# Engineering atomic polarization with microwave-assisted optical pumping

A. Tretiakov  Dept. of Physics, University of Alberta, Edmonton AB, Canada T6G
2E1  C. A. Potts  Dept. of Physics, University of Alberta, Edmonton AB, Canada
T6G 2E1  Y. Y. Lu  Dept. of Physics, University of Alberta, Edmonton AB,
Canada T6G 2E1  J. P. Davis  Dept. of Physics, University of Alberta, Edmonton
AB, Canada T6G 2E1  L. J. LeBlanc  Dept. of Physics, University of Alberta,
Edmonton AB, Canada T6G 2E1

###### Abstract

Polarized atomic ensembles play a crucial role in precision measurements. We
demonstrate a novel method of creating atomic polarization in an alkali vapor
in a continuous-wave regime. The method relies on a combination of optical
pumping by a laser beam and microwave transitions due to a cavity-enhanced
magnetic field. With this approach, atomic internal angular momentum can be
oriented along a static magnetic field at an arbitrary angle with respect to
the laser beam. Furthermore, the atomic polarization depends on the microwave
parameters, which can be used for microwave-to-optical transduction and
microwave-controlled nonlinear magneto-optical rotation.

Light-atom interactions, radio-over-fiber communication, microwave signal
manipulation

††preprint: APS/123-QED

Throughout atomic physics, the light-matter interaction is used to control
quantum states of matter, with results that range from revealing fundamental
physics [1] to enabling precision timekeeping [2]. The advent of the laser
meant that the electric-dipole transitions (with their stronger couplings)
could be readily accessed and precisely controlled within the atom. In
parallel, microwave transitions, enabled by the weaker magnetic-dipole
interactions such as in the “clock” transitions between hyperfine ground
states of alkali metal vapours, can be independently and simultaneously
manipulate atomic states, providing a path towards a microwave-to-optical
interface using atomic media [3].

As a platform for quantum technologies, alkali-metal atomic ensembles, which
naturally support microwave and optical transitions, provide a good
opportunity for microwave-to-optical quantum transduction [4, 5, 6, 7, 8, 9]
and for storing and retrieving quantum information on demand [10, 11, 12]. The
nonlinear effects of microwave interactions on optical properties of alkali
vapors also include the transduction of classical signals [13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23], compact atomic clocks [24, 25, 26], microwave
electrometry [27], and static and microwave magnetometry [28, 29, 30, 31, 32].

In many cases, the microwave’s effect on the optical properties in atomic
vapors is due to _atomic polarization_ : a particular distribution of the
atoms in the ensemble among the available quantum states, which generally have
well-defined angular momenta or “spin,” allowing for the definition of a
direction along which the polarization points and the equivalent nomenclature
of a _spin polarization_. Polarized atomic ensembles like these are essential
components of optically pumped atomic magnetometers [33, 34, 35, 36, 37];
indeed, some approaches require atomic polarization to be perpendicular to the
probe laser beam [33, 35], which is non-trivial task for a single-laser setup
[38].

Figure 1: Principle of the optical pumping (a) and microwave-assisted optical
pumping (b) in a four-level system.

In this Letter, we introduce a novel technique for engineering spin
polarization in atomic ensembles based on microwave-assisted optical pumping
(MAOP). MAOP can be understood as an extension of microwave-optical double-
resonance and similar to optically detected magnetic resonance (ODMR) methods
in solid-state systems [39, 40], except here, at least one of the two ground-
state levels requires several sublevels [Fig. 1]. MAOP leads to an uneven
distribution of atomic populations, resulting in atomic polarization . We
observe this polarization as a microwave-induced optical birefringence and
dichroism. The dichroism can be used to implement a polarization-selective
microwave-to-optical interface, while the birefrengince leads to microwave-
controlled nonlinear magneto-optical rotation [41] that can be applied for
signal transduction based on pulse carving [42].

The atomic polarization emerging from MAOP results from the particular
configuration of dark states that emerge in the presence of combined optical
and microwave fields. To illustrate this principle, consider a four-level toy
model shown in Fig. 1. Here, an optical field (“pump”) couples ground level
$\left|3\right\rangle$ to excited level $\left|e\right\rangle$ [Fig. 1(a)].
Ground states $\left|1\right\rangle$ and $\left|2\right\rangle$ are separated
from state $\left|3\right\rangle$ by a microwave magnetic-dipole transition.
We assume that the frequency separation is sufficient to make optical
excitation of $\left|1\right\rangle$ and $\left|2\right\rangle$ by the pump
field negligible, rendering both states “dark.” Additionally assuming that
there is no relaxation between the ground state when this system interacts
only with the optical field, optical pumping transfers the atomic population
into the dark states $\left|1\right\rangle$ and $\left|2\right\rangle$. When
an additional microwave field couples states $\left|2\right\rangle$ and
$\left|3\right\rangle$, $\left|1\right\rangle$ remains dark and the atomic
population accumulates here [Fig. 1(b)], thus polarizing the ensemble.

