1. Introduction

Optical pumping, to produce strong atomic polarization, is used in many areas, partially including fundamental physics research [1,2], biomagnetism [3], and inertial navigation [4]. Unfortunately, in an optically thick medium, the strong absorption of pump light produces a significant polarization gradient and imposes a limit on the efficiency of optical pumping [5]. Recently, a hybrid spin-exchange optical-pumping (HSEOP) approach that uses a mixture of K and Rb was developed to overcome this difficulty, where the optically pumped low-density alkali metal was used to polarize the other high-density alkali metal via spin-exchange collisions [6,7]. In spin-exchange relaxation free (SERF) magnetometers and comagnetometers, the condition to achieve optimal sensitivity, demanding the optical pumping rate equaling the spin relaxation rate, cannot be maintained throughout the optical thick cell, not even across the intersection of the pump-probe beams [8, 9]. Using HSEOP, the optimal sensitivity can be achieved throughout the vapor by the virtue of uniform polarization. This large sensitive area is useful to yield multi-channel biomagnetic field measurements with fine spatial resolution [10,11]. Besides, in the spin-exchange optical pumping of noble gas such as ³He and ²¹Ne, HSEOP increases the pumping efficiency by nearly an order of magnitude, compared to the pure alkali-metal pumping method [7].

The main challenge for HSEOP is that both the homogeneity of polarization and the pumping efficiency are determined by the density ratio of the used hybrid alkali metals. The higher the density of the optically pumped alkali metal is, the lower is the polarization homogeneity. On the other hand, the lower the density is of the optically pumped alkali metal, the more pumping power is required. The motivation to accurately determine the density ratio of hybrid alkali metals vapor is to find the optimal density ratio that is necessary to improve both the sensitivity of the magnetometer [10,11] and the pumping efficiency in hyperpolarizing noble gas [7].

At present, the commonly used methods to measure atomic density in vapor cells are laser- absorption-spectroscopy (LAS) [12], Faraday-rotation [13], and measuring-spin-exchange-relaxation [14, 15]. The LAS method, where the density is derived from fitted absorption spectrum, is difficult to use with dense alkali vapor cells. This is because it is not only very challenging to directly measure the extremely weak transmitted light near the resonance, but also hard to indirectly infer the absorption line-shape from measurements in far-detuned wings due to near-complete attenuation of the light within many linewidths of resonance. Even at lower densities, the accuracy is limited by the broadening and distortion of the line-shape. In contrast, the magnetic-field-induced and polarization-induced Faraday-rotation methods can be used to monitor the density of dense vapor [13,16,17]. However, a very strong magnetic field (∼1 T) is needed for the magnetic-field-induced Faraday-rotation method [13], which would magnetize the magnetic field shields in some devices. Besides, the polarization-induced Faraday-rotation method requires an additional absolute measurement of the polarization [16,17]. The recently proposed measuring-spin-exchange-relaxation method measures the spin-exchange relaxation rate of alkali atoms, in the condition of a moderate magnetic field and low spin polarization, to derive the density [14, 15]. Unfortunately, measurement accuracy is affected by the spin polarization, and it cannot be used under exactly the normal operation of many devices with higher spin polarization. Furthermore, this method cannot measure the density in a hybrid alkali cell due to fast spin-exchange collisions.

In this paper, a new method is used to directly measure the density ratio of K-Rb in hybrid alkali vapor based on the mixture of the K and Rb light shifts caused by fast spin-exchange collisions. Firstly, the mixture of light shifts of K and Rb atoms in magnetometer and comagnetometer is described based on Bloch equations, and further explained by fast spin-exchange collisions with an intuitive model. We establish the relationship between the density ratio and the mixed light shifts, and present the measurement procedure of this method. Next, measurements of the density ratio using this method were performed with a K-Rb SERF magnetometer and a K-Rb-²¹Ne SERF comagnetometer at different temperatures, pump light powers and mole fractions of K. As a further check, the measured density ratios were compared with those measured using the LAS method. In the last section, we discuss limitations of this new method. This new method has the advantage that it does not need additional strong magnetic field and absolute polarization measurement, which are necessary for magnetic-field-induced and polarization-induced Faraday-rotation methods respectively, to directly measure the density ratio of the optically-thick hybrid alkali atoms. It also has the advantage of in-situ measuring the density ratio under exactly normal operations, thus the errors caused by the strong pump light heating-effect and the long-term temperature drift can be real-time monitored by this method. Besides, the cell temperature and alkali atoms densities in the K-Rb cell could be further determined by measuring the density ratio based on our method.

