Naskar, Sardar, Deb and Agarwal [J. Phys. B 52, 245204 (2019)] proposed to restore a bound state in the continuum (BIC) of ultracold two-body collisions — predicted earlier by Deb and Agarwal under the assumption of no spontaneous emission — by using vacuum-induced coherence (VIC) to cancel the spontaneous decay of two microwave-dressed excited molecular states. In their idealized model the BIC exists only when the VIC exactly cancels emission, and a deviation of the VIC condition by as little as 0.1% destroys it. The open problem is whether this VIC-protected BIC can survive experimentally realistic conditions and be observed: (i) quantitative tolerance of the BIC (quasi-BIC width and lifetime) to deviation from the exact VIC cancellation condition, including partial orthogonality of the two transition dipole moments; (ii) the effect of additional spontaneous decay channels and the neglected vacuum-induced (Lamb-like) level shifts on BIC existence and stability; and (iii) the deferred variant based on orbital Feshbach resonances between ground-state and metastable atoms. A 2026 experiment realized a cold-atom BIC by a different interference mechanism (two Feshbach resonances coherently coupled by periodic Floquet driving), leaving the VIC-based scheme's feasibility unquantified.

Background

A bound state in the continuum (BIC) is a spatially localized quantum state whose energy lies inside the band of extended scattering states yet which does not decay, because destructive interference cuts its coupling to the continuum to zero — making it a resonance of vanishing width. The interference mechanism dates back to Friedrich and Wintgen's work on overlapping resonances [Phys. Rev. A 32, 3231 (1985)].

In ultracold atomic gases, the relevant continuum is that of two colliding atoms. A magnetic Feshbach resonance arises when a bound state of a 'closed' internal channel (a spin configuration whose dissociation threshold lies higher) is magnetically tuned near the collision energy of the 'open' channel, strongly modifying the scattering. Photoassociation (PA) uses lasers to couple colliding atom pairs to excited electronic molecular bound states. Deb and Agarwal [Phys. Rev. A 90, 063417 (2014)] showed theoretically that magneto-optical manipulation of low-energy scattering resonances and atom-molecule transitions can create a BIC of two ultracold atoms — a two-body multichannel state above the collision threshold with zero width — provided there is no spontaneous emission.

Spontaneous emission is, however, unavoidable: excited molecular states decay radiatively, and Naskar, Sardar, Deb and Agarwal [J. Phys. B 52, 245204 (2019)] showed that even an exceedingly small decay from the excited molecular states spoils this BIC. Their remedy is vacuum-induced coherence (VIC): when two excited states decay to one common lower state via two indistinguishable spontaneous-emission pathways, the two decay amplitudes interfere destructively and the emission is partially or fully cancelled. Effective VIC requires (a) the two transition dipole moments to be non-orthogonal — for molecular bound states, achievable with two vibrational levels sharing the same rotational quantum number — and (b) the spacing of the two decaying states to be comparable to or smaller than the geometric meanγ1γ2\sqrt{\gamma_1\gamma_2}of their spontaneous linewidthsγ1\gamma_1,γ2\gamma_2. The authors propose to satisfy (b) by dressing the two excited states with a microwave field whose Rabi coupling is set toΩm≃γ1γ2\Omega_m \simeq \sqrt{\gamma_1\gamma_2}, and they treat the system with a Feshbach projection-operator technique — eliminating the scattering and photon continua to obtain an effective non-Hermitian Hamiltonian (one with complex decay terms) for the bound states alone.

The BIC then exists only when the VIC cancels spontaneous emission exactly. Writingη=γVIC/ΓF\eta = \gamma_{\rm VIC}/\Gamma_F(the VIC strength in units ofΓF\Gamma_F, the width of the Feshbach resonance) andγ~n=γn/ΓF\tilde{\gamma}_n = \gamma_n/\Gamma_F, the exact-cancellation condition — Eq. (20) of arXiv:1903.02355 — isη2=γ~1γ~2\eta^2 = \tilde{\gamma}_1\tilde{\gamma}_2, which becomesη=1\eta = 1in the equal-linewidth scaling used in the paper's figures; the authors demonstrate extreme sensitivity — if the VIC condition deviates by even 0.1%, the BIC is destroyed (the quasi-BIC spectral line, observable as a vanishing-width feature in the Fano-shaped, i.e. asymmetric and interference-distorted, PA spectrum, broadens by orders of magnitude). The model also assumes both dressed excited states decay predominantly to a single common lower state, neglects all other spontaneous decay pathways and all vacuum-induced level shifts, and assumes equal laser phases so thatγVIC\gamma_{\rm VIC}is independent of laser phase.