In our experiment, we observe MAOP in thermal 87Rb vapor, which is contained
by a vapor cell enclosed in a microwave cavity [Fig. 2(a)]. We use the TE011
cavity mode tuned to the ground-state hyperfine transition $\left|F=1,m_{\rm
F}=0\rightarrow F=2,m_{\rm F}=0\right\rangle$ at $6.834~{}682~{}610$ GHz. The
cavity quality factor for this mode is $Q\approx 27,000$. The microwave field
is provided by a microwave source, with a power of 100 $\mu$W at the source
output, without additional amplification. The cavity has a pair of holes
providing optical access for the pump and probe laser beams. The pump and
probe beams are provided by two different lasers. For the optical pumping we
use $\left|F=2\right\rangle\rightarrow\left|F^{\prime}=2\right\rangle$
transition [Fig. 2(b)] within the D2 line at around $780.2$ nm with a typical
laser power of $1-2$ mW.

Figure 2: (a) Schematics of the experimental setup. Microwave magnetic field
vector is parallel to $x$ (b) Transitions used in the experiment. Optical
transitions correspond to the D2 line. (c) Possible microwave transitions
between Zeeman sublevels labelled $\left|F,m_{\rm F}\right\rangle$ of the 87Rb
ground state. The transitions whose lines cross (in an “X”) have the same
frequency. A rich assortment of atomic polarizations that can be achieved by
MAOP in this level manifold [43].

For the probe, we use a typical power between $10-100\ \mu$W, and we set its
frequency according to the particular measurement: in the absorptive regime,
it is resonant with
$\left|F=1\right\rangle\rightarrow\left|F^{\prime}=0\right\rangle$ transition
of the D2 line, and in the dispersive case, the probe is red-detuned from this
transition. The counterpropagating probe and pump beams are overlapped inside
the cell (Fig. 2a). We use a wave plate (WP) (either a quarter-wave plate or a
half-wave plate) to control the probe polarization as it enters the cell. To
analyze the MAOP effect on the probe polarization, a polarizing beamsplitter
(PBS2) splits the probe between two photodetectors (PD1 and PD2). The static
magnetic field inside the cell is controlled by three pairs of Helmholtz coils
along mutually perpendicular axes.

The 87Rb ground state has nine possible microwave transitions [Fig. 2(c)] with
three different polarizations: $\pi$, $\sigma^{+}$, and $\sigma^{-}$. A static
magnetic field $\mathbf{B}$ inside the cell affects both the frequency
difference between the microwave transitions and the strength of transitions
corresponding to a particular polarization of the microwave field. The
magnitudes of the static magnetic fields in our experiment are within the
regime of the linear Zeeman effect, giving the frequency separation between
two neighbouring transitions by the Larmor frequency $\omega_{\rm L}=\mu_{\rm
B}|\mathbf{B}|/2\hbar$, where $\mu_{\rm B}$ is the Bohr magneton and $\hbar$
is the reduced Planck constant. Some of the transitions are shifted by the
same frequency, giving two pairs of degenerate transitions. The typical values
of the Larmor frequency in our experiments are on the order of $100$ kHz,
which is much less than the width of the D2 line (on the order of MHz),
meaning that both probe and pump couple to all Zeeman sublevels of the
corresponding hyperfine levels simultaneously. We define the quantization axis
as parallel to $\mathbf{B}$, so the polarization of the microwave field with
respect to the quantization axis is determined by the angle between the
microwave magnetic field vector $\mathbf{B}_{\mu}$ and $\mathbf{B}$. Here, the
$\pi$ transitions correspond to the microwave vector component parallel to
$\mathbf{B}$, while $\sigma^{\pm}$ transitions correspond to the microwave
vector orthogonal to $\mathbf{B}$. In our setup, the microwave field vector is
parallel to the cavity axis, which is fixed, so we control the microwave
polarization with respect to the quantization axis by rotating $\mathbf{B}$
with the Helmholtz coils.

Figure 3:  Absorptive measurement of MAOP. (a) Optical signal during a
microwave frequency scan detected by PD1 (red dotted) and PD2 (green solid)
for circularly polarized probe in case when the static magnetic field
$\mathbf{B}$ is parallel to the probe beam. Inset shows the orientation of the
static magnetic-field vector and the polarization of the probe electric-field
vector $\mathbf{E}$ with respect to $x$ and $y$ axes from Fig. 2(a). The probe
polarization lies in the $xz$ plane. The microwave detuning is given with
respect to the clock transition. (b) Schematics of the microwave (blue arrows)
and probe (yellow arrow) transitions, and the expected relative distribution
of the atomic population between Zeeman substates corresponding to peaks I and
II from (a). The probe polarization is indicated with respect to the static
magnetic field. The pump field is not shown. (c) Absorption parameter [43]
calculated for $\hbar=1$, $\Gamma_{\rm OP}=2\pi\times 10^{-3}$, $\omega_{\rm
L}=2\Gamma_{\rm OP}\times 10^{2}$, $\Omega_{0}=\Gamma_{\rm OP}\times 10^{-2}$,
$\theta=\pi/2$, $g_{\rm C}=0.1\omega_{\rm L}$ , and $\Gamma_{\rm
th}=\Gamma_{\rm OP}$. (d-f) are same as (a-c), except the static magnetic
field is perpendicular to $xy$-plane. Microwave power at the source output is
100 $\mu$W. The frequency sweeping rate is $35.5$ Hz. The Larmor frequency
$\omega_{\rm L}$ was estimated from the peak separation as $160$ kHz in (a),
and $259$ kHz in (d).

