Abstract
Ultracold polar molecules offer strong electric dipole moments and rich internal structure, which makes them ideal building blocks to explore exotic quantum matter^{1,2,3,4,5,6,7,8,9}, implement quantum information schemes^{10,11,12} and test the fundamental symmetries of nature^{13}. Realizing their full potential requires cooling interacting molecular gases deeply into the quantumdegenerate regime. However, the intrinsically unstable collisions between molecules at short range have so far prevented direct cooling through elastic collisions to quantum degeneracy in three dimensions. Here we demonstrate evaporative cooling of a threedimensional 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 bluedetuned 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 elastictoinelastic collision ratio allows us to cool the molecular gas to 21 nanokelvin, corresponding to 0.36 times the Fermi temperature. Such cold and dense samples of polar molecules open the path to the exploration of manybody phenomena with strong dipolar interactions.
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Main
The field of ultracold polar molecules is on the verge of entering an exciting phase^{14,15}. The possibilities of quantum simulation that come within experimental reach range from pwave superfluids^{1,2,3,4}, supersolids^{5,6} and Wigner crystals^{7,8}, to novel spin systems and extended Hubbard models^{9}. Many of these proposals require a deeply degenerate quantum gas of molecules at tens of nanokelvin with strong dipolar interactions in three dimensions (3D).
Although noninteracting quantumdegenerate gases of polar molecules have been produced by assembling degenerate atomic mixtures^{16,17,18}, active cooling to the quantumdegenerate regime has remained challenging. The main hurdle towards efficient evaporative cooling of ultracold molecules is posed by their rapid collisional loss. In particular, when the molecules are polarized by an electric field in threedimensional traps they tend to collapse in attractive headtotail collisions^{19}. It turns out that even molecules that are nominally stable against chemical reactions^{20} undergo inelastic collisions when they reach short range. The exact nature of these loss processes is still not fully understood and remains under investigation^{21,22,23,24,25,26}.
Collisional loss in shortrange encounters between molecules can be suppressed by engineering repulsive interactions using external fields^{27,28,29,30,31,32}. Recently, a molecular gas of ^{40}K^{87}Rb was stabilized and evaporatively cooled to below the Fermi temperature by applying a strong d.c. electric field and confining the motion of the molecules to two dimensions (2D), which effectively prevented attractive collisions along the third dimension^{33}. For molecules in 3D, reduced collisional loss has been demonstrated by using a specific d.c. electric field^{34} to bring rotational states of colliding molecules in resonance with each other. However, attempts to produce a degenerate quantum gas of polar molecules through collisional cooling has so far remained unsuccessful in 3D owing to a low elastictoinelastic collision ratio as well as a relatively low initial phasespace density^{34,35}. Repulsive barriers between molecules can also be engineered by applying bluedetuned circularly polarized microwaves^{36,37}, which has been successfully used recently to reduce collisional loss between two calcium monofluoride molecules in an optical tweezer^{38}.
In this work, we induce fast elastic dipolar collisions in a quantum gas of fermionic ^{23}Na^{40}K 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 labframe 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 threedimensional neardegenerate sample of molecules to 21(5) nK, which corresponds to 36(9)% of the Fermi temperature T_{F}, deep in the quantumdegenerate regime. At such low temperatures, the inelastic collision rate between molecules becomes negligible, leading to a lifetime of the degenerate molecular sample of up to 0.6 s.
Microwave shielding and dipolar elastic collisions
To realize shielding in 3D, the two lowest rotational states of the molecules can be coupled via a bluedetuned circularly polarized microwave^{36}. The dressed state is a timedependent superposition of rotational states that features an induced rotating dipole moment, which follows the strong a.c. 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.