Problem Statement

The source work establishes that, within its idealized model, a cold-atom bound state in the continuum exists only when vacuum-induced coherence exactly cancels spontaneous emission —η=γVIC/ΓF\eta = \gamma_{\rm VIC}/\Gamma_Fsatisfying Eq. (20) of arXiv:1903.02355,η2=γ~1γ~2\eta^2 = \tilde{\gamma}_1\tilde{\gamma}_2withγ~n=γn/ΓF\tilde{\gamma}_n = \gamma_n/\Gamma_F(that is,η=1\eta = 1in the paper's illustrative equal-linewidth scaling), with microwave dressing set toΩm≃γ1γ2\Omega_m \simeq \sqrt{\gamma_1\gamma_2}and two non-orthogonal transition dipoles — and that a deviation of the VIC condition by as little as 0.1% destroys the BIC. The open problem is to determine whether, and under what quantified conditions, the VIC-protected BIC can exist, remain stable, and be observed in an actual cold-atom experiment. Concretely, the unresolved questions left by Naskar et al. are:

  1. Robustness to experimental imperfections: quantify how the BIC degrades — quasi-BIC spectral width, lifetime, and PA-spectroscopic signature — under deviation from the exact VIC cancellation conditionη2=γ~1γ~2\eta^2 = \tilde{\gamma}_1\tilde{\gamma}_2, including partial orthogonality (reduced non-orthogonality) of the two transition dipole moments, to which the source analysis shows extreme sensitivity, and derive the tolerances an experiment must meet for the BIC (or a long-lived quasi-BIC) to be observable.
  2. Neglected decay structure: quantify the impact of additional spontaneous decay channels beyond the assumed single common lower state, competing decay routes, and the neglected vacuum-induced (Lamb-like) level shifts, on the existence and stability of the BIC.
  3. Deferred platform variant: analyze the variant the authors deferred to future work, based on the orbital Feshbach resonance between ground-state and metastable atoms — how metastability and the different coupling structure modify the VIC requirements, spontaneous-emission budgets, and practical controllability.

These three facets are all mandatory, and every quantity each of them lists must be reported. Together they share one overarching objective: a quantitative feasibility assessment of the vacuum-induced-coherence-based cold-atom BIC scheme under experimentally realistic conditions. If that assessment finds the scheme feasible, an enabling analysis is also required: specify the experimental design and preparation protocol — the fields, microwave dressing, dipole choice, and detection scheme — that meets the tolerances reported under facet 1.

References

  1. Somnath Naskar, Dibyendu Sardar, Bimalendu Deb, and G. S. Agarwal, 'Suppressing deleterious effects of spontaneous emission in creating bound states in cold atom continuum', Journal of Physics B: Atomic, Molecular and Optical Physics 52, 245204 (2019). DOI: 10.1088/1361-6455/ab4ef2; arXiv:1903.02355. https://doi.org/10.1088/1361-6455/ab4ef2
  2. Bimalendu Deb and G. S. Agarwal, 'Creation and manipulation of bound states in the continuum with lasers: Applications to cold atoms and molecules', Physical Review A 90, 063417 (2014). DOI: 10.1103/PhysRevA.90.063417. https://doi.org/10.1103/PhysRevA.90.063417
  3. H. Friedrich and D. Wintgen, 'Interfering resonances and bound states in the continuum', Physical Review A 32, 3231 (1985). DOI: 10.1103/PhysRevA.32.3231. https://doi.org/10.1103/PhysRevA.32.3231
  4. Bing-Kuan Lyu, Bo-Wen Si, Zong-Han Yu, Gao-Ren Wang, and Shu-Lin Cong, 'The two-body collision controlled by the magnetic field and laser field near magnetic Feshbach resonance', Frontiers in Physics 11, 1198477 (2023). DOI: 10.3389/fphy.2023.1198477. https://doi.org/10.3389/fphy.2023.1198477
  5. Lucas Happ and Pascal Naidon, 'Stabilization of three-body resonances to bound states in a continuum', Physical Review A 113, 032203 (2026). DOI: 10.1103/9nrl-qv75; arXiv:2503.02037. https://doi.org/10.1103/9nrl-qv75
  6. Alexander Guthmann, Louisa Marie Kienesberger, Felix Lang, Eleonora Lippi, and Artur Widera, 'Engineering a Bound State in the Continuum via Quantum Interference', arXiv:2602.07809 (2026). https://arxiv.org/abs/2602.07809