Three-body bound states in a harmonic waveguide with cylindrical symmetry

TL;DR

This paper investigates the energy spectrum of a three-body system in a cylindrically symmetric waveguide, revealing deviations from 1D models and predicting new universal bound states influenced by mass ratio and scattering length.

Contribution

It provides a detailed three-dimensional analysis of three-body bound states in a harmonic waveguide, including the effects of transverse excitations and identifying new universal states for large mass ratios.

Findings

Full 3D energies deviate from 1D models when scattering length is small.

Existence of new universal bound states with |M_rel|=1 for large mass ratios.

Transverse excitations significantly influence the energy spectrum.

Abstract

Highly-elongated quasi-one-dimensional cold atom samples have been studied extensively over the past years experimentally and theoretically. This work determines the energy spectrum of two identical fermions and a third distinguishable particle as functions of the mass ratio $\kappa$ and the free-space $s$-wave scattering length $a_{3\text{D}}$ between the identical fermions and the distinguishable third particle in a cylindrically symmetric waveguide whose symmetry axis is chosen to be along the $z$-axis. We focus on the regime where the mass of the identical fermions is equal to or larger than that of the third distinguishable particle. Our theoretical framework accounts explicitly for the motion along the transverse confinement direction. In the regime where excitations in the transverse direction are absent (i.e., for states with projection quantum number $M_{\text{rel}}=0$), we determine the binding energies for states with odd parity in $z$. These full three-dimensional energies deviate significantly from those obtained within a strictly one-dimensional framework when the $s$-wave scattering length is of the order of or smaller than the oscillator length in the confinement direction. If transverse excitations are present, we predict the existence of a new class of universal three-body bound states with $|M_{\text{rel}}|=1$ and positive parity in $z$. These bound states arise on the positive $s$-wave scattering length side if the mass ratio $\kappa$ is sufficiently large. Implications of our results for ongoing cold atom experiments are discussed.