In our experiment, the microwave field is generated by a helical antenna, as illustrated in Fig. 1a. The antenna emits a mainly σ^{−}polarized microwave that couples the rotational ground state \( \,J=0,{m}_{J}=0,{\rm{\delta }}{N}_{{\rm{p}}}=0\rangle \) 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 \({\rm{\delta }}{N}_{{\rm{p}}}\) is the change in the number of photons in the microwave field. The frequency of the microwave is approximately given by the rotational constant \({B}_{{\rm{rot}}}=h\times 2.822\,{\rm{GHz}}\) (ref. ^{39}) and the detuning from resonance Δ as \(2{B}_{{\rm{rot}}}/h+\varDelta /(2{\rm{\pi }})\), where h is Planck's constant. Coupling the rotational states creates the dressed states^{40}
with the mixing angle \(\phi =\arctan ((\varDelta +\sqrt{{\varDelta }^{2}+{{\Omega }}^{2}})/\varOmega )\), where Ω is the Rabi frequency. Rotationally excited states with m_{J} = 0 and m_{J} = 1, to which the microwave does not couple, remain as spectator states \(0\rangle \). On resonance, that is, at Δ = 0, the coupling strength \(\hbar \varOmega \approx h\times 11\,{\rm{MHz}}\) (where ħ is the reduced Planck's constant) defines the splitting of \(\,+\,\rangle \) and \(\,\,\rangle \). We typically choose a blue detuning Δ on the order of Ω to prepare the molecules in the \(\,+\,\rangle \) state (for details on the microwave transition, the coupling strength and the dressedstate preparation, see Methods and Supplementary Information).
Molecules that interact in the upper dressed state through their induced rotating dipoles are described by the collisional channel \(\,++\,\rangle \), shown in Fig. 1c,d. At long range, their timeaveraged interaction energy is given by^{40}
where d_{0} ≈ 2.7 Debye is the intrinsic dipole moment of NaK, ε_{0} is the vacuum permittivity, R is the intermolecular distance and θ denotes the angle between the rotation axis of the induced dipole and the intermolecular axis. Molecules thus acquire an effective dipole moment \({d}_{{\rm{eff}}}={d}_{0}/\sqrt{12(1+{(\varDelta /\varOmega )}^{2})}\). It is noted that the sign of the interaction energy is inverted compared with the interaction between two regular dipoles with dipole moment d_{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^{41}. The reorientation is made possible by contributions of the spectator states \(0\rangle \). As 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^{42}. In fact, the remaining twobody loss is dominated by nonadiabatic transitions to lowerlying fielddressed states, such as \(\,+\,0\rangle \), which is accompanied by an energy release on the order of ħΩ and suffices to eject molecules from the optical dipole trap.
The performance of evaporative cooling is ultimately limited by the ratio γ of elastictoinelastic twobody collision rates. We characterize the inelastic collision rate coefficient β_{in} by measuring the twobody 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^{43} (see Methods 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 × 10^{11} cm^{−3}. The twobody loss is relatively high for hot and dense molecules, which helps us to distinguish it from onebody loss. The latter is mainly caused by the coupling to other dressed states owing to the phase noise of the microwave, which results in an exponential decay with a time constant of about 600 ms when the twobody loss is small (Methods). We determine the elastic collision rate coefficient β_{el} by applying parametric heating to the molecular sample along the vertical direction and measuring the crossdimensional thermalization of effective temperatures T_{h} and T_{v}, as shown in Fig. 2b (see also Methods). T_{h} and T_{v} are defined along the horizontal (x and y) and the vertical (z) directions, respectively. The rate coefficients β_{el} and β_{in} are extracted from the time evolution of the measured molecule number N and of the temperatures T_{h} and T_{v} by fitting a set of coupled differential equations that model the molecule losses and the crossdimensional rethermalization (Methods). The results of the measurements are compared with coupledchannel calculations for a thermal sample at T = 800 nK and for a degenerate sample at 30 nK = 0.4T_{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 β_{in} = 7.7(5) × 10^{−11} cm^{3} s^{−1} for T = 800 nK, which is in reasonable agreement with the calculated value of 4.9 × 10^{−11} cm^{3} s^{−1}. The shielding is most efficient at Δ = 2π × 8 MHz, where β_{in} drops to 6.2(4) × 10^{−12} cm^{3} s^{−1}, corresponding to an order of magnitude suppression of the twobody losses, as illustrated in Fig. 2a.
