  We study a model describing $N$ identical bosonic atoms trapped in a
double-well potential together with a single impurity atom, comparing and
contrasting it throughout with the Dicke model. As the boson-impurity coupling
strength is varied, there is a symmetry-breaking pitchfork bifurcation which is
analogous to the quantum phase transition occurring in the Dicke model. Through
stability analysis around the bifurcation point, we show that the critical
value of the coupling strength has the same dependence on the parameters as the
critical coupling value in the Dicke model. We also show that, like the Dicke
model, the mean-field dynamics go from being regular to chaotic above the
bifurcation and macroscopic excitations of the bosons are observed. Overall,
the boson-impurity system behaves like a poor man's version of the Dicke model.
