1. Introduction

Preparing atomic populations into a specific Zeeman state is essential for quantum information science and precision measurements, including quantum memory [1–4], quantum manipulation [5,6], single-photon generation [7], atomic magnetometry [8–10], and atomic clock [11–14]. Accumulating the population in the desired Zeeman sublevel increases the optical density (OD) of the atomic medium according to the Clebsch-Gordan coefficient of the selected atomic transition. This approach mitigates energy loss for manipulating light retrieval [3,4,15,16]. Researchers have prepared populations in two specific ground Zeeman states to increase strong interactions between two light fields, forming motionless light pulses via quantum interference [5]. Such ultrahigh OD systems have been proposed for achieving a phase shift of $\pi$ by a single photon [17]. Additionally, an increase in population in the clock state ($m_F=0$), a magnetically insensitive state, effectively reduces noise in atomic clocks. [13]. Optical pumping (OP) is a standard method to pump all atoms into an uncoupled dark state. Stimulated Raman adiabatic passage (STIRAP) enables coherent population transfer between quantum states using two coherent light pulses [18]. Combining optical and microwave (MW) pumping, researchers prepared the population in a clock state with a purity of $83{\% }$ [13]. With appropriate polarization configurations and laser power, $96{\% }$ [19] or $97{\% }$ [20] state purities have been reported.

Currently, various methods are employed to prepare and detect the population distribution in atomic systems. These methods include measuring (1) the electromagnetically-induced-transparency-based transmission [19], (2) the transition between different Zeeman states with the same quantum number $F$ [9,21], (3) the hyperfine state transition ($\Delta F=\pm 1$) driving by a microwave [11,22–26], and (4) the Ramsey atom interferometer [13,27]. Traditionally, detecting atomic population distribution involved using reflected microwave power spectra analyzed with real-time nondestructive imaging systems [28]. The technique of microwave spectroscopy has been previously explored. For instance, in Ref. [20], the atomic distribution was measured using microwave pulses in a two-step procedure. The microwave spectroscopy scheme in Ref. [23] exhibited limitation, as it was restricted to inducing only $\Delta _m=0$ transitions, hindering the measurement of populations in the rightmost/leftmost Zeeman states. However, our innovative approach represents a significant breakthrough, performing a more comprehensive analysis of the atomic population distribution across different Zeeman states in a single microwave spectrum. This advancement offers a powerful and efficient means to investigate and characterize atomic states, facilitating a deeper understanding of atomic interactions.

We employed microwave spectroscopy as a highly suitable technique for measuring high optical densities [3] compared to all-optical methods, enabling us to measure the population distribution accurately. We elucidate the key features of the microwave (MW) spectrum and perform individual determinations of the population distribution, polarization, and MW Rabi frequency. High-precision microwave pulses are essential for ground-state manipulation and Ramsey interference studies [13,27,29–31]. Microwave spectroscopy also proved valuable in compensating the environmental magnetic field with a precision below two mG. Once the population distribution was known, we applied the optical pumping method to efficiently pump the population into the desired quantum state with high purity, reaching up to $96(2){\% }$ or $98(1){\% }$. An estimation based on the far-off-resonant transitions, which involves pumping the atoms away from the dark state, shows an upper limit of $99.8{\% }$. Experimentally, the impure polarization of light and the non-uniform magnetic field affect the state purity. Therefore, our study presents a powerful approach utilizing a single MW spectrum for determining all unknown parameters. Integrating this method with our theoretical analysis enhances our study’s capabilities and insights while extensively discussing and comparing the main features of microwave spectroscopy.

