Critical exponents for an impurity in a bosonic Josephson junction: Position measurement as a phase transition

We use fidelity susceptibility to calculate quantum critical scaling exponents for a system consisting of $N$ identical bosons interacting with a single impurity atom in a double well potential (bosonic Josephson junction). Above a critical value of the boson-impurity interaction energy there is a spontaneous breaking of $\mathbb{Z}_2$ symmetry corresponding to a second order quantum phase transition from a balanced to an imbalanced number of particles in either the left or right hand well. We show that the exponents match those in the Lipkin-Meshkov-Glick and Dicke models suggesting that the impurity model is in the same universality class. The phase transition can be interpreted as a measurement of the position of the impurity by the bosons.