## Abstract

An optically pumped K–Rb hybrid atomic magnetometer can be a useful tool for biomagnetic measurements due to the high spatial homogeneity of its sensor property inside a cell. However, because the property varies depending on the densities of potassium and rubidium atoms, optimization of the densities is essential. In this study, by using the Bloch equations of K and Rb and considering the spatial distribution of the spin polarization, we confirmed that the calculation results of spin polarization behavior are in good agreement with the experimental data. Using our model, we calculated the spatial distribution of the spin polarization and found that the optimal density of K atoms is 3 × 10^{19} m^{−3} and the optimal density ratio is *n*_{K}*/n*_{Rb} ~ 400 to maximize the output signal and enhance spatial homogeneity of the sensor property.

© 2016 Optical Society of America

## 1. Introduction

Optically pumped atomic magnetometers (OPAMs) in the spin exchange relaxation free (SERF) condition are of interest due to their high sensitivity to magnetic fields comparable or surpassing that of superconducting quantum interference devices (SQUIDs) [1]. In contrast to SQUIDs, OPAMs do not require cryogenic cooling for their operation. Therefore, OPAMs are expected to become alternatives to SQUIDs and have been studied actively by many research groups in the recent years [2–5]. The theoretical potential sensitivity is estimated to be 2 × 10^{−18} T/Hz^{1}^{/}^{2} in the spin-exchange relaxation free regime [1], and Romalis group achieved 1.6 × 10^{−16} T/Hz^{1}^{/}^{2} in a gradiometer arrangement [6]. To apply OPAMs to biomagnetic measurements, a multichannel sensor is required to obtain magnetic field distributions as well as to improve the sensitivity of the OPAMs.

There are several ways to fabricate the multichannel sensor, such as the integration of individual OPAMs [7–9] and realizing many sensing regions in a sensor cell [10–13]. We have adopted the latter method because it is easy to obtain identical sensor properties for each sensing region. However, in this method, the sensor properties fluctuate along the pump beam direction due to the absorption of the pump beam by alkali metal atoms.

We have developed OPAMs using hybrid cells that contain two types of alkali metal atoms such as K and Rb. In the OPAMs, the type of alkali metal atoms that has low density is optically pumped by the pump beam and spin polarization arises [14, 15]. The spin polarization is transferred to the other type of alkali metal atoms that has high density by spin exchange collisions. By this technique, we can obtain a spatially homogeneous sensor with a sensitivity of several tens of fT/Hz^{1}^{/}^{2} and have demonstrated simultaneous multilocation measurements [14, 16, 17]. However, the sensor properties depend on the densities of K and Rb atoms; therefore, optimization of these densities is required for applying the magnetometer to simultaneous multilocation measurements.

In this study, we investigated the optimal densities of K and Rb atoms by establishing the calculation model considering the spatial distribution of the spin polarization inside the sensor cell.

The rest of this paper is organized as follows. In Sec. 2, we describe the principles of response signals of the OPAMs using hybrid cells. In Sec. 3, we describe the experimental procedure and numerical calculations of the dependency of response signals on pump-beam densities for different K and Rb density ratios. In Sec. 4, we discuss the optimal densities of K and Rb with regard to response signals and spatial homogeneity of the spin polarization. Finally, we present our conclusions in Sec. 5.

## 2. Principle of an OPAM using a potassium and rubidium hybrid cell

Figure 1 shows the principle of OPAMs using K and Rb hybrid cells. In a hybrid cell, both K and Rb atoms are enclosed. In this paper, we designate Rb as pump atoms and K as probe atoms. The wavelength of the circularly-polarized pump beam is tuned to the D1 resonance of Rb and penetrates the cell along the *z* direction. In contrast, the wavelength of the linearly polarized probe beam is slightly detuned from the D1 resonance of K and passes through the cell along the *x* direction.

