE}^{KK}) S^{K},

where S denotes the electron spin polarization, γ^e denotes the gyromagnetic ratio of electrons, q denotes the slowing-down factor, D denotes the diffusion coefficient, B denotes the magnetic field,

R_{SE}^{SS}

denotes the spin exchange rate attributed to the spin exchange collisions with atoms of its own species (S),

R_{SE}^{{SS}^{'}}

denotes the spin exchange rate attributed to the spin exchange collisions with atoms of other species (S′), R_OP denotes the pumping rate, and R_SD denotes the spin destruction rate attributed to the spin destruction collisions [13]. Superscripts indicate the atom with respect to the vectors and coefficients. The pumping rate R_OP is expressed in terms of the absorption cross section σ and photon flux Ψ as follows:

R_{OP} = \int σ (ν) Ψ (ν) d ν,

Ψ (ν) = \frac{2 I_{pump} \sqrt{π ln 2}}{A_{pump} π ν_{0}^{Rb} δ ν} exp [- \frac{4 {(ν - ν_{0}^{Rb})}^{2} ln 2}{δ ν^{2}}],

where I_pump and A_pump denote the intensity and the cross section of the pump beam, respectively. Further, ν denotes the frequency of the pump beam,

ν_{0}^{Rb}

denotes the resonant frequency of the pumped atoms, and δν denotes the linewidth of the pump beam. In addition, the pumping rate R_OP changes because of the absorption of the pump beam. This variation can be expressed as shown in the following equation by considering the propagation direction of the pump beam as z:

\frac{d R_{OP} (z)}{d z} = - n_{Rb} σ (ν) R_{OP} (z) (1 - 2 S_{z}^{Rb} (z)) .

In this equation, n_Rb denotes the density of Rb.

In addition, $S_{x}^{K}$ causes the optical rotation of the polarization plane of the probe beam by the magneto-optical effect. When the rotation angle θ becomes sufficiently small, the output signal of OPMs S_out can be described as follows [11]:

S_{out} ≃ 2 η I_{probe} e^{- α l} θ,

θ = \frac{n_{K} c r_{e} f (ν_{probe} - ν_{0}^{K})}{{(ν_{probe} - ν_{0}^{K})}^{2} + {(\frac{Γ_{k}}{2})}^{2}},

where η denotes the efficiency of conversion from light intensity to voltage, I_probe denotes the light intensity of the probe beam, α is the absorption coefficient, θ denotes the magneto-optical rotation angle, n_K denotes the density of the K atoms,

ν_{0}^{K}

denotes the D1 resonance frequency of K, ν_probe denotes the frequency of the probe beam, c denotes the light velocity, r_e denotes the classical electron radius, f denotes the oscillator strength, and Γ_K denotes the pressure linewidth of the K atoms.

3. Methods

3.1. Experimental setup

The experimental setup of OPMs is illustrated in Fig. 1. A vertical-cavity surface-emitting laser is used as the pump beam and the pump laser is tuned to the D1 transition of Rb. A diode laser is used as the probe beam, and the probe laser is slightly detuned from the D1 transition of K. The intensity and wavelength of both the pump beam and the probe beam are set to maximize the signal-to-noise ratio of the output signal of OPMs. The cell was placed in an oven, and the oven was heated by a wire heater using the AC current. The AC current frequency was set as 100 kHz to reduce the low-frequency noise. The probe beam penetrated at a position 1-cm under the top of the cell to be close to the subject in the case of biomagnetic-field measurements. Furthermore, the three-axis compensation coils were placed in the magnetic shield to reduce the residual magnetic field.

Fig. 1: The experimental setup of OPMs. The hybrid cell and three pairs of coils were placed in a three-layered mu-metal magnetic shield.

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The hybrid cell is a cubic Pyrex glass (5 × 5 × 5 cm), and it encloses the Rb and K atoms in the He and N₂ buffer gases that are present in a ratio of 10:1 and at a total pressure of 1.5 atm at room temperature. For performing the measurements, we fabricated five hybrid cells, and their densities are specified in Table 1.

