Residual two-body loss and the road from microwave-shielded degeneracy to topological $p$-wave superfluidity in fermionic polar-molecule gases

The 2022 demonstration of evaporative cooling of a microwave-shielded gas of fermionic NaK molecules to $0.36\,T_{F}$ posed two related open problems: understand and suppress the residual short-range two-body loss that shielding mitigates but does not eliminate, and extend evaporation to $T\lesssim0.1\,T_{F}$ with strong tunable dipolar interactions, where dipolar many-body phases — above all topological $p$-wave superfluidity — are predicted. Since 2022 the enabling-technology threads have been largely resolved (ultralow-phase-noise, polarization-pure microwave apparatus; double microwave shielding; molecular Bose–Einstein condensates of NaCs and NaRb; Fermi-surface deformation observed at $0.23\,T_{F}$), but the core survives: the microscopic origin of universal complex-mediated short-range loss remains unexplained, no fermionic molecular gas has reached $T\lesssim0.1\,T_{F}$ (best: $0.23\,T_{F}$ NaK, $0.25\,T_{F}$ KRb, now Pauli-blocking-limited), and topological $p$-wave superfluidity of dipolar fermions remains unobserved.

Background

Ultracold polar molecules, with permanent electric dipole moments of order a Debye, interact through strong, tunable, anisotropic dipole–dipole forces — a long-standing proposed platform for quantum simulation, quantum computation, and exotic superfluidity. A key obstacle is that when two ground-state bialkali molecules collide and reach short intermolecular distance, they are almost always lost from the trap, even for species that are nominally non-reactive: the two molecules form a transient collision complex (a 'sticky collision') whose dense manifold of resonant states makes close-coupling scattering calculations intractable, and the complex mediates inelastic loss. The measured lifetimes of these complexes exceed statistical (RRKM-type) predictions by orders of magnitude, and this 'sticky collision' puzzle is still unresolved. Microwave shielding circumvents the problem without solving it: a blue-detuned circularly polarized microwave field couples the rotational ground state to an excited rotational state and engineers a repulsive long-range interaction barrier, so that pairs of molecules almost never reach short range. The same dressing induces a large effective dipole moment, so elastic (heat-transferring) collisions become very fast. In 2022, Schindewolf et al. (Nature 607, 677 (2022)) used microwave shielding to evaporatively cool a three-dimensional gas of fermionic ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$ molecules to 21 nK, i.e. $T/T_{F}=0.36(9)$, where $T_{F}$ is the Fermi temperature (the temperature scale below which quantum statistics dominate a Fermi gas), with a measured elastic-to-inelastic collision ratio $\gamma=460(110)$. The paper itself identified what remains open: (i) the microscopic origin of short-range two-body loss is 'not fully understood'; (ii) one-body loss from microwave phase noise (measured $1/e$ lifetime 590(100) ms) and residual polarization ellipticity (about 6° in their setup) cap performance, while theory predicts $\gamma\gtrsim1000$ and up to $\approx5000$ for ideal polarization; (iii) when elastic collisions become so frequent that the mean free path is shorter than the cloud, rethermalization saturates at the hydrodynamic limit (about $\bar{\omega}/2\pi\approx120$ Hz for their trap, with $\bar{\omega}$ the geometric mean trap frequency), so evaporation protocols must be redesigned for this regime; (iv) the coldest cloud showed a small central optical-density excess relative to a Fermi–Dirac fit, which could not be attributed unambiguously to imaging noise or to dipolar interaction effects; and (v) many-body phases — most notably a topological $p$-wave superfluid, in which identical fermions pair via the anisotropic dipolar attraction and the resulting order parameter hosts topological (Majorana-like) excitations, and a Bose–Einstein condensate (BEC) of four-molecule 'tetramer' bound states — require $T\lesssim0.1\,T_{F}$ and stronger interactions than achieved. Later theory adds a further constraint: Pauli blocking, the suppression of elastic collisions between identical fermions occupying the same internal state near degeneracy, itself throttles rethermalization during the final evaporation stage.