2. Basic principle

The energy levels of an atom can be shifted by a far-off-resonant laser beam, even in the absence of light absorption, which can be classified into scalar, vector, and tensor contributions [18]. Circularly-polarized light generates a vector light shift, while linearly-polarized light produces scalar and tensor shifts. The vector light shift affects the individual Zeeman sublevels separately, like a fictitious magnetic field along the light-propagation direction, while a scalar light shift shifts the Zeeman sublevels collectively. The tensor light-shift is negligible compared to the vector light shift [19]. In this work, we concentrate on the effect of the vector light shift (which we refer to simply as “light shift” from now on) of circularly polarized light. This study is performed on a K-Rb SERF magnetometer and a K-Rb-²¹Ne SERF comagnetometer. The K-Rb magnetometer consists of a vapor cell filled with a small mixed droplet of K-Rb and ⁴He gas (buffer gas) as well as N₂ gas (quenching gas). The mole fraction of K in the hybrid K-Rb cell is kept small to obtain a relative low K density at high temperature, where the density ratio of K to Rb, D_r = n_K/n_Rb, is typically on the order of 0.01. Here, n_K and n_Rb are the densities of the K and Rb atoms, respectively. The low-density K atoms, whose absorption of pump light along the propagation direction is much smaller than that of a pure K vapor cell at the same temperature, are uniformly polarized along the Z-axis by circularly polarized D1 resonant light of K. The high-density Rb atoms are polarized through spin-exchange collisions with the polarized K atoms to obtain uniform polarization [6]. The main difference between the K-Rb SERF magnetometer and the K-Rb-²¹Ne SERF comagnetometer is that the ⁴He buffer gas is replaced by ²¹Ne gas with a nonzero nuclear magnetic moment. Furthermore, ²¹Ne nuclear spins are polarized along the Z-axis by spin-exchange collisions with alkali atoms. When the frequency of the circularly polarized pump light is detuned away from the resonance frequency of the K D1 line, light shifts for K and Rb atoms occur, which are referred to as L^K and L^Rb, respectively [20].

The properties of the comagnetometer are described by the Bloch equations [4,20], coupling K electron, Rb electron, ²¹Ne nuclear polarizations P^K, P^Rb, and Pⁿ with spin-exchange interactions - see Eqs. (1)–(3). The K-Rb magnetometer can be described by Eqs. (1) and (2) with the parameters related to ²¹Ne atoms omitted [10],

When the external magnetic field is relatively small compared to the effective spin-destruction rate and pump rate, the equilibrium polarization of alkali atoms along the Z-axis for both magnetometer and comagnetometer is given by [20]:

To extract the excitation signals, the transverse polarization of Rb spins in both magnetometer and comagnetometer, which are induced by either a magnetic field or inertia rotation, are measured using linearly polarized probe light that is detuned towards the red side of the Rb D1 line using optical rotation [4]. When the input excitations are relatively small, the Bloch equations can be linearized, and the quasi-static signals of the magnetometer and comagnetometer are given by [9,20]:

According to Eq. (7), the effective light shift experienced by the Rb atoms is a mixture of the light shifts of Rb and K [20]. This can be explained by the rapid spin-exchange collision between K and Rb atoms - see Fig. 1. Initially, the transverse polarizations $P_x^K$ and $P_x^{\mathit{Rb}}$ are aligned, and the total transverse polarization is P_x. When applying the light shifts $L_z^K$ and $L_z^{\mathit{Rb}}$, $P_x^K$ and $P_x^{\mathit{Rb}}$ are rotated by their own light shift during the short period between spin-exchange collision, which leads to $P_x^{\prime K}$ and $P_x^{\prime \mathit{Rb}}$ respectively. Then, the spin-exchange collision can re-orient $P_x^{\prime K}$ and $P_x^{\prime \mathit{Rb}}$ together in a new direction P′_x. Therefore, the precession of one spin species around their own light shift can be transferred to the other due to rapid spin-exchange collision. Thus, the K and Rb atoms can experience a mixture of the light shifts that is related to the density ratio.

 

Fig. 1 Vector diagram illustrating the mixture of light shifts due to rapid spin-exchange collision.

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Near the D1 line of K, the light shift of Rb, $L_z^{\mathit{Rb}}$, is nearly independent of the light frequency. The direction and amplitude of the light shift of K, $L_z^K$, however, changes significantly with the light frequency. Thus, the effective light shift L_z can become zero by changing $L_z^K$ to compensate $L_z^{\mathit{Rb}}$. According to Eq. (7), at this L_z = 0 point, the density ratio D_r is given by

Moreover, under typical experimental conditions, the frequency shifts of the D1 and D2 resonance frequencies of Rb (${\nu }_{D1}^{\mathit{Rb}}$,${\nu }_{D2}^{\mathit{Rb}}$) from the resonance frequencies in vacuum (${\nu }_{0D1}^{\mathit{Rb}}$, ${\nu }_{0D2}^{\mathit{Rb}}$) are usually on the order of several GHz, and the pump light frequency ν is usually detuned several GHz away from the D1 resonance frequency of K in vacuum (${\nu }_{0D1}^K$). Because these are negligible compared to the frequency differences $\nu -{\nu }_{D1}^{\mathit{Rb}}$ and $\nu -{\nu }_{D2}^{\mathit{Rb}}$ (on the order of several THz), $\nu -{\nu }_{D1}^{\mathit{Rb}}$ and $\nu -{\nu }_{D2}^{\mathit{Rb}}$ in Eq. (10) can be approximated by ${\nu }_{0D1}^K-{\nu }_{0D1}^{\mathit{Rb}}$ and ${\nu }_{0D1}^K-{\nu }_{0D2}^{\mathit{Rb}}$ respectively. In addition, both ${\nu }_{D1}^K$ and ${\mathrm{\Gamma }}_{D1}^K$ can be extracted by fitting the dispersion curve for the measured effective light shift with Eqs. (7)–(9). Therefore, we only need to scan the pump light frequency ν around the resonance frequency and measure the corresponding L_z to find the ν for L_z = 0. We then fit the measured L_z with Eqs. (7)–(9) to obtain ${\nu }_{D1}^K$ and ${\mathrm{\Gamma }}_{D1}^K$, and substitute the frequency detune $\delta \nu =\nu -{\nu }_{D1}^K$ and linewidth ${\mathrm{\Gamma }}_{D1}^K$ into Eq. (10) to determine the density ratio D_r.

The effective light shift in the K-Rb magnetometer can be measured using the cross-modulation method, which is usually applied to zero residual magnetic field along the Z-axis [23]. When a small field modulation is applied along the X-axis, with a slow modulation frequency, the response to the field can be simplified due to the small residual magnetic field inside the magnetic shield :

The effective light shift in the K-Rb-²¹Ne comagnetometer can be measured using the B_z magnetic-field-zeroing method [24]. According to Eq. (6), the response of a B_y square wave modulation is given by:

To further verify the accuracy of this measurement method, the density ratios measured with this method for different conditions were compared to those measured using the conventional LAS method and Raoult’s law. According to Raoult’s law, the density of K is n_K = f_Kn_K0, where n_K0 is the saturated vapor density for pure K alkali metal, and f_K is the mole fraction of K in the mixed alkali metals [6]. Because at a typical high temperature, around 463 K, the LAS method becomes inaccurate due to strong light-absorption, it cannot be used to measure the high density at high temperatures directly. However, the LAS method can be used to measure the K density at lower temperature to determine the mole fraction of K, f_K, and then to derive the density of K at high temperature using Raoult’s law. The basic principle of the LAS method is briefly described below [23]. For linear polarized light, transmitting through alkali vapor, the optical depth OD is given as:

