# Evaporation of microwave-shielded polar molecules to quantum degeneracy

Andreas Schindewolf  Roman Bause  Xing-Yan Chen  Marcel Duda  Max-Planck-
Institut für Quantenoptik, 85748 Garching, Germany  Munich Center for Quantum
Science and Technology, 80799 München, Germany  Tijs Karman  Institute for
Molecules and Materials, Radboud University, Heijendaalseweg 135, 6525 AJ
Nijmegen, Netherlands  Immanuel Bloch  Max-Planck-Institut für Quantenoptik,
85748 Garching, Germany  Munich Center for Quantum Science and Technology,
80799 München, Germany  Fakultät für Physik, Ludwig-Maximilians-Universität,
80799 München, Germany  Xin-Yu Luo  e-mail<EMAIL_ADDRESS>Max-Planck-
Institut für Quantenoptik, 85748 Garching, Germany  Munich Center for Quantum
Science and Technology, 80799 München, Germany

###### Abstract

Ultracold polar molecules offer strong electric dipole moments and rich
internal structure, which makes them ideal building blocks to explore exotic
quantum matter [1, 2, 3, 4, 5, 6, 7, 8], implement novel quantum information
schemes [9, 10, 11], or test fundamental symmetries of nature [12]. Realizing
their full potential requires cooling interacting molecular gases deeply into
the quantum degenerate regime. However, the complexity of molecules which
makes their collisions intrinsically unstable at the short range, even for
nonreactive molecules [13, 14, 15, 16, 17, 18], has so far prevented the
cooling to quantum degeneracy in three dimensions. Here, we demonstrate
evaporative cooling of a three-dimensional gas of fermionic sodium-potassium
molecules to well below the Fermi temperature using microwave shielding. The
molecules are protected from reaching short range with a repulsive barrier
engineered by coupling rotational states with a blue-detuned circularly
polarized microwave. The microwave dressing induces strong tunable dipolar
interactions between the molecules, leading to high elastic collision rates
that can exceed the inelastic ones by at least a factor of $460$. This large
elastic-to-inelastic collision ratio allows us to cool the molecular gas down
to $21$ nanokelvin, corresponding to $0.36$ times the Fermi temperature. Such
unprecedentedly cold and dense samples of polar molecules open the path to the
exploration of novel many-body phenomena, such as the long-sought topological
$p$-wave superfluid states of ultracold matter.

The field of ultracold polar molecules is currently on the verge of entering
an exciting phase of showing its full potential in quantum sciences [19, 20].
The tunable anisotropic long-range interactions and rich internal degrees of
freedom of polar molecules open up the possibility to study exotic quantum
phases ranging from $p$-wave superfluids [1, 2, 3], supersolids [4, 5], and
Wigner crystals [6, 7] to novel spin systems and extended Hubbard models [8]
and can also be utilized for quantum computation [9, 10, 11]. Many of these
proposals require a deeply degenerate quantum gas of molecules at tens of
nanokelvin with strong dipolar interactions in three dimensions (3D).

While non-interacting polar molecules can partially inherit low entropy from
degenerate atomic mixtures [21, 22, 23], active cooling to the quantum
degenerate regime has remained challenging. The main hurdle towards efficient
evaporative cooling of ultracold molecules is posed by their rapid collisional
loss. Especially when the molecules are polarized by an electric field in 3D
traps they tend to collapse in attractive head-to-tail collisions [24]. It
turned out that even molecules that are nominally stable against chemical
reactions [25] undergo inelastic collisions when they reach short range. The
exact nature of these loss processes is still not fully understood and remains
under investigation [13, 14, 15, 16, 17, 18].

Collisional loss in short-range encounters between molecules can be suppressed
by engineering repulsive interactions using external fields [26, 27, 28, 29,
30]. Recently, a molecular gas of 40K87Rb was stabilized and evaporatively
cooled to below the Fermi temperature by applying a strong dc electric field
and confining the motion of the molecules to two dimensions (2D), which
effectively prevented attractive collisions along the third dimension [31].
For molecules in 3D, reduced collisional loss has been demonstrated by using a
specific dc electric field [32] to bring rotational states of colliding
molecules in resonance with each other. However, attempts to produce a
degenerate quantum gas of polar molecules via collisional cooling so far
remained unsuccessful in 3D due to a low elastic-to-inelastic collision ratio
as well as a relatively low initial phase space density [33, 32]. Repulsive
barriers between molecules can also be engineered by applying blue-detuned
circularly polarized microwaves [34], which was successfully employed recently
to reduce collisional loss between two CaF molecules in an optical tweezer
[35].

In this work, we induce fast elastic dipolar collisions in a quantum gas of
fermionic 23Na40K molecules dressed by a circularly polarized microwave field
while strongly suppressing inelastic collisions in all three dimensions. The
elastic collision rate increases dramatically with the effective lab-frame
dipole moment, which can be adjusted with the microwave power and detuning.
Under appropriate conditions, we find that the elastic collision rate can be
about $500$ times larger than the inelastic collision rate, and even exceed
the trap frequencies such that the molecular gas enters the hydrodynamic
regime. We evaporate a 3D near-degenerate sample of molecules down to $21(5)$
nK, which corresponds to $36(9)$% of the Fermi temperature $T_{\text{F}}$,
deep in the quantum degenerate regime. At such low temperatures, the inelastic
collision rate between molecules becomes negligible, leading to a long
lifetime of the degenerate molecular sample of up to $0.6$ s, mainly limited
by residual one-body loss induced by the technical noise of the microwave.

