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

摘要

The universal three-body dynamics in ultra-cold binary gases confined toone-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 thelow-energy limit by zero-range potentials. The critical values of the massratio $m/m_1$, at which the three-body states arise and the (2 + 1)-scatteringlength equals zero, are determined both for zero and infinite interactionstrength $\lambda_1$ of the identical particles. A number of exact results areenlisted and asymptotic dependences both for $m/m_1 \to \infty$ and $\lambda_1\to -\infty$ are derived. Combining the numerical and analytical results, aschematic diagram showing the number of the three-body bound states and thesign of the (2 + 1)-scattering length in the plane of the mass ratio andinteraction-strength ratio is deduced. The results provide a description of thehomogeneous and mixed phases of atoms and molecules in dilute binary quantumgases.