3. Experimental setup and results

The experimental setup is shown in Fig. 2, which is similar to the one used in our earlier studies [25, 26]. The spherical cell was heated by a 110 kHz AC electrical heater, which was situated within a boron nitride ceramic oven and installed in a PEEK vacuum vessel. The vessel, cooled by the water-cooling tube, was enclosed by a ferrite barrel and a 5-layer cylindrical μ-metal shield to shield ambient magnetic fields and reduce inner magnetic noise. In addition, a three-axis coil was used to compensate the residual magnetic fields. The pump light, produced by an external cavity diode laser, was tuned to the D1 resonance of K, amplified by a tapered amplifier to 1.5 W (Toptica TApro), and expanded to cover the cell to polarize K atoms along the Z-axis. The transverse component of the Rb spin polarization was measured by a linear-polarized probe light, propagating along X-axis from a distributed feedback (DFB) laser, whose wavelength was red-detuned (0.4 nm) from the D1 resonance of Rb. The probe light was modulated by a photo-elastic modulator (Hinds Instruments PEM100) with the amplitude of about 0.08 rad and a frequency of about 50 kHz. It was demodulated with a lock-in amplifier (Zurich Instruments HF2LI). The wavelength of the light was monitored by a wavelength meter with a measurement accuracy of 60 MHz (High Finesse WS7). In this work, four different cells were measured: two spherical cells with a diameter of 25 mm, which were filled with the same 2660 Torr ⁴He, 50 Torr N₂ gases but with a different K-Rb mole fraction to be used in the magnetometer (referred to as cells Mag1 and Mag2). Another two cells with a diameter of 12 mm were filled with the same 2665 Torr ²¹Ne, 52 Torr N₂ gases but with a different K-Rb mole fraction to be used in the comagnetometer (referred to as cells Comag1 and Comag2).

 

Fig. 2 Schematic of the experimental setup. TA, tapered amplifier; BE, beam expander; GT, Glan-Thompson polarizer; PD, photo detector; PBS, polarization beam splitter; PEM, photo elastic modulator.

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Firstly, we simulated the light shifts in a K-Rb SERF magnetometer. At typical experimental conditions, the cell is filled with 2660 Torr ⁴He and 50 Torr N₂. Thus, the frequency shifts of the K D1 line, the Rb D1 line and the D2 line are 10.67 GHz, 9.06 GHz, and 0.51 GHz respectively. The linewidth of the K D1 line, due to pressure broadening, is 41.28 GHz. In Fig. 3, the light shift for Rb with a pump light power of 200 mW/cm² is calculated using Eq. (9) and plotted on the left-Y-axis-bottom-X-axis coordinates (black curve). The zero point of the bottom-X-axis is the D1 line of K in vacuum, 770.108 nm. Around the D1 line of K, the $L_z^{\mathit{Rb}}$ is nearly constant, with a variation below 1%. According to Eqs. (7)–(9), the effective light shifts L_z for a density ratio of D_r = 0.01 are simulated for different pump-light powers and drawn using the left-Y-axis-bottom-X-axis coordinates (red, blue and pink curves). These effective light shift curves intersect at L_z = 0 with a detune frequency of $\delta \nu =\nu -{\nu }_{D1}^K=5.43\hspace{0.17em}\text{GHz}$. For different density ratios, the frequency detune δν for L_z = 0 is different, and their relationship can be simulated using Eq. (10) and plotted using the right-Y-axis-top-X-axis coordinates (green curve). The smaller the density ratio is, the larger is the frequency detune δν for L_z = 0. Thus, it can be used to measure the density ratio directly.

 

Fig. 3 Simulation of light shifts in the K-Rb magnetometer under different conditions.

