Evaporation of microwave-shielded polar molecules to quantum degeneracy

Ultracold polar molecules offer strong electric dipole moments and rich internal structure, which makes them ideal building blocks to explore exotic quantum matter1–9, implement quantum information schemes10–12 and test the fundamental symmetries of nature13. Realizing their full potential requires cooling interacting molecular gases deeply into the quantum-degenerate 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 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 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 many-body phenomena with strong dipolar interactions.

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 Supplementary Fig. 1a. In contrast to the situation in CaF [51], 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,Na , m i,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,Na and m i,K are the projections of the nuclear spins of Na and K onto the magnetic field axis, respectively. At 72.35 G a σ − -polarized microwave couples mainly to the |J = 1, m J = −1⟩ state with transition frequency 5.643 4137 GHz. In absence of electric fields, the transition dipole moment (TDM) is already 0.96 d 0 / √ 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 / √ 3.
Unfortunately, we cannot directly measure the coupling strength Ω 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 Ω 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 Supplementary Fig. 1b. 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 Ω eff shown in Supplementary Fig. 1c we deduce Ω ≈ 2π × 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 [52]. Fortunately, the strong ac electric field of the microwave redefines the quantization axis of the molecules so that we only have to consider σ + and σ − 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 Supplementary Fig. 1a, are reasonably isolated. The measurements, shown in Supplementary  Fig. 1(d-f), are performed similarly to the measurements of Ω 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 2π × 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 σ + , π, and σ − transitions are 0.875 d 0 / √ 3, 0.789 d 0 / √ 3, and 0.989 d 0 / √ 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 σ + /E σ − = 0.169(8) and E π /E σ − = 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) • and 29.0(16) • with respect to the magnetic field axis. In the microwave frame the ellipticity angle is then given by the electric field amplitudes E ′ σ + and E ′ σ − as ξ = arctan E ′ σ + /E ′ σ − and has a value between 11.5(5) • and 5.9(6) • . To calculate the potential curves in Fig. 1 of the main text and the rate co-  efficients in Fig. 2a of the main text we assume ξ = 6.2 • . The ellipticity of the microwave polarization has a significant effect on the inelastic collision rate and therefore on the shielding efficiency, as illustrated in Supplementary Fig. 2a. 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. [52][53][54] 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 min = 30 a 0 to R max = 60,000 a 0 , imposing a capture boundary condition at R min and the usual scattering boundary conditions at R 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 [55] in the absence of external fields, which is in reasonable but not perfect agreement with experimental loss rates, 4.9 × 10 −11 cm 3 /s versus 7.7(5) × 10 −11 cm 3 /s at T = 800 nK, respectively. We performed scattering calculations for nine values of the collision energy spaced logarithmically between 0.1 k B T and 10 k 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 Supplementary Fig. 2a. Here we truncated the hyperfine basis by including functions with ∆m i = ±1 only, i.e., functions that differed at most one quantum from the initial state. Convergence tests with ∆m i = ±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 θ, 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. [52]. This omits the centrifugal kinetic energy, which is not well defined for fixed θ, and it has neglected hyperfine interactions for clarity of Fig. 1 of the main text.