Table 1:. The densities of K and Rb.

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In the experiment, a sinusoidal magnetic field of 48 pT and 10 Hz was applied using the Helmholtz coil (y-axis) to verify the properties of OPMs. The resonant frequency of OPMs was tuned to 10 Hz by adjusting the bias magnetic field in the direction of the pump beam.

3.2. Numerical calculation

The numerical calculation model is depicted in Fig. 2. The pump beam is irradiated at a circle with a diameter of 5 cm, and the probe beam is irradiated at a cross section of 2 × 2 mm. To simulate the experimental condition, the probe beam passes through the cell at y = 4.0 cm. Here, we have assumed that the pump and probe power intensities were uniformly irradiated; further, we have performed the numerical calculation using the measured power intensity value of the laser. Rb has two isotopes ⁸⁵Rb and ⁸⁷Rb, and the natural abundance ratio of these isotopes is 72.2:27.8. In this study, for simplicity, all the Rb atoms were considered as ⁸⁵Rb for performing the calculation.

Fig. 2: The numerical calculation model.

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The inside of the cell was considered to be the computational domain, which was divided into 51 × 51 × 51 grid points. At each grid point, the spin polarizations of Rb and K were calculated as follows. First, to consider a steady state without the application of a sinusoidal magnetic field, we calculated the optical pumping rate and the spin polarization of K and Rb using Eqs. (1), (2), and (4) and using an iteration method that considers the boundary condition of the wall in a cell for which the spin polarization is zero. Further, we obtained $S_{z}^{Rb}$ , $S_{z}^{K}$ , and R_OP. The relevant parameters that are used for performing the numerical calculation are presented in Table 2. Second, we considered the state in which the sinusoidal magnetic field is applied and ignored the diffusion of spin polarization. We used Eqs. (1), (2), (6), and (7) and adopted the Runge–Kutta method; further, we obtained the spin polarizations S^Rb, S^K, and S_out.

Table 2:. The parameters used for performing the numerical calculation.

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The spin exchange rate $R_{SE}^{{SS}^{'}}$ corresponding to the spin exchange collisions with atoms (S) of other species (S′) can be expressed in terms of the mean thermal velocity v̄ and the spin exchange cross section $σ_{SE}^{{SS}^{'}}$ .

R_{SE}^{{SS}^{'}} = n_{S^{'}} σ_{SE}^{{SS}^{'}} \bar{ν}

The spin exchange rate

R_{SE}^{SS}

corresponding to the spin exchange collisions with atoms of its own species (S) can be expressed in terms of the mean thermal velocity v̄ and the spin exchange cross section

σ_{SE}^{SS}

.

R_{SE}^{SS} = {(\frac{γ^{e} B_{z}}{q^{S}})}^{2} \frac{{(q^{s})}^{2} - {(2 I + 1)}^{2}}{2 R_{SE}^{' SS}}

R_{SE}^{' SS} = n_{S} σ_{SE}^{SS} \bar{ν}

In this equation, I denotes the angular momentum of the nuclear spin of the alkali metal atoms, and B_z denotes the bias magnetic field in the z-axis direction. The spin destruction rates

R_{SD}^{S}

caused by He and N₂ can be expressed as follows [13]:

R_{SD}^{S} = 2 (n_{S} σ_{SD}^{SS} \bar{ν} + n_{S'} σ_{SD}^{{SS}^{'}} \bar{ν} + n_{He} σ_{SD}^{SHe} \bar{ν} + n_{N_{2}} σ_{SD}^{{SN}_{2}} \bar{ν}),

where σ_SD denotes the spin destruction cross section; further, n_He and n_N₂ denote the densities of each atom. The diffusion coefficients D corresponding to the diffusion of the alkali atoms to the wall can be obtained as follows [11]:

D = {[{(D_{He}^{S} \frac{{(T / 273)}^{3 / 2}}{P_{He} / 1 atm})}^{- 1} + {(D_{N_{2}}^{S} \frac{{(T / 273)}^{3 / 2}}{P_{N_{2}} / 1 atm})}^{- 1}]}^{- 1},

where

D_{He}^{S}

and

D_{N_{2}}^{S}

denote the diffusion constants of the alkali metal atoms with buffer and quench gas, respectively, P_He, and P_N₂ is the partial pressure. In addition, the pumping rate R_OP was calculated from Eqs. (3) and (4) using the measured value of the linewidth of the pump beam, the intensity of the pump beam, and the cross section of the pump beam. The slow-down factor q^K is equal to

\frac{6 + {(2 S_{z}^{K})}^{2}}{1 + {(2 S_{z}^{K})}^{2}}

, and the slow-down factor q^Rb is equal to

\frac{38 + 52 {(2 S_{z}^{Rb})}^{2} + 6 {(2 S_{z}^{Rb})}^{4}}{3 + 10 {(2 S_{z}^{Rb})}^{2} + 3 {(2 S_{z}^{Rb})}^{4}}

.

3.3. Vapor density

The saturated density of the pure alkali metal vapor at temperature T can be given by the following equation [14]:

n_{K}^{0} = \frac{10^{26.268 - \frac{4453}{T}}}{T}

n_{Rb}^{0} = \frac{10^{26.188 - \frac{4040}{T}}}{T}

The saturated vapor density for pure alkali metal atoms can be given as

n_{Rb}^{0} ≫ n_{K}^{0}

at a given temperature according to the above equation. However, the hybrid cells contain potassium and rubidium; hence, it is necessary to be vaporized from the alkali metal mixtures. For estimating the vapor density, we assume that the vapor densities obey Raoult’s law [15]:

n_{K} = f_{K} n_{K}^{0},

n_{Rb} = f_{Rb} n_{Rb}^{0},

where f_K, f_Rb denotes the mole fraction of the alkali metal atoms. Table 3 presents the relation between the mole fraction of K and temperature when n_K/n_Rb = 2.11. As denoted in Table 3, there is little fluctuation in the mole fraction of K. Therefore, we determined the mole fraction from the experimental results; further, the vapor densities were also estimated.

Table 3:. The mole fraction of K (n_K/n_Rb = 2.11 (No.2)).

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4. Results and discussion

Figures 3(a) and 3(b) depict the experimental and numerically calculated signal intensities as functions of the temperature of the hybrid cell. We used five different hybrid cells in the experiments. The results demonstrate that the output signals of OPMs increase with the temperature, and the amount of increase in signal intensity of OPMs differs by the density ratio of K and Rb atoms. However, the experimental values are not in good agreement with the calculated values. These discrepancies are attributed to the spin relaxation of the absorption of the probe beam using the probed atoms. The probe beams are slightly detuned from the D1 resonance of the K atoms, and they are absorbed by the alkali metal atoms when they pass through the cell.

Fig. 3: The temperature dependence of the output signal of OPMs. (a) depicts the experimental results as a function of the temperature of the hybrid cell, and (b) depicts the numerical calculation results plotted against the temperature based on Eqs. (1), (2), (6), and (7).

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Here, we consider the fourth term in Eq. (2). These relaxation terms are expressed as shown in Eqs. (8) and (11). Table 1 indicates that $σ_{SE}^{KRb}$ is larger than any other spin destruction cross section; hence, $R_{SE}^{KRb}$ has a considerable influence on the fourth term when the vapor densities for Rb and K can be given as n_Rb < n_K. It can be deduced that the relaxation caused by the absorption of the probe beam is dominant when the Rb density is low, i.e., the density ratio n_K/n_Rb is high.

We can observe from Fig. 3 that the calculated results are considerably larger than the experimental results when n_K/n_Rb = 178 at high temperature, and these results indicate that the relaxation due to the absorption of the probe beam influences the output signal of OPMs. When n_K/n_Rb = 28.4, the calculated results are smaller than the experimental results at a low temperature. This can be explained by the probe beam pumping effect [16] and the measurement error of the density of Rb by absorption spectroscopy because the density of Rb is small.