For spinpolarized fermionic polar molecules, the elastic collision rate is dominated by dipolar scattering. The corresponding scattering crosssection scales approximately as \({d}_{{\rm{eff}}}^{4}\) (Supplementary Information) and can therefore be tuned over multiple orders of magnitude with the detuning of the microwave. In the regime of weak interactions, that is, at large detunings, the rethermalization rate is proportional to β_{el}. However, for Δ ≤ 2π × 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 × 10^{11} cm^{−3}. In this hydrodynamic regime, the rethermalization rate is limited to about \(\bar{\omega }/(2{\rm{\pi }})\approx 120\,{\rm{Hz}}\), where \(\bar{\omega }\) is the geometric mean trap frequency^{44}. Consequently, measured values of β_{el} saturate near the socalled hydrodynamic limit \({N}_{{\rm{col}}}\bar{\omega }/(2{\rm{\pi }}{n}_{0})\) where N_{col} ≈ 2 is the average number of collisions required for rethermalization in our system (Methods). This also limits the maximum measured value of γ 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 β_{el} and γ for T = 800 nK. Our calculations show that γ can exceed 1,000 for ideal values of Δ. In the future, it should be possible to improve the shielding and achieve γ ≈ 5,000 by optimizing the purity of the microwave polarization (Supplementary Information).
Evaporative cooling to deep quantum degeneracy
With γ ≳ 500 at the optimum shielding detuning Δ = 2π × 8 MHz, the evaporative cooling of our molecular sample is straightforward. We start with a lowentropy but nonthermalized sample of about 2.5 × 10^{4} molecules produced from a densitymatched degenerate atomic mixture^{18}. The power of the two horizontally propagating laser beams, which hold the molecules against gravity and thereby define the effective trap depth U_{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 through elastic dipolar collisions, effectively reducing T and T/T_{F}. During evaporation, the calculated elastic collision rate is about 500 Hz, which is much higher than the trap frequencies. Therefore, the rethermalization rate saturates to around \(\bar{\omega }/(2{\rm{\pi }})\approx 60\,{\rm{Hz}}\). 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.
We characterize the evaporation by varying the final trap depth. Figure 3a shows T and T/T_{F} against the number of remaining molecules N after 150 ms of forced evaporation. The values of T and T/T_{F} are deduced from a polylogarithmic fit to the momentum distribution of the sample, which is imaged after 10 ms time of flight (Methods). 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 photonrecoil transfer from the STIRAP pulses. Damping of such collective excitations and particle loss substantially reduce the phasespace density of the sample. After 150 ms holding at the initial trap depth, we obtain 1.43(5) × 10^{4} molecules at a temperature of 176(5) nK with T/T_{F} = 1.00(3) (Fig. 3d). If we instead evaporate to 3.6(3) × 10^{3} molecules by reducing the final trap depth to about k_{B} × 250 nK, where k_{B} is the Boltzmann constant, we reach a temperature of 38(2) nK with T/T_{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_{h} in the trap to make use of plain evaporation, as shown in Fig. 3b,c. The plain evaporation lasts for about 100 ms. Thereafter, the molecule number exhibits an exponential decay with a 1/e lifetime of about 600 ms. At t_{h} = 150 ms, we measure a temperature of 21(5) nK with T/T_{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 (Methods). Interestingly, the optical density in a small region of the cloud centre is higher than the Fermi–Dirac fit in the coldest sample III. However, the low signaltonoise ratio does not allow us to conclusively establish whether this is owing to imaging noise or potential dipolar interaction effects. In the future, it will be interesting to investigate the underlying mechanism.