## Microwave shielding and dipolar elastic collisions

To realize shielding in 3D, the two lowest rotational states of the molecules
can be coupled via a blue-detuned circularly polarized microwave [34]. The
mixed state features an induced rotating dipole moment that follows the strong
ac electric field of the microwave. Molecules prepared in this state exhibit
an effective dipolar interaction at long range. When they approach each other,
the direct coupling between the induced dipoles starts to dominate, forming a
repulsive barrier that effectively shields the molecules from detrimental
inelastic collisions at short range.

Figure 1: Microwave shielding. a, Sketch of the experimental setup. The
molecules are confined by up to three optical dipole traps with the indicated
wavelengths. A helical antenna emits the rotating electric field $\vec{E}$. A
dc magnetic offset field $\vec{B}$ is generated by a pair of coils (not
shown). b, Preparation of the molecules in the dressed state $|{+}\rangle$.
The microwave detuning $\Delta$ is typically comparable with the Rabi
frequency $\Omega$. c, Interaction potentials for $\Delta=2\pi\times 8$ MHz
and $\Omega=2\pi\times 11$ MHz. The solid and dashed lines show the potential
energy of molecules colliding along ($\theta=0$) and orthogonal
($\theta=90^{\circ}$) to the microwave propagation direction, respectively.
The centrifugal barrier is omitted for clarification. Here, $a_{0}$ is the
Bohr radius. d, Close-up of the shielded collisional channel $|{++}\rangle$.
At intermediate range the molecules align with respect to the intermolecular
axis.

In our experiment, the microwave field is generated by a helical antenna, as
illustrated in Fig. 1a. The antenna emits a mainly $\sigma^{-}$-polarized
microwave that couples the rotational ground state $|J=0,m_{J}=0,\delta
N_{\text{p}}=0\rangle$ to the excited state $|1,-1,-1\rangle$, as shown in
Fig. 1b. Here, $J$ is the rotational quantum number, $m_{J}$ is its projection
on the magnetic field axis, and $\delta N_{\text{p}}$ is the change in the
number of photons in the microwave field. The frequency of the microwave is
approximately given by the rotational constant $B_{\text{rot}}=h\times 2.822$
GHz [36] and the detuning from resonance $\Delta$ as
$2B_{\text{rot}}/h+\Delta/(2\pi)$. Coupling the rotational states creates the
mixed dressed states $|{+}\rangle$ and $|{-}\rangle$. Rotationally excited
states with $m_{J}=0,\,1$, to which the microwave does not couple, remain as
spectator states $|{0}\rangle$. On resonance, i.e. at $\Delta=0$, the coupling
strength $\hbar\Omega\approx h\times 11$ MHz defines the splitting of
$|{+}\rangle$ and $|{-}\rangle$. We typically choose a blue detuning $\Delta$
on the order of $\Omega$ to prepare the molecules in the $|{+}\rangle$ state
(for details on the microwave transition, the coupling strength, and the
dressed-state preparation see Supplementary Information).

Molecules that interact in the upper dressed state via their induced rotating
dipoles are described by the collisional channel $|{++}\rangle$, shown in Fig.
1c and 1d. At long range, their time-averaged interaction energy is given by
[37]

$V_{\text{dd}}=-\frac{d_{0}^{2}}{4\pi\varepsilon_{0}}\frac{1-3\cos^{2}{\theta}}{12(1+(\Delta/\Omega)^{2})R^{3}},$
(1)

where $d_{0}\approx 2.7$ Debye is the intrinsic dipole moment of NaK, $R$ is
the intermolecular distance and $\theta$ denotes the angle between the
rotation axis of the induced dipole and the intermolecular axis. Molecules
thus acquire an effective dipole moment
$d_{\text{eff}}=d_{0}/\sqrt{12(1+(\Delta/\Omega)^{2})}$. Note that the sign of
the interaction energy is inverted compared to the interaction between two
regular dipoles with dipole moment $d_{\text{eff}}$.

At intermediate range, the dipole-dipole interaction dominates, so that the
molecules orient themselves with respect to the intermolecular axis, rather
than along the rotating electric field [38]. The reorientation is made
possible by contributions of the spectator states $|0\rangle$. Since the
molecules are prepared in the upper dressed state, they couple at this point
to the repulsive branch of the dipole-dipole interaction, which shields the
molecules from reaching short range regardless of their initial angle of
approach [39]. In fact, the remaining two-body loss is dominated by non-
adiabatic transitions to lower-lying field-dressed states, such as
$|{+}0\rangle$, which is accompanied by an energy release on the order of
$\hbar\Omega$ and suffices to eject molecules from the optical dipole trap.

Figure 2: Elastic and inelastic collisions. a, Rate coefficients for elastic
(orange) and inelastic (blue) scattering events. The solid and dashed lines
show coupled channel calculations for a thermal sample with $T=800$ nK and a
degenerate sample with $T=30\,\text{nK}=0.4\,T_{\text{F}}$, respectively. The
data points show the measurement results for $T=800$ nK. The shaded area
indicates the limit for measurements of the elastic collision rate imposed by
the hydrodynamic regime. On resonance, the Rabi frequency is $\Omega\approx
2\pi\times 11$ MHz. The triangular markers on the side indicate the calculated
inelastic rate coefficients in absence of a microwave field. The error bars
are the standard deviation from the fit to the differential equations. b,
Examples of cross-thermalization measurements at $\Delta=2\pi\times 30$ MHz
(bright) and $\Delta=2\pi\times 80$ MHz (dark). In the latter case,
$\beta_{\text{el}}$ is consistent with zero within the error bars. The error
bars are the standard error of the mean of 4 repetitions. The lines are fits
of a coupled differential equation system modelling the collision rates
(Supplementary Information). c, The ratio $\gamma$ of elastic to inelastic
collision rates based on the measurements and calculations presented in a.