When n_K/n_Rb = 4.85, we measured the transmitted probe beam power output signal of the OPMs (Fig. 4) by changing the intensity of the probe beam. Figure 4(a) depicts the transmitted probe beam power as a function of the incident probe beam power. We considered the probe beams of two wavelengths (770.01 and 769.76 nm). The transmitted probe beam powers were proportional to the incident probe beam powers for both the wavelengths, which can be explained using Eq. (6). When the magneto-optical rotation becomes constant, the transmitted light intensity of the probe beam and the output signal of OPMs are observed to become proportional. Figure 4(b) depicts the relation between the intensity of the probe beam and output signal of OPMs. The incident light intensity is proportional to the transmitted probe beam intensity; however, the output signals of the OPMs are not proportional to the incident light intensity for the 770.01-nm probe beam. Further, both the values are proportional to the incident probe beam power for the 769.76-nm probe beam. Therefore, the output signals of OPMs are affected by the probe beam when the probe beam wavelength is detuned from the D1 transition of K. From Eq. (6), it can be observed that the magneto-optical rotation decreases because of the relaxation of the absorption of the probe beam by the probed atoms when the probe beam power becomes large. Therefore, the relaxation due to the absorption of the probe beam is set to R_PR = C (C :constant), and this relaxation is added to the fourth term in Eq. (2) as follows:

\frac{d}{d t} S^{K} = D^{K} \nabla^{2} S^{K} + \frac{γ^{e}}{q^{K}} S^{K} \times B + \frac{1}{q^{K}} R_{SE}^{K} S^{Rb} - \frac{1}{q^{K}} (R_{SD}^{K} + R_{SE}^{KRb} + R_{SE}^{KK} + R_{PR}) S^{K}

The output signal intensity of the cell with n_K/n_Rb = 4.85 that was measured and calculated from the above equation is denoted in Fig. 5. The calculated result with R_PR = 1.5 × 10³ s⁻¹ agreed with the measured data. The value of the relaxation R_PR estimated from the probe beam intensity was 6.9 × 10² s⁻¹, and it can be observed that the relaxation due to the absorption of the probe beam influenced the output signal intensity of the OPMs.

Fig. 4: (a) depicts the dependence on the probe beam power of transmitted light intensity, and (b) depicts the output signal of OPMs as a function of the probe beam power.

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Fig. 5: Measured data of n_K/n_Rb = 4.85 and calculated results considering the relaxation of the absorption of the probe beam plotted against the temperature. The relaxation rate was varied and the calculated result with R_PR = 1.5 × 10³ s⁻¹ agreed with the measured data.

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Furthermore, in a previous study, the probed atoms absorbed the pump beam, and the absorption aided the relaxation of the probed atoms when the power of the pump beam became large [12]. The relaxation due to the absorption of the probe beam by the probed atoms is set to R_K, and the relaxation is approximated as $R_{K} = 4 R_{OP} \frac{δ ν}{Γ_{K}}$ [15], where δν is a large frequency detuned from the D1 transition of Rb. According to the calculation results, R_K may be negligible.

Figure 6 depicts the numerically calculated output signal intensity as well as OPM sensitivity, assumed to be achievable at each temperature, from the result as a function of the temperature of the cell when n_K/n_Rb = 4.85. Because the magnetic noise, which remains constant regardless of the temperature, is dominant in the noise of the measuring data, the average value of the frequency spectrum ranging from 9 to 11 Hz, excluding 10 Hz, was used for obtaining the noise level. Furthermore, the wavelength of the probe beam was set to maximize the OPM output signals at each temperature. Figure 6 depicts that the output signal of the OPMs increases as the temperature increases to 270°C and that it rapidly decreases above this temperature. This is because the spin polarization along the pump beam becomes low when the density of the Rb atom is large at high temperatures. Hence, the spatial homogeneity of the sensor properties becomes low with high temperature. Furthermore, it is estimated that a sensitivity of 1.6 fT/Hz^1/2 can be achieved by increasing the temperature to 270°C. However, our sensor cells were made of Pyrex glass, whose maximum continuous operating temperature is 260°C. Therefore, we must change the material of the sensor cell to increase the temperature. In addition, heat insulators must be thick although the sensor should be close to subjects for biomagnetic-field measurements. Thus, the temperature should be selected considering the cell materials and application. In the experimental configuration considered in this study, the temperature can only be increased to 180°C; therefore, it is considered to be preferable to perform measurements at temperatures higher than 180°C in the future study. In addition, the performance of the OPMs is ultimately limited by the quantum noise [17]; therefore, the sensitivity of the OPMs can be further improved by reducing the environmental magnetic field noise.

