Universal bound states of one-dimensional bosons with two- and three-body attractions

When quantum particles are confined into lower dimensions, an effective three-body interaction inevitably arises and may cause significant consequences. Here we study bosons in one dimension with weak two-body and three-body interactions, predict the existence of two three-body bound states when both interactions are attractive, and determine their binding energies as universal functions of the two-body and three-body scattering lengths. We also show that an infinitesimal three-body attraction induces an excited bound state only for 3, 39, or more bosons. Our findings herein have direct relevance to a broad range of quasi-one-dimensional systems realized with ultracold atoms.

Figure 1

Binding energies of three-body bound states $E=-{\kappa }^2/m$ in the forms of $a_3\kappa$ (top panel) and $a_2\kappa$ (bottom panel) as functions of the three-body to two-body scattering length ratio $a_3/a_2$. The upper (lower) solid curve corresponds to the ground (excited) state and the dotted line indicates $\kappa =2/a_2$ for the McGuire trimer. The shaded region in the top panel indicates the atom-dimer continuum where $\kappa <\theta \left(a_2\right)/a_2$.

Figure 2

Scattering length ${\alpha }_{1,N-1}$ between an atom and an $(N-1)$-body bound state induced by an infinitesimal three-body attraction in the form of ${\beta }_{1,N-1}\equiv a_{2}ln(a_3/a_2)/\left(\sqrt{3}\pi {\alpha }_{1,N-1}\right)$. It turns out to be positive for $N=3$ and $N\ge 39$ but negative for $4\le N\le 38$ as indicated by the different colors.