2. Scheme and experimental setup

We perform the experiment using $^{87}$Rb atoms in a typical magneto-optical-trap (MOT) system, including three pairs of trapping beams ($F=2\rightarrow F=3'$ transition with -14 MHz detuning) and one repumping beam ($F=1\rightarrow F=2'$ transition). The magnetic field gradients in the X, Y, and Z directions of 1.4, 3.1, and 5.0 Gauss/cm/A were produced by a pair of rectangular anti-Helmholtz coils (Qpole) with a current of 2.0 Amp. The trapped atom cloud had a dimension of $7\times 3 \times 3$ mm$^3$, containing $2\times 10^8$ atoms. Additionally, the atomic temperature was in the range of a few hundred microKelvin. The setup schematic, relevant energy levels, and laser excitations are shown in Figs. 1(a) and 1(b). The population can be prepared by trapping, repumping, and optical pumping fields. The pumping fields OP$_1$ and OP$_2$ deplete the population from $F=1$ and $F=2$, respectively, and accumulate atoms in the uncoupled hyperfine state. The beam size of OP$_1$ was 6 mm and the power level was varied from 1 $\mu$W to 6 mW, while the size of OP$_2$ was 2 mm and the power level was set to a few mW. Figure 1(c) shows the timing sequence. The repetition time was 436 ms, including 430 ms for the cooling process and 6 ms for preparing and detecting the population. Denote the time of switching on the microwave as $t=0$. We typically turned off the Qpole field, repumping, and trapping beams of the MOT at $t=-6$ ms, $-1.12$ ms, and $-0.52$ ms in sequence. To characterize the state-preparation efficiency, we use microwave spectroscopy to measure the population distribution on the Zeeman sublevels of the ground state $F = 1$, as shown in Fig. 1(d). The microwave lasted for 80 $\mu$s, but other measurements varied the period. When turning off the microwave, the trapping beams were turned on again to serve as the image beams. The fluorescence signals were detected by a mono-color CCD camera (The Imaging Source: DMK 22BUC03) with an exposure time of 122 $\mu$s.

 

Fig. 1. (a) Schematic of the experimental setup. The Qpole field used for capturing the cold atoms is not shown in the figure. WP: waveplate. (b) Relevant energy levels for $^{87}$Rb atoms and laser excitations. Optical pumping fields OP$_1$ and OP$_2$ deplete the population from hyperfine states $F=1$ and $F=2$, respectively, and accumulate atoms in the uncoupled hyperfine state. The trapping field, which captures atoms, also serves as an image field. A mono-color CCD camera was employed to detect the fluorescence signals. A microwave field was frequency scanned around 6.835 GHz to obtain the microwave spectrum. (c) Timing sequence for the cooling process, population preparation, and population detection. (d) A typical microwave spectrum displays seven major peaks under a magnetic field in an arbitrary direction. (e) Microwave transitions between the Zeeman sublevels. The energy splittings for $F=2$ and $F=1$ are $m_F\times$0.7 MHz/G and $m_F\times$(-0.7) MHz/G, respectively.

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After cooling, the repumping and trapping fields were turned off sequentially to pump the cold atoms to three Zeeman states of $F=1$. Under a magnetic field, the energy splittings of Zeeman sublevels for $F=2$ and $F=1$ states are $m_F\times$0.7 MHz/G and $m_F\times$(-0.7) MHz/G, respectively. Choose the direction of the magnetic field as the quantization axis. The polarization of the microwave was random in an arbitrary magnetic field. As shown in Fig. 1(e), the microwave can induce $\alpha _i$, $\beta _i$, and $\gamma _i$ transitions involving populations in states $\alpha$, $\beta$, and $\gamma$, respectively, where $i = 1, 2, 3$ corresponds to $\sigma ^+$, $\pi$, $\sigma ^-$ transitions. Therefore, a typical microwave spectrum displays seven major peaks, among which $\alpha _3$ and $\beta _1$ (as well as $\beta _3$ and $\gamma _1$) exhibit the same resonance frequency. The 6.835 GHz microwave was generated by a synthesizer (Rigol DSG830) combined with a frequency multiplier (Minicircuits: ZX90-3-812-S+), power amplifier (Minicircuits: ZVE-3W-183+), and waveguide adapter (Woken: 0060WA2KOBB01X). The microwave power was around 10 dBm in the vicinity of the atom cloud. We scanned the microwave at a speed of 0.6 kHz per repetition time, and at each MW frequency point, we recaptured the atom cloud.