Fig. 6: Numerically calculated output signal and OPM sensitivity as a function of the temperature of the hybrid cell (No. 3)

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5. Conclusion

In this study, we fabricated five hybrid cells of K and Rb with different densities and performed various measurements to investigate the temperature dependence of the output signal intensity of OPMs. Furthermore, we compared the experimental results with the numerically obtained results. The densities of K and Rb increased with an increase in temperature obeying the Raoult’s law, even though the atom for probing was less than the saturated density. The experimental results denoted that the OPM signal intensities increased with an increase in temperature, and had different temperature characteristics depending on the density ratio of K and Rb atoms. In addition, the experimental results showed good agreement with the results of the numerical calculations based on the Bloch equation after the introduction of the relaxation term due to the absorption of the probe beam. Furthermore, the densities of alkali metal atoms increase as temperature increases, but the signal intensity of OPMs has a maximum because the pumping rate becomes low when the density of the Rb atom is high at high temperatures. These results indicate that we can estimate the optimal temperatures in hybrid OPMs with the given pump beam intensity, enabling us to improve their sensitivity.

In future, we intend to calculate the optimal densities of the K and Rb atoms and their density ratios for the hybrid cell that may be helpful to achieve the highest sensitivity for the hybrid OPM.

Funding

Ministry of Education, Culture, Sports, Science and Technology (15H01813); Nakatani Foundation for Advancement of Measuring Technologies in Biomedical Engineering.

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Cell number	Density ratio n_K/n_Rb	Atom	Temperature (°C)	Density (m⁻³)
1	0.421 (150°C)	K	150	4.93×10¹⁸
1	0.421 (150°C)	Rb	150	1.17×10¹⁹
2	2.11 (180°C)	K	180	7.22×10¹⁸
2	2.11 (180°C)	Rb	180	3.42×10¹⁸
3	4.85 (180°C)	K	180	1.07×10¹⁹
3	4.85 (180°C)	Rb	180	2.21×10¹⁸
4	28.4 (180°C)	K	180	1.13×10¹⁸
4	28.4 (180°C)	Rb	180	3.98×10¹⁶
5	178 (180°C)	K	180	5.49×10¹⁸
5	178 (180°C)	Rb	180	3.08×10¹⁶

Parameter	Value
Diffusion constant of K in He: $D_{He}^{K}$	0.35 cm²/s [11]
Diffusion constant of K in N₂: $D_{N_{2}}^{K}$	0.20 cm²/s [11]
Diffusion constant of Rb in He: $D_{He}^{Rb}$	0.50 cm²/s [11]
Diffusion constant of Rb in N₂: $D_{N_{2}}^{Rb}$	0.19 cm²/s [11]
Spin exchange cross section of K–K: $σ_{SE}^{KK}$	1.8 × 10⁻¹⁸ m² [11]
Spin exchange cross section of Rb–Rb: $σ_{SE}^{RbRb}$	1.9 × 10⁻¹⁸ m² [11]
Spin exchange cross section of K–Rb: $σ_{SE}^{KRb}$	$σ_{SE}^{KK} \frac{n_{K}}{n_{K} + n_{Rb}} + σ_{SE}^{RbRb} \frac{n_{Rb}}{n_{K} + n_{Rb}}$ [9]
Spin destruction cross section of K–K: $σ_{SD}^{KK}$	1.0 × 10⁻²² m² [11]
Spin destruction cross section of Rb–Rb: $σ_{SD}^{RbRb}$	1.6 × 10⁻²¹ m² [11]
Spin destruction cross section of K–Rb: $σ_{SD}^{KRb}$	4.0 × 10⁻²² m² [11]
Spin destruction cross section of K–He: $σ_{SD}^{KHe}$	8.0 × 10⁻²⁵ m² [11]
Spin destruction cross section of K–N₂: $σ_{SD}^{{KN}_{2}}$	7.9 × 10⁻²⁷ m² [11]
Spin destruction cross section of Rb–He: $σ_{SD}^{RbHe}$	9.0 × 10⁻²⁸ m² [11]
Spin destruction cross section of Rb–N₂: $σ_{SD}^{{RbN}_{2}}$	1.0 × 10⁻²⁶ m² [11]
The efficiency of conversion from light intensity to voltage: η	4.5 × 10⁵ V/W

