Also at ]Center for Quantum Technology, Hsinchu 30013, Taiwan

# Optimizing Population Accumulation in Quantum States Using Microwave
Spectroscopy

Jia-You Liou  Chi-En Wu  Hsuan-Jui Su  Yi-Hsin Chen  [
<EMAIL_ADDRESS>Department of Physics, National Sun Yat-Sen
University, Kaohsiung 80424, Taiwan

###### Abstract

We present an all-optical method for efficiently preparing cold atoms in a
desired Zeeman state, either on the magnetically insensitive clock state
($m_{F}=0$) or a particular state suitable for processing or storing quantum
information. By applying the theoretical fitting model to a single microwave
spectrum, we can individually determine the population distribution, microwave
polarization ratio, and microwave Rabi frequency. We can dynamically track the
population distribution during the optical pumping process using this real-
time microwave spectrum. In a steady-state condition, a simplified model,
which considers resonant and off-resonant transitions, indicates that there is
an upper limit to the purity under a weak optical pumping field. The
population purity up to $96(2)\%$ or $98(1)\%$ on the desired quantum state
has been achieved after optimizing the intensity and polarization of the
optical pumping field. Our study provides valuable information and potential
applications in precision measurement and quantum computation research.

††preprint: APS/123-QED

##  I Introduction

Preparing atomic populations into a specific Zeeman state is essential for
quantum information science and precision measurements, such as quantum memory
[1, 2, 3, 4], quantum manipulation [5, 6], single-photon generation [7],
atomic magnetometry [8, 9, 10], and atomic clock [11, 12, 13, 14].
Accumulating the population in the desired Zeeman sublevel can increase the
optical density (OD) of the atomic medium according to the Clebsch-Gordan
coefficient of the selected atomic transition, and can avoid energy loss in
the manipulation of the light retrieval [15, 16, 3, 4]. To make two light
fields strongly interact, the researchers prepared the population in the two
specific ground Zeeman states, forming two motionless light pulses via the
quantum interference [5]. A phase shift of $\pi$ by a single photon has been
proposed in such an ultrahigh OD system [17]. In addition, increasing the
population in the clock state ($m_{F}=0$), which is a magnetically insensitive
state, leads to an apparent reduction in the noise for the atomic clock [13].
Optical pumping (OP) is a standard method to pump all atoms into an uncoupled
dark state. By applying the stimulated Raman adiabatic passage (STIRAP), one
can coherently transfer the population between two quantum states via two
coherent light pulses [18]. Moreover, through combining optical pumping with
microwave pumping, the researchers prepared the population in a clock state
with a purity of $83\%$ [13]. With appropriate polarization configuration and
laser power, state purities of more than $96\%$ [19] or $97\%$ [20] were
achieved.

Figure 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 87Rb atoms and laser excitations. Optical pumping
fields OP1 and OP2 were applied to pump the population away from hyperfine
states $F=1$ and $F=2$, respectively. 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.

At present, the methods to prepare and detect the population distribution
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 [22, 11, 23, 24, 25, 26], and (4) the Ramsey
atom interferometer [27, 13]. In addition, the atomic population distribution
can be detected via the reflected microwave power spectrum using a real-time
nondestructive imaging system [28]. In this work, we applied microwave
spectroscopy to measure the population distribution and provided an analytical
solution for atomic state detection. The Rabi frequency of the applied
microwave was individually determined from the detection of the population
cycling transition. Microwave pulses with high accuracy would be helpful in
the atomic ground-state manipulation and for the Ramsey interference study
[29, 27, 13, 30, 31]. The environment magnetic field can be compensated with a
precision of below two mG from microwave spectroscopy. After knowing the
population distribution, the optical pumping method was employed to pump the
population to the desired quantum state with high purity 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. The main feature and time evolution
during the OP of microwave spectroscopy will be discussed and compared with
our theoretical predictions.

##  II Scheme and experimental setup

We perform the experiment using 87Rb atoms in a typical magneto-optical-trap
(MOT) system, including three pairs of trapping beams and one repumping beam.
The magnetic field gradients in the X, Y, and Z directions of 3.06, 1.44, and
4.97 Gauss/cm were produced by a pair of anti-Helmholtz coils (Qpole) with a
current of 2.0 Amp. The trapped atom cloud has a dimension of $7.5\times
5.7\times 3.7$ mm3 containing $2\times 10^{8}$ atoms. The schematic of the
setup, 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 OP1 and OP2 drove the population away from
$F=1$ and $F=2$, respectively. 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. In order 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 the period was varied in other measurements. 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.

