Bound states and scattering lengths of three two-component particles with zero-range interactions under one-dimensional confinement

TL;DR

This paper investigates the universal three-body bound states and scattering properties of a binary mixture of particles confined to one dimension, providing analytical and numerical insights into their dependence on mass ratio and interaction strength.

Contribution

It offers new analytical and numerical results on three-body energies and scattering lengths in 1D binary gases with zero-range interactions, including critical mass ratios and phase diagrams.

Findings

Critical mass ratios for three-body states are identified.

Exact asymptotic behaviors for large mass ratios and strong interactions are derived.

A schematic phase diagram of bound states and scattering lengths is constructed.

Abstract

The universal three-body dynamics in ultra-cold binary gases confined to one-dimensional motion are studied. The three-body binding energies and the (2 + 1)-scattering lengths are calculated for two identical particles of mass $m$ and a different one of mass $m_1$, which interactions is described in the low-energy limit by zero-range potentials. The critical values of the mass ratio $m/m_1$, at which the three-body states arise and the (2 + 1)-scattering length equals zero, are determined both for zero and infinite interaction strength $\lambda_1$ of the identical particles. A number of exact results are enlisted and asymptotic dependences both for $m/m_1 \to \infty$ and $\lambda_1 \to -\infty$ are derived. Combining the numerical and analytical results, a schematic diagram showing the number of the three-body bound states and the sign of the (2 + 1)-scattering length in the plane of the mass ratio and interaction-strength ratio is deduced. The results provide a description of the homogeneous and mixed phases of atoms and molecules in dilute binary quantum gases.