Absorption. We begin with MAOP measurements in absorptive regime [Fig. 3]. For
these measurements, the probe is circularly polarized with respect to its
wavevector, in which case the probe power splits equally between the two
photodetectors. Fig. 3(a) shows the AC part of the photovoltages from both
detectors during a linear microwave frequency scan when $\mathbf{B}=B\hat{y}$
is parallel to the probe wavevector. In this case, the probe is
$\sigma^{+}$-polarized with respect to the quantization axis, while the
microwave field, whose vector is orthogonal to $\mathbf{B}=B\hat{y}$, is an
equal superposition of $\sigma^{+}$ and $\sigma^{-}$ polarizations. As we scan
the microwave frequency, we see an increase in the probe transmission whenever
the microwave is on resonance with a transition allowed by the microwave
polarization. The asymmetry in the signal in Fig. 3(a) results from the fact
the microwave field creates different atomic polarizations at different
frequencies. Fig. 3(b) illustrates the population distribution for peaks I and
II obtained with corresponding to opposite values of the microwave detuning
with respect to the clock transition
$\left|1,0\right\rangle\rightarrow\left|2,0\right\rangle$. In the first case,
the microwave field simultaneously drives
$\left|1,-1\right\rangle\rightarrow\left|2,0\right\rangle$ and
$\left|1,0\right\rangle\rightarrow\left|2,-1\right\rangle$, in which case the
MAOP transfers the atomic population to $\left|1,1\right\rangle$. Because only
the $\left|1,-1\right\rangle\rightarrow\left|0,0\right\rangle$ transition is
allowed for a $\sigma^{+}$ probe, its transmission is increased due to the
reduced population of $\left|1,-1\right\rangle$. In the case of peak II, the
MAOP moves the atomic population to $\left|1,-1\right\rangle$, where the probe
can be absorbed, making the observed transmission smaller than in the first
case. If we reverse the bias field direction, the asymmetry of the observed
signal is reversed (not shown) since in this case probe is $\sigma^{-}$
polarized and it is absorbed more strongly when the atomic population is
transferred to $\left|1,1\right\rangle$.

Note that the simplified concept of MAOP introduced in Fig. 1 does not fully
explain the signals observed in Fig. 3(a). For example, instead of peak II, we
would expect to see a dip in the optical transmission since the atomic
polarization, in this case, provides a higher optical density compared to an
unpolarized atomic ensemble. To resolve this, we introduced a comprehensive
theoretical model describing MAOP in an ensemble with thermal relaxation [43].
It also includes the contribution to the absorption from the velocity classes
resonant with
$\left|F=1\right\rangle,\rightarrow\left|F^{\prime}=1\right\rangle$ and
$\left|F=1\right\rangle\rightarrow\left|F^{\prime}=2\right\rangle$ transitions
due to the Doppler effect. With this model, we calculate the steady-state
optical absorption in terms of parameter $\alpha_{\rm p}$, where $p$ denotes
the particular probe polarization. Parameter $\alpha_{\rm p}$ is proportional
to the absorption, and it captures the qualitative dependence on the microwave
field characteristics. Fig. 3(c) shows the absorption coefficient calculated
for an ensemble dominated by thermal relaxation between the ground-state
levels, and it is in a good agreement with the observed signal.

To confirm that the direction of the atomic polarization is defined by the
bias magnetic field vector, we perform the same measurement with the bias
field applied out-of-page ($\mathbf{B}=B\hat{z}$) [Fig. 3(d-e)], in which
case, it is perpendicular to both the microwave magnetic field and the probe
wave vector. While the microwave field is still $\sigma^{\pm}$ polarized, the
probe polarization now has components parallel and orthogonal to the
quantization field and couples to all three sublevels of the lower-energy
ground state. Because the probe power is divided equally between $\sigma^{+}$
and $\sigma^{-}$ polarization components, the atomic polarizations
corresponding to opposite microwave detunings have the same effect on optical
absorption, and so the observed signal is symmetric.

Figure 4:  Dispersive measurement of MAOP. (a) Optical rotation during the
microwave frequency scan detected as a change in the photovoltage on detectors
PD1 and PD2. The green curve corresponds to the sum of the two voltages.
Microwave detuning is given with respect to the clock transition. The
microwave source is set to $-10$ dBm, the microwave frequency is scanned at a
rate of $35$ Hz. The estimated optical rotation angle corresponding to peak I
is around $0.025$ mrad. (b) Schematics of the microwave (blue arrows) and
probe (yellow arrow) transitions, and the expected relative distribution of
the atomic population between Zeeman substates corresponding to peaks I and II
from (a). The pump is not shown.