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Next, we measured the density ratio using this method in the K-Rb SERF magnetometer. To study the validity of this method for different pump light intensities, the density ratios of the cell Mag1 with different pump light intensities at 463 K were measured. Using the cross-modulation method from above, the residual magnetic field $B_z^{\mathit{res}}$ is the intercept of the line $B_z=-B_z^{\mathit{res}}-k{I}_z$ with the vertical coordinate, 0.73 nT - see the inset in Fig. 4. L_z at different pump light frequencies and incident intensities were measured - see Fig. 4. After fitting the measured L_z with Eqs. (7)–(9), we can determine the light shifts for Rb $L_z^{\mathit{Rb}}$, the linewidths ${\mathrm{\Gamma }}_{D1}^K$ and the frequency detunes δν for L_z = 0 point. The fitted light shifts for Rb $L_z^{\mathit{Rb}}$ are shown in Fig. 5(a) as red squares. They can be fitted with $L_z^{\mathit{Rb}}=0.0039I_z+0.00014$ (blue solid line), as predicted by Eq. (9). The slope of the fitted line is slightly smaller than the theoretically calculated line $L_z^{\mathit{Rb}}=0.0072I_z$ (black dashed line). This observed difference is due to the absorption of pump light along the propagation direction and inhomogeneity of the light intensity. The fitted linewidths ${\mathrm{\Gamma }}_{D1}^K$ and the frequency detunings δν for L_z = 0 are shown in Fig. 5(b) as green circular and blue triangular symbols respectively. The fitted linewidths are larger than the theoretically calculated pressure-broadened linewidth (47.9 GHz) using the filled gas pressure. This can be explained by the power broadening of the high light intensity [20,27]. Substituting the measured ${\mathrm{\Gamma }}_{D1}^K$ and δν into Eq. (10), the density ratios D_r at these light intensities can be calculated respectively - see the red squares in Fig. 5(b). The mean value and the standard deviation of these measured density ratios at different light intensities are 0.0436 and 0.0033, respectively. That means the measured D_r at different light intensities agree with each other, thus this method is valid for different pump light intensities.

 

Fig. 4 The effective light shift as a function of pump light frequency detuning. The point data were obtained at different light intensities, while the corresponding curves are fitted lines. The zero point of the horizontal axis is the D1 line of K in vacuum, 770.108 nm. Inset: The compensating magnetic field B_z as a function of light intensity I_z.

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Fig. 5 (a) The light shift of Rb $L_z^{\mathit{Rb}}$ as a function of light intensity, (b) Fitted linewidths and the frequency detunes for L_z = 0, plotted against the left Y-axis, while the plotted frequency detunes are scaled up 10-fold for clarity. The calculated density ratios are plotted against the right Y-axis.