Problem Statement

For a three-dimensional gas of fermionic polar molecules stabilized by microwave shielding, the source experiment (Schindewolf et al., Nature 607, 677 (2022)) reached $T/T_{F}=0.36(9)$ and posed the question of what ultimately limits and how one goes decisively beyond this regime. The open problem is: (1) determine, at the microscopic level, the origin of the residual short-range two-body loss that microwave shielding suppresses but does not eliminate — including why essentially every collision complex formed at short range is lost and why complex lifetimes exceed statistical (RRKM-type) predictions by orders of magnitude — so that this loss can be understood rather than merely fenced out; and (2) evaporate a strongly interacting 3D fermionic polar-molecule gas to $T\lesssim0.1\,T_{F}$ — which requires reduced one-body loss from microwave phase noise and two-body loss from microwave polarization ellipticity (including control of how these losses scale with the shielding parameters), increased initial phase-space density, and evaporation strategies that remain efficient when rethermalization saturates in the hydrodynamic regime and when Pauli blocking suppresses elastic collisions between identical fermions — sufficiently deep in degeneracy and with sufficiently strong tunable dipolar interactions to observe the predicted many-body phases, most notably topological $p$-wave superfluidity (and the predicted tetramer Bose–Einstein condensate), and to characterize emergent dipolar interaction effects in the degenerate cloud. These are parallel, physically related open questions, and the two parts may be answered independently: reducing the residual loss is one enabling route toward the deeper-evaporation target, not a precondition for explaining it. Within the second part, the loss-reduction measures, the increased phase-space density, the evaporation strategies, and each of the observational goals may likewise be pursued and resolved separately.