To further demonstrate the emergence of atomic polarization due to the
microwave field, we explore effects in the dispersive regime. In this case,
the MAOP induced atomic polarization leads to circular birefringence,
resulting in the rotation of the probe polarization. To observe the optical
rotation, we use a linearly polarized probe whose frequency is set to the
lower-frequency edge of the Doppler-broadened absorption peak of the D2-line
originating in the $F=1$ level. In this case, the probe is red-detuned from
the $\left|F=1\right\rangle\rightarrow\left|F^{\prime}=0\right\rangle$
transition to the extent where its absorption is reduced and the dispersion is
enhanced compared to the resonant case. In addition,
$\left|F^{\prime}=1\right\rangle$ and $\left|F^{\prime}=2\right\rangle$
resonant classes experience an even higher detuning, and their contribution to
the total signal is reduced. With a half-wave plate, we set the probe
polarization at $45^{\circ}$ with respect to the $x$-axis ($-45^{\circ}$ with
respect to the $z$-axis), in which case PBS2 splits the probe power equally
between the two photodetectors. In this setup, the optical rotation will
result in opposite changes in the photovoltage of the two detectors without
affecting the total optical power. The bias magnetic field is applied parallel
to the probe, so the probe electric-field vector with respect to the
quantization axis is an equal superposition of $\sigma^{+}$\- and
$\sigma^{-}$-polarized fields.

The optical rotation is a result of circular birefringence due to unbalanced
populations of states $\left|1,1\right\rangle$ and $\left|1,-1\right\rangle$.
Assuming that the single-atom refraction index is the same for each
transition, the effective refractive index for each transition is proportional
to the population at the corresponding level. As a result, the phase
difference between the two polarization components leaving the ensemble is
given by

$\displaystyle\Delta\phi=kL(n_{\sigma^{+}}-n_{\sigma^{-}})=kLn[N_{-1}-N_{+1}],$
(1)

where $k$ is the wavenumber, $L$ and the distance through which the light
travels in the medium, $n$ is the single-atom refraction index, and $N_{-1}$
and $N_{+1}$ are the numbers of atoms in states
$\left|F=1,m_{F}=-1\right\rangle$ and $\left|F=1,m_{F}=1\right\rangle$ states,
respectively. This phase difference determines the angle by which the linear
optical polarization is rotated.

Fig. 4(a) shows the relative photovoltage from PD1 and PD2 measured in the
dispersive regime during a linear sweep of the microwave frequency. Whenever
the microwave field is on resonance with an allowed hyperfine transition, it
results in opposite voltage changes on the two detectors, indicating the
optical rotation. The sum of the two voltages is small but not exactly zero,
which indicates that some part of the signal corresponds to the absorptive
regime. The fact that the signal from each detector is antisymmetric with
respect to the clock-transition frequency means that the direction of the
polarization rotation depends on the sign of the microwave detuning. According
to Eq. 1, this is a consequence of a change in the sign of the population
difference between levels $\left|1,1\right\rangle$ and
$\left|1,-1\right\rangle$, and thus a change in the “direction” of the atomic
polarization. Fig. 4(b) schematically shows the population distribution for
peaks I and II from Fig. 4(a). Due to a higher population of
$\left|1,1\right\rangle$ in the case of negative microwave detuning, the
$\sigma^{-}$-polarization component experiences a larger refractive index and
acquires a phase delay relative to the $\sigma^{+}$-polarization component, so
the linear polarization rotates towards the horizontal axis, and more optical
power goes to PD2. For the positive microwave detuning, the population is
transferred to $\left|1,-1\right\rangle$, so the $\sigma^{+}$ component is
delayed, and the rotation direction is reversed. If we turn off the pump light
or set its frequency off-resonance, the observed signal feature disappears,
which suggests that the observed circular birefringence of the vapor results
from MAOP and not from the non-linear interaction between the microwave field
and the probe itself. We also highlight that when the microwave field is not
on resonance with any transition, no optical rotation is observed.

Finally, we emphasize that the optical rotation due to polarization of atomic
vapor discussed above corresponds to _paramagnetic_ Faraday rotation [44],
which differs from diamagnetic Faraday rotation, where the optical rotation
angle is proportional to the magnitude of a strong static magnetic field
applied parallel to the optical beam [45]. In the case of diamagnetic Faraday
rotation, transitions corresponding to different circular polarizations
experience different frequency shifts due to the interaction with the static
magnetic field leading to the circular birefringence.

To summarize, we introduced the technique of microwave-assisted optical
pumping, which enables atomic-spin polarization engineering by combining
microwave magnetic-dipole transitions with optical pumping. We demonstrated
that the engineered atomic-spin polarization depends on the microwave
frequency by observing its effect on optical circular birefringence and
dichroism in a thermal atomic vapor in the dispersive and absorptive regimes,
respectively. While the current experiments were performed in a thermal atomic
vapor, and we expect better performance in cold ensembles, where collisional
and spin-exchange relaxation between the ground-state levels is reduced. By
engineering atomic polarization engineering via MAOP-tailored optical
absorption and dispersion, we have demonstrated a suite of techniques for
domain of microwave-to-optical control [3], including possibilities for a
microwave-controlled polarization-selective optical switch that could encode
polarization states of light, where applying a microwave field makes the
atomic vapor transparent for light with a particular polarization, with
applications to microwave-to-optical transduction.

