Field-linked resonances of polar molecules

Abstract

Scattering resonances are an essential tool for controlling the interactions of ultracold atoms and molecules. However, conventional Feshbach scattering resonances¹, which have been extensively studied in various platforms^1-7, are not expected to exist in most ultracold polar molecules because of the fast loss that occurs when two molecules approach at a close distance^8-10. Here we demonstrate a new type of scattering resonance that is universal for a wide range of polar molecules. The so-called field-linked resonances^11-14 occur in the scattering of microwave-dressed molecules because of stable macroscopic tetramer states in the intermolecular potential. We identify two resonances between ultracold ground-state sodium-potassium molecules and use the microwave frequencies and polarizations to tune the inelastic collision rate by three orders of magnitude, from the unitary limit to well below the universal regime. The field-linked resonance provides a tuning knob to independently control the elastic contact interaction and the dipole-dipole interaction, which we observe as a modification in the thermalization rate. Our result provides a general strategy for resonant scattering between ultracold polar molecules, which paves the way for realizing dipolar superfluids¹⁵ and molecular supersolids¹⁶, as well as assembling ultracold polyatomic molecules.

Fig. 1. Interaction potentials and bound states of microwave-dressed ground-state molecules.

a–c, The cut-open three-dimensional surfaces illustrate the interaction potentials U(r) including the p-wave centrifugal potential between two molecules in the x–y plane for different ellipticity angles, ξ, of the field polarization: ξ = 0° (a), ξ = 12° (b) and ξ = 37° (c). Below, a projection of the same potential is shown. The interaction potential along the z direction is always repulsive (not shown). The microwave is on resonance (Δ = 0). The shaded areas in b and c show the radial wavefunction of the bound states. The insets visualize the rotating electric field vector E, and sketch the interaction between the rotating dipoles colliding along the x or y direction. The markers on the colour bar denote the potential depths for the three cases. d,e, Coupled-channel calculations of the energy of the bound states as a function of Δ for values of ξ = 19° (d) and ξ = 45° (e). In all panels the Rabi frequency is set to Ω = 2π × 10 MHz. Source data

Fig. 2. FL resonances.

a, Inelastic collision rate coefficient β_in between microwave-dressed NaK molecules as a function of the microwave detuning Δ for various microwave polarizations with the ellipticity angle ξ = 6(2)° (green), 19(2)° (blue) and 37(2)° (orange) at the Rabi frequency Ω ≈ 2π × 10 MHz. The solid lines show the corresponding theory calculations. The shaded regions show the calculations within the uncertainty of ξ. The grey dashed line denotes the theoretical universal value of β_in and the grey dotted line denotes the single-channel unitarity limit. The coloured error bars show the standard deviation of the fit results and the black error bar illustrates the exemplary common systematic uncertainty. b,c, Colour density map of the experiment data (b) and the theory calculation (c) of the inelastic rate coefficient as a function of microwave detuning and ellipticity. The triangles on the right axis of b mark ellipticity for the data shown in a. Source data

Fig. 3. Temperature dependence of the inelastic scattering.

a,b, Experimental results (a) and calculations (b) of the inelastic collision rate coefficient β_in as a function of the microwave detuning Δ at 700 nK (orange), 230 nK (blue) and 20 nK (purple). The coloured error bars show fitting errors, and the black error bar additionally contains the common systematic uncertainty. The solid lines are coupled-channel calculations and the dashed lines are the unitarity limit at the corresponding temperatures. The molecules are dressed by microwaves with ellipticity ξ = 19(2)° and Rabi frequency Ω ≈ 2π × 10 MHz. Source data

Fig. 4. Elastic scattering.

a, The grey data points show the thermalization rate Γ_th normalized by the mean in situ density n as a function of the microwave detuning Δ at an ellipticity angle ξ = 19(2)° and a Rabi frequency Ω ≈ 2π × 10 MHz. The temperature is 230 nK. The error bars are the standard error of the mean of 7 to 16 repetitions. For comparison, the solid blue line shows the corresponding theory calculation of the elastic collision rate coefficient β_el. The uncertainty of ξ is taken into account by the shaded area. The dashed line is the Born approximation of the background collision rate coefficient, which holds for detunings Δ ≳ 2π × 8 MHz (Methods). The insets show the normalized linear density $\tilde{n}$ along the lattice axis averaged over ten repetitions as a function of the molecule velocity v in the lattice direction for Δ = 2π × 10 MHz and 2π × 25 MHz. b, The coupled-channel calculations of the energy-dependent scattering length with the same microwave parameters and a fixed collision energy of k_BT, with T = 230 nK. The solid (dashed) lines are the real (imaginary) part of the scattering lengths in the channel p_x (orange), p_y (blue) and p_z (green). The dotted line is a fit of equation (3). The inset illustrates the partial waves of the scattering channels. k_B is the Boltzmann constant. Source data

Extended Data Fig. 1. Microwave setup.

a, Electronic setup used to generate and control the microwave field. Individually, the two antenna feeds produce mainly linear polarized fields parallel to each feed, respectively. The voltage-controlled attenuators are used to balance the fields and to adiabatically ramp the field intensity. The phase shifters allow to tune the polarization. b, Half-section view of the waveguide antenna that shows the inside of the waveguide and the transition from the coaxial cable (red jacket) to the feed.

Extended Data Fig. 2. Calibration of the field polarization.

a, The Rabi frequencies of rotational σ⁺ (green), π (blue), and σ⁻ (orange) transitions at low microwave power as a function of the phase shift ϕ between the antenna feeds. The error bars show the fitting error of the Rabi oscillations. The solid lines are fits to equation (4). b, Ellipticity of the microwave field in the frame of the microwave. The data points show the ellipticity angle ξ calulated from the data in a. The error bars denote the uncertainty of ξ that originates from the projection from the frame of the magnetic offset field to the frame of the microwave field due to the unknown phase relation between the three field components. The gray band is calculated from the fit functions shown in a and its width considers the uncertainty of the microwave orientation and the uncertainty of the offset phases ϕ₀. Source data

Extended Data Fig. 3. One- and two-body loss.

a, Example molecule loss at ξ ≈ 19° with detuning on resonance Δ = 2π × 10 MHz (bright) and away from resonance Δ = 2π × 25 MHz (dark). The lines are fits to the differential-equation model. b, Loss at low initial densities from which the one-body loss rate Γ₁ is determined. The line is an exponential fit function. The error bars show the standard deviation for repeated experiments. Source data

Extended Data Fig. 4. Thermalization model.

a, The blue line shows the diffraction pattern ñ(v) from a single experiment run at Δ = 2π × 10 MHz. The red line is a fit to equation (8). The gray curve describes the thermalized part c_thñ_th(v). b, The same measurement as in Fig. 4a but here a simplified model is used to determine β_el from the experimental data. The error bars show the standard error of the mean of 7–16 repetitions. Source data

Extended Data Fig. 5. Field-linked resonance near circular microwave polarization.

Coupled-channel calculations of the elastic (orange) and inelastic (blue) scattering rate coefficient at T = 20 nK, ξ = 1°, and Δ = 2π × 1 MHz. Source data