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

Impurity in a bosonic Josephson junction: swallowtail loops, chaos, self-trapping and the poor man's Dicke model