An atom chip whose coplanar waveguide radiates multi-gauss microwave near fields recently enabled fast coherent manipulation of the hyperfine state of magnetically trapped sodium quantum gases, with Rabi frequencies up to 2π×6.92\pi\times6.9 MHz. The source work (M. Ballu et al., Phys. Rev. A 110, 053312 (2024)) left two coupled objectives open: (i) coherent two-photon microwave plus radio-frequency transfer from the trapped ∣F=1,mF=−1⟩|F=1,m_F=-1\rangle state into the trapped ∣F=2,mF=1⟩|F=2,m_F=1\rangle 'clock' state, whose matched second-order Zeeman shift at a static field of 0.68 G promises long coherence, under the constraint that the two-photon Rabi coupling stay below the approximately 500 kHz Zeeman splittings to untrapped states; and (ii) the determination, location, and quantitative characterization of microwave-induced Fano-Feshbach resonances in ultracold sodium — the magnet-free interaction-tuning mechanism proposed by Papoular, Shlyapnikov, and Dalibard, whose resonance width scales as the square of the microwave amplitude and is predicted to reach 2π×1.4 kHz/G22\pi\times1.4\ \mathrm{kHz/G^2} for sodium. A 2025 follow-up from the same apparatus has measured the required Na2_2 least-bound molecular-state energies to the ~10 kHz level and demonstrated ac-Zeeman-compensated fields near 8 G, but no microwave-induced Fano-Feshbach resonance has been observed in any species, and the coherent two-photon transfer has not been demonstrated on the sodium platform. The problem is to determine both capabilities and quantitatively characterize the resulting interaction tunability, including inelastic losses, through a written analysis and design rather than an experimental realization.

Background

An atom chip is a microfabricated device whose current-carrying wires create magnetic traps a few hundred micrometers from the surface, where ultracold atoms confined in low-field-seeking spin states can be evaporatively cooled to Bose-Einstein condensation. The platform of the source work (M. Ballu, B. Mirmand, T. Badr, H. Perrin, and A. Perrin, Phys. Rev. A 110, 053312 (2024)) traps around 10610^6 sodium-23 atoms on a chip that also carries a coplanar waveguide (CPW), a planar microwave transmission line that, when driven near the 1.77 GHz hyperfine splitting of the sodium 3S1/23S_{1/2} ground state, radiates a strong oscillating magnetic near field. At the closest trap position the microwave amplitude exceeds 5 G and drives Rabi oscillations (coherent population oscillations at the Rabi frequency Ω\Omega) between hyperfine states at up to 2π×6.92\pi\times6.9 MHz on the strongest transition — fast enough to flip the atomic spin in 72 ns. The measured amplitude distribution over the cloud has a relative width of about 16%, which, together with the spread of static-field orientations, dephases the oscillations and motivates questions about near-field homogeneity and waveguide losses (on-chip transmission −5.2±0.4-5.2\pm0.4 dB).

Microwave manipulation, however, reaches mostly untrapped states. In the ground-state hyperfine manifolds F=1F=1 and F=2F=2 (with magnetic sublevels mFm_F), magnetic trapping confines only weak-field seekers; from the trapped state ∣F=1,mF=−1⟩|F=1,m_F=-1\rangle, one-photon microwave transitions couple to F=2F=2 states that are all untrapped except ∣F=2,mF=1⟩|F=2,m_F=1\rangle. That state is reachable by a two-photon transition combining a microwave and a radio-frequency photon, and it has a special property: at a static field of about 0.68 G its second-order Zeeman shift equals that of ∣F=1,mF=−1⟩|F=1,m_F=-1\rangle, so the two-photon transition frequency is first-order insensitive to field fluctuations — the same 'clock' insensitivity exploited in rubidium atom-chip work by P. Böhi et al. (Nat. Phys. 5, 592 (2009)), who demonstrated coherent two-photon microwave manipulation and long-lived superpositions on a chip. The price is a weak-coupling regime: the two-photon Rabi coupling must stay below the approximately 500 kHz Zeeman splittings to neighboring untrapped states at the optimal field, so the transfer is slow and exposed to dephasing from near-field inhomogeneity.

The deeper prospect is interaction control. A Feshbach resonance occurs when a bound molecular state crosses the collision threshold of two atoms, making the s-wave scattering length aa — the parameter setting the mean-field interaction of a dilute gas — tunable; magnetically tuned Feshbach resonances are the workhorse of the field, but sodium's usable resonances are narrow and inconvenient. D. J. Papoular, G. V. Shlyapnikov, and J. Dalibard (Phys. Rev. A 81, 041603(R) (2010)) proposed a magnet-free variant: a microwave photon couples an incoming pair of atoms in one hyperfine collision channel to a bound state of another hyperfine manifold. Because the bound state is embedded in, and coupled to, the scattering continuum, the resonance has Fano character, and its frequency width grows as the square of the microwave amplitude — precisely the quantity the chip near field provides in large measure. The ingredients are now in place for sodium: microwave photoassociation (using microwave photons to associate atom pairs into weakly bound molecules) has been demonstrated in rubidium (C. Maury et al., Phys. Rev. Research 5, L012020 (2023)), and the same apparatus as the source work has since measured the energies of the Na2_2 least-bound molecular states to the ~10 kHz level, observed predissociation (the natural decay of a bound state into the scattering continuum), characterized the ac Zeeman shift (the level shift induced by strong off-resonant microwave dressing, compensated using a second, frequency-symmetric microwave signal to reach stable amplitudes near 8 G), and calculated a resonance width scaling of 2π×1.4 kHz/G22\pi\times1.4\ \mathrm{kHz/G^2} for sodium (M. Ballu et al., Phys. Rev. A 112, 013312 (2025)). At 8 G this predicts a width of order 90 kHz, well above a typical gas energy spread (chemical potential μ≃h×20\mu\simeq h\times20 kHz), so the equilibrium properties of the gas should respond observably to the microwave-modified scattering length. Related dressed-resonance physics has been realized by other routes — Floquet-engineered Feshbach resonances in 6^6Li via strong megahertz magnetic-field modulation (A. Guthmann et al., Science Advances 11, eadw3856 (2025)) and a modulation-induced Feshbach resonance in cesium driven by intensity-modulated far-detuned light (T. Wang et al., Phys. Rev. Lett. 135, 203201 (2025)) — but not by the microwave near-field scheme, and not in sodium.

