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# A Feshbach resonance in collisions between triplet ground-state molecules

## Abstract

Collisional resonances are important tools that have been used to modify interactions in ultracold gases, for realizing previously unknown Hamiltonians in quantum simulations1, for creating molecules from atomic gases2 and for controlling chemical reactions. So far, such resonances have been observed for atom–atom collisions, atom–molecule collisions3,4,5,6,7 and collisions between Feshbach molecules, which are very weakly bound8,9,10. Whether such resonances exist for ultracold ground-state molecules has been debated owing to the possibly high density of states and/or rapid decay of the resonant complex11,12,13,14,15. Here we report a very pronounced and narrow (25 mG) Feshbach resonance in collisions between two triplet ground-state NaLi molecules. This molecular Feshbach resonance has two special characteristics. First, the collisional loss rate is enhanced by more than two orders of magnitude above the background loss rate, which is saturated at the p-wave universal value, owing to strong chemical reactivity. Second, the resonance is located at a magnetic field where two open channels become nearly degenerate. This implies that the intermediate complex predominantly decays to the second open channel. We describe the resonant loss feature using a model with coupled modes that is analogous to a Fabry–Pérot cavity. Our observations provide strong evidence for the existence of long-lived coherent intermediate complexes even in systems without reaction barriers and open up the possibility of coherent control of chemical reactions.

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## Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.

## Code availability

The codes used to generate the results are available from the corresponding author on reasonable request.

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## Acknowledgements

We thank J. Bohn for valuable discussions. We acknowledge support from the National Science Foundation (NSF) through the Center for Ultracold Atoms and grant no. 1506369 and from the Air Force Office of Scientific Research (MURI, grant no. FA9550-21-1-0069). Some of the analysis was performed by W.K. at the Aspen Center for Physics, which is supported by NSF grant PHY-1607611. J.J.P. acknowledges further support from the Samsung Scholarship. T.V.T. gratefully acknowledges support from the NSF CAREER award no. 2045681.

## Author information

Authors

### Contributions

J.J.P. carried out the experimental work. All authors contributed to the development of models, data analysis and writing the manuscript.

### Corresponding author

Correspondence to Juliana J. Park.

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The authors declare no competing interests.

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## Extended data figures and tables

### Extended Data Fig. 1 Density and temperature-dependent loss rate.

a, Density dependency of molecular decay rate at 334.92 G. The initial decay rates are plotted as a function of initial molecule mean density. The green, blue and red dashed lines show the behaviour expected for single-molecule decay, two-body and three-body collisions, respectively. b, Threshold laws of molecule–molecule collisions. Initial rate coefficients are plotted as a function of the initial temperature of NaLi molecules. Blue data points are measurements near the centre of the resonance and red data points are measurements away from resonance near 745 G. The lines show the linear dependence expected for p-wave collisions. Data points were acquired from five to eight different hold times at each bias field; three to six measurements at a given hold time were averaged. Error bars represent one standard deviation of a fitted decay parameter.

### Extended Data Fig. 2 Ground-state hyperfine structure in an external magnetic field.

The dashed vertical blue line indicates the position of the Feshbach resonance (about 334.92 G). The subplot shows the Zeeman energies of NaLi hyperfine states from 0 to 1,000 G, whereas the main plot is zoomed into where there are nine near-degenerate hyperfine states (between 300 and 400 G).

### Extended Data Fig. 3 Ground-state hyperfine structure near 334.92 G.

State a in red is the lower stretched hyperfine state of NaLi molecules. States b1, b2 and b3 in blue are other hyperfine states that are energetically close to state a near 334.92 G.

### Extended Data Fig. 4 Radial dependence of NaLi–NaLi interactions.

Radial dependence of the dipole–dipole, dipole–quadrupole and quadrupole–quadrupole interactions of NaLi(a3Σ+) molecules. The p-wave centrifugal barrier is also shown (dashed line). The upper and lower bounds on the experimental collision energies (4.2 μK and 1.8 μK) are marked by green horizontal lines. The turning points at the centrifugal barrier for these collision energies are Rb = 89.3 and 136.4 nm, respectively.

### Extended Data Fig. 5 Matrix elements of NaLi–NaLi interactions.

Matrix elements of the NaLi–NaLi interaction at R = 100 nm as a function of the channel index labelling the basis states |γAγBlmlη. The initial channel is |aa, l = 1, ml = 0 and the total angular momentum projection Mtot = −7. The channel index labels closed channels, in which one or both NaLi molecules are in their N ≥ 1 excited rotational states. Only the matrix elements with the absolute magnitude exceeding 1 Hz are plotted. The magnetic field B = 333 G is tuned near the crossing between the |a and |b1 hyperfine-Zeeman levels. Inset, histogram of direct coupling matrix elements between the incident channel and lower-lying open channels, in which both NaLi molecules are in the ground N = 0 rotational states.

### Extended Data Fig. 6 Degeneracy-induced resonance model.

a, Schematic of the resonance model with two open channels and a p-wave bound state trapped behind a centrifugal barrier. b, Inelastic rate $${g}_{2}\left(\widetilde{\Delta }\right)$$ (in arbitrary units) plotted as a function of $$\widetilde{\Delta }/{\gamma }_{1}$$ for the different values of detuning from resonance normalized by γ1, δE/γ1. Note that for $$\widetilde{\Delta } < 0$$, the channel |2 becomes closed and thus $${g}_{2}\left(\widetilde{\Delta }\right)=0$$.

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Park, J.J., Lu, YK., Jamison, A.O. et al. A Feshbach resonance in collisions between triplet ground-state molecules. Nature 614, 54–58 (2023). https://doi.org/10.1038/s41586-022-05635-8

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• DOI: https://doi.org/10.1038/s41586-022-05635-8