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To further verify this method, the density ratios for the cell Mag1 at different temperatures and identical incident pump light intensity were measured and compared with the results of the LAS method. The mole fraction of K in the cell Mag1 was calibrated using a pure K cell (filled with the same gases) by the LAS method. As shown in the inset of Fig. 6, the density of K, $n_K^p$, in a pure K cell was measured to be 1.217×10¹³ cm⁻³ using the LAS method with a temperature set to 423 K. This value approaches the saturated vapor density of 1.222×10¹³ cm⁻³ at 423 K, which indicates that the temperature of the inner cell is consistent with the set value. At 443 K, the mole fraction for K in the cell Mag1 $f_K=n_K/n_K^p$ was measured to be 0.204 ± 0.021. Thus, the density ratios at higher temperatures could be calculated using Raoult’s law with an estimation error about 7.4% - see the black circles and shaded area in Fig. 7(a). With the same pump light intensity (166 mW/cm²) but different temperatures (443 K, 453 K, 463 K, and 473 K), the density ratios were measured using the new method - see Fig. 6. These measured effective light shifts, L_z, were fitted using Eq. (7)–(9). The L_z of the higher-temperature cell is smaller than that of the lower-temperature cell, because the L_z is proportional to the pump light intensity according to Eqs. (7)–(9) and the pump light intensity in the probe area of the higher-temperature cell is smaller due to the stronger absorption of the identical incident pump light at higher-temperature. The fitted linewidths ${\mathrm{\Gamma }}_{D1}^K$ and frequency detunes δν for L_z = 0 are shown in Fig. 7(a) using blue and green triangles, respectively. The linewidths at different temperatures are consistent with each other, which suggests that the linewidths are not significantly dependent on temperature [28]. Using Eq. (10), the density ratios at these temperatures were calculated and drawn in Fig. 7(a) with red squares. The density ratios measured with our method (red squares) are consistent with those using the LAS method (black circles). The mean relative error of these two methods is within 10%. The density ratios for these two methods all increase with temperature as predicted by Raoult’s law. The density ratios of our method are slightly larger than those of the LAS method, which is attributed to the heating of the cell by the strong pump light under the normal operation of K-Rb magnetometer. The measurement of density ratio by our method was performed under the normal operation of K-Rb magnetometer (with strong pump light), while the measurement of density ratio by the LAS method was performed with a weak linearly polarized light, thus the error caused by the heating-effect of the strong pump light under the normal operation of K-Rb magnetometer was considered in our method compared to the LAS method. To verify the heating effect, we monitored the absorption of the probe light when the pump light was turned on or off. The intensity of the transmitted probe light decreased significantly within several minutes when the pump light was turned on and slowly heated the cell.

 

Fig. 6 The effective light shift as a function of the pump light frequency detuning. The data points were measured at different temperatures, while the corresponding curves are fitted lines. The inset shows the laser absorption spectra of the pure K vapor cell (black triangles) and the cell Mag1 (red squares). The zero point of the horizontal axis is the D2 line of K in vacuum 766.701 nm, while the vertical axis represents the log base e of the transmitted light intensity. The black triangles and red squares are fitted with y = 4119/[(x − 0.3039)² + 30.72²] − 1.811 and y = 1315/[(x + 0.0709)² + 31.25²] − 1.796 based on Eq. (14).

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Fig. 7 Density ratios at different temperatures plotted against the left Y-axis. The fitted linewidths and the frequency detunes for L_z = 0 at different temperatures plotted against the right Y-axis. The shaded areas are the estimation errors of the calculated density ratios. (a): Data for the cell Mag1. The plotted frequency detunes are scaled up 10-fold for clarity. (b) Data for the cell Mag2. The plotted frequency detunes are scaled by a factor of five for clarity.

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Using the K-Rb SERF magnetometer, another K-Rb cell (cell Mag2) with a smaller mole fraction of K was also used to verify this method. Likewise, the density ratios of the cell Mag2 were measured at different temperatures (443 K, 453 K 463 K and 473 K) with identical pump light intensity using our method. They were then compared with the results of the LAS method. The mole fraction of K in the cell Mag2 was calibrated to be 0.076 ± 0.057 at 423 K, using the LAS method, which is nearly three times smaller than that of the cell Mag1. Using Raoult’s law, the density ratios were calculated at these temperatures with an estimation error about 6.1% - see the black circles and shaded area in Fig. 7(b). Using our method, the effective light shifts L_z were measured at these temperatures and fitted with Eqs. (7)–(9). The fitted linewidths and frequency detunes for L_z = 0 are shown in Fig. 7(b) with blue and green triangles. The density ratios calculated with our method for these temperatures are marked as red squares in Fig. 7(b). There is a good agreement between the results of these two methods, with a mean relative error below 5%.