The pump beam induces optical pumping and causes the electron spin polarization for Rb atoms to *z* direction
${\mathbf{S}}^{\text{Rb}}={S}_{z}^{\text{Rb}}\mathbf{z}$. Then, the spin polarization is transferred to the K atoms by spin exchange collisions [18]. The spin polarization of the K atoms is designated as
${\mathbf{S}}^{\mathrm{K}}={S}_{z}^{\mathrm{K}}\mathbf{z}$. By applying the magnetic field to be measured **B**_{m} = *B _{y}*

**y**, the spin polarizations

**S**

^{Rb}and

**S**

^{K}are rotated by the torque of

**S**×

**B**

_{m}[19]. This makes

**S**

^{K}to have an

*x*component ${S}_{x}^{\mathrm{K}}$, which causes the optical rotation of the polarization plane of the probe beam passing through the cell by the magneto-optical effect. The rotation angle

*θ*depends on ${S}_{x}^{\mathrm{K}}$; therefore, we can measure the component

*B*by measuring the rotation angle.

_{y}In addition, by applying the bias field **B**_{0} = *B*_{0}**z**, the spin polarization precesses around the *z* axis. Then, the center frequency of the frequency characteristic of the OPAM shifts to *f*_{0} = *γ*^{K}*B*_{0}/2*π*. Here *γ*^{K} is the gyromagnetic ratio of electrons in K atoms.

To assess the sensor properties of the OPAMs using K and Rb hybrid cells, we have developed a simulation model of the spin-polarization behavior in the hybrid cell [20]. Since the spin polarization is diminished by diffusion, spin exchange and spin destruction collisions during the above process, we can express the Bloch equations of the spin polarization of the K and Rb atoms as follows:

**B**=

**B**

_{m}+

**B**

_{0}.

*D*is the diffusion coefficient,

*q*is the slowing-down factor [1],

*R*

_{OP}is the optical pumping rate,

*R*

_{SD}is the spin-destruction relaxation rate, and

*R*

_{SE}is the spin-exchange rate.

In addition, in the cell, the optical pumping rate *R*_{OP} varies spatially due to the absorption of the pump beam by pump atoms. Therefore, we consider the spatial change in the optical pumping rate to analyze the spatial distribution of the spin polarization in the cell. This change is described by the absorption cross-section *σ* (*ν*) for light with the frequency *ν*:

*n*

_{Rb}is the density of Rb atoms, ${v}_{0}^{\text{Rb}}$ is the D1 resonant frequency of Rb atoms, and the optical pumping rate at the incident plane is $\sigma \left({v}_{0}^{\text{Rb}}\right){I}_{\text{pump}}/h{v}_{0}^{\text{Rb}}$, where

*I*

_{pump}is the incident pump beam power density. Typically, the cell is filled with a buffer gas to a pressure of 1 – 3 atm so that the absorption spectrum is dominated by pressure broadening that follows the Lorentz function. Therefore,

*σ*(

*ν*) is expressed by

*c*is the light velocity,

*r*

_{e}is the classical electron radius, Γ is the absorption linewidth of the buffer gas, and

*f*is the oscillator strength. For D1 transitions,

*f*is approximately 1/3.

Considering the spatial distribution of
${S}_{x}^{\mathrm{K}}$, *θ* is described as

*ν*

_{probe}is the frequency of the probe beam, ${v}_{0}^{\mathrm{K}}$ is the D1 resonant frequency of K, and

*l*is light path length of the probe beam. By measuring

*θ*with a polarimeter, which usually consists of a polarizing beam splitter and two photodetectors, we can obtain the response signal:

*η*is the conversion factor between light and voltage,

*φ*is the initial angle of the plane of the polarized probe beam, and

*I*

_{out}is the probe-beam intensity after passing through the cell.

*I*

_{out}is expressed as

*θ*is sufficiently small and

*φ*=

*π/*4, Eq. (6) gives Thus, the response signal is proportional to

*θ*correlating to ${S}_{x}^{\mathrm{K}}$.

## 3. Experimental procedure and numerical calculation

#### 3.1. Experimental procedure

Figure 2 shows our experimental setup. The measurements were operated in a three-layer magnetic shield with a shielding factor of more than 10^{6} at 10 Hz. To cancel residual fields, field coils were installed in the shield. The sensor cells had a volume of 50 × 50 × 50 mm^{3} in which K and Rb were enclosed with He as buffer gas and N_{2} as quenching gas. He gas suppresses the spin relaxation caused by wall collisions. Colliding with excited alkali metal atoms and absorbing the energy, N_{2} gas restrains radiative deexcition of the excited alkali metal atoms. The ratio of He and N_{2} was 10:1 and the total pressure was 150 kPa at room temperature. The cell was heated to 453 K in an oven to vaporize the alkali-metal atoms. Table 1 shows the densities of K and Rb at 453 K in the four cells used in the experiments.