The performance of evaporative cooling is ultimately limited by the ratio
$\gamma$ of elastic to inelastic two-body collision rates. We characterize the
inelastic collision rate coefficient $\beta_{\text{in}}$ by measuring the two-
body decay of the molecules in a thermal gas at temperatures $T$ around 800
nK. The molecules are initially formed from ultracold atoms by means of a
magnetic Feshbach resonance and subsequent stimulated Raman adiabatic passage
(STIRAP) to their absolute ground state [40] (see Supplementary Information
for more details on the experimental conditions and the preparation of the
molecular samples). For most measurements the initial average density $n_{0}$
is about $3.0\times 10^{11}\,\text{cm}^{-3}$. The two-body loss is relatively
high for hot and dense molecules, which helps us to distinguish it from one-
body loss caused by phase noise of the microwave (Supplementary Information).
We determine the elastic collision rate coefficient $\beta_{\text{el}}$ by
applying parametric heating to the molecular sample along the vertical
direction and measuring the cross-dimensional thermalization of effective
temperatures $T_{\text{h}}$ and $T_{\text{v}}$, as shown in Fig. 2b (see also
Supplementary Information). $T_{\text{h}}$ and $T_{\text{v}}$ are defined
along the horizontal ($x$ and $y$) and the vertical ($z$) directions,
respectively. The rate coefficients $\beta_{\text{el}}$ and
$\beta_{\text{in}}$ are extracted from the time evolution of the measured
molecule number $N$ and of the temperatures $T_{\text{h}}$ and $T_{\text{v}}$
by fitting a set of coupled differential equations that model the molecule
losses and the cross-dimensional rethermalization (Supplementary Information).
The results of the measurements are compared with coupled-channel calculations
for a thermal sample at $T=800$ nK and for a degenerate sample at
$30\,\text{nK}=0.4\,T_{\text{F}}$, as shown in Fig. 2. The calculations
account for a residual ellipticity of the microwave polarization
(Supplementary Information).

In absence of the microwave field, we find $\beta_{\text{in}}=7.7(5)\times
10^{-11}\,\text{cm}^{3}/\text{s}$ for $T=800$ nK, which is in reasonable
agreement with the calculated value of $4.9\times
10^{-11}\,\text{cm}^{3}/\text{s}$. The shielding is most efficient at
$\Delta=2\pi\times 8$ MHz, where $\beta_{\text{in}}$ drops to $6.2(4)\times
10^{-12}\,\text{cm}^{3}/\text{s}$, corresponding to an order of magnitude
suppression of the two-body losses, as illustrated in Fig. 2a.

For spin-polarized fermionic polar molecules, the elastic collision rate is
dominated by dipolar scattering. The corresponding scattering cross section
scales approximately as $d_{\text{eff}}^{4}$ (Supplementary Information) and
can therefore be tuned over multiple orders of magnitude with the detuning of
the microwave. In the regime of weak interactions, i.e., at large detunings,
the rethermalization rate is proportional to $\beta_{\text{el}}$. However, for
$\Delta\leq 2\pi\times 10$ MHz the elastic collisions become so frequent that
the mean free path of the molecules is less than the size of the molecular
cloud, even though here, we intentionally reduced the initial density to
$n_{0}=0.7\times 10^{11}\,\text{cm}^{-3}$. In this hydrodynamic regime, the
rethermalization rate is limited to about $\bar{\omega}/(2\pi)\approx 120$ Hz,
where $\bar{\omega}$ is the geometric mean trap frequency [41]. Consequently,
measured values of $\beta_{\text{el}}$ saturate near the so-called
hydrodynamic limit $N_{\text{col}}\bar{\omega}/(2\pi n_{0})$ where
$N_{\text{col}}\approx 2$ is the average number of collisions required for
rethermalization in our system (Supplementary Information). This also limits
the maximal measured value of $\gamma$ to 460(110), as illustrated in Fig. 2c.
Away from the hydrodynamic regime, we find excellent agreement between the
experimentally determined and the calculated values of $\beta_{\text{el}}$ and
$\gamma$ for $T=800$ nK. Our calculations show that $\gamma$ can exceed $1000$
for ideal values of $\Delta$. In the future, it should be possible to improve
the shielding and achieve $\gamma\approx 5000$ by optimizing the purity of the
microwave polarization (Supplementary Information).

## Evaporative cooling to a deeply degenerate Fermi gas of molecules

With $\gamma\gtrsim 500$ at the optimum shielding detuning $\Delta=2\pi\times
8$ MHz, the evaporative cooling of our molecular sample is straightforward. We
start with a low-entropy but non-thermalized sample of about $2.5\times
10^{4}$ molecules produced from a density-matched degenerate atomic mixture
[23]. The power of the two horizontally propagating laser beams, which hold
the molecules against gravity and thereby define the effective trap depth
$U_{\text{trap}}$, is lowered exponentially over the course of $150$ ms. The
hottest molecules can then escape the dipole trap in the direction of gravity.
The remaining molecules rethermalize via elastic dipolar collisions,
effectively reducing $T$ and $T/T_{\text{F}}$. During evaporation, the
calculated elastic collision rate is about 500 Hz, much higher than the trap
frequencies. Therefore, the rethermalization rate saturates to around
$\bar{\omega}/(2\pi)\approx 60$ Hz. To maintain a high rethermalization rate
while lowering the trap depth, we reinforce the horizontal confinement by
exponentially ramping up the power of an additional beam propagating along the
vertical direction.