Discussion
As a result of the large effective dipole moment of the microwavedressed NaK molecules, our highest measured value of γ of about 500 is a factor of 40 larger than the ratio realized in previous experiments in 3D^{34} and almost twice as large compared with experiments in 2D^{33}. Such optimal collisional parameters facilitate the efficient evaporation of NaK molecules to below 0.4T_{F}.
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^{45}. In the near future, intriguing dipolar manybody phenomena such as modifications of collective excitation modes^{46}, distortion^{45} or the collapse^{47} of the Fermi sea should be observable in suitable trap geometries and with improved detection of the cloud expansion.
In ref. ^{18}, based on particle loss, we estimated the initial temperature of the nonthermalized groundstate molecules to be 0.52T_{F}. However, damping of collective excitations during the rethermalization in combination with more particle loss leads to a sample temperature of about 1T_{F} after 150 ms if we do not force evaporation. The performance of the evaporation can consequently be further improved by implementing the following measures: first, the initial phasespace 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 twobody losses during evaporation. Finally, new strategies of evaporation in the hydrodynamic regime need to be explored to accelerate the thermalization process^{44}. With these upgrades, it should be possible to reach temperatures below 0.1T_{F}, where many intriguing quantum phases are expected^{1,2,3,4,5,6,7,8,9}. In particular, fermionic polar molecules can pair up and form a superfluid with an anisotropic order parameter and even a Bose–Einstein condensate of tetramers^{1,2,4}. 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
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 bluedetuned circularly polarized microwave, achieving very low temperatures together with strong tunable dipolar interactions. The simplicity of the technical setup makes our method directly applicable in a wide range of ultracoldmolecule experiments. Our results point to an exciting future of longlived degenerate polar molecules for investigating quantum manybody phases with longrange anisotropic interactions and for other applications in quantum sciences.
Methods
Sample preparation
To create our molecular samples, we first prepared a densitymatched doubledegenerate mixture of ^{23}Na and ^{40}K atoms. The atoms were subsequently associated to weakly bound molecules by means of a magnetic Feshbach resonance. Finally, the molecules were transferred to their absolute ground state via STIRAP. Details about the preparation process are described in refs. ^{18,43}. At the beginning of the measurements described in the main text, the molecules were trapped by the 1,064nm and the 1,550nm beam shown in Fig. 1a at a d.c. magnetic field of 72.35 G.
For the measurements of the collision rates, the microwave transition strength and to characterize the onebody loss, we worked with thermal molecules and sometimes reduced the molecule number to suppress interactions. For the collision rate measurements, the trap frequencies were (ω_{x}, ω_{y}, ω_{z}) = 2π × (67, 99, 244) Hz. For the evaporation, however, we started with neardegenerate molecules at (ω_{x}, ω_{y}, ω_{z}) = 2π × (45, 67, 157) Hz and ended up, for example, at (ω_{x}, ω_{y}, ω_{z}) = 2π × (52, 72, 157) Hz in case I or at (ω_{x}, ω_{y}, ω_{z}) = 2π × (42, 56, 99) Hz in case II and case III (Fig. 3).
To measure the crossdimensional thermalization, we heated the weakly bound molecules along the vertical direction after we separated them from unbound atoms and before STIRAP was applied. For this purpose, we used parametric heating by modulating the intensity of the 1,064nm beam at twice the vertical trap frequency.
Microwavefield generation
It is essential that the phase noise of the microwave source does not induce transitions between the dressed states. We generated the microwave with a vector signal generator (Keysight E8267D). The microwave passes through a voltagecontrolled attenuator (General Microwave D1954) before it is amplified with a 10W power amplifier (KUHNE electronic KU PA 510590 – 10 A). At 10MHz carrier offset, we measured −150 dBc Hz^{−1} phasenoise density from the signal generator and no significant enhancement from the amplifier. The microwave is emitted by a fiveturn helical antenna (customized by Causemann Flugmodellbau) whose top end is about 2.2 cm away from the molecular sample.