Problem Statement

On the strong-microwave sodium atom-chip platform — or any source of multi-gauss near-field microwave amplitudes acting on an ultracold sodium gas — determine and quantitatively characterize the two coupled capabilities left open by the source work, through a written analysis and design computed from the platform parameters and the cited theory rather than an experimental realization:

(1) Coherent two-photon microwave plus radio-frequency transfer into the trapped clock state: couple the trapped ∣F=1,mF=−1⟩|F=1,m_F=-1\rangle state to the trapped ∣F=2,mF=1⟩|F=2,m_F=1\rangle state at the optimal static field near 0.68 G — where the two states' second-order Zeeman shifts match, making the two-photon transition frequency first-order insensitive to magnetic-field fluctuations — with the two-photon Rabi coupling held below the approximately 500 kHz Zeeman splittings to untrapped neighboring states, and characterize the transfer by driven Rabi oscillations and/or Ramsey fringes (population oscillations observed with two separated drive pulses) between the two trapped states, reporting its coherence time; at least one of the two protocols is required.

(2) Microwave-induced Fano-Feshbach resonance in sodium: determine, locate, and quantitatively characterize at least one such resonance — the resonance positions in microwave frequency near the now precisely known least-bound molecular transitions; the resonance width as a function of microwave amplitude, to be compared with the predicted quadratic scaling of coefficient 2π×1.4 kHz/G22\pi\times1.4\ \mathrm{kHz/G^2}; and the resulting tunability of the ss-wave scattering length, including its imaginary part since strong microwave dressing mixes hyperfine channels and opens inelastic loss — for instance through changes of equilibrium or collective properties of the gas, together with the associated two- and three-body loss rates. All of the characterizations listed for component (2) are required for completion.

The common objective, which the source work calls its most promising application, is to convert the demonstrated large near-field amplitudes into coherent control of long-lived trapped states and into interaction control of the trapped sodium gas. The auxiliary technical questions the source also raises — spatial homogeneity of the microwave near field and compensation of dephasing, on-chip coplanar-waveguide and packaging losses, and microwave adiabatic potentials in and beyond the rotating-wave approximation (RWA, the standard simplification of the atom-field interaction that neglects fast-oscillating terms) at these amplitudes — support the two components but do not by themselves constitute completion.

References

  1. M. Ballu, B. Mirmand, T. Badr, H. Perrin, A. Perrin, 'Fast manipulation of a quantum gas on an atom chip with a strong microwave field', Physical Review A 110, 053312 (2024). DOI: 10.1103/PhysRevA.110.053312; arXiv:2405.07583; https://arxiv.org/abs/2405.07583
  2. M. Ballu, Z. Yao, B. Mirmand, D. J. Papoular, H. Perrin, A. Perrin, 'Microwave spectroscopy of ultracold-sodium least-bound molecular states', Physical Review A 112, 013312 (2025). DOI: 10.1103/qv5s-71ll; arXiv:2503.20330; https://arxiv.org/abs/2503.20330
  3. D. J. Papoular, G. V. Shlyapnikov, J. Dalibard, 'Microwave-induced Fano-Feshbach resonances', Physical Review A 81, 041603(R) (2010). DOI: 10.1103/PhysRevA.81.041603; arXiv:0909.4633; https://arxiv.org/abs/0909.4633
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  5. A. Guthmann, F. Lang, L. M. Kienesberger, S. Barbosa, A. Widera, 'Floquet engineering of Feshbach resonances in ultracold gases', Science Advances 11, eadw3856 (2025). DOI: 10.1126/sciadv.adw3856; arXiv:2503.05454; https://arxiv.org/abs/2503.05454
  6. A. Guthmann, L. M. Kienesberger, F. Lang, E. Lippi, A. Widera, 'Engineering a Bound State in the Continuum via Quantum Interference', arXiv:2602.07809 (2026). https://arxiv.org/abs/2602.07809
  7. T. Wang, Y. Liu, J. Wang, Y. Huang, W. Chen, Z. Zhang, J. Hu, 'Observation of Modulation-Induced Feshbach Resonance', Physical Review Letters 135, 203201 (2025). DOI: 10.1103/qrz6-gjp6; arXiv:2505.06871; https://arxiv.org/abs/2505.06871
  8. P. Böhi, M. F. Riedel, J. Hoffrogge, J. Reichel, T. W. Hänsch, P. Treutlein, 'Coherent manipulation of Bose–Einstein condensates with state-dependent microwave potentials on an atom chip', Nature Physics 5, 592–597 (2009). DOI: 10.1038/nphys1329; https://doi.org/10.1038/nphys1329
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  10. M. Ballu, R. Dubessy, A. Perrin, H. Perrin, A. Minguzzi, 'Soliton turbulence of a strongly driven one-dimensional Bose gas', Physical Review A 114, 023310 (2026). DOI: 10.1103/vbsx-xyjm; arXiv:2603.24832; https://arxiv.org/abs/2603.24832