Figure 3: Evaporation. a, $T/T_{\text{F}}$ and $T$ (inset) against the
remaining number of molecules $N$ after $150$ ms evaporative cooling for
various final trap depths. b, Plain evaporation during the hold time
$t_{\text{h}}$ after $150$ ms of forced evaporation. The dashed line indicates
the one-body decay and the inset sketches the evolution of the effective trap
depth $U_{\text{trap}}$. c, $T/T_{\text{F}}$ and $T$ (inset) during the hold
time $t_{\text{h}}$. The error bars of $N$ in a and b are the standard error
of the mean of 5–20 repetitions. The error bars of $T$ in a and c are the
standard deviation from the fit to the averaged images. d–f, Azimuthally
averaged optical density of the samples after $10$ ms time of flight. The
samples are prepared under the evaporation conditions I (d), II (e), and III
(f). The images of the samples are averages from $5$ (d), $20$ (e) or $6$
individual images (f). The black lines show polylogarithmic fit functions,
while the orange lines are fits of a Gaussian to the thermal wings of the
sample.

We characterize the evaporation by varying the final trap depth. Figure 3a
shows $T$ and $T/T_{\text{F}}$ against the number of remaining molecules $N$
after 150 ms forced evaporation. The values of $T$ and $T/T_{\text{F}}$ are
deduced from a polylogarithmic fit to the momentum distribution of the sample,
which is imaged after $10$ ms time of flight (Supplementary Information). If
the trap depth is not reduced, the interacting molecules will thermalize but
are not forced to evaporate. Initially the molecules exhibit a sloshing motion
in the trap due to photon-recoil transfer from the STIRAP pulses. This motion
damps out while the molecules thermalize, which effectively reduces the phase
space density of the sample. After $150$ ms holding at the initial trap depth,
we obtain $1.43(5)\times 10^{4}$ molecules at a temperature of $176(5)$ nK
with $T/T_{\text{F}}=1.00(3)$ (Fig. 3d). If we instead evaporate down to
$3.6(3)\times 10^{3}$ molecules by reducing the final trap depth to about
$k_{\text{B}}\times 250$ nK, where $k_{\text{B}}$ is the Boltzmann constant,
we reach a temperature of $38(2)$ nK with $T/T_{\text{F}}=0.47(2)$ (Fig. 3e).
However, the evaporation has not stopped at this point. We can hold the
molecular sample for an additional hold time $t_{\text{h}}$ in the trap to
make use of plain evaporation, as shown in Fig. 3b and 3c. The plain
evaporation lasts for about $100$ ms. Thereafter, the loss of molecules is
dominated by one-body loss caused by phase noise of the microwave, which
results in a $1/e$ lifetime of $590(100)$ ms (Supplementary Information). At
$t_{\text{h}}=150$ ms, we measure a record low temperature of $21(5)$ nK with
$T/T_{\text{F}}=0.36(9)$ (Fig. 3f).

We checked that the temperatures obtained from the Fermi–Dirac fit are
consistent with those deduced from a Gaussian fit to the thermal wing, in
which a possible influence of the dipolar interactions is relatively small
(Supplementary Information). Interestingly, the optical density in a small
region of the cloud center is higher than the Fermi–Dirac fit in the coldest
sample III. However, the low signal-to-noise ratio does not allow us to
conclusively establish whether this is due to imaging noise or potential
dipolar interaction effects. In the future, it will be interesting to
investigate the underlying mechanism with improved evaporation and detection.

## Discussion

Our highest measured value of $\gamma$ of about $500$ is a conservative
estimate due to the limitations of our measurement method in the hydrodynamic
regime. Our numerical models, which agree well with the data outside of the
hydrodynamic regime, predict a ratio of $\gamma\geq 1000$. This is almost a
factor of 100 higher than the ratio realized in previous experiments in 3D
[32] and almost an order of magnitude higher compared to experiments in 2D
[31]. Such optimal collisional parameters facilitate the efficient evaporation
of NaK molecules to below 0.4 $T_{\text{F}}$ compared to previous evaporation
results reaching 1.4 $T_{\text{F}}$ in 3D [32] and 0.6 $T_{\text{F}}$ in 2D
[31].

With the coldest samples realized in our experiment, the dipolar interaction
in the system corresponds to about $5$% of the Fermi energy. This is three
times higher than what was reached in degenerate Fermi gases of magnetic atoms
[42]. In the near future, intriguing dipolar many-body phenomena such as
modifications of collective excitation modes [43], distortion [42] or the
collapse [44] of the Fermi sea should be observable in suitable trap
geometries and with improved detection of the cloud expansion.

The reduced temperature $T/T_{\text{F}}$ reached here is only a factor of
about four higher than the predicted critical temperature of topological
$p$-wave superfluidity with strong dipolar interactions [1, 2, 3]. The
performance of the evaporation can be further improved by implementing the
following measures: first, the initial phase space density can be increased by
optimizing the STIRAP transfer. Second, lower phase noise and better
polarization purity of the microwave field should lead to reduced one- and
two-body losses during evaporation. Finally, new strategies of evaporation in
the hydrodynamic regime need to be explored to accelerate the thermalization
process [41]. With these upgrades, it should be possible to reach temperatures
below 0.1 $T_{\text{F}}$, where many intriguing quantum phases are expected
[1, 2, 3, 5, 4, 6, 7, 8]. Especially, fermionic polar molecules can pair up
and form a superfluid with anisotropic order parameter and even a
Bose–Einstein condensate of tetramers [1, 3]. Up to now, such scenarios have
rarely been theoretically investigated because it was believed that polar
molecules could not be sufficiently stable under conditions where both
attractive and repulsive interactions play an important role.

## Conclusion

In conclusion, we have demonstrated a general and efficient approach to
evaporatively cool ultracold polar molecules to deep quantum degeneracy in 3D
by dressing the molecules with a blue-detuned circularly polarized microwave,
achieving record low temperatures together with strong tunable dipolar
interactions. The simplicity of the technical setup makes our method directly
applicable in a wide range of ultracold molecule experiments. Our results
point to an exciting future of long-lived degenerate polar molecules for
investigating novel quantum many-body phases with long-range anisotropic
interactions and for other applications in quantum sciences.