References

1. A. Schindewolf, R. Bause, X.-Y. Chen, M. Duda, T. Karman, I. Bloch, X.-Y. Luo, 'Evaporation of microwave-shielded polar molecules to quantum degeneracy', Nature 607, 677–681 (2022). DOI: 10.1038/s41586-022-04900-0, https://doi.org/10.1038/s41586-022-04900-0 (preprint arXiv:2201.05143, https://arxiv.org/abs/2201.05143)
2. N. Bigagli, C. Warner, W. Yuan, S. Zhang, I. Stevenson, T. Karman, S. Will, 'Collisionally stable gas of bosonic dipolar ground-state molecules', Nature Physics 19, 1579–1584 (2023). DOI: 10.1038/s41567-023-02200-6, https://doi.org/10.1038/s41567-023-02200-6 (preprint arXiv:2303.16845, https://arxiv.org/abs/2303.16845)
3. N. Bigagli, W. Yuan, S. Zhang, B. Bulatovic, T. Karman, I. Stevenson, S. Will, 'Observation of Bose–Einstein condensation of dipolar molecules', Nature 631, 289–293 (2024). DOI: 10.1038/s41586-024-07492-z, https://doi.org/10.1038/s41586-024-07492-z (preprint arXiv:2312.10965, https://arxiv.org/abs/2312.10965)
4. T. Karman, N. Bigagli, W. Yuan, S. Zhang, I. Stevenson, S. Will, 'Double Microwave Shielding', PRX Quantum 6, 020358 (2025). DOI: 10.1103/b8pm-3prn, https://doi.org/10.1103/b8pm-3prn (preprint arXiv:2501.08095, https://arxiv.org/abs/2501.08095)
5. Z. Shi, Z. Huang, F. Deng, W.-J. Jin, S. Yi, T. Shi, D. Wang, 'Bose–Einstein condensate of ultracold sodium–rubidium molecules with tunable dipolar interactions', Nature Physics (published 9 July 2026). DOI: 10.1038/s41567-026-03362-9, https://doi.org/10.1038/s41567-026-03362-9 (preprint arXiv:2508.20518, https://arxiv.org/abs/2508.20518)
6. S. Zhang, W. Yuan, N. Bigagli, H. Kwak, T. Karman, I. Stevenson, S. Will, 'Observation of self-bound droplets of ultracold dipolar molecules', Nature 651, 601–606 (2026). DOI: 10.1038/s41586-026-10245-9, https://doi.org/10.1038/s41586-026-10245-9 (preprint arXiv:2507.15208, https://arxiv.org/abs/2507.15208)
7. I. Stevenson, S. Singh, A. Elkamshishy, N. Bigagli, W. Yuan, S. Zhang, C. H. Greene, S. Will, 'Three-Body Recombination of Ultracold Microwave-Shielded Polar Molecules', Physical Review Letters 133, 263402 (2024). DOI: 10.1103/PhysRevLett.133.263402, https://doi.org/10.1103/PhysRevLett.133.263402 (preprint arXiv:2407.04901, https://arxiv.org/abs/2407.04901)
8. J. Dutta, B. Mukherjee, J. M. Hutson, 'Universality in the microwave shielding of ultracold polar molecules', Physical Review Research 7, 023164 (2025). DOI: 10.1103/PhysRevResearch.7.023164, https://doi.org/10.1103/PhysRevResearch.7.023164 (preprint arXiv:2501.03170, https://arxiv.org/abs/2501.03170)
9. R. R. W. Wang, S. Biswas, S. Eppelt, F. Deng, X.-Y. Luo, J. L. Bohn, 'Simulations of evaporation to deep Fermi degeneracy in microwave-shielded molecules', Physical Review A 110, 043309 (2024). DOI: 10.1103/PhysRevA.110.043309, https://doi.org/10.1103/PhysRevA.110.043309 (preprint arXiv:2407.14466, https://arxiv.org/abs/2407.14466)
10. R. R. W. Wang, J. L. Bohn, 'Prospects for thermalization of microwave-shielded ultracold molecules', Physical Review Research 6, L022033 (2024). DOI: 10.1103/PhysRevResearch.6.L022033, https://doi.org/10.1103/PhysRevResearch.6.L022033 (preprint arXiv:2310.17812, https://arxiv.org/abs/2310.17812)
11. R. R. W. Wang, J. L. Bohn, 'Collisional relaxation in shielded dipolar molecular gases', arXiv:2510.17020 (2025). https://arxiv.org/abs/2510.17020
12. R. Bause, A. Christianen, A. Schindewolf, I. Bloch, X.-Y. Luo, 'Ultracold Sticky Collisions: Theoretical and Experimental Status', The Journal of Physical Chemistry A 127, 729–741 (2023). DOI: 10.1021/acs.jpca.2c08095, https://doi.org/10.1021/acs.jpca.2c08095 (preprint arXiv:2211.10223, https://arxiv.org/abs/2211.10223)
13. K. B. Xu, J. L. Bohn, 'Limits of Statistical Models of Ultracold Complex Lifetimes', arXiv:2604.12063 (2026). https://arxiv.org/abs/2604.12063
14. J. Lin, G.-D. Chen, M. Jin, Z. Shi, F. Deng, W. Zhang, G. Quéméner, T. Shi, S. Yi, D. Wang, 'Microwave Shielding of Bosonic NaRb Molecules', Physical Review X 13, 031032 (2023). DOI: 10.1103/PhysRevX.13.031032, https://doi.org/10.1103/PhysRevX.13.031032 (preprint arXiv:2304.08312, https://arxiv.org/abs/2304.08312)
15. J. Lin, A. N. Carroll, P. Martin, C. Miller, R. R. W. Wang, K. Xu, J. L. Bohn, T. de Jongh, J. Ye, 'Fermi gas of polar molecules in the Pauli-blocked regime', arXiv:2606.14135 (2026). https://arxiv.org/abs/2606.14135
16. S. Biswas, S. Eppelt, W. Tian, W. Zhang, F. Deng, C. Frank, T. Shi, I. Bloch, X.-Y. Luo, 'Controlled symmetry breaking of the Fermi surface in ultracold polar molecules', arXiv:2602.22447 (2026). https://arxiv.org/abs/2602.22447
17. S. Biswas, S. Eppelt, C. Buchberger, X.-Y. Chen, A. Schindewolf, M. Hani, E. Biebl, I. Bloch, X.-Y. Luo, 'Generation of strong ultralow-phase-noise microwave fields with tunable ellipticity for ultracold polar molecules', arXiv:2512.03007 (2025). https://arxiv.org/abs/2512.03007
18. J.-L. Li, G. M. Koutentakis, M. Hrast, M. Lemeshko, A. Schindewolf, R. Alhyder, 'Tunable Field-Linked $s$-wave Interactions in Dipolar Fermi Mixtures', arXiv:2506.23318 (2025). https://arxiv.org/abs/2506.23318