3. Microwave spectrum analysis

In our study, we elucidate the key features of the microwave (MW) spectrum and perform individual determinations of the population distribution, polarization, and MW Rabi frequency. The oscillation structure observed in the spectrum for a given MW period allows us to derive the Rabi frequency. We initially measured the MW strength from a specific transition to streamline the fitting process. Consider a two-level system with states $|1\rangle$ and $|2\rangle$ and assume that the population is initially in state $|1\rangle$, as shown in Fig. 2(a). The population time evolution is a function of the MW detuning $\Delta$, MW period $t$, and MW Rabi frequency $\Omega$. The population in state $|2\rangle$, $P_2$, is given by

The MW Rabi frequency can be estimated by varying the MW transition period and tracking the population oscillation. The MW frequency was fixed for $\alpha _1$ transition, i.e., $\Delta =0$, and the normalized population signals are shown in Fig. 2(b). In this measurement, states $|1\rangle$ and $|2\rangle$ represent $|F=1, m_F=1 \rangle$ and $|F=2, m_F=2 \rangle$, respectively. After considering the decay [32], the measured oscillation gives the Rabi frequency for the $\alpha _1$ transition of $2\pi \times 6.0$ kHz.

 

Fig. 2. (a) Energy structure of a two-level system. The microwave field is tuned to the transition frequency at approximately 6.835 GHz with a certain detuning $\Delta$. (b) Directly determine the Rabi frequency from the specific transition, e.g., $\alpha _1$ transition $|F=1, m_F=1 \rangle \rightarrow |F=2, m_F=2 \rangle$, where the MW detuning $\Delta =0$. According to Eq. (1), the Rabi frequency is $2\pi \times$6.0 kHz.

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Below we show the procedure to extract all of the unknown parameters from a seven peaks microwave spectrum (e.g., Fig. 1(d)), including the population distribution $P_\alpha$, $P_\beta$, $P_\gamma$, the polarization component of the microwave, and the Rabi frequency for each transition. Here we define $\Omega _1$ as the MW Rabi frequency for the dipole matrix element between two states of one. The dipole matrix elements of $\sigma ^+$ ($\sigma ^-$) transitions are ($C_{\gamma _1(\alpha _3)}$, $C_{\beta _1 (\beta _3)}$, $C_{\alpha _1(\gamma _3)}$) = ($\sqrt {1/12}$, $\sqrt {1/4}$, $\sqrt {1/2}$). The ones for $\pi$ transitions are $(C_{\gamma _2}, C_{\beta _2}, C_{\alpha _2}) = (\sqrt {1/4}, \sqrt {1/3}, \sqrt {1/4})$ for $\gamma _2$, $\beta _2$, and $\alpha _2$ transitions, respectively. We can derive the bias magnetic field from the peak splittings. The MW resonant frequencies were shifted due to the Zeeman effect, and therefore, the detuning $\Delta$ in Eq. (1) needs to be replaced by $\Delta -\Delta _0$, where $\Delta _0$ is the MW resonant frequency. For the spectrum shown in Fig. 1(d), $\Delta _0$ was $\pm$ $j\times$ 532.5 kHz for $j=1,2,3$.