## Acknowledgements

We thank Y. Bao, L. Anderegg, A. Pelster and A. Balaž, and T. Shi for
stimulating discussions, B. Braumandl for the simulation of the microwave
field, F. Deppe for lending the ultra low-noise microwave signal generator,
and T. Hilker for critical reading of the manuscript. We gratefully
acknowledge support from the Max Planck Society, the European Union (PASQuanS
Grant No. 817482) and the Deutsche Forschungsgemeinschaft under Germany’s
Excellence Strategy – EXC-2111 – 390814868 and under Grant No. FOR 2247. A.S.
acknowledges funding from the Max Planck Harvard Research Center for Quantum
Optics.

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## Supplementary Information

### Sample preparation

To create our molecular samples, we first prepare a density-matched double-
degenerate mixture of 23Na and 40K atoms. The atoms are subsequently
associated to weakly bound molecules by means of a magnetic Feshbach
resonance. Finally, the molecules are transferred to their absolute ground-
state via STIRAP. Details about the preparation process are described in Refs.
[23, 40]. At the beginning of the measurements described in the main text, the
molecules are trapped by the 1064-nm and the 1550-nm beam shown in Fig. 1a at
a dc magnetic field of $72.35$ G.

For the measurements of the collision rates, the microwave transition
strength, and to characterize the one-body loss, we work with thermal
molecules and sometimes reduce the molecule number to suppress interactions.
For the collision rate measurements, the trap frequencies are
$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(67,99,244)$ Hz. For the
evaporation, on the other hand, we start with near degenerate molecules at
$2\pi\times(45,67,157)$ Hz and end up, e.g., at $2\pi\times(52,72,157)$ Hz in
case I or at $2\pi\times(42,56,99)$ Hz in case II and III (see Fig. 3).

To measure the cross-dimensional thermalization, we heat the weakly bound
molecules along the vertical direction after we separated them from unbound
atoms and before STIRAP is applied. For this purpose, we use parametric
heating by modulating the intensity of the 1064-nm beam at twice the vertical
trap frequency.

### Microwave field generation

It is essential that the phase noise of the microwave source does not induce
transitions between the dressed states. We generate the microwave with a
vector signal generator (Keysight E8267D). The microwave passes through a
voltage controlled attenuator (General Microwave D1954) before it is amplified
with a $10$-W power amplifier (KUHNE electronic KU PA 510590 – 10 A). At $10$
MHz carrier offset we measure $-150\,\text{dBc}/\text{Hz}$ phase noise density
from the signal generator and no significant enhancement from the amplifier.
The microwave is emitted by a $5$-turn helical antenna (customized by
Causemann Flugmodellbau) whose top end is about $2.2$ cm away from the
molecular sample.

With the voltage controlled attenuator we can adiabatically prepare the
molecules in the dressed state by ramping the power attenuation linearly
within $100$ µs over a range of $65$ dB.

### Imaging and thermometry

To image the molecules, we transfer them back into the non-dressed absolute
ground-state by ramping down the microwave power. Subsequently, the dipole
traps are turned off and return STIRAP pulses are applied to bring the
molecules back into the weakly bound state. After time of flight, typically
$10$ ms, the atoms are dissociated by ramping the magnetic field back over the
Feshbach resonance. The magnetic field has to cross the Feshbach resonance
slowly to minimize the release energy. In the end, the dissociated molecules
are imaged via absorption imaging. We estimate that the derived temperature of
the molecular sample could be overestimated by about $7$ nK due to the
residual release energy. Note that the values of $T$ and $T/T_{\text{F}}$
reported in the main text do not account for the release energy.

To obtain the temperature of the molecular sample, we fit the absorption
images with the Fermi–Dirac distribution

$n_{\mathrm{FD}}(x,z)=n_{\mathrm{FD,0}}\,\mathrm{Li}_{2}\left[-\zeta\exp(-\frac{x^{2}}{2\sigma_{x}^{2}}-\frac{z^{2}}{2\sigma_{z}^{2}})\right],$
(S1)

where $n_{\text{FD},0}$ is the peak density, $\mathrm{Li}_{2}(x)$ is the
dilogarithmic function, $\zeta$ is the fugacity and $\sigma_{i={x,z}}$ are the
cloud widths in the $x$\- and $z$-direction. Given a cloud width $\sigma_{i}$,
we can calculate the temperature $T_{i}$ by

$\sigma_{i}=\frac{\sqrt{1+\omega_{i}^{2}t_{\text{TOF}}^{2}}}{\omega_{i}}\sqrt{\frac{k_{\text{B}}T_{i}}{m}},$
(S2)

where $\omega_{i}$ is the trapping frequency in the $i$-th direction,
$t_{\text{TOF}}$ is the time of flight and $m$ is the mass of the molecules.
The fugacity can be associated to the ratio of the temperature $T$ and the
Fermi temperature $T_{\text{F}}$ with the relation

$\left(\frac{T}{T_{\text{F}}}\right)^{3}=-\frac{1}{6\,\mathrm{Li}_{3}(-\zeta)},$
(S3)

where $\mathrm{Li}_{3}(x)$ is the trilogarithmic function.
$T_{\text{F}}=(6N)^{1/3}\hbar\bar{\omega}/k_{\text{B}}$ is given by the
molecule number $N$ and the geometric mean trap frequency
$\bar{\omega}=(\omega_{x}\omega_{y}\omega_{z})^{1/3}$. By rewriting $\zeta$
and fixing $T_{\text{F}}$, we are only left with the fitting parameters
$n_{\text{FD},0}$, $T_{x}$, and $T_{z}$. We note that the temperature in the
direction of the imaging beam $T_{y}$ is assumed to be equal to
$T_{x}=T_{\text{h}}$.