## Acknowledgements

This work was supported by the University of Alberta; the Natural Sciences and
Engineering Research Council, Canada (Grants No. RGPIN-04523-16, No.
RGPIN-2021-02884, and No. CREATE-495446-17); the Alberta Quantum Major
Innovation Fund; Alberta Innovates; and the Canada Research Chairs (CRC)
Program. As researchers at the University of Alberta, we acknowledge that we
are located on Treaty 6 territory, and that we respect the histories,
languages, and cultures of First Nations, Métis, Inuit, and all First Peoples
of Canada, whose presence continues to enrich our vibrant community.

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Supplemental Materials for
Engineering atomic polarization with microwave-assisted optical pumping

##  I Atomic populations, microwave transitions, and microwave polarizations

To understand the principles of microwave-assisted optical pumping (MAOP),
Fig. S1 illustrates distribution of atomic populations under various
conditions. First, we consider the distribution of populations among the two
hyperfine ground states, $\left|F=1\right\rangle$ and
$\left|F=2\right\rangle$. For the hyperfine splitting in 87Rb of $\omega_{\rm
HFS}/2\pi=6.834$ GHz, the energy associated with room temperature ($T\approx
300$ K) is far greater than the energy associated with this splitting (e.g.,
the Boltzmann factor $\exp(-\hbar\omega_{\rm HFS}/k_{\rm B}T)\approx 1$, and
we assume that the two levels are occupied according to their degeneracy
factors (associated with the $F$ quantum number), with $5/8$ of the population
in $F=2$ and $3/8$ in $F=1$ [Fig. S1(a)].

Next, when introducing optical pumping (as per the scheme shown in Fig. 2(b)
where the optical pumping light is resonant with a transition from ground
state $\left|F=2\right\rangle$ to excited state
$\left|F^{\prime}=2\right\rangle$, we assume that all population accumulate in
the lower-energy $\left|F=1\right\rangle$ hyperfine ground state and is
equally distributed among Zeeman sublevels [Fig. S1(b)].

Third, upon introducing a resonant microwave field, we must consider its
polarization with respect to the external magnetic field $\mathbf{B}$ in a
direction that sets the quantization axis for the Zeeman sublevels. If the
thermal equilibration rate is negligible compared to the MAOP rate, a resonant
microwave magnetic field $\mathbf{B}_{\rm\mu}(t)$ removes the population from
the optically coupled Zeeman substates, and we assume here that it splits
equally between the uncoupled states [Fig. S1(c-e)]. This allows us to
engineer atomic polarization by deliberately choosing the microwave
parameters. For example, when the microwave magnetic field
$\mathbf{B}_{\rm\mu}(t)$ is parallel to the quanitizaiton field $\mathbf{B}$,
only the $\pi$-polarized microwave transitions are possible. By tuning the
microwave frequency to one of these transitions, we can empty the population
from the connected ground Zeeman level [Fig. S1(c)]. Similarly, a
perpendicular microwave field effects only $\sigma^{+}$ and/or $\sigma^{-}$
transitions [Fig. S1(d,e)], and the particular transition can be selected by
frequency tuning. Note that for the low-field linear Zeeman splitting regime
in which we work, there are two sets of degenerate transitions among the nine
possible transitions, yielding seven independent transition frequencies.

Figure S1:  Atomic population before and after the MAOP. (a) Atomic population
in thermal equilibrium. (b) atomic population after optical pumping without a
microwave field. (c-d) Possible atomic polarization created by the MAOP for
different arrengements of an unmodulated microwave field.(e) Modulating the
microwave field at twice the Larmor frequency allows to transfer the atomic
population to $\left|F=1,m_{\rm F}=0\right\rangle$.

##  II Modelling MAOP with relaxation and Doppler broadening

The model for MAOP depicted in FIG.S1(c-e) assumes a simple configuration in
which the relaxation between the ground-state levels is negligible compared to
the MAOP rate, so the whole atomic population accumulates in the states
uncoupled from the microwave field. Our experiments are performed in a room-
temperature vapor where atomic collisions with the cell walls and thermal
motion through the cell region illuminated by the pump light introduce a
significant relaxation between the states. In addition, due to significant
Doppler broadening, several velocity classes in the atomic ensemble are
resonant with different optical transitions that contribute differently to the
total absorption signal observed experimentally. Here, we account for these
effects, as well as experimental parameters like cavity linewidth, using a
comprehensive model that treats optical pumping as a relaxation process to
predict the relative optical absorption of a microwave-assisted optically
pumped system of room-temperature atoms.