To determine the population distribution, $(P_{\gamma }, P_{\beta }, P_{\alpha })$, we focus initially on the $\pi$ transition peaks, i.e., the second ($\gamma _2$), fourth ($\beta _2$), and sixth ($\alpha _2$) peaks. The fluorescence signals from these peaks are proportional to the total number of atoms and the sensitivity of the CCD camera, which impact each peak equally. Thus, we need to account for the factors $(P_{\gamma }, P_{\beta }, P_{\alpha })$ in Eq. (1), where $P_{\gamma }+P_{\beta }+P_{\alpha }=1$, to accurately interpret the fluorescence peak heights. In Eq. (1), we replace $\Omega$ with $\Omega _1\times$($C_{\gamma _2}$, $C_{\beta _2}$, $C_{\alpha _2}$), and the population distribution is derived from these three peaks. The analysis reveals that the population is almost equally distributed among the three states. Next, we determine the polarization component of the microwave from the leftmost ($\sigma ^-$-polarization) and the rightmost ($\sigma ^+$- polarization) peaks. The quantization axis is aligned with the applied magnetic field, allowing the MW to drive $\sigma ^+$, $\sigma ^-$, and $\pi$ transitions. From comparing the determined Rabi frequencies that fit the data and the derived ones from the dipole matrix element, we obtain the ratios of polarization components to be ($14{\% }, 80{\% }, 5.4{\% }$) for ($\sigma ^-$, $\pi$, $\sigma ^+$) polarizations. To verify the accuracy of the determined parameters, we simulated the third ($\gamma _1$, $\beta _3$) and fifth ($\beta _1$, $\alpha _3$) peaks using the obtained parameters. The simulated spectrum exhibits excellent agreement with the observed data, including the relevant peak heights and oscillation structure, validating the reliability and precision of our analysis.

4. Results

We now discuss how to improve the purity of the population accumulation by applying optical pumping. Assume the laser polarization $\hat {Z}_L$ differs from the magnetic field direction $\hat {Z}_B$. In that case, the laser field can drive $\Delta m_F = \pm 1$ or $\Delta m_F=0$ transitions, causing the pumping process to be more complicated. The effective magnetic field $\bf {B_{eff}}$ = $\bf {B_{coil}}+\bf {B_0}$, where $\bf {B_0}$ is the stray field from the environment, and $\bf {B_{coil}}$ can be adjusted by the three pairs of the Helmholtz coils. Since the Zeeman splitting is proportional to the strength of $\bf {B_{eff}}$, we minimized the stray magnetic field on the three orthogonal directions by measuring the frequency shift for various $\bf {B_{eff}}$. Then, we can define the quantization axis on the $\hat {Z}_B$ direction and purify the optical pumping field transition. The frequency splitting for two nearby peaks was obtained by extracting the detuning $\Delta _0$ for several magnetic field strengths along a fixed direction. The spectrum’s peak resolution is better than 5 kHz, determined by the full width at half maximum of the peak. Figures 3(a), 3(b), and 3(c) show the extracted frequency splittings as a function of the magnetic field on X-, Y-, and Z-axes, respectively, when the stray magnetic field was almost canceled. The red fitting lines (a linear or a quadratic function) show that the DC magnetic fields, $B_{coil}$, were 0.80, 0.83, 1.11 Gauss/A on the X-, Y-, and Z-axes. In addition, the fitting lines’ intercept points or the minimum points give the best setting current to compensate for the ${B_0}$ and the lowest achievable magnetic field with a resolution of below two mG for each axis.

 

Fig. 3. The magnetic field strength derived from the Zeeman splittings as a function of the applied coil current in the X (Fig. (a)), Y (b), and Z (c) axes after minimizing the stray field. The red linear fitting lines give that the DC magnetic fields, $B_{coil}$, were 0.80, 0.83, 1.1 Gauss/Amp. By extrapolating both linear fits to the magnetic field, we derive the resolution to minimize the magnetic field below two mG.

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Once the stray magnetic field had been compensated, we applied an additional magnetic field as the quantization axis. The polarization components of the optical pumping and microwave fields will be varied according to the direction of the quantization axis. To determine the polarization purity, a minor magnetic field was varied along the X-axis in Fig. 4(a), while a magnetic field of 65 mG was set along the Y-axis. We have determined the purity of the MW polarization from theoretical fits to the MW spectra, and 97${\% }$ purity has been achieved for the $\pi$ transition. Figure 4(b) depicts the corresponding microwave spectrum, showing three main peaks. Several factors, including population, MW power, MW transition period, and the matrix elements of the relevant transitions, determine peak height. The parameters have all been determined independently, and only the population remains. According to the best fit (solid red line), the populations $(P_{\gamma }, P_{\beta }, P_{\alpha })$ are (32${\% }$, 36${\% }$, 32${\% }$) almost equally distributed. When we set the magnetic field of 65 mG along the Z-axis, the microwave polarization was perpendicular to this quantization axis, and it drove the $\sigma ^+$ and $\sigma ^-$ transitions. The four peaks in Fig. 4(c) correspond to $\gamma _3$, $\gamma _1 + \beta _3$, $\beta _1 + \alpha _3$, and $\alpha _1$ transitions, from the left to the right peaks. According to the best fits to the data, the red lines show that the population was equally distributed in the present measurements. The polarization purity, period, and microwave power are crucial for manipulating $\pi$-pulse transition, such as the Ramsey interference study.