In addition, we independently determine the temperatures of the molecular
samples from the time-of-flight images by fitting the thermal wings of the
cloud to a Gaussian distribution

$n_{\mathrm{th}}(x,z)=n_{\text{th},0}\exp(-\frac{x^{2}}{2\sigma_{x}^{2}}-\frac{z^{2}}{2\sigma_{z}^{2}}),$
(S4)

where $n_{\text{th},0}$ is the peak density. Similar to Ref. [31], we first
fit a Gaussian distribution to the whole cloud. We then constrain the Gaussian
distribution to the thermal wings of the cloud by excluding a region of
$1.5\sigma$ around the center of the image. We find that by excluding
$1.5\sigma$, the ratio of signal to noise allows for the fit to converge for
all datasets in Fig. 3a.

The temperatures extracted from fitting the Fermi–Dirac distribution and
fitting the Gaussian distribution to the thermal wings are compared in Fig.
S1.

Fig. S1: Thermometry. Temperatures extracted from a Fermi–Dirac distribution
(dark) and a Gaussian distribution to the thermal wings (bright) against the
number of molecules $N$ after $150$ ms evaporative cooling for various final
trap depths. The temperatures are extracted from averaged images. The error
bars of $N$ are the standard error of the mean of 5–20 repetitions. The error
bars in $T$ are the standard deviation from the fit.

### Model for elastic and inelastic collisions

The elastic and inelastic collision rate coefficients $\beta_{\text{el}}$ and
$\beta_{\text{in}}$ are experimentally determined from the time evolution of
the measured molecule number $N$, the average temperature
$(2T_{\text{h}}+T_{\text{v}})/3$ and the differential temperature
$T_{\text{v}}-T_{\text{h}}$ by numerically solving the differential equations
[24, 32]

$\frac{dN}{dt}=\left(-K\frac{2T_{\text{h}}+T_{\text{v}}}{3}n-\Gamma_{1}\right)N,$
(S5)
$\frac{dT_{\text{h}}}{dt}=\frac{1}{12}KT_{\text{v}}T_{\text{h}}n+\frac{\Gamma_{\text{th}}}{3}(T_{\text{v}}-T_{\text{h}}),$
(S6)
$\frac{dT_{\text{v}}}{dt}=\frac{1}{12}K(2T_{\text{h}}-T_{\text{v}})T_{\text{v}}n-2\frac{\Gamma_{\text{th}}}{3}(T_{\text{v}}-T_{\text{h}}),$
(S7)

with the mean density

$n=\frac{N}{8\sqrt{\pi^{3}k_{\text{B}}^{3}T_{\text{h}}^{2}T_{\text{v}}/m^{3}\bar{\omega}^{6}}}.$
(S8)

Here, $K$ is the temperature-independent two-body loss coefficient, averaged
for simplicity over all collision angles, and

$\Gamma_{\text{th}}=\frac{n\sigma_{\text{el}}v}{N_{\text{col}}}$ (S9)

is the rethermalization rate with the elastic scattering cross section
$\sigma_{\text{el}}$ and the thermally averaged collision velocity

$v=\sqrt{16k_{\text{B}}(2T_{\text{h}}+T_{\text{v}})/(3\pi m)}.$ (S10)

The average number of collisions per rethermalization is taken from Ref. [45]
as

$N_{\text{col}}=\bar{\mathcal{N}}_{z}(\phi)=\frac{112}{45+4\cos(2\phi)-17\cos(4\phi)}$
(S11)

where $\phi$ is the tilt of the dipoles in the trap, which, in our case,
corresponds to the tilt of the microwave wave vector with respect to the dc
magnetic field. Following our characterization of the microwave polarization,
we assume $\bar{\mathcal{N}}_{z}(29^{\circ})=2.05$.

The anti-evaporation terms, i.e., the first terms in Eqs. S6 and S7, assume a
linear scaling of the two-body loss rate with temperature. Our calculations
predict that this assumption does not hold for small detunings
($\Delta<2\pi\times 20$ MHz), as illustrated in Fig. 2. Our results, however,
do not significantly change when we instead assume no temperature dependence
in this regime.

Finally, after determining $\sigma_{\text{el}}$ and $K$, the elastic and
inelastic collision rate coefficients

$\beta_{\text{el}}=\sigma_{\text{el}}v$ (S12)

and

$\beta_{\text{in}}=K(2T_{\text{h}}+T_{\text{v}})/3$ (S13)

are plotted in Fig. 2 assuming a fixed temperature
$T=T_{\text{h}}=T_{\text{v}}$.

Fig. S2: One- and two-body loss. a, Molecule loss with (dark) and without
(bright) microwave at high initial densities. The gray dashed line shows the
one-body contribution. The lines are fits to the differential-equation model.
b, Loss at low initial densities from which the one-body loss rate
$\Gamma_{1}$ is determined. The lines are exponential fit functions. In both
subfigures, bright blue data are taken without a microwave field, while dark
blue data are taken with a microwave field at $\Delta=2\pi\times 8$ MHz. The
error bars are the standard error of the mean of two (a) or three (b)
repetitions.