We also verified this method with the K-Rb-²¹Ne comagnetometer. The effective light shift was measured using the B_z magnetic field zeroing method from above: Scanning δB_z, the response signals $\mathrm{\Delta }{S}_x^e$ of a B_y square wave modulation were measured. The measured $\mathrm{\Delta }{S}_x^e$ were fitted with Eq. (12) to acquire L_z. The measured $\mathrm{\Delta }{S}_x^e$ at different pump light frequencies for cell Comag1 are shown in Fig. 8, and fitted using Eq. (12) to obtain L_z. Similarly, the light shift for Rb, $L_z^{\mathit{Rb}}$, the linewidth ${\mathrm{\Gamma }}_{D1}^K$, and frequency detune δν for L_z = 0 could be acquired by fitting the measured L_z with Eqs. (7)–(9) - see Fig. 9(a). The fitted light shift of Rb $L_z^{\mathit{Rb}}$ is 1.75 nT, which is consistent with the theoretically calculated result (1.77 nT) using pump light of 242 mW/cm². The fitted linewidth is ${\mathrm{\Gamma }}_{D1}^K=34.8\hspace{0.17em}\text{GHz}$, larger than the theoretically calculated pressure broadened linewidth of K, 26.9 GHz. This can be explained by power broadening of the strong pump light, similar to the K-Rb magnetometer. The frequency detune for L_z = 0 is δν = 1.9 GHz. Substituting ${\mathrm{\Gamma }}_{D1}^K$ and δν into Eq. (10), the density ratio is 0.0191 with an operating temperature of 463 K. Similarly, to verify the accuracy of this method, the measured density ratio with our method was compared to the LAS method. As shown in the inset of Fig. 9(a), the density of the K atoms was measured at a lower temperature (443 K) with the LAS method, resulting in the mole fraction f_K = 0.095 ± 0.009 for K. Thus, the density ratio is 0.0166 at 463 K with an estimation error about 8.6% according to Raoult’s law. This value is consistent with the result obtained with our method with a relative error of 15.1%. This difference is due to the optical heating-effect and the measurement error of the mole fraction of K both.

 

Fig. 8 The response signals $\mathrm{\Delta }{S}_x^e$ of a B_y square wave modulation for different δB_z and pump light frequencies. The dots are measured signals, while the solid lines are fitted lines.

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Fig. 9 Effective light shift L_z as a function of the pump light frequency detuning. The zero point of the horizontal axis is the K D1 line in vacuum, 770.108 nm. Inset: The absorption curve of K atoms in the cell at 443 K. The zero point of the horizontal axis is the D2 line of K in vacuum (766.701 nm), while the vertical axis is the log base e of the transmitted light intensity. (a) Data for cell Comag1. The black dots in the inset are fitted using y = 284.7/[(x +7.411)² +14.52²]−1.597 based on Eq. (14). (b) Data for cell Comag2. The black dots in the inset are fitted using y = 195.9/[(x +6.511)² + 16.94²] − 1.425 based on Eq. (14).

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To further verify the accuracy of our method, another K-Rb-²¹Ne spherical cell (cell Comag2) with a smaller mole fraction of K, was measured with our method and compared with the LAS method. As shown in the inset of Fig. 9(b), the mole fraction of K was calibrated to be f_K = 0.034 ± 0.002 using the LAS method at 453 K. This value is about 2.8 times smaller than that of the cell Comag1. The corresponding density ratio at 463 K is D_r = 0.0056 with an estimation error about 5.7% using Raoult’s law. The effective light shift was also measured with our method - see Fig. 9(b). The fitted linewidth is 29.13 GHz, and the fitted light shift of Rb $L_z^{\mathit{Rb}}$ is 1.76 nT. These values are consistent with the theoretically calculated results. The frequency detune for L_z = 0 is 4.43 GHz, which means the density ratio is D_r = 0.0062 at 463 K. This is consistent with the result measured using the LAS method. The relative error between these two results is 10.7%.