 

Fig. 4. (a) The microwave polarization measurements as a function of the magnetic field strength along the X-axis under an additional magnetic field of 65 mG along the Y-axis, which had the same direction as the MW polarization. A 97${\% }$ polarization purity for microwave $\pi$ transition has been achieved. (b) and (c) represent the spectral measurements obtained when applying an external magnetic field along the Y-axis (where MW drove $\pi$ transitions) and the Z-axis (where MW drove $\sigma ^\pm$ transitions), respectively. The solid lines are the best fits by applying Eq. (1) and considering all possible MW transitions. The best fits show that the populations $(P_{\gamma }, P_{\beta }, P_{\alpha })$ are (32${\% }$, 36${\% }$, 32${\% }$) in (b) and (34${\% }$, 34${\% }$, 32${\% }$) in (c), which are almost equally distributed.

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We then applied optical pumping to accumulate the populations to a Zeeman state and improve state purity. The optical pumping fields, OP$_1$ and OP$_2$, were set at the resonant frequencies of $F=1\rightarrow F'=1$ and $F=2\rightarrow F'=2$ transitions, respectively. First, the target state is the clock state $|F=1,m_F=0\rangle$. We applied the external magnetic field along the Z-axis as the quantization axis. The polarizations of the MW, OP$_1$, and OP$_2$ fields were $\sigma ^\pm$, $\pi$, and $\sigma ^\pm$, respectively. Initially, the population was almost equally distributed among the three Zeeman states. When the trapping beams were turned off at $t=-330 ~\mu s$, OP$_1$ and OP$_2$ were switched on simultaneously and lasted for 300 $\mu$s. As the shown energy levels and laser excitations, represent as the green dashed-dotted (OP$_1$) and blue dotted (OP$_2$) lines in the left of Fig. 5(a), the dark state is $|F=1,m_F=0\rangle$. After optimizing the laser polarization and intensity, the MW spectrum displayed only two major peaks corresponding to $\beta _1$ and $\beta _3$ transitions. The best fit shows that the fraction of atoms on this clock state increases to $98(1){\% }$. Note that the small peak at the center, with $\Delta =0$, corresponds to a $\pi$-polarized MW transition, with only $3{\% }$ polarization impurity. Similarly, we can pump the population in another clock state $|F=2,m_F=0\rangle$ by changing the polarization of the pumping fields, e.g., $\pi$ polarization for OP$_1$ and $\sigma ^\pm$ polarization for OP$_2$. Here we observed the dips instead of the peaks in Fig. 5(b). The baseline signals of the fluorescence showed the number of atoms in $F=2$ states. The fluorescence signals were taken by turning on the trapping beam again as the image beam ($F=2\rightarrow F=3'$ transition) after the population was initially prepared in the $F=2$ state. However, the resonant microwave drove some of the population to the $F=1$ state, thereby reducing the strength of the resonant fluorescence.