With the voltagecontrolled 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 transferred them back into the nondressed absolute ground state by ramping down the microwave power. Subsequently, the dipole traps were turned off and return STIRAP pulses were applied to bring the molecules back into the weakly bound state. After time of flight, typically 10 ms, the atoms were 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 were imaged by absorption imaging. We estimated that the derived temperature of the molecular sample could be overestimated by about 7 nK owing to the residual release energy. It is noted that the values of T and T/T_{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
where n_{FD,0} is the peak density, Li_{2}(x) is the dilogarithmic function, ζ is the fugacity and σ_{i=x,z} are the cloud widths in the x and z directions. Given a cloud width σ_{i}, we can calculate the temperature T_{i} by
where ω_{i} is the trapping frequency in the ith direction, t_{TOF} is the time of flight and m is the mass of the molecules. The fugacity can be associated with the ratio of the temperature T and the Fermi temperature T_{F} with the relation
where Li_{3}(x) is the trilogarithmic function. \({T}_{{\rm{F}}}={(6N)}^{1/3}\hbar \bar{\omega }/{k}_{{\rm{B}}}\) 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 ζ and fixing T_{F}, we are left with only the fitting parameters n_{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_{h}.
In addition, we independently determine the temperatures of the molecular samples from the timeofflight images by fitting the thermal wings of the cloud to a Gaussian distribution
where n_{th,0} is the peak density. Similar to ref. ^{33}, 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 1.5σ around the centre of the image. We find that by excluding 1.5σ, 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 Extended Data Fig. 1.
Model for elastic and inelastic collisions
The elastic and inelastic collision rate coefficients β_{el} and β_{in} are experimentally determined from the time evolution of the measured molecule number N, the average temperature (2T_{h} + T_{v})/3 and the differential temperature T_{v} − T_{h} by numerically solving the differential equations^{19,34}
with the mean density
Here, K is the temperatureindependent twobody loss coefficient, averaged for simplicity over all collision angles, and
is the rethermalization rate with the elastic scattering crosssection σ_{el} and the thermally averaged collision velocity
The average number of elastic collisions per rethermalization is taken from ref. ^{48} as
where ϕ 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 d.c. magnetic field. Following our characterization of the microwave polarization, we assume \({\bar{{\mathscr{N}}}}_{z}({29}^{\circ })=2.05\).
The antievaporation terms, that is, the first terms in equations (8) and (9), assume a linear scaling of the twobody loss rate with temperature. Our calculations predict that this assumption does not hold for small detunings (Δ < 2π × 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 σ_{el} and K, the elastic and inelastic collision rate coefficients
and
are plotted in Fig. 2 assuming a fixed temperature T = T_{h} = T_{v}.
Example data of the loss measurements, performed to determine β_{in}, are shown in Extended Data Fig. 2a. At high densities, twobody loss is the dominant contribution, whereas at low densities, the exponential shape of the loss curve shows that onebody effects outweigh inelastic collisions. To limit the number of freefit parameters, we determine Γ_{1} = 1.7(4) Hz in independent measurements at low densities, as shown in Extended Data Fig. 2b. To suppress confounding effects from inelastic collisions, we reduce the initial molecule number to about 2,000 for these measurements. Under these conditions, the 1/e lifetime is 570(100) ms without shielding, which is still mostly limited by residual twobody collisions. Turning on the shielding results in a similar 1/e lifetime of about 590(100) ms. The lifetime reduces to 300(50) ms when a microwave source with a 3 dB higher phasenoise density (Rohde & Schwarz SMF100A) is used. If we isolatethe molecules by loading them into a threedimensional optical lattice, we measure a lifetime of 8.0(1.2) s in absence of a microwave field, as shown in Extended Data Fig. 2c. Turning on the microwave field (using the Rohde & Schwarz SMF100A) results in a fast exponential decay to about half of the initially detected molecules, followed by a slow exponential decay. Assuming that the particles are isolated on individual lattice sites and that the faster decay is a result of mixing of two dressed states by phase noise of the microwave, we fit the data with the function
where N_{0} is the initial number of molecules. We find a onebody loss time τ_{0} = 4.4(1.4) s and a statemixing time τ_{MW} = 210(90) ms, which is in reasonable agreement with the lifetime measurements in the bulk. The slightly faster decay might be caused by molecules in higher bands, leading to residual collisions in the lattice, which are not accounted for in equation (16). The scaling of the lifetime with the microwave phase noise in the bulk and the measurements in the lattice indicate that Γ_{1} is currently limited by the noise power spectral density in the dressedstate transition (that is, at around 2π × 10 MHz offset from the carrier), even at a level of −150 dBc Hz^{−1} (ref. ^{38}). In addition, we find that the 1,550nm light, which is used to trap the molecules in the bulk (Fig. 1a), contributes with about 0.5 Hz to the onebody decay. The underlying loss mechanism is under investigation.