Example data of the loss measurements, performed to determine
$\beta_{\text{in}}$, are shown in Fig. S2a. At high densities, two-body loss
is the dominating contribution, while at low densities, the exponential shape
of the loss curve shows that one-body effects outweigh collisions. In order to
limit the number of free fit parameters, we determine $\Gamma_{1}=1.7(4)$ Hz
in independent measurements at low densities, as shown in Fig. S2b. To avoid
confounding effects from collisions, we reduce the initial molecule number to
about 2000 for these measurements. Under these conditions, the $1/e$ lifetime
is $570(100)$ ms without shielding. Turning on the shielding results in a
similar $1/e$ lifetime of about $590(100)$ ms. The lifetime reduces by a
factor two when a microwave source with a 3 dB higher phase-noise density is
used. This observation is consistent with the assumption that the noise power
spectral density around $2\pi\times 10$ MHz offset from the carrier is a
limiting factor of the one-body lifetime of the dressed state, even at a level
of $-150\,\text{dBc}/\text{Hz}$ [35].

Fig. S3: Microwave transitions. a, Zeeman diagram of the microwave transition
frequencies from the absolute ground state with energy $E_{0}$ to the
hyperfine manifold of the exited rotational state ($J=1$). The color scales
show the transition dipole moment (TDM) for the different microwave
polarizations, assuming that the ground state has the nuclear spin projections
$(m_{i,\text{Na}},m_{i,\text{K}})=(3/2,-4)$, which is a good approximation for
magnetic fields larger than $3$ G. The state energies and TDMs are calculated
with the code from Ref. [46] and the parameters from Ref. [36]. The circle
marks the $\sigma^{-}$ transition that is used for microwave shielding. The
diamond, triangle, and square mark the microwave transitions that are used to
probe the polarization of the microwave field. b, Example for off-resonant
Rabi oscillations at full microwave power with a detuning of $2\pi\times 2$
MHz from the $\sigma^{-}$ transition at $72.35$ G. The data show the number of
molecules in the absolute ground state $N_{0}$ and the error bars are the
standard error of the mean of two repetitions. The line is a sinusoidal fit
function that yields an effective Rabi frequency $2\pi\times 11.157(6)$ MHz.
c, Effective Rabi frequency $\Omega_{\text{eff}}$ at full microwave power
versus detuning from the $\sigma^{-}$ transition at $72.35$ G. d–f, Examples
of resonant Rabi oscillations at the $\sigma^{+}$ (d), $\pi$ (e), and
$\sigma^{-}$ (f) transitions indicated in a at $135$ G. The sinusoidal fit
functions yield Rabi frequencies of $2\pi\times 2.96(8)$ kHz, $2\pi\times
2.58(13)$ kHz, and $2\pi\times 9.9(4)$ kHz, respectively. The power of the
microwave is attenuated by $55$ dB (d) or $61$ dB (e and f).

### Microwave transitions

Figure 1 of the main text shows only the most essential parts of the energy
level structure. The full hyperfine structure of the excited rotational state
$J=1$ is presented in Fig. S3a. In contrast to the situation in CaF [35], NaK
has no fine structure in the electronic ground state. Consequently the
rotationally excited states that can couple to the absolute ground state,
which have $(m_{i,\text{Na}},m_{i,\text{K}})=(3/2,-4)$ character, are spread
over just a few hundred kilohertz, much less than the microwave detunings used
for shielding. Here, $m_{i,\text{Na}}$ and $m_{i,\text{K}}$ are the
projections of the nuclear spins of Na and K onto the magnetic field axis,
respectively. At $72.35$ G a $\sigma^{-}$-polarized microwave couples mainly
to the $|J=1,m_{J}=-1\rangle$ state with transition frequency $5.643\,4137$
GHz. In absence of electric fields, the transition dipole moment (TDM) is
already $0.96\,d_{0}/\sqrt{3}$. In a strong microwave field, the nuclear spin
projections are further purified bringing the TDM even closer to the maximum
value of $d_{0}/\sqrt{3}$.

Unfortunately, we cannot directly measure the coupling strength $\Omega$ at
full microwave power by driving resonant Rabi oscillations. Coupling to weaker
transitions would lead to beat signals with the main Rabi oscillations.
However, we can measure the effective Rabi frequency $\Omega_{\text{eff}}$ at
detunings that are large enough to suppress coupling to unwanted transitions.
For these measurements we temporarily switch off the optical dipole traps to
avoid ac Stark shifts from the trapping light. We then generate a rectangular
microwave pulse with a fast microwave switch that controls the input of our
microwave amplifier. The microwave pulse drives Rabi oscillations between the
rotational states, as shown in Fig. S3b. We ignore the oscillations during the
first $5$ µs of the microwave pulse, as the amplifier requires some time to
reach full output power. These measurements are performed at low molecule
density in order to suppress dephasing and inelastic collisions between the
molecules. From the measurements of $\Omega_{\text{eff}}$ shown in Fig. S3c we
deduce $\Omega\approx 2\pi\times 11$ MHz.