 

Fig. 5. Timing sequence, the relevant energy levels, and laser excitations. (a), (b), and (c) are the microwave spectra for atomic preparation in states $|F=1,m_F=0\rangle$, $|2, 0\rangle$, and $|1,1\rangle$, respectively. Take (a) as an example. The polarizations of the MW, OP$_1$, and OP$_2$ fields were $\sigma ^\pm$, $\pi$, and $\sigma ^\pm$, respectively. After turning off the repumping and trapping beams, we switched on OP$_1$ and OP$_2$ simultaneously and lasted for 300 $\mu$s. As the shown energy levels and laser excitations, represent as the green dashed-dotted (OP$_1$) and blue dotted (OP$_2$) lines in the left of Fig. 5(a), the dark state is $|F=1,m_F=0\rangle$. For a different target state, the polarizations of the pumping fields were varied. The best purity of population in these states were $98(1){\% }$, $98(2){\% }$, and $96(2){\% }$, from (a) to (c). The red, black, and blue circles in the inset represent the populations $P_\alpha$, $P_\beta$, and $P_\gamma$ as a function of the orientation angle of a zero-order $\lambda /4$ plate for OP$_1$ beam. These measurements were taken at 65 mG homogeneous magnetic field. (d) A simplified set of atomic energy levels involving one dark state $|Dg\rangle$, one bright state $|Bg\rangle$, and two excited states, $|e_{1(2)}\rangle$. These states are coupled by a resonant $R(0)$ and a far-off resonant $R(\delta )$ driving fields.

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Finally, we accumulated the population in the state $|F=1,m_F=1\rangle$ as it exhibits the strongest coupling strength between two hyperfine states, resulting in the highest optical density. The magnetic field was set along the X-axis, while OP$_1$ with $\sigma ^+$ polarization and OP$_2$ with $\sigma ^\pm$ polarization were utilized. The MW duration was adjusted based on the coupling strength to achieve a $\pi$-pulse transition for the $\alpha _1$ peak. As shown in Fig. 5(c), the linewidth of the $\alpha _1$ peak was broader than the fitting curve because of the non-uniform magnetic field across the center and the edge of the atomic cloud [33]. This broadening effect was observed only when the magnetic field was aligned along the X-axis. A longer atomic cloud was observed along the X-axis relative to the Z-axis, making the magnetic field’s homogeneity crucial. To optimize the purity of each desired Zeeman state, we well-adjusted the orientation of a zero-order $\lambda /4$ plate for the OP$_1$ (or OP$_2$) beam. Only the measurements for state $|1,1\rangle$ are shown in the inset of Fig. 5(c). The presence of polarization impurity causes a population to be pumped into bright states, leading to distinct spectral peaks that can be attributed to the bright-state atoms. The purity has been optimized up to $96(2){\% }$. Compared to the purity of states $|1,0\rangle$ and $|2,0\rangle$, the lower purity and larger error in state $|1,1\rangle$ are mainly due to the broader transition peaks caused by the non-uniform magnetic field.

Off-resonant transitions hinder the population purity despite having only one dark state in the optical pumping process. These transitions induce a heating effect attributed to random photon scattering. A detailed description of this effect on optical pumping is presented in Ref. [23]. Here we consider a simplified set of atomic energy levels, as depicted in Fig. 5(d). The populations in the dark state ($|Dg\rangle$) and bright state ($|Bg\rangle$) are $N_0(t)$ and $N_1(t)$, respectively. We assume the branching ratios of the radiative decay from the excited state to the ground states are equal, i.e., $\Gamma /2$ ($\Gamma$ is the total nature decay rate of the excited state). The absorption rate $R$ depends on the detuning and laser intensity [34],

In the steady-state condition, the population purity in the dark state is

Therefore, consider the detunings of the near hyperfine levels of the $^{87}\rm {Rb}$ $D_2$-line transitions, $\delta =12\Gamma$ and $24\Gamma$. We estimate the upper limit of the purity approach $99.8{\% }$ at low intensities, where $I\ll I_{sat}$. The limitation can be further improved by selecting the $^{87}\rm {Rb}$ $D_1$ transitions, which have a more significant splitting between the two hyperfine levels.