Data availability
The experimental data that support the findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.
Code availability
All relevant codes are available from the corresponding author upon reasonable request.
References
You, L. & Marinescu, M. Prospects for pwave paired Bardeen–Cooper–Schrieffer states of fermionic atoms. Phys. Rev. A 60, 2324–2329 (1999).
Baranov, M. A., Mar’enko, M. S., Rychkov, V. S. & Shlyapnikov, G. V. Superfluid pairing in a polarized dipolar Fermi gas. Phys. Rev. A 66, 013606 (2002).
Cooper, N. R. & Shlyapnikov, G. V. Stable topological superfluid phase of ultracold polar fermionic molecules. Phys. Rev. Lett. 103, 155302 (2009).
Shi, T., Zhang, J.N., Sun, C.P. & Yi, S. Singlet and triplet Bardeen–Cooper–Schrieffer pairs in a gas of twospecies fermionic polar molecules. Phys. Rev. A 82, 033623 (2010).
Wu, Z., Block, J. K. & Bruun, G. M. Liquid crystal phases of twodimensional dipolar gases and Berezinskii–Kosterlitz–Thouless melting. Sci. Rep. 6, 19038 (2016).
Schmidt, M., Lassablière, L., Quéméner, G. & Langen, T. Selfbound dipolar droplets and supersolids in molecular Bose–Einstein condensates. Phys. Rev. Res. 4, 013235 (2022).
Bruun, G. M. & Taylor, E. Quantum phases of a twodimensional dipolar Fermi gas. Phys. Rev. Lett. 101, 245301 (2008).
Matveeva, N. & Giorgini, S. Liquid and crystal phases of dipolar fermions in two dimensions. Phys. Rev. Lett. 109, 200401 (2012).
Baranov, M. A., Dalmonte, M., Pupillo, G. & Zoller, P. Condensed matter theory of dipolar quantum gases. Chem. Rev. 112, 5012–5061 (2012).
DeMille, D. Quantum computation with trapped polar molecules. Phys. Rev. Lett. 88, 067901 (2002).
Yelin, S. F., Kirby, K. & Côté, R. Schemes for robust quantum computation with polar molecules. Phys. Rev. A 74, 050301 (2006).
Ni, K.K., Rosenband, T. & Grimes, D. D. Dipolar exchange quantum logic gate with polar molecules. Chem. Sci. 9, 6830–6838 (2018).
Safronova, M. S. et al. Search for new physics with atoms and molecules. Rev. Mod. Phys. 90, 025008 (2018).
Carr, L. D., DeMille, D., Krems, R. V. & Ye, J. Cold and ultracold molecules: science, technology and applications. New J. Phys. 11, 055049 (2009).
Bohn, J. L., Rey, A. M. & Ye, J. Cold molecules: progress in quantum engineering of chemistry and quantum matter. Science 357, 1002–1010 (2017).
De Marco, L. et al. A degenerate Fermi gas of polar molecules. Science 363, 853–856 (2019).
