# Few-body states of bosons interacting with a heavy quantum impurity

Shuhei M. Yoshida$^*$, Zhe-Yu Shi$^*$, Jesper Levinsen, Meera M. Parish

December 2018
### Abstract

We consider the problem of a fixed impurity coupled to a small number $N$ of
noninteracting bosons. We focus on impurity-boson interactions that are mediated by a
closed-channel molecule, as is the case for tuneable interatomic interactions in
cold-atom experiments. We show that this two-channel model can be mapped to a boson
model with effective boson-boson repulsion, which enables us to solve the three-body
($N=2$) problem analytically and determine the trimer energy for impurity-boson
scattering lengths $a>0$. By analyzing the atom-dimer scattering amplitude, we find a
critical scattering length $a^∗$ at which the atom-dimer scattering length diverges
and the trimer merges into the dimer continuum. We furthermore calculate the tetramer
energy exactly for $a>0$ and show that the tetramer also merges with the continuum at
$a^∗$. Indeed, since the critical point $a^∗$ formally resembles the unitary point
$1/a=0$, we find that all higher-body bound states (involving the impurity and
$N>1$ bosons) emerge and disappear at both of these points. We show that the behavior
at these “multibody resonances” is universal, since it occurs for any model with an
effective three-body repulsion involving the impurity. Thus we see that the
fixed-impurity problem is strongly affected by a three-body parameter even in the
absence of the Efimov effect.

Publication

Physical Review A **98**, 062705 (2018)