A Ti-Sapphire laser was used for the pump beam. The cross-section of the pump beam was expanded to 3 × 1 cm^{2}. A diode laser was used for the probe beam. The probe beam whose cross-section had a radius of approximately 1 mm entered the cell 25 mm away from the pump beam window. The wavelength of the pump beam was tuned to 794.98 nm, which corresponds to the D1 resonant frequency of Rb. The intensity of the probe beam was set to 10 mW/cm^{2}. The wavelength of the probe beam was slightly detuned from 770.11 nm, which is the wavelength of the D1 resonance of K to maximize the response signal.

In the experiment, the OPAMs were tuned to the resonant frequency of 10 Hz by adjusting the bias field. The sensor properties of the OPAMs were tested by measuring a test sinusoidal magnetic field of 48 pT and 10 Hz while changing the pump beam power density.

#### 3.2. Numerical calculation

The three-dimensional finite difference method was applied to our model. The parameters used for calculations are listed in Table 2. The harmonic mean of the diffusion coefficients in He and N_{2} is used for *D* in the calculations. The rate coefficients of the collisions between different kinds of alkali metal atoms are unknown. Therefore, we assumed that those are the average of the rate coefficients of the collisions between same kinds of alkali metal atoms. In Eqs. (1) and (2), *R*_{SD} is the sum of the spin-destruction relaxation caused by He, N_{2}, K and Rb:

The computational domain was inside the cell, which was divided into 61 × 61 × 61 grid points. The spin polarization at each grid point was calculated as follows. First, to determine the steady state without the testing signal, we calculated Eqs. (1)–(3) by iteration under the condition that the time-derivative term is zero, and obtained
${S}_{z}^{\text{Rb}}$,
${S}_{z}^{\mathrm{K}}$, and *R*_{OP}. Secondly, we considered the case of the presence of the testing signal of 48 pT at 10 Hz. Here because the rotation of the spin polarization is minute, we can consider
${S}_{z}^{\text{Rb}}$ to be time-invariant even when the testing signal was applied. This suggests that *R*_{OP} was also time-invariant. Furthermore, because the optical pumping was the dominant factor for the spatial distribution of the spin polarization, for simplicity, we calculated the response signals by solving Eqs. (1) and (2) by ignoring the diffusion terms and using the calculated
${S}_{z}^{\text{Rb}}$,
${S}_{z}^{\mathrm{K}}$, and *R*_{OP} as initial conditions. The other parameters were set to the experimental values. The response signal was calculated with Eqs. (5) and (8) with the probe beam passing through at a distance of 25 mm from the inlet of the pump beam to assess the strength of the response signal quantitatively.

## 4. Results and discussion

#### 4.1. Comparison of calculation with experiment

Figure 3 shows experimental and calculated response signals as a function of the pump-beam intensity. The experimental results agree well with the calculations. In cases of *n*_{K}*/n*_{Rb} = 0.14 and 0.37, the experimental and calculated results slightly differ in the weaker pump-beam intensity region. This is caused by the residual circular-polarization component of the probe beam [28]. However, the effect is negligible in the strong pump-beam intensity region; therefore, the behavior of the OPAMs using K and Rb hybrid cells is well described by the model. Figure 4 shows typical noise levels for *n*_{K}*/n*_{Rb} = 0.14 and 15.5. The noise level for *n*_{K}*/n*_{Rb} = 0.14 was about 200 fT/Hz^{1}^{/}^{2} and that for *n*_{K}*/n*_{Rb} = 15.5 was about 30 fT/Hz^{1}^{/}^{2}. The best sensitivities in the experiments were in the order of 10^{−13} T/Hz^{1}^{/}^{2} in cases of *n*_{K}*/n*_{Rb} = 0.14 and 0.37, and those were in the order of 10^{−14} T/Hz^{1}^{/}^{2} for *n*_{K}*/n*_{Rb} = 5.8 and 15.5. Since the system noise was dominant when the alkali metal atoms were spin-polarized sufficiently in this experiment, the sensitivity depends on the response signal.

