Title: First order dipolar phase transition in the Dicke model with infinitely coordinated frustrating interaction,
Abstract: We found analytically a first order quantum phase transition in the Cooper
pair box array of $N$ low-capacitance Josephson junctions capacitively coupled
to a resonant photon in a microwave cavity. The Hamiltonian of the system maps
on the extended Dicke Hamiltonian of $N$ spins one-half with infinitely
coordinated antiferromagnetic (frustrating) interaction. This interaction
arises from the gauge-invariant coupling of the Josephson junctions phases to
the vector potential of the resonant photon field. In $N \gg 1$ semiclassical
limit, we found a critical coupling at which ground state of the system
switches to the one with a net collective electric dipole moment of the Cooper
pair boxes coupled to superradiant equilibrium photonic condensate. This phase
transition changes from the first to second order if the frustrating
interaction is switched off. A self-consistently `rotating' Holstein-Primakoff
representation for the Cartesian components of the total superspin is proposed,
that enables to trace both the first and the second order quantum phase
transitions in the extended and standard Dicke models respectively.