As a further check, we measured the heating-effect of the strong light with different intensities by the LAS method. With the electric heater temperature set to 443 K, the absorptions of the weak linearly polarized the K D2 light (along X-axis) in a K-Rb-²¹Ne cell were measured with the strong linearly polarized Rb D1 light (along Z-axis) turned off or on in Fig. 10(a). Fitting these absorption curves with Eq. (14), we obtained the densities of K. And then the densities were used to derive the corresponding true in-situ temperatures based on the Rauolt’s law and the saturated vapor density. The temperature increments between the true in-situ temperature and the setting temperature of electric heater are plotted on the left-Y-axis-bottom-X-axis in Fig. 10(b), which increase significantly with the increasing of light intensities. Optical heating via the absorption of light has been studied previously [29,30]. A 200 mW laser light (λ = 915 nm) was used to heat the cell to 97 °C [29], while a cell temperature of 150 °C was achieved with a 140 mW laser light (λ = 1.5 um) absorbed by absorption filters attached to the cell windows [30]. Compared with these previous works, the optical heating effect in our experiment is much smaller. This is because the optical heating efficiency strictly depends on the size and absorption element of the cell, and thermal isolation [29, 30]. The relationship between the saturated vapor density ratio D_r0 = n_K0/n_Rb0 and temperature is simulated and plotted on the right-Y-axis-top-X-axis in Fig. 10(b). In our experiment, the pump light of about 100 mW/cm² − 700 mW/cm² causes an increment of cell temperature about 5 − 10 °C resulting in an increment of the saturated vapor density ratio smaller than 4%. Therefore, it would cause an increment of the density ratio smaller than 4% according to D_r ≈ D_r0f_K/(1 − f_K).

 

Fig. 10 Measurement of the optical heating effect. (a). The absorption curves of K atoms at 443 K with different light intensities. The zero point of the horizontal axis is the D2 line of K in vacuum (766.701 nm), while the vertical axis is the log base e of the transmitted light intensity. The dots are measured data, and solid curves are corresponding fitting curves based on Eq. (14). (b). The cell temperature increments derived from the absorption curves is plotted on the left-Y-axis-bottom-X-axis. The simulated relationship between the saturated vapor density ratio D_r0 and temperature is plotted on the right-Y-axis-top-X-axis.

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4. Discussions and conclusion

We investigated a new method to directly measure the density ratio of an alkali metal mixture. The mixture of the light shifts was formulated using Bloch equations and explained by the fast spin-exchange interaction. The relationship between the density ratio and the frequency detune of L_z = 0 from the resonance was both formulated and simulated. We described and executed the procedure to measure the effective light shifts, and we found the L_z = 0 point for both the K-Rb SERF magnetometer and the K-Rb-²¹Ne SERF comagnetometer. The density ratios of different K-Rb magnetometer and K-Rb-²¹Ne comagnetometer cells were measured using our method for different temperatures, pump light powers, and mole fractions of K. We compared these with the results measured using the conventional LAS method. The measured density ratios of these two methods are consistent. When operating at different temperatures and mole fractions of K, the density ratios measured with our method are all slightly larger than the results measured by the LAS method, which is due to the heating of the cell by the strong pump light and the measurement errors of f_K both.

This method is limited, when the density ratio of K to Rb is too small, because when the density of K is too small the light shift of K cannot compensate the light shift of Rb, which means the L_z = 0 point cannot be found. This limitation depends on the gas composition in the cell, and the minimum density ratio that can be measured is about 1/500 for typical operating conditions of the K-Rb SERF magnetometer and the K-Rb-²¹Ne SERF comagnetometer. However, this problem can be well solved by pumping Rb instead of K. Thus the light shift of Rb can be changed to compensate that of K to find the L_z = 0 point.

This method can be used to directly measure the density ratio of optical-thick hybrid alkali atoms with no need for additional strong magnetic field and absolute polarization measurement, which are necessary for magnetic-field-induced and polarization-induced Faraday-rotation methods respectively. With the advantage of in-situ measuring the density ratio under exactly normal operations, the errors caused by the strong pump light heating-effect and the long-term temperature drift can be real-time monitored by this method. This method can also be used for K-Cs and Rb-Cs hybrid cells. Precise determination of the density ratio is useful to find the optimal density ratio to decrease the gradient of polarization, increase the sensitive area, and improve the sensitivity of both magnetometer and comagnetometer, as well as to increase the pumping efficiency to hyperpolarize noble gases.