### Microwave polarization

It is crucial that the microwave field has a high degree of polarization
purity in order to achieve efficient shielding. Especially for small
detunings, unwanted polarization components can couple to excited states with
different $m_{J}$ character which significantly reduces the shielding effect
[39]. Fortunately, the strong ac electric field of the microwave redefines the
quantization axis of the molecules so that we only have to consider
$\sigma^{+}$ and $\sigma^{-}$ components in the microwave frame. We can use
different microwave transitions of the molecules to characterize the
polarization of the microwave field in situ. However, in order to resolve the
individual transitions, only weak microwave fields can be applied, i.e., the
microwave polarization can only be characterized in the frame of the dc
magnetic field. We probe the microwave field polarization at $135$ G, where we
can still stabilize the dc magnetic field and where the used transitions,
marked in Fig. S3a, are reasonably isolated. The measurements, shown in Fig.
S3(d–f), are performed similarly to the measurements of $\Omega_{\text{eff}}$,
described earlier. However, here we measure on resonance and the microwave
power is attenuated by $55$–$61$ dB. The microwave power has to be low enough
to avoid off-resonant coupling to neighbouring transitions but strong enough
to realize Rabi oscillations of at least $2\pi\times 2$ kHz, because we can
only turn off the dipole traps for about $1$ ms before we start losing
molecules. The TDMs of the selected $\sigma^{+}$, $\pi$, and $\sigma^{-}$
transitions are $0.875\,d_{0}/\sqrt{3}$, $0.789\,d_{0}/\sqrt{3}$, and
$0.989\,d_{0}/\sqrt{3}$, respectively. From the measured Rabi frequencies, the
relative microwave power, and the TDMs, we can determine the ratio of the
electric field amplitudes $E_{\sigma^{+}}/E_{\sigma^{-}}=0.169(8)$ and
$E_{\pi}/E_{\sigma^{-}}=0.462(30)$. Although we do not know the phase relation
between the measured ac electric field components in the frame of the dc
magnetic field, we can deduce that the wave vector of the microwave is tilted
somewhere between $21.5(12)^{\circ}$ and $29.0(16)^{\circ}$ with respect to
the magnetic field axis. In the microwave frame the ellipticity angle is then
given by the electric field amplitudes $E_{\sigma^{+}}^{\prime}$ and
$E_{\sigma^{-}}^{\prime}$ as
$\xi=\arctan(E_{\sigma^{+}}^{\prime}/E_{\sigma^{-}}^{\prime})$ and has a value
between $11.5(5)^{\circ}$ and $5.9(6)^{\circ}$. To calculate the potential
curves in Fig. 1 of the main text and the rate coefficients in Fig. 2a of the
main text we assume $\xi=6.2^{\circ}$.

Fig. S4: Scaling of the collision rates. a, Calculations of the elastic
(orange) and inelastic (blue) collision rate coefficients for various
microwave ellipticity angles $\xi$. Further simulation parameters are $T=800$
nK and $\Omega=2\pi\times 11$ MHz. The markers are calculations including
hyperfine interactions for $\xi=6.2^{\circ}$. b, The circles and the solid
line are the measured values and the calculations of the elastic rate
coefficients, as shown in Fig. 2 of the main text for $T=800$ nK. The dashed
line is a simplified calculation of the elastic scattering rate based on the
effective dipole moment $d_{\text{eff}}$. The error bars are the standard
deviation from the fit to the differential equations.

The ellipticity of the microwave polarization has a significant effect on the
inelastic collision rate and therefore on the shielding efficiency, as
illustrated in Fig. S4a. For large enough ellipticity even inelastic
scattering resonances can arise when the undesired circular-polarization
component contributes to the coupling at small detunings. In our setup the
purity of the circular polarization can partially be optimized by moving and
rotating a cylindrical metal sheet that surrounds the helical antenna.

### Coupled-channels calculations

We perform coupled-channels scattering calculations using the framework
developed in Refs. [34, 39, 47] and here we summarize numerical details of
these calculations.

The Hamiltonian describes the NaK molecules as rigid rotors with electric
dipole moments that interact with one another as well as with the microwave
electric field. Furthermore, the molecules have nuclear spins that couple with
one another and with a static magnetic field. The channel basis was truncated
by including only the lowest two rotational states $J=0,1$ and partial waves
$L=1,3,5$. We propagated the scattering wavefunctions from
$R_{\mathrm{min}}=30\,a_{0}$ to $R_{\mathrm{max}}=60\,000\,a_{0}$, imposing a
capture boundary condition at $R_{\mathrm{min}}$ and the usual scattering
boundary conditions at $R_{\mathrm{max}}$, from which the $S$-matrix and
collision cross sections are obtained. This short-range boundary results in
loss rates given by the universal loss model [48] in the absence of external
fields, which is in reasonable but not perfect agreement with experimental
loss rates, $4.9\times 10^{-11}\,\text{cm}^{3}/\text{s}$ versus $7.7(5)\times
10^{-11}\,\text{cm}^{3}/\text{s}$ at $T=800$ nK, respectively. We performed
scattering calculations for nine values of the collision energy spaced
logarithmically between $0.1\,k_{\text{B}}T$ and $10\,k_{\text{B}}T$, and
subsequently cross sections are multiplied by the velocity and averaged over
the Maxwell–Boltzmann or Fermi–Dirac distribution to obtain collision rates
compared to experiment in Fig. 2 of the main text.

Scattering rates presented in Fig. 2 of the main text were obtained neglecting
hyperfine interactions, and using the microwave polarization determined from
the experiment, which is elliptical and tilted with respect to the magnetic
field. We have also performed calculations including hyperfine interactions,
but their effect is small, as can be seen in Fig. S4a. Here we truncated the
hyperfine basis by including functions with $\Delta m_{i}=\pm 1$ only, i.e.,
functions that differed at most one quantum from the initial state.
Convergence tests with $\Delta m_{i}=\pm 2$ were also performed. This figure
also contains scattering rates for microwave polarizations of varying
ellipticity. In this case, the polarization lies in the plane perpendicular to
the magnetic field axis.

Potential energy curves shown in Fig. 1 of the main text were obtained by
diagonalizing the Hamiltonian excluding kinetic energy for fixed $\theta$, the
angle between the direction of approach of the colliding molecules and the
microwave propagation direction, and for fixed $R$, the distance between the
two molecules, as described in Ref. [39]. This omits the centrifugal kinetic
energy, which is not well defined for fixed $\theta$, and it has neglected
hyperfine interactions for clarity of Fig. 1 of the main text.

The elastic cross section can, to a reasonable degree, be approximated by
$(32\pi/15)(\mu d_{\text{eff}}^{2}/(\hbar^{2}4\pi\varepsilon_{0}))^{2}$ with
the reduced mass $\mu=m/2$ [49]. A comparison with our coupled-channel
calculation is shown in Fig. S4b.