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 a Cooper pair box array of $N$ low-capacitance Josephson junctions capacitively coupled to resonant photons in a microwave cavity. The Hamiltonian of the system maps on the extended Dicke Hamiltonian of $N$ spins $1/2$ with infinitely coordinated antiferromagnetic (frustrating) interaction. This interaction arises from the gauge-invariant coupling of the Josephson-junction phases to the vector potential of the resonant photons field. In the $N\ensuremath{\gg}1$ semiclassical limit, we found a critical coupling at which the ground state of the system switches to one with a net collective electric dipole moment of the Cooper pair boxes coupled to a super-radiant 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 one to trace both the first- and the second-order quantum phase transitions in the extended and standard Dicke models, respectively.