###  II.1 Theoretical model

In our model, the Hamiltonian includes eight levels corresponding to the
Zeeman sublevels of the ground states of 87Rb interacting with the microwave
field. The diagonal part of the Hamiltonian in the rotating frame (rotating at
the applied microwave frequency), after applying the rotating-wave
approximation, is

$\displaystyle\hat{H}_{0}=\hbar\sum_{m_{\rm F}}\left(\dfrac{\delta}{2}-m_{\rm
F}\omega_{\rm L}\right)\left|F=1,m_{\rm F}\right\rangle\left\langle F=1,m_{\rm
F}\right|+\hbar\sum_{\widetilde{m}_{\rm F}}\left(\widetilde{m}_{\rm
F}\omega_{\rm L}-\dfrac{\delta}{2}\right)\left|F=2,\widetilde{m}_{\rm
F}\right\rangle\left\langle F=2,\widetilde{m}_{\rm F}\right|,$

where the $\omega_{\rm L}={\mu_{\rm B}B}/{2\hbar}$ is the Larmor frequency in
a static magnetic field $\mathbf{B}$ with magnitude $B$ (already taking into
account the $g_{\rm F}$ factors for 87Rb’s hyperfine ground states), and
$\delta$ is the detuning of the microwave field from the clock
($\left|F=1,m_{\rm F}=0\right\rangle\rightarrow\left|F=2,m_{\rm
F}=0\right\rangle$) transition. The interaction part is given by

$\displaystyle\hat{V}=\dfrac{\hbar}{2}\sum_{\widetilde{m}_{\rm F}}\sum_{m_{\rm
F}}\Omega_{m_{\rm F},\widetilde{m}_{\rm F}}\left|F=1,m_{\rm
F}\right\rangle\left\langle F=2,\widetilde{m}_{\rm F}\right|+{\rm h.c.},$

where $\Omega_{m_{\rm F},\widetilde{m}_{\rm F}}$ is the Rabi frequency for
coupling between between hyperfine states $\left|F=1,m_{\rm F}\right\rangle$
and $\left|F=2,\widetilde{m}_{\rm F}\right\rangle$. If $\theta$ is the angle
between the microwave magnetic field $\mathbf{B}_{\mu}$ and the quantizing
field $\mathbf{B}$, the coupling strength for the $\pi$-transitions ($m_{\rm
F}\rightarrow m_{\rm F}$) is given by

$\displaystyle\Omega_{m_{\rm F},m_{\rm F}}=\Omega_{0}\cos\theta\sqrt{1-g_{\rm
F}^{2}m_{\rm F}^{2}},$

where $\Omega_{0}={\mu_{\rm B}B_{\mu}}/{\hbar}$, $B_{\mu}$ is the magnitude of
the microwave magnetic field $\mathbf{B}_{\mu}$, and $g_{\rm F}=1/2$ is the
Lande factor for 87Rb. For the $\sigma^{\pm}$-transitions ($m_{\rm
F}\rightarrow m_{\rm F}\pm 1$), the coupling strength is given by

$\displaystyle\Omega_{m_{\rm F},m_{\rm F}\pm
1}=\frac{\Omega_{0}\sin\theta}{2I+1}\sqrt{\left(I\mp m_{\rm
F}\right)^{2}-\dfrac{1}{4}},$

where $I=3/2$ is the nuclear spin of 87Rb.

The finite linewidth of the cavity is taken into account in this model by
modifying the coupling strength to be dependent on detuning, and assuming this
dependence has a Lorentzian character, such that

$\Omega_{m_{\rm F},\widetilde{m}_{\rm F}}(\delta)=\left(\frac{g_{\rm
C}^{2}}{g_{\rm C}^{2}+\delta^{2}}\right)\Omega_{m_{\rm F},\widetilde{m}_{\rm
F}},$

where $g_{\rm C}$ represents the cavity linewidth.

The dynamics of the system are described by the Lindblad master equation

$\displaystyle\dfrac{d}{dt}\hat{\rho}=-\dfrac{i}{\hbar}\left[\hat{H},\hat{\rho}\right]+\dfrac{1}{2}\sum_{n}\left(2\hat{L}_{n}\hat{\rho}\hat{L}_{n}^{\dagger}-\hat{\rho}\hat{L}_{n}^{\dagger}\hat{L}_{n}-\hat{L}_{n}^{\dagger}\hat{L}_{n}\hat{\rho}\right),$
(S1)

where $\hat{\rho}$ is the density matrix, $\hat{H}=\hat{H}_{0}+\hat{V}$ is the
total Hamiltonian, and $\hat{L}_{n}$ is the collapse operator due to a
relaxation process $n$.

We treat optical pumping, which in practice operates through an excited state
level, as a relaxation process from ground state $\left|F=2,\widetilde{m}_{\rm
F}\right\rangle$ to ground state $\left|F=1,m_{\rm F}\right\rangle$, and
assign to it the collapse operator

$\displaystyle\hat{L}_{\rm OP}(m_{\rm F},\widetilde{m}_{\rm
F})=\sqrt{\Gamma_{\rm OP}}\left|F=1,m_{\rm F}\right\rangle\left\langle
F=2,\widetilde{m}_{\rm F}\right|,$

where $\Gamma_{\rm OP}$ is the optical pumping rate, which we assume is the
same for all pairs of $m_{\rm F}$ and $\widetilde{m}_{\rm F}$.