Tobias, W. G. et al. Thermalization and subPoissonian density fluctuations in a degenerate molecular Fermi gas. Phys. Rev. Lett. 124, 033401 (2020).
Duda, M. et al. Transition from a polaronic condensate to a degenerate Fermi gas of heteronuclear molecules. Preprint at https://arxiv.org/abs/2111.04301 (2021).
Ni, K.K. et al. Dipolar collisions of polar molecules in the quantum regime. Nature 464, 1324–1328 (2010).
Żuchowski, P. S. & Hutson, J. M. Reactions of ultracold alkalimetal dimers. Phys. Rev. A 81, 060703 (2010).
Mayle, M., Quéméner, G., Ruzic, B. P. & Bohn, J. L. Scattering of ultracold molecules in the highly resonant regime. Phys. Rev. A 87, 012709 (2013).
Christianen, A., Zwierlein, M. W., Groenenboom, G. C. & Karman, T. Photoinduced twobody loss of ultracold molecules. Phys. Rev. Lett. 123, 123402 (2019).
Gregory, P. D., Blackmore, J. A., Bromley, S. L. & Cornish, S. L. Loss of ultracold ^{87}Rb^{133}Cs molecules via optical excitation of longlived twobody collision complexes. Phys. Rev. Lett. 124, 163402 (2020).
Liu, Y. et al. Photoexcitation of longlived transient intermediates in ultracold reactions. Nat. Phys. 16, 1132–1136 (2020).
Bause, R. et al. Collisions of ultracold molecules in bright and dark optical dipole traps. Phys. Rev. Res. 3, 033013 (2021).
Gersema, P. et al. Probing photoinduced twobody loss of ultracold nonreactive bosonic ^{23}Na^{87}Rb and ^{23}Na^{39}K molecules. Phys. Rev. Lett. 127, 163401 (2021).
Avdeenkov, A. V., Kajita, M. & Bohn, J. L. Suppression of inelastic collisions of polar ^{1}Σ state molecules in an electrostatic field. Phys. Rev. A 73, 022707 (2006).
Gorshkov, A. V. et al. Suppression of inelastic collisions between polar molecules with a repulsive shield. Phys. Rev. Lett. 101, 073201 (2008).
Micheli, A. et al. Universal rates for reactive ultracold polar molecules in reduced dimensions. Phys. Rev. Lett. 105, 073202 (2010).
GonzálezMartínez, M. L., Bohn, J. L. & Quéméner, G. Adimensional theory of shielding in ultracold collisions of dipolar rotors. Phys. Rev. A 96, 032718 (2017).
Xie, T. et al. Optical shielding of destructive chemical reactions between ultracold groundstate NaRb molecules. Phys. Rev. Lett. 125, 153202 (2020).
Matsuda, K. et al. Resonant collisional shielding of reactive molecules using electric fields. Science 370, 1324–1327 (2020).
Valtolina, G. et al. Dipolar evaporation of reactive molecules to below the Fermi temperature. Nature 588, 239–243 (2020).
Li, J.R. et al. Tuning of dipolar interactions and evaporative cooling in a threedimensional molecular quantum gas. Nat. Phys. 17, 1144–1148 (2021).
Son, H., Park, J. J., Ketterle, W. & Jamison, A. O. Collisional cooling of ultracold molecules. Nature 580, 197–200 (2020).
Karman, T. & Hutson, J. M. Microwave shielding of ultracold polar molecules. Phys. Rev. Lett. 121, 163401 (2018).
Lassablière, L. & Quéméner, G. Controlling the scattering length of ultracold dipolar molecules. Phys. Rev. Lett. 121, 163402 (2018).
Anderegg, L. et al. Observation of microwave shielding of ultracold molecules. Science 373, 779–782 (2021).
Will, S. A., Park, J. W., Yan, Z. Z., Loh, H. & Zwierlein, M. W. Coherent microwave control of ultracold ^{23}Na^{40}K molecules. Phys. Rev. Lett. 116, 225306 (2016).