#### 4.2. Optimal densities of potassium and rubidium

For optimal use of the OPAMs, we consider the following two points. The first point is the intensity of the response signal. For the application to biomagnetic measurements, a quite high signal-to-noise ratio is required. Therefore, the optimal atomic densities, which maximize the response signals, should be obtained. The second point is the spatially homogeneous sensor properties. For simultaneous multilocation measurements within a sensor cell, the homogeneous sensor properties in the cell is required to obtain spatially undistorted signals. Therefore, ${S}_{x}^{\mathrm{K}}$ should be spatially homogeneous. Moreover, the sensing region should coincide with the region where the pump and probe beams cross. From these two points, the ideal spatial distribution of ${S}_{x}^{\mathrm{K}}$ is considered to be the upper limit of ${S}_{x}^{\mathrm{K}}$ anywhere in the region the pump beam passes through.

The response signal depends not only on
${S}_{x}^{\mathrm{K}}$ but also on the density of K according to Eqs. (5) and Eq. (8). The solid line in Fig. 5 shows the response signal depending on the density of K, assuming an ideal distribution of
${S}_{x}^{\mathrm{K}}$. The maximum response signal is obtained when the density is approximately 3 × 10^{19} m^{−3} because *θ* decreases when the density is lower than the optimal value according to Eq. (5) and *I*_{out} decreases when the density is higher than the optimal value according to Eq. (7).

First, for simplicity, we assumed that the density of K is spatially homogeneous and the value is 3 × 10^{19} m^{−3}. In this case, we calculated the response signal as a function of the density ratio of K and Rb. Figure 6 shows the calculated results of the maximum response signal as a function of the density ratio of K and Rb and the required pump beam power density. We plotted the response signal for the pump beam power density of 10 W/cm^{2} because we could not find a maximum response signal by changing the pump beam power density from 0 to 10 W/cm^{2}. The maximum response signal increased with an increase in the density ratio of K and Rb, and it reached a maximum when the ratio was around 400. Then, it decreased when the ratio was higher than 400. The required pump beam power density was minimum when the ratio was approximately 10, and we could not find a local maximum of the response signal when the ratio was larger than 400.

To evaluate the difference of the sensor cells from the view point of spatial distribution, we introduce a *g* value as follows:

*i*indicates each calculated point, ${S}_{i}^{\prime}$ is the calculated ${S}_{x}^{\mathrm{K}}$, and

*S*is the ideal ${S}_{x}^{\mathrm{K}}$ that is homogeneous in the region where the pump beam passes through.

_{i}*S*is normalized by the maximum value of ${S}_{i}^{\prime}$. We used 61 × 61 × 61 points in the cell. Figure 7 shows the

_{i}*g*value plotted as a function of the density ratio of K and Rb. The

*g*value increases with an increase in the density ratio when the density ratio is smaller than 10. Beyond that, the

*g*value is more than 70%, and the spatial homogeneity is comparably high.

Figure 8 shows distributions of *R*_{OP},
${S}_{z}^{\text{Rb}}$,
${S}_{z}^{\mathrm{K}}$, and
${S}_{x}^{\mathrm{K}}$ for the pump and probe beams crossing plane in the cases of (a), (b), and (c) in Figs. 6 and 7. The maximum value of the spin polarization is 0.5 due to the spin quantum number of 1/2. In Fig. 8(a), *R*_{OP} changed spatially because the pump beam was absorbed by the dense Rb. This led to an inhomogeneity of
${S}_{z}^{\text{Rb}}$ followed by one of
${S}_{z}^{\mathrm{K}}$. Therefore, the case of Fig. 8(a) is not the optimal density ratio from the viewpoints of strength of the response signal and spatial homogeneity of the sensor.

By increasing the density ratio or by decreasing *n*_{Rb}, the absorption of the pump beam becomes weaker and the homogeneity of spatial distribution of Rb becomes larger. A decrease of *n*_{Rb} causes a decrease of the spin relaxation and an increase of the strength of the response signal. However, it also causes a decrease of the spin polarization rate by spin-exchange collisions. Therefore, stronger pump beam power density is required for larger response signal.