Atom	Temperature (°C)	The mole fraction
K	120	0.127
	140	0.114
	160	0.133
	180	0.119

Cell number	Density ratio n_K/n_Rb	Atom	Temperature (°C)	Density (m⁻³)
1	0.421 (150°C)	K	150	4.93×10¹⁸
1	0.421 (150°C)	Rb	150	1.17×10¹⁹
2	2.11 (180°C)	K	180	7.22×10¹⁸
2	2.11 (180°C)	Rb	180	3.42×10¹⁸
3	4.85 (180°C)	K	180	1.07×10¹⁹
3	4.85 (180°C)	Rb	180	2.21×10¹⁸
4	28.4 (180°C)	K	180	1.13×10¹⁸
4	28.4 (180°C)	Rb	180	3.98×10¹⁶
5	178 (180°C)	K	180	5.49×10¹⁸
5	178 (180°C)	Rb	180	3.08×10¹⁶

Parameter	Value
Diffusion constant of K in He: $D_{He}^{K}$	0.35 cm²/s [11]
Diffusion constant of K in N₂: $D_{N_{2}}^{K}$	0.20 cm²/s [11]
Diffusion constant of Rb in He: $D_{He}^{Rb}$	0.50 cm²/s [11]
Diffusion constant of Rb in N₂: $D_{N_{2}}^{Rb}$	0.19 cm²/s [11]
Spin exchange cross section of K–K: $σ_{SE}^{KK}$	1.8 × 10⁻¹⁸ m² [11]
Spin exchange cross section of Rb–Rb: $σ_{SE}^{RbRb}$	1.9 × 10⁻¹⁸ m² [11]
Spin exchange cross section of K–Rb: $σ_{SE}^{KRb}$	$σ_{SE}^{KK} \frac{n_{K}}{n_{K} + n_{Rb}} + σ_{SE}^{RbRb} \frac{n_{Rb}}{n_{K} + n_{Rb}}$ [9]
Spin destruction cross section of K–K: $σ_{SD}^{KK}$	1.0 × 10⁻²² m² [11]
Spin destruction cross section of Rb–Rb: $σ_{SD}^{RbRb}$	1.6 × 10⁻²¹ m² [11]
Spin destruction cross section of K–Rb: $σ_{SD}^{KRb}$	4.0 × 10⁻²² m² [11]
Spin destruction cross section of K–He: $σ_{SD}^{KHe}$	8.0 × 10⁻²⁵ m² [11]
Spin destruction cross section of K–N₂: $σ_{SD}^{{KN}_{2}}$	7.9 × 10⁻²⁷ m² [11]
Spin destruction cross section of Rb–He: $σ_{SD}^{RbHe}$	9.0 × 10⁻²⁸ m² [11]
Spin destruction cross section of Rb–N₂: $σ_{SD}^{{RbN}_{2}}$	1.0 × 10⁻²⁶ m² [11]
The efficiency of conversion from light intensity to voltage: η	4.5 × 10⁵ V/W

Temperature characteristics of K-Rb hybrid optically pumped magnetometers with different density ratios

Abstract

1. Introduction

2. Principle

3. Methods

3.1. Experimental setup

3.2. Numerical calculation

3.3. Vapor density

4. Results and discussion

5. Conclusion

Funding

References

Cited By

Figures (6)

Tables (3)

Equations (17)

Optics Express