Karman, T., Yan, Z. Z. & Zwierlein, M. Resonant and firstorder dipolar interactions between ultracold ^{1}Σ molecules in static and microwave electric fields. Phys. Rev. A 105, 013321 (2022).
Yan, Z. Z. et al. Resonant dipolar collisions of ultracold molecules induced by microwave dressing. Phys. Rev. Lett. 125, 063401 (2020).
Karman, T. & Hutson, J. M. Microwave shielding of ultracold polar molecules with imperfectly circular polarization. Phys. Rev. A 100, 052704 (2019).
Bause, R. et al. Efficient conversion of closedchanneldominated Feshbach molecules of ^{23}Na^{40}K to their absolute ground state. Phys. Rev. A 104, 043321 (2021).
Ma, Z.Y., Thomas, A. M., Foot, C. J. & Cornish, S. L. The evaporative cooling of a gas of caesium atoms in the hydrodynamic regime. J. Phys. B 36, 3533–3540 (2003).
Aikawa, K. et al. Observation of Fermi surface deformation in a dipolar quantum gas. Science 345, 1484–1487 (2014).
Wächtler, F., Lima, A. R. P. & Pelster, A. Lowlying excitation modes of trapped dipolar Fermi gases: from the collisionless to the hydrodynamic regime. Phys. Rev. A 96, 043608 (2017).
Veljić, V., Pelster, A. & Balaž, A. Stability of quantum degenerate Fermi gases of tilted polar molecules. Phys. Rev. Res. 1, 012009 (2019).
Wang, R. R. W. & Bohn, J. L. Anisotropic thermalization of dilute dipolar gases. Phys. Rev. A 103, 063320 (2021).
Acknowledgements
We thank Y. Bao, L. Anderegg, A. Pelster, A. Balaž and T. Shi for discussions; B. Braumandl for the simulation of the microwave field; F. Deppe and B. Wang for lending the ultralownoise microwave signal generators; and T. Hilker for reading of the manuscript. We acknowledge support from the Max Planck Society, the European Union (PASQuanS grant number 817482) and the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy – EXC2111 – 390814868 and under grant number FOR 2247. A.S. acknowledges funding from the Max Planck Harvard Research Center for Quantum Optics.
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All authors contributed substantially to the work presented in this manuscript. A.S., R.B. and X.Y.C. carried out the experiments and improved the experimental setup. A.S., R.B. and M.D. analysed the data. T.K. performed the theoretical calculations. I.B. and X.Y.L. supervised the study. All authors worked on the interpretation of the data and contributed to the final manuscript.
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Extended data figures and tables
Extended Data Fig. 1 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.
Extended Data Fig. 2 One and twobody loss.
The bright blue data are taken without a microwave field, while the dark blue and dark green data are taken at Δ = 2π × 8 MHz using a Keysight E8267D and a Rohde & Schwarz SMF100A, respectively. a, Molecule loss at high initial densities. The grey dashed line shows the onebody contribution. The lines are fits to the differentialequation model. b, Loss at low initial densities. The onebody loss rate Γ_{1} is determined from the measurement with shielding using the Keysight E8267D (dark blue). The lines are exponential fit functions. c, Loss in a 3D optical lattice. The data without (with) microwave are fitted using an exponential (a doubleexponential) fit function. The error bars are the standard error of the mean of two (a) or three (b) repetitions.
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Supplementary Information
This file contains Supplementary Methods comprising the characterization of the microwave Rabi frequency, the characterization of the microwave polarization, the microwave transitions used for these characterizations and the coupledchannel calculations. Included are Supplementary Figs. 1 and 2 with legends.
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Schindewolf, A., Bause, R., Chen, XY. et al. Evaporation of microwaveshielded polar molecules to quantum degeneracy. Nature 607, 677–681 (2022). https://doi.org/10.1038/s41586022049000
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DOI: https://doi.org/10.1038/s41586022049000
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