Coherence transition of small Josephson junctions coupled to a single-mode resonant cavity: Connection to the Dicke model

We calculate the thermodynamic properties of a collection of N small Josephson junctions coupled to a single-mode resonant electromagnetic cavity, at finite temperature T, using several approaches. In the first approach, we include all the quantum-mechanical levels of the junction, but treat the junction-cavity interaction using a mean-field approximation developed previously for T=0. In the other approaches, the junctions are treated including only the two lowest energy levels per junction, but with two different Hamiltonians. The first of these maps onto the Dicke model of quantum optics. The second is a modified Dicke model which contains an additional XY-like coupling between the junctions. The modified Dicke model can be treated using a mean-field theory, which in the limit of zero XY coupling gives the solution of the Dicke model in the thermodynamic limit using Glauber coherent states to represent the cavity. In all cases, for an N-independent junction-cavity coupling, there is a critical junction number N above which there is a continuous transition from incoherence to coherence with decreasing T. If the coupling scales with N so as to give a well-behaved thermodynamic limit, there is a critical minimum coupling strength for the onset of coherence. In all three models, the cavity photon occupation numbers have a non-Bose distribution when the system is coherent.