As found in Eqs. (2)–(4), the pumping field intensity is another factor influencing purity. The duration of the optical pumping process plays a similar role in the population accumulation that we did not discuss here. Figure 6(a) illustrates the fluorescence signals for identifying the optimal population accumulation in state $|1,0\rangle$. The signal in the absence of an optimized OP field is represented by the black line, while the red line represents the signal with an optimized OP field and the green line represents the signal with the strongest OP field. A stronger fluorescence signal indicates a larger population. Increasing the power of the optical pumping (OP$_1$), more population is transferred to the hyperfine state with $F=2$ through off-resonant transitions. As a result, the baseline level of the MW spectrum increases. We then extract the peak heights ($\beta _1$ transitions) and baseline levels from the spectrum as a function of the OP power, as shown in Fig. 6(c). The black dotted line is the peak height without the OP field. Population (black circles) can be concentrated in a specific state with proper pumping power. A purity of $98(1){\% }$ has been achieved from the systematic measurements. Similarly, results for state $|1, 1\rangle$ are shown in Figs. 6(b) and 6(d). Both figures demonstrate that the population $P_\beta$ in (c) or $P_\alpha$ in (d) increases with higher OP power. However, increasing the power beyond a certain point results in a decrease in population. The fluorescence signal of the $\alpha _1$ transition has been improved by a factor of three, corresponding to a three-fold population increase. In this case, population accumulation was optimized to $98(1){\% }$ in the $|1,0\rangle$ state and $96(2){\% }$ in the $|1,1\rangle$ state, with limitations imposed by the inhomogeneous magnetic field and far-off-resonant transitions. Once the population is accumulated in a single Zeeman state, people can apply the stimulated-Raman adiabatic passage (STIRAP) procedure, a two-photon adiabatic transition in a three-level system, to pump the population to any other desired state [18,35].

 

Fig. 6. (a) and (b) illustrate the fluorescence signals, taken at 50 mG homogeneous magnetic field, for identifying the optimal population accumulation in states $|1,0\rangle$ and $|1,1\rangle$, respectively. The black line represents signals without the OP field, the red one with an optimized OP field, and the green one with the strongest OP field. We extract the peak heights ($\beta _1$ and $\alpha _1$) and baseline levels as a function of OP power (OP$_1$) in (c) and (d). The dashed lines are the guide to the eyes. The dotted lines are the extracted peak heights without the OP field. Vertical axes are displayed in arbitrary units. Both measurements show that the population $P_\beta$ in (c) or $P_\alpha$ in (d) was enhanced with increasing pumping power; however, further increasing the power reduced the population. Meanwhile, the baseline level increased with increasing pumping power, so more populations were pumped to another hyperfine state $F=2$ via off-resonant transitions. The fluorescence signal for the $\alpha _1$ transition has improved by a factor of 3, corresponding to the 3-fold population increment. Therefore, the population $P_\beta$ in (c) or $P_\alpha$ in (d) can be optimized by applying a proper OP power (or intensity).

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5. Conclusions

Our study presents a groundbreaking and comprehensive approach to achieving precise population accumulation in any desired Zeeman state. By utilizing a theoretical fitting model, we can individually determine the population distribution, microwave (MW) polarization ratio, and MW Rabi frequency, all from a single MW spectrum under an arbitrary magnetic field. Using MW spectroscopy, we successfully compensate for the environment’s stray magnetic field with high precision, reaching below two mG. We achieve impressive state purities by carefully optimizing optical pumping (OP) polarization, period, and intensity. A simplified model considering resonant and off-resonant transitions in steady-state conditions reveals an upper limit to the purity under a weak optical pumping field. Experimentally, we account for the influence of impure polarization of light, non-uniform magnetic fields, and significant off-resonant transitions, which may affect the attained state purity. We managed to achieve state purities of 98(1)${\% }$ in $|1,0\rangle$, 98(2)${\% }$ in $|2,0\rangle$, and 96(2)${\% }$ in $|1,1\rangle$ states, with the slight decrease in purity attributed to the inhomogeneous magnetic field. Importantly, our approach allows us to accurately track the population distribution during the optical pumping process, providing a powerful tool for controlling and manipulating multilevel atomic systems. The implications of this technique will impact various quantum-information processing applications and advance the field of precision measurement, particularly in atomic clocks.