After the cooling process, the trapping fields were applied 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 was scanned at a
speed of 0.6 kHz per repetition time.

Figure 2: (a) Energy structure of a two-level system. The microwave field is
tuned to the frequency of the transition 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
$|F=1,m_{F}=1\rangle\rightarrow|F=2,m_{F}=2\rangle$, which gives the Rabi
frequency of $2\pi\times$6.0 kHz.

##  III Theoretical model

We theoretically explain the main feature of the microwave (MW) spectrum and
individually determine the population distribution, polarization, and MW Rabi
frequency. First, we measured the MW strength. Consider a two-level system
with states $|1\rangle$ and $|2\rangle$, and assume the population is
initially in $|1\rangle$. 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

$P_{2}(t,\Delta)=\\\ \frac{\Omega^{2}}{\Omega^{\prime
2}}\left(1-\text{Cos}(\Omega^{\prime}t)\right),$ (1)

where

$\Omega^{\prime}=\sqrt{\Omega^{2}+\Delta^{2}}.\\\ $ (2)

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, and the fluorescence 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. Considering the
population decay, the measured oscillation gives the Rabi frequency for the
$\alpha_{1}$ transition of $2\pi\times 6.0$ kHz.

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 first focus on the $\pi$ transition peaks, i.e., the second, fourth, and
sixth peaks. The fluorescence signals are proportional to the total number of
atoms and the sensitivity of the CCD camera, which have an equal impact on
each peak. The factors $(P_{\gamma},P_{\beta},P_{\alpha})$, where
$P_{\gamma}+P_{\beta}+P_{\alpha}=1$, can vary the fluorescence peak height and
need to be taken into account in Eq. (1). Here, $\Omega$ in Eq. (1) is
replaced by $\Omega_{1}\times$($C_{\gamma_{2}}$, $C_{\beta_{2}}$,
$C_{\alpha_{2}}$). The population distribution was derived from these three
peaks of seven, showing that the population was almost equally distributed. We
next determine the polarization component of the microwave from the most-left
($\sigma^{-}$-polarization) and the most-right ($\sigma^{+}$\- polarization)
peaks. The quantization axis is aligned with the applied magnetic field,
allowing the MW to drive $\sigma^{+}$, $\sigma^{-}$, and $\pi$ transitions.
From the comparison of the determined Rabi frequencies that fit the data and
the derived ones from the dipole matrix element, we obtain the ratios of
polarization components are ($14\%,80\%,5.4\%$) for ($\sigma^{-}$, $\pi$,
$\sigma^{+}$) polarizations. In order to verify the accuracy of the above-
determined parameters, the third and the fifth peaks were simulated with the
above-given parameters. The feature of the spectrum fitted quite well,
including the relevant peak height and the oscillation structure.

Figure 3: The magnetic field strength derived from the Zeeman splittings as a
function of the applied coil current in 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.

##  IV Results

We now discuss how to improve the purity of the population accumulation by
applying optical pumping. First, if the laser polarization $\hat{Z}_{L}$
differs from the magnetic field direction $\hat{Z}_{B}$, 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 (either 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 intercept points or the minimum points of the fitting
lines give the best setting current to compensate the $\bf{B_{0}}$ and the
lowest achievable magnetic field with a resolution of below two mG for each
axis.

Figure 4: (a) The microwave polarization measurements as a function of the
magnetic field strength on the X axis under an additional magnetic field of 65
mG on the Y axis, which had the same direction as the MW polarization. A
97$\%$ polarization purity for $\pi$ transition has been achieved. (b) and (c)
are the spectral measurements when the external magnetic field was applied on
the Y and Z axes, respectively. The solid lines are the best fits by applying
Eq. (1) and considering all possible MW transitions. The best fits show that
the population was equally distributed in the present measurements.

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. The magnetic field of 65 mG was set on the Y axis,
and another minor field on the X axis was varied to determine polarization
purity. Based on theoretical fits to the MW spectra, we can determine the
purity of the MW polarization. A purity of 97$\%$ for the $\pi$ transition has
been achieved, as demonstrated in the corresponding microwave spectrum shown
in Fig. 4(b). When we set the magnetic field of 65 mG on the X 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.

Figure 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. 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 OP1
beam. (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.