In Fig. 8(b), the effect of the absorption of the pump beam is negligible.
${S}_{z}^{\text{Rb}}$ and
${S}_{z}^{\mathrm{K}}$ are homogeneous in the region of the pump beam passing through. At the outer side where the pump beam passes through, there are regions where
${S}_{x}^{\mathrm{K}}$ is large. This is caused by the spin polarization diffusing from the region where the pump beam passes through. In this region,
${S}_{z}^{\text{Rb}}$ is not affected by the optical pumping rate and is rotated by magnetic fields. This causes large
${S}_{x}^{\mathrm{K}}$. Furthermore, at *n*_{K}*/n*_{Rb} = 400, the spin polarization and relaxation rate are balanced, so that, the response signal becomes maximum. The *g* value is approximately 75% because a difference between calculated and ideal distributions of the spin polarization is only present at the boundary of the region where the pump beam passes through. For this condition, relatively high *g* values and maximized response signal are achieved, although the diffusion of the spin polarization causes a misregistration between the region that the pump beam passes through and the sensing region. Moreover it is assumed that *n*_{K}*/n*_{Rb} ~ 400 is the optimal condition.

Figure 8(c) shows the region with *n*_{K}*/n*_{Rb} = 3000. In this region, the spin polarization rate is quite small due to a small *n*_{Rb}. This causes a smaller response signal and requires an extremely high pump beam power density for obtaining the maximum response signal. The *g* value was approximately 87%, which was higher than that at *n*_{K}*/n*_{Rb} = 400 because
${S}_{x}^{\mathrm{K}}$ was weaker than in case of *n*_{K}*/n*_{Rb} = 400.

These distributions are quite different from those of a single alkali metal OPAM [29] because the pump beams orient the spin polarization in the *z* direction and diminish its *x* component. In the case of the hybrid cells, the pump beams no longer affect the spin polarization of the probed atoms. This is a promising feature for multilocation measurements and gradiometer configurations [30] to improve the signal-to-noise ratio, and demonstrates an advantage of the K-Rb hybrid atomic magnetometers.

Finally, these numerical calculations were performed under the condition of *n*_{K} = 3 × 10^{19} m^{−3}. To confirm that the density is optimal, we calculated the response signal by changing. In Fig. 5, the solid squares reveal the calculation results. Those two calculation results agree well; therefore, it is expected that we can obtain nearly an ideal
${S}_{x}^{\mathrm{K}}$ distribution like in Fig. 8(b) in the hybrid cell by controlling *n*_{Rb} and the pump beam power density optimally.

Taken together, the results indicate that the conditions of *n*_{K} ~ 3 × 10^{19} m^{−3} and *n*_{K}*/n*_{Rb} ~ 400 are optimal considering the spatial distribution of
${S}_{x}^{\mathrm{K}}$.

## 5. Conclusion

In this study, we investigated the sensor properties of an OPAM with K and Rb hybrid cells by using a numerical model based on the Bloch equations considering the spin polarization distribution in the cell to determine optimal conditions for the densities of K and Rb. For optimal use of the OPAMs, we considered the strength of the response signal and the spatial homogeneity of the sensor. First, to validate our model, we compared the calculated and measured signal strengths of the OPAM using K and Rb hybrid cells and found that those data are in good agreement. Then, using the model, we calculated the spatial distribution of the spin polarization. When *n*_{K}*/n*_{Rb} is smaller than 10, the absorption of the pump beam causes a spatial inhomogeneity of the spin polarization. When *n*_{K}*/n*_{Rb} is greater than 10, the spatial homogeneity is comparably high. Finally, we found the optimal conditions by considering the strength of the response signal and the spatial homogeneity of the sensor to be *n*_{K} ~ 3 × 10^{19} m^{−3} and *n*_{K}*/n*_{Rb} ~ 400, respectively.

In future work, we plan to fabricate a sensor cell with the optimal condition and improve its sensitivity to achieve multichannel sensors for magnetocardiograms and magnetoencephalograms.

## Acknowledgments

This work was supported in part by Grant-in-Aid for Scientific Research (A) under Grant 15H01813, (C) under Grant 15K06106, and the Grant-in-Aid for Challenging Exploratory Research under Grant 16K13114, through the Ministry of Education, Culture, Sports, Science, and Technology, Japan.

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