In the thermal equilibrium, we assume that all Zeeman sublevels of the ground-
state hyperfine levels are equally occupied. This can be achieved by modeling
the thermal relaxation with the following collapse operator:

$\displaystyle\hat{L}=\sqrt{\Gamma_{\rm th}}\left|i\right\rangle\left\langle
j\right|$

where $\left|i\right\rangle$ and $\left|j\right\rangle$ are any two different
Zeeman sublevels, and $\Gamma_{\rm th}$ is the thermal relaxation rate.

###  II.2 Simulating steady-state optical absorption

To model our system, we numerically solve Eq. S1 using the computational
package for Python, QuTip [S1]. To begin with, we test the concept of the MAOP
by looking at the populations $\rho_{m_{\rm F}}=\left\langle F=1,m_{\rm
F}\right|\hat{\rho}\left|F=1,m_{\rm F}\right\rangle$ in the absence of thermal
relaxation. Figs. S2 (a) and (b) show the populations corresponding to the
MAOP configurations from Figs. S1(c) and (d), respectively. These simulations
show that a resonant microwave field clears out the addressed sublevels,
moving the atomic population to the uncoupled sublevels.

Figure S2:  Simulation of the steady-state population of $F=1$ sublevels under
MAOP and the corresponding change in the absorption parameters $\alpha_{+}$
and $\alpha_{\rm m}$. The microwave detuning $\delta$ is given with the
respect to the clock transition $\left|F=1,m_{\rm
F}=0\right\rangle\rightarrow\left|F=2,m_{\rm F}=0\right\rangle$. The steady
state is found using the inverse-power method [., S2] The simulation
parameters are $\hbar=1$, $\Gamma_{\rm OP}=2\pi\times 10^{-3}$, $\omega_{\rm
L}=2\Gamma_{\rm OP}\times 10^{5}$, $\Omega_{0}=\Gamma_{\rm OP}\times 10^{-2}$.
(a,c) The microwave magnetic field is parallel to the quantization axis. (b,d)
The microwave magnetic field is perpendicular to the quantization axis.

In order to simulate the optical transmission shown in Fig. 3 of the main
text, we first need to examine how the optical absorption depends on the
distribution of the atomic population between the ground-state sublevels. For
a dilute vapor, the absorption is given by

$\displaystyle A(\omega)=1-e^{-nl\sigma(\omega)}\approx nl\sigma(\omega),$
(S2)

where $n$ is the atomic density, $l$ is the length of the optical path in the
medium and $\sigma(\omega)$ is the frequency-dependent absorption cross-
section. For a particular $\left|F,m_{\rm
F}\right\rangle\rightarrow\left|F^{\prime},m_{\rm F}^{\prime}\right\rangle$
transition, the cross-section is given by

$\displaystyle\sigma(\omega)=A_{0}N(\omega)\|d\|^{2}$

where $A_{0}$ is a parameter which is the same for all transitions within the
D2 line, $N(\omega)$ is the fraction of atoms undergoing this transition, and
$\|d\|$ is the dipole matrix element corresponding to this transition. In a
thermal ensemble, due to the Doppler effect, the value of $N(\omega)$ is a
product of the fraction of atoms in the velocity class resonant with the
transition at frequency ($\omega$) and the probability of an atom from this
velocity class to be in the state $\left|F,m_{\rm F}\right\rangle$

$\displaystyle N(\omega)=\sqrt{\dfrac{m}{2\pi
k_{B}T}}\exp\left[-\left(\dfrac{c(\omega-\omega_{0})}{\omega}\right)^{2}\dfrac{m}{2k_{B}T}\right]\rho_{m_{\rm
F}},$

where $m$ is the atomic mass, $T$ is the ensemble temperature, $k_{\rm B}$ is
the Boltzmann constant, $c$ is the speed of light, $\omega$ is the optical
laser frequency, and $\omega_{0}$ is the optical transition frequency in the
atomic reference frame. For the D2-line, the transition matrix element given
in multiples of $\left\langle J=1/2\|d\|J^{\prime}=3/2\right\rangle$ is

$\displaystyle\|d\|=\sqrt{S_{FF^{\prime}m_{\rm F}m_{\rm
F}^{\prime}}}\left\langle J=1/2\|d\|J^{\prime}=3/2\right\rangle,$

where $S_{FF^{\prime}m_{\rm F}m_{\rm F}^{\prime}}$ is the relative transition
strength for the transition from $\left|F,m_{\rm F}\right\rangle$ to
$\left|F^{\prime},m^{\prime}_{\rm F}\right\rangle$.

