Stabilization of three-body resonances to bound states in a continuum

TL;DR

This work addresses stabilizing three-body resonances into bound states in a continuum (BIC) using a two-channel Feshbach-like model. It shows that a single resonance embedded in a single continuum can acquire an infinite lifetime when a transition element vanishes, with the tunable relative momentum $p_{ ext{rel}}$ acting as a control. The authors demonstrate the mechanism in a 1D mass-imbalanced system and in 3D Efimov-relevant physics, where an external magnetic field $B$ provides the tuning; the two-channel description reproduces the vanishing width and reveals multiple stabilization points, suggesting broad applicability. The work opens pathways to long-lived trimers and enhanced control of three-body interactions in cold-atom platforms, with potential extensions to nuclear or hadronic few-body systems. Overall, it provides a general, parametric route to bound-state-in-continuum formation for unstable few-body systems and lays groundwork for experimental exploration of long-lived three-body states.

Abstract

Three-body resonances are ubiquitous in quantum few-body physics and are characterized by a finite lifetime before decaying into continuum states of their composing subsystems. In this work we present a theoretical study on the possibility to stabilize three-body resonances to so-called bound states in a continuum: resonances with vanishing width that do not decay. Within a two-channel approach we unveil the underlying mechanism and show how the lifetime can be made infinitely long by a continuous tuning of system parameters. The validity of our theory is illustrated in two different examples: a mass-imbalanced system in one dimension and a system of three identical bosons in three dimensions, relevant to Efimov physics. Crucially, for the latter we find that one of the parameters that can be tuned to achieve a three-body bound state in a continuum is an external magnetic field, a common tunable variable in cold-atom experiments. Due to the generality of this stabilization effect, it is expected to be applicable to a wide range of unstable few-body systems, opening new perspectives for fundamental studies as well as technical applications.

Figures (4)

• Figure 1: Oscillations in the width of three-body resonances in 1D revealing multiple BIC locations. Position $E_\mathrm{R}^{(3)}$ and width $\Gamma$ of a three-body resonance in a 1D, 2+1 boson system as a function of the relative momentum $p_\mathrm{rel}$, Eq. \ref{['eq:krel']}. All energies are presented in units of the characteristic two-body energy $B_2 = \hbar^2/2\mu_{bx}z_0^2$. Accordingly, $p_{\mathrm{rel}} = \frac{\hbar}{z_0} \, \sqrt{(1+\beta)[E^{(3)} - E^{(2)}]/B_2}$ is given in units of $\hbar/z_0$, and the momentum is scanned by varying the mass ratio in the range $1/20\leq\beta\leq20$. Upper panel: The three-body resonance (solid red line) is located in the atom-dimer continuum induced by the two-body ground state at energy $E_g^{(2)}$ (dashed black line). The resonance width (shaded area) shows oscillatory behavior with several zero-points (green crosses). For better visibility $\Gamma$ is scaled by the mass ratio $\beta$. Lower panel: resonance width in log-scale as a function of $p_\mathrm{rel}$, for the full three-body calculation (red line) similar to Ref. happ2024 and the BO two-channel model (blue line). Lower bounds of $\Gamma$ are limited by numerical accuracy.
• Figure 2: Magnetic-field-induced BIC formation in the Efimov scenario. Position $E_\mathrm{R}^{(3)}$ and width $\Gamma$ of a resonance of three identical bosons in 3D, as a function of the magnetic field $B$. Energies are given in units of $E^\star = \hbar^2/m {R^\star}^2$. Upper panel: The location, where the resonance width vanishes is marked by a green cross. Additionally, three curves characterizing the two-body Feshbach resonance are displayed: the bare bound state energy $e_\mathrm{b}$ (dashed dark blue line), the open-channel bound state energy $e_\mathrm{o}$ (dash-dotted orange line), and the dressed two-body states $e_\mathrm{d}$ (solid black lines). Lower panel: width $\Gamma$ of the three-body resonance in log-scale for the full three-body calculation (red line), and the hyperspherical two-channel model (blue line).
• Figure 3: Robustness of BIC existence against variation of the background scattering length. BIC-location (green line) for three identical bosons in 3D, in the parameter space of the inverse background scattering length $1/a_\mathrm{bg}$ and the magnetic field $B$. The scattering length is given in units of $R^\star$. Additionally, the three-body binding threshold, $E_\mathrm{R}^{(3)}=0$, is displayed (dashed purple line).
• Figure 4: Hyperradial potential curves with dimer and trimer spectrum. Hyperradial potentials $W_{n}(\rho)$ for the two-channel contact model with $e_{\mathrm{b}}=-0.1R^{\star-2}$ and $a_{\mathrm{bg}}=20R^{\star}$. The energies of the dimer and trimer states are indicated by horizontal dashed lines.