We then applied optical pumping to accumulate the populations to a Zeeman
state and improve state purity. The optical pumping fields, OP1 and OP2, were
set at the resonant frequencies of $F=1\rightarrow F^{\prime}=1$ and
$F=2\rightarrow F^{\prime}=2$ transitions, respectively. First, the target
state is the clock state $|F=1,m_{F}=0\rangle$. We applied the external
magnetic field on the Z axis as the quantization axis. The polarizations of
the MW, OP1, and OP2 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
$t=-330~{}\mu s$, OP1 and OP2 were turned on simultaneously and lasted for 300
$\mu$s. As the shown energy levels and laser excitations in 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 OP1 and
$\sigma^{\pm}$ polarization for OP2. Here we observed the dips instead of the
peaks. The baseline signals of the fluorescence showed the number of atoms in
$F=2$ states. The fluorescence signals were taken using an image beam
($F=2\rightarrow F=3^{\prime}$ transition frequency with a specified detuning)
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.

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 on the X
axis, while OP1 with $\sigma^{+}$ polarization and OP2 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. Note
that 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 [32]. This broadening effect was observed only when the
magnetic field was aligned along the X axis. A longer atomic cloud was
observed on 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 OP1
(or OP2) beam. Only the measurements for state $|1,1\rangle$ are shown in the
inset of Fig. 5(c). The polarization impurity causes population to be pumped
to the bright states, resulting in clear peaks on the spectrum which are
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
wider 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. 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 [33],

$R(\delta)=C_{ij}\frac{\Gamma}{2}\frac{I/I_{sat}}{1+4(\delta/\Gamma)^{2}+(I/I_{sat})},$
(3)

where $I$ and $I_{sat}$ are the laser intensity and saturation intensity, and
$\delta$ is the detuning of the laser field from the atomic resonance,
$C_{ij}$ denotes the Clebsch-Gordan coefficient involving the transition. The
population transfer between the dark and bright states involves two steps: (1)
atoms are excited either from the dark state $|Dg\rangle$ to the off-resonant
excited state $|e_{1}\rangle$ or from the bright state $|Bg\rangle$ to the
resonant excited state $|e_{2}\rangle$, and (2) they then spontaneously decay
either from $|e_{1}\rangle$ to $|Bg\rangle$ or from $|e_{2}\rangle$ to
$|Dg\rangle$. The time evolution of the population can be described as

$\frac{dN_{0}(t)}{dt}=\frac{1}{2}N_{1}(t)R(0)-\frac{1}{2}N_{0}(t)R(\delta).$
(4)

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

$\frac{N_{0}}{N_{1}+N_{0}}=\frac{R(0)}{R(0)+R(\delta)}.$ (5)

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.

Figure 6: (a) and (b) illustrate the fluorescence signals 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
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).

As found in Eqs. (3)–(5), 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. By increasing the intensity of the optical
pumping, 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. The 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, which corresponds to a three-fold increase in population. 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, which is a two-
photon adiabatic transition in a three-level system, to pump the population to
any other desired state [18, 34].

##  V Conclusions

We present a comprehensive approach to population accumulation on any desired
Zeeman states. Using a theoretical fitting model, we can individually
determine the population distribution, the MW polarization ratio, and the MW
Rabi frequency by applying it to a single real-time MW spectrum under an
arbitrary magnetic field. The environment’s stray magnetic field can be
compensated below two mG from the MW spectroscopy. The state purities were
optimized by adjusting OP polarization, period, and intensity. In a steady-
state condition, a simplified model, which considers resonant and off-resonant
transitions, indicates that there is an upper limit to the purity under a weak
optical pumping field. Experimentally, the impure polarization of light, non-
uniform magnetic field, and significant off-resonant transitions affect the
state purity. The purities reached up to 98(1)$\%$ in $|1,0\rangle$ and
98(2)$\%$ in $|2,0\rangle$ states after the optimization. The population
accumulated in $|1,1\rangle$ state was $96(2)\%$ mainly due to the
inhomogeneous magnetic field. Using this real-time microwave spectrum, we can
dynamically track the population distribution in the optical pumping process.
This technique can significantly impact quantum-information processing in
multilevel atomic systems, and our investigation will advance future
applications in precision measurement, such as atomic clocks.

###### Acknowledgements.

This work was supported by Grants Nos. 109-2112-M-110-008-MY3 and
111-2123-M-006-001 of the National Science and Technology Council, Taiwan. The
authors thank Prof. Cheng Chin for the valuable comments on the study.

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