As in the measurements from the main text, we consider the case for which the
probe is tuned to the
$\left|F=1\right\rangle\rightarrow\left|F^{\prime}=0\right\rangle$ transition.
Because of Doppler broadening, three velocity classes contribute to the
absorption: the zero-velocity atoms are resonant to the
$\left|F=1\right\rangle\rightarrow\left|F^{\prime}=0\right\rangle$ transition
itself, which we represent by the frequency $\omega_{10}$, and non-zero
velocity classes are resonant to the other two transitions allowed by
selection rules:
$\left|F=1\right\rangle\rightarrow\left|F^{\prime}=1\right\rangle$ and
$\left|F=1\right\rangle\rightarrow\left|F^{\prime}=2\right\rangle$. Because
the Doppler shift for the microwave field is negligible, the MAOP equally
affects the three velocity classes. However, due to different coefficients
$S_{FF^{\prime}m_{\rm F}m_{\rm F}^{\prime}}$ representing the different
strengths of these three transitions, each velocity class contributes to the
total absorption differently. To take into account the cumulative effect that
the population distribution $\rho_{m_{\rm F}}$ (in the three velocity classes)
has on the total absorption, we introduce a coefficient $\alpha(\omega)$, such
that $A(\omega)\propto\sigma(\omega)\propto\alpha(\omega)$, with
$\alpha(\omega)$ given by

$\displaystyle\alpha(\omega)=\frac{1}{I}\left[I_{+}\alpha_{+}(\omega)+I_{\pi}\alpha_{\pi}(\omega)+I_{-}\alpha_{-}(\omega)\right],$

where $I$ is the total probe intensity and $I_{\pm}$ and $I_{\pi}$ are
intensities of the $\sigma^{\pm}$ and $\pi$ polarization components, and

$\displaystyle\alpha_{+}(\omega_{10})=\sum_{m_{\rm F}}\rho_{m_{\rm
F}}\sum_{F^{\prime}}S_{1F^{\prime}m_{\rm F}m_{\rm
F}+1}\exp\left\\{-\frac{m}{2k_{B}T}\left[\frac{c(\omega_{10}-\omega_{1F^{\prime}})}{\omega_{10}}\right]^{2}\right\\},$
$\displaystyle\alpha_{-}(\omega_{10})=\sum_{m_{\rm F}}\rho_{m_{\rm
F}}\sum_{F^{\prime}}S_{1F^{\prime}m_{\rm F}m_{\rm
F}-1}\exp\left\\{-\frac{m}{2k_{B}T}\left[\frac{c(\omega_{10}-\omega_{1F^{\prime}})}{\omega_{10}}\right]^{2}\right\\},$
$\displaystyle\alpha_{\pi}(\omega_{10})=\sum_{m_{\rm F}}\rho_{m_{\rm
F}}\sum_{F^{\prime}}S_{1F^{\prime}m_{\rm F}m_{\rm
F}}\exp\left\\{-\frac{m}{2k_{B}T}\left[\frac{c(\omega_{10}-\omega_{1F^{\prime}})}{\omega_{10}}\right]^{2}\right\\},$

where $\omega_{FF^{\prime}}$ is the frequency of the
$\left|F\right\rangle\rightarrow\left|F^{\prime}\right\rangle$ transition.

In our MAOP configuration, the atomic population depends on the microwave
detuning $\delta$, as does, therefore, the absorption, which we now represent
as $\alpha(\omega,\delta)$. Fig. S3 shows how $\alpha(\omega_{10},\delta)$
changes due to MAOP with the microwave field detuned by $\delta$ from the
clock transition. The simulation is done for the two cases of probe
polarization corresponding to the experimental data from Fig. 3 in the main
text: first, a $\sigma^{+}$-polarized probe, in which case
$\alpha=\alpha_{+}$, and second, a mixed polarization with

$\displaystyle\alpha_{\rm
m}(\omega,\delta)=\dfrac{1}{4}\alpha_{+}(\omega,\delta)+\dfrac{1}{4}\alpha_{-}(\omega,\delta)+\dfrac{1}{2}\alpha_{\pi}(\omega,\delta),$

where half of the probe power corresponds to $\pi$-polarization and the other
half is split equally between $\sigma^{+}$ and $\sigma^{-}$. As seen in [Fig.
S3(a)], when there is no relaxation between the ground-state levels, MAOP
should result in an antisymmetric absorption feature for a
$\sigma^{+}$-polarized probe, and a symmetric feature for the probe with mixed
polarization. In both cases, there is an increase in the absorption
coefficient for some values of the microwave detuning, which is not what we
observe experimentally, where the absorption decreases whenever the microwave
field is on resonance. Upon adding thermal relaxation to the simulation
results, we find the results seen in Fig. S3(b), which is similar to what we
observe experimentally.

Figure S3:  Simulating the change in the steady-state absorption parameter
$\alpha$ as a function of. Simulation is done with the inverse-power solution
method. The simulation parameters are $\hbar=1$, $\Gamma_{\rm OP}=2\pi\times
10^{-3}$, $\Omega_{0}=\Gamma_{\rm OP}\times 10^{-2}$, $\theta=\pi/2$,
$\omega_{\rm L}=2\Gamma_{\rm OP}\times 10^{2}$, $\Gamma_{\rm th}=\Gamma_{\rm
OP}$, and $g_{C}=0.1\omega_{\rm L}$.

[S1] J. R. Johansson, P. D. Nation, and F. Nori, QuTiP 2: A Python framework
for the dynamics of open quantum systems, Comput. Phys. Commun.184, 1234
(2013).

[S2] P. D. Nation, Steady-state solution methods for open quantum optical
systems, arxiv.org , 1504.06768 (2015) .