Confinement-induced resonances for the creation of quasi-one-dimensional ultra cold gases of alkali--alkaline-earth dimers

TL;DR

The paper tackles the formation of weakly bound dimers in quasi-1D ultracold atomic mixtures by exploiting confinement-induced resonances (CIRs). It develops a two-body, quasi-1D framework with a regularized zero-range interspecies potential, deriving the effective 1D coupling $g_{1D}$ and the 1D scattering length $a_{1D}$ from the 3D scattering length $a$, and analyzes how mismatched transverse traps introduce center-of-mass–relative-motion coupling that yields an additional narrow CIR. Numerical results for the $^{87}$Rb–$^{87}$Sr mixture reveal a main CIR consistent with Olshanii’s result and a second narrow CIR arising from excited cm states when trap frequencies differ; in the Rb-Sr case, a pole in $a_{1D}$ occurs near experimentally accessible trap frequencies, suggesting a practical ramp protocol to form weakly bound dimers. The study provides concrete parameter guidance for experimentally realizing CIR-assisted molecule formation and lays groundwork for extending CIR concepts to other alkali–alkaline-earth mixtures and to quasi-2D geometries.

Abstract

We theoretically investigate the role of confinement-induced resonances (CIRs) in low-dimensional ultracold atomic mixtures for the formation of weakly bound dimers. To this end, we examine the scattering properties of a binary atomic mixture confined by a quasi-one-dimensional (quasi-1D) potential. In this regime, the interspecies two-body interaction is modeled as an effective 1D zero-range pseudopotential, with a coupling strength $g_\mathrm{1D}$ derived as a function of the three-dimensional scattering length $a$. This framework enables the study of CIRs in harmonically confined systems, with particular attention to the case of mismatched transverse trapping frequencies for the two atomic species. Finally, we consider the Bose-Fermi mixture of $^{87}$Rb and $^{87}$Sr, and identify values of the experimentally accessible parameters for which CIRs can be exploited to create weakly bound molecules.

Figures (4)

• Figure 1: (a) Scheme of the system: a mixture of two species with masses $m_1$ and $m_2$ confined by quasi-one-dimensional potentials with different trapping frequencies. The interspecies interaction is represented by the regularized zero-range pseudopotential $V(r)$, with $r$ being the interparticle distance, $\mu$ the reduced mass and $a$ the 3D scattering length. (b) Harmonic potentials along direction $\xi=x,y,z$ for the two species $i=1,2$. Direction $z$ is the direction of weak confinement.
• Figure 2: Energy spectrum of $H_\mathrm{rel}$ as a function of the 1D scattering length. Thick black lines: energies for even $n$ approaching the noninteracting values (gray dashed line) at large negative or positive values of $a_\mathrm{1D}$. Thin green lines: energies for odd $n$. Units are rescaled with respect to the energy $\hbar\omega$ and length $\ell_\mathrm{rel}=\sqrt{\hbar/(\mu\omega)}$ of the relative harmonic oscillator.
• Figure 3: Confinement induced resonances for different frequency ratios. (a) Interaction strength $g_\mathrm{1D}$ as a function of scattering length. (b) Values $a_\mathrm{res}$ of scattering length $a$ for which the main (black circles) and second resonance (gray diamonds) appear. Dashed lines indicate the analytical value obtained for $\omega_{\perp,2}=\omega_{\perp,1}$. The position of the resonances shown in (a) are highlighted with the corresponding color. Dotted lines are only a guide to the eye.
• Figure 4: Results for the 87Rb-87Sr mixture. (a) Spectrum in the 1D limit. The upper axis indicates the transverse strontium frequency $\omega_{\perp,1}$ considering the fixed interspecies scattering length $a_\mathrm{Sr-Rb}=1420\,a_0$ ($a_0$ the Bohr radius). Horizontal dashed lines correspond to the asymptotic noninteracting states for different values of $(n_1,n_2)$ as indicated on the right of the plot, with $n_1$ and $n_2$ the principal quantum numbers of the longitudinal harmonic oscillator of strontium and rubidium, respectively. Purple lines are for $n_2=0$, blue lines for $n_2=1$ and green lines for $n_2=2$. The vertical dotted line indicates the position of the resonance of $a_\mathrm{1D}$ appearing at $\omega_{\perp,1}=12.418\,\mathrm{kHz}$ or $a=0.77647\,\ell_{\perp,1}$ ($\ell_{\perp,1}$ is the length associated with $\omega_{\perp,1}$). The red arrow indicates a possible pathway for dimer formation: the system is prepared on the right side of the resonance and adiabatically ramped across it to populate a bound state. (b) 1D scattering length $a_\mathrm{1D}=-1/(\mu g_\mathrm{1D})$ in units of $\ell_{\perp,1}$. The bottom axis indicates the 3D scattering length in units of $\ell_{\perp,1}$. (c) 1D scattering length on a broader range. (d) 1D interaction strength $g_\mathrm{1D}